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0000000000000000000000000000000000000000..e4bedb70719fef1bc486d22c448a301d2703a669 --- /dev/null +++ b/1NAzT4oBgHgl3EQf8v6L/content/tmp_files/2301.01909v1.pdf.txt @@ -0,0 +1,1735 @@ +arXiv:2301.01909v1 [math-ph] 5 Jan 2023 +Solid-solid phase transitions in the ‘near-liquid’ limit +Yury Grabovsky∗ +Lev Truskinovsky† +January 6, 2023 +Abstract +In this paper, dedicated to the memory of J. Ericksen, we address the fundamental +difference between solid-solid and liquid-liquid phase transitions while remaining within +the Ericksen’s nonlinear elasticity paradigm. To this end we assume that rigidity is +weak and explore the nature of solid-solid phase transitions in a ‘near-liquid’ limit. In +the language of calculus of variations we probe limits of quasiconvexity in an ’almost +liquid’ solid by comparing the thresholds for cooperative (laminate based) and non- +cooperative (inclusion based) nucleation. We consider a 2D problem and work with a +prototypical two-phase Hadamard material. Using these two types of nucleation tests +we obtain for this material surprisingly tight two-sided bounds on the elastic binodal +without computing the quasi-convex envelope. +1 +Introduction +In 1975 J. Ericksen posed the problem of equilibrium for solids undergoing first order phase +transitions in the framework of nonlinear elasticity theory. In this way he effectively reformu- +lated the classical problem of physics into a problem of vectorial calculus of variations. The +contemporaneous physical theory viewed non-hydrostatically stressed solids as metastable +and therefore did not distinguish between solid-solid and liquid-liquid phase transitions. J. +Ericksen realized that at normal conditions the assumption of complete relaxation of non- +hydrostatic stresses is impractical and his pioneering research program of studying materials +with non-rank-one convex energies revolutionized elasticity theory. The goal of this paper is +to elucidate the difference between solid-solid and liquid-liquid phase transitions within the +Ericksen’s nonlinear elasticity paradigm. +From the perspective of elasticity theory, the main difference between liquids and solids +is that liquids do not resist shear [6, 10]. This degeneracy in the elastic constitutive structure +of liquids is responsible for their peculiar behavior during first order phase transitions vis a +vis the behavior of solids, characterized by finite rigidity [17]. While in both cases reaching +phase equilibrium usually leads to the formation of phase mixtures, in the case of solids +∗Department of Mathematics, Temple University, Philadelphia, PA 19122, USA +†PMMH, CNRS – UMR 7636, ESPCI, PSL, 75005 Paris, France +1 + +the knowledge of phase fractions carries considerably more information about the geometry +of the resulting microstructure than in the case of liquids. More specifically, if the phase +organization in liquid phase transitions is largely controlled by surface tension, in solid phase +transitions the dominance of elastic long-range interactions leaves to surface tension only a +minor role of a scale selection. +First order phase transitions in liquids are well understood at both physical and mathe- +matical level [32, 9]. The reason is that the scalar problem confronted in the liquid case is +fully solvable [7]. Instead, despite many dedicated efforts, largely inspired by the pioneering +contributions of J. Ericksen himself [12, 11, 13, 14, 15], the mathematical understanding of +elastic phase transitions in solids is still far from being complete as the underlying nonconvex +vectorial problems of the calculus of variations remain highly challenging. +To set the stage, we recall that in nonlinear elasticity the energy functional can be written +in the form E[y] = +� +Ω W(F )dx, where F = ∇y and y : Ω → Rn is the deformation. For +the energy minimizing configurations the conventional physically informed energy density +W(F ) can be replaced by a relaxed one QW(F ) = infφ∈C∞ +0 (D;Rn) |D|−1 � +D W(F + ∇φ)dx +which is known as quasiconvexification of W(F ) [8]. To construct the function QW(F ) +one must know the energy minimizing phase microstructures. +In the case of liquids the +geometry of such microstructures is irrelevant and the construction of QW(F ) reduces to +convexification. In solids the task of finding the equilibrium microstructures in a generic +case is hardly tractable [4, 1]. +With the aim of building a bridge between elastic phase transitions in liquids and solids, +we consider a special limit of ‘near-liquid’ solids which are characterized by an arbitrarily +weak resistance to shear. While we pose the general question of how in such a limit the tight +control on the geometry of optimal microstructures by elastic interactions is progressively +lost, we address a simpler problem of describing in this limit the boundary of the set of +stable single-phase configurations. In the case of liquid-liquid phase transitions the incipient +microstructures do not have any special features. The problem also simplifies in the case +of ‘strongly-solid’ elastic phase transitions when the equilibrium microstructures are just +simple laminates [24]. The goal of the present paper is to understand the opposite, ‘weakly- +solid’ limit, when some of the simplest laminate-based microstructures are proved to be +suboptimal. +In the physics of phase transformations, the Maxwell-Gibbs critical/equilibrium condi- +tions [34, 16], defining the incipient transitions in liquids, are designed to account for the +possibility of phase nucleation. In other words, their role is to delimit the homogeneous +configurations that are unstable to perturbations that are small only in extent and the set of +such configurations is known in physics as the binodal region [36]. From the perspective of +the mathematical theory of elastic phase transitions the analog of the binodal region would +incorporate the homogeneous states that fail to be strong minima of the energy functional. +Therefore, the binodal region is a subset in the configurational space of strain measures +where the quasi convex envelope lays below the energy density. Locating the boundaries +of the binodal region (known simply as a binodal) in the ’near-liquid’ limit constitutes the +main task of the present paper. While remaining nontrivial, this task appears, a priori as +more tractable than the task of constructing the actual quasiconvex envelope. +In our prior work we have developed a general method for identifying the subsets of the +2 + +d1 +d2 +d +h(d) +Figure 1: Double-well structure of the energy density h. +binodal supporting the laminate type energy minimizing configurations [18, 23, 25]. Behind +this method is the study of stability of the jump set—a codimension one variety in the phase +space that has a dual nature. On the one hand it determines the set of pairs F± that could +be the traces of the deformation gradient at the phase boundary in a stable configuration. +On the other, the jump set consists of points that are at most marginally stable in the sense +that their every neighborhood contains points where quasiconvexity fails. Therefore, if one +can prove quasiconvexity at a point on the jump set, then this point must lie on the binodal. +In addition, we have also developed tools to constrain the location of the binodal by means +of addressing nucleation phenomenon directly [19]. As we show in this paper, combined +together, these two types of approaches can produce in the ’near-liquid’ limit a rather good +practical understanding of the whole structure of the binodal, and even allow one to obtain +the exact formulas for the quasiconvex envelope. +To highlight ideas we focus here only on the simplest family of non-quasiconvex energy +densities known as Hadamard materials [27, 28]: W(F ) = µ +2|F |2 + h(det F ). Specifically, +we’ll be interested in the case of two space dimensions and assume that the function h(d) +describes a generic double-well potential modeling isotropic-to-isotropic phase transitions +(see Fig. 1). The main advantage of this class of elastic materials is that one can identify +a single parameter µ, scaling the effective rigidity; by varying this parameter we can study +the entire range of intermediate rigidity responses from ’strong’ (µ ≫ 1) to ’weak’ (µ ≪ 1). +A notable feature of the Hadamard materials is that the phase with the larger value of det F +(smaller density) is characterized by a larger tangential (effective) rigidity than the phase +with the smaller value of det F (larger density). As a result, the latter is more ’liquid-like’ +than the former and therefore the incipient phase transformation induced by compression can +be expected to be different from the incipient phase transformation induced by stretching. +As we show in what follows, this asymmetry leads to a coexistence of ’strongly-solid’ and +’weakly-solid’ responses inside a single material model as, even in the absence of hysteresis, +the direct and reverse solid-solid phase transitions proceed according to morphologically +different transformation mechanisms. +While for an Hadamard material the double well energy structure is described by the +simplest scalar potential h(d), the results of relaxation of W(F ) are nontrivial due to the +inherent incompatibility of the energy wells [3]. +We recall that W(F ) is quasiconvex if +and only if h(d) is convex [2]. The relaxation of W(F ) with non-convex h(d) is known for +3 + +the ‘infinitely-weak’ solids (effectively fluids) with µ = 0, where QW(F ) = h∗∗(det F ) [7]. +Previously we explicitly constructed the quasiconvex envelope for W(F ) in the ‘strongly- +solid’ limit assuming that the shear modulus µ is sufficiently large and the corresponding +quadratic term dominates the double-well term. In this case the formula for QW(F ) couples +|F | and det F and the relaxed energy is sandwiched between W(F ) above and U(F ) = +µ +2|F |2 + h∗∗(det F ) [24]. +In this paper we show that the constraint on µ in [24] was not a technical limitation, +and that, as µ decreases, our formula for QW(F ) ceases to be valid in the subsets of the +binodal region close to the ’liquid-like’ phase with smaller rigidity. In the limit of small µ, +we show that the relaxation of W(F ) goes through a chain of structural transitions with +simple lamination persisting only in the vicinity of the pure ’solid-like’ phase, being replaced +by very different phase arrangements close to the ’liquid-like’ phase. +Our main technical approach is to generate bounds on the binodal surface. +The simplest bounds is obtained by probing the binodal by means of nucleating first +rank laminates. Their optimality is proved by establishing their polyconvexity (and therefore +quasi-convexity). In this setting this is an algebraic problem, because the supporting null- +Lagrangians can be constructed explicitly, [25]. In contrast with the strongly solid regime of +large µ analyzed in [24], in the near liquid regime of small µ, not all of the first rank laminate +bounds are optimal. +These bounds are then improved by nucleating second rank laminates. However, as shown +in [26], the second rank laminate bounds are not optimal either, and are further improved +for hydrostatic strains by means of nucleating a bounded circular inclusion in the infinite +plane. We conjecture that this bound is optimal. If our conjecture is true, then the values +of the deformation gradient in the exterior of the circular nucleus would provide a bound +on the binodal from the outside of the binodal region. Another consequence of the assumed +optimality of the inclusion-based nucleation bound is the explicit formula for the quasiconvex +envelope QW(F ) at all hydrostatic strains. +By juxtaposing the hypothetical bound provided by the study of bounded inclusions and +unbounded second rank laminates we derive tight two-sided bounds on the binodal. As we +demonstrate in [20], both bounds remain tight in the full range of parameters for which the +bounds are meaningful. Moreover, the hypothetical bound being in complete agreement with +the numerically computed rank-one convex binodal. +The paper is organized as follows. In Section 2 we recall some general results from the +calculus of variations for nonconvex vectorial problems, used in the rest of the paper. In +Section 3 we specialize these results for the Hadamard material and present the numerical +illustrations of the obtained bounds. Analytical results for the limiting case µ → 0 are pre- +sented in Section 4 where we also compare them with numerical computations. In Section 5 +we demonstrate the far reaching consequences of the assumed optimality of the nucleation +bound. The paper ends with a general discussion and conclusions in Section 6. +4 + +2 +Preliminaries +Binodal region. Hyperelastic materials in a d-dimensional space have the following form of +the energy stored in the deformed elastic body +E[y] = +� +Ω +W(∇y(x))dx, +where Ω ⊂ Rd is the reference configuration, and y : Ω → Rd is the deformation. +In +order to understand the stable (i.e. experimentally observable) configurations of the body +it is often necessary to replace the energy density W(F ) with a relaxed one QW(F ), called +quasiconvexification. Even though, there is a formula for QW(F ) [8]: +QW(F ) = +inf +φ∈C∞ +0 (D;Rn) +1 +|D| +� +D +W(F + ∇φ)dx, +(2.1) +there is no systematic approaches to compute it. A simpler, but just as useful an object, is +the elastic binodal. +Definition 2.1. An elastic binodal is the boundary of the binodal region +B = {F : W(F ) < QW(F )}. +(2.2) +Definition 2.2. We say that the matrix F is stable, if W(F ) = QW(F ). +Thus, the binodal is the boundary separating the binodal region from the set of stable +points. +Jump set. While we acknowledge that there could be rank-one convex, non quasiconvex +functions, most cases of practical interest in elastic phase transitions feature multiwell ener- +gies that are not rank-one convex and possess a non-trivial jump set, stable points of which +form a part of the binodal (or the entire binodal, if one is lucky). The jump set is the set of +solutions F = F− of the equations + + + + + + + + + +F+ = F− + a ⊗ n, +[[P ]]n = 0, +[[P T ]]a = 0, +[[W]] − ⟨{{P }}, [[F ]]⟩ = 0, +(2.3) +where a ̸= 0 and |n| = 1 are thought to be excluded from the above system resulting in a +single scalar equation for F . We refer the reader to [26] for a discussion of the geometry of +the solution set of (2.3). Here we used the standard notations +P± = WF (F±), +[[F ]] = F+ − F−, +{{P }} = P+ + P− +2 +, +⟨A, B⟩ = Tr (ABT), +where WF indicates the matrix of partial derivatives Pij = ∂W/∂Fij. +5 + +The points on the jump set belong either to the binodal or to the binodal region B, [18]. +Hence, the jump set always represents a bound on the binodal region from within. One of +the easy ways to detect the unstable parts of the jump set is to use the Weierstrass condition, +which is necessary for stability. +W ◦(F , b ⊗ m) ≥ 0, +∀b ∈ Rn, |m| = 1, +(2.4) +where +W ◦(F , H) = W(F + H) − W(F ) − ⟨WF (F ), H⟩. +We have proved in [22] that the pairs of points F± on the jump set are either both stable +or both unstable. Hence, a point F+ satisfying (2.4) can be still classified as unstable, if F− +fails (2.4). While there are other conditions of stability that don’t follow from (2.4) (see [23]) +we will only make use of an easily verifiable corollary of(2.4) that restricts the rank-one test +fields b ⊗ m to an infinitesimally small neighborhood of [[F ]] = a ⊗ n. +Currently, the only general tool for establishing stability is polyconvexity, which is suf- +ficient but rather far from necessary. In two dimensions it reduces to finding a constant +m ∈ R, such that +W ◦(F , H) − m det H ≥ 0, +∀H ∈ R2×2. +(2.5) +If (2.5) holds, then F is stable in the sense of Definition 2.2. For points F± on the jump set, +however, the only value of m that could possibly work is, as shown in [25], +m = ⟨[[P ]], cof[[F ]]⟩ +|[[F ]]|2 +. +(2.6) +Secondary jump set. An improved bound on the binodal is provided by the secondary +jump set corresponding to the nucleation of a rank-two laminate in the infinite homoge- +neously strained space. Thus, the secondary jump set is defined by the system of equations + + + + + + + + + + + +F = F + b ⊗ m, +P m = P m, +P Tb = P +Tb, +W(F ) − W = P m · b, +(2.7) +where the pair F±, to be determined, is assumed to satisfy the primary jump set equations +(2.3), while +W = λW(F+) + (1 − λ)W(F−), +P = λP+ + (1 − λ)P−, +(2.8) +for some λ ∈ [0, 1], which also plays the role of a variable to be solved for in (2.7), along +with F , b ̸= 0, and |m| = 1. Once again, the secondary jump set represents a bound on the +binodal region from within. +Nucleation criterion. Let us now recall another method of probing the binodal: nucleation +of inclusions either of a prescribed shape [5, 33, 31] or of an optimal inclusion, whose shape +must be determined [29, 35, 30]. The theory justifying why these tests probe the binodal +6 + +was developed in [19]. In the case of “nucleation of a bounded inclusion”, the criterion for +F0 to be “marginally stable”, i.e. to lie in the closure of B, is the existence of a field +φ ∈ S = {φ ∈ L2 +loc(Rd) : ∇φ ∈ L2(Rd; Rd)}, +such that +∇ · P (F0 + ∇φ) = 0, +∇ · P ∗(F0 + ∇φ) = 0 +(2.9) +in the sense of distribution in Rd, where +P (F ) = WF (F ), +P ∗(F ) = W(F )Id − F TP (F ). +We also need to verify the non-degeneracy of the solution φ: +� +Rd W ◦ +F (F0, ∇φ)dx ̸= 0. +(2.10) +In the case of nucleation of an actual inclusion ω with smooth boundary the verification of +(2.9) consists in verifying that the field φ ∈ S solves ∇ · P (F0 + ∇φ) = 0 both inside and +outside of ω, together with the condition that the traces F±(x) = F0 + ∇φ±(x) on the two +sides of ∂ω form a corresponding pair on the jump set for each x ∈ ∂ω. If, in addition, we +can somehow prove that F + ∇φ(x) is stable in the sense of Definition 2.2, for each x ∈ Rd, +then F0 must lie on the binodal. Conversely, if it is known that that at some x0 ∈ Rd the +matrix F0 + ∇φ(x0) is unstable, then F0 must lie in the interior of B. +3 +Hadamard material +In this paper we focus our attention on a particularly simple, yet nontrivial energy +W(F ) = µ +2|F |2 + h(d), +F ∈ {F ∈ GL(n) : det F > 0}, +d = det F , +(3.1) +where h(d) is a C2(0, +∞) function with a double-well shape. In our explicit computations +and illustrations we use the quartic double-well energy1 +h(d) = (d − d1)2(d − d2)2, +(3.2) +which affords certain simplification of general formulas. +Jump set. We recall (see [24]) that in two dimensions the jump set of (3.1) consists of +matrices F±, whose two singular values labelled ε0 and ε± satisfy the equations +ε0[[h′]] + µ[[ε]] = 0, +[[h]] − {{h′}}[[d]] = 0, +d± = det F± = ε0ε±. +(3.3) +The notation reflects that for each pair F± on the jump set there is a frame in which +both matrices are diagonal and share the same singular value ε0 with the same eigenvector. +1Formula (3.2) only needs to hold in an arbitrary neighborhoodof [d1, d2]. The potential h(d) can be +modified outside of that neighborhood arbitrarily, as long as h∗∗(d) = h(d) there. In particular, the singular +behavior of h(d) as d → 0+, required in nonlinear elasticity, can be easily assured. +7 + +Equations (3.3) can be used to derive the semi-explicit parametric equations of the jump set, +where, say d+ = ε0ε+, can serve as a parameter. Given d+ we can use the second equation in +(3.3) to compute d− = D(d+). Then, multiplying the first equation in (3.3) by ε0 we obtain +the parametric equations + + + +ε0(d+) = +� +−µ[[d]] +[[h′]] , +ε+(d+) = +d+ +ε0(d+). +In the case of potential (3.2) we obtain +[[h]] − {{h′}}[[d]] = [[d]]3(d1 + d2 − d+ − d−). +Hence, d− = d1 + d2 − d+ = D(d+). It follows that +{{h′}} = 0, +ε+ + ε− = d1 + d2 +ε0 +. +(3.4) +In particular, we can eliminate h′(d±) from our formulas by means of (3.3) and (3.4): +h′(d±) = {{h′}} ± 1 +2[[h′]] = ∓µ +2 +[[ε]] +ε0 +. +(3.5) +For quartic energy (3.2) we can also write the equation of the jump set explicitly as +ε± = ε±(ε0). Indeed, ε± = d±/ε0, while d± solves +(d± − d1)(d± − d2) = − µ +4ε2 +0 +(3.6) +The two roots of (3.6) are the values of d±, where, by convention, we denote by d+ the +larger root. Equation (3.6) has exactly two real roots whenever ε0 > √µ/(d2 − d1). Hence, +explicitly, +ε± = +1 +2ε0 +� +d1 + d2 ± +� +(d2 − d1)2 − µ +ε2 +0 +� +. +(3.7) +In our calculations we will use equations (3.5) to eliminate all occurrences of h′(d±) and +equations (3.7) to eliminate ε±, since the pair ε± is uniquely determined by a single parameter +ε0. +Numerical illustrations. When µ is large we have shown in [24] that the jump set [18] +comprises the entire binodal, each point of which corresponds to the nucleation of a simple +laminate, leading to an explicit formula for the relaxation QW(F ). As the shear modulus +µ decreases, parts of the jump set will become unstable. The jump set will then undergo +a topological change at µ = µtop and in the limit µ → 0, which is the main focus of this +paper, a specific portion of it will remain stable, as we will show using methods from [25]. +Fig. 2 shows the jump sets and indicates their unstable parts for four different values of the +shear modulus µ. The values of µ in Fig. 2 are chosen to be µ = 0, µtop/3, 0.9µtop, and +1.5µtop. Dotted lines indicate “convexification hyperbolas”, i.e., hyperbolas ε2 = d1/ε1 and +ε2 = d2/ε1, where the interval [d1, d2] is the interval on which h(d) differs from its convex hull. +8 + +1 +2 +3 +1 +0.5 +1 +1.5 +2 +2.5 +3 +2 + = 0 +0.5 +1 +1.5 +2 +2.5 +1 +0.5 +1 +1.5 +2 +2.5 +2 + = 2.8168 +0.5 +1 +1.5 +2 +1 +0.5 +1 +1.5 +2 +2 + = 7.6617 + = 12.6757 +1 +2 +3 +1 +0.5 +1 +1.5 +2 +2.5 +3 +2 +Figure 2: Jump sets for h(d) given by (3.2) with d1 = 1, d2 = 3, and different values of µ. +All points outside of the region bounded by the convexification hyperbolas are well-known +to be stable (see e.g. [9]), since they are obviously polyconvex. +W-points. In [23] we have shown that the easily computable corollary of the Weierstrass +condition (2.4) for the energy (3.1) has the form +ε0 ≥ ε±. +(3.8) +In [24] we have shown that this condition is always satisfied for large values of µ as is evident +from the lower right panel in Fig. 2, while it has an obvious geometric interpretation in the +two panels in which the part of the jump set failing (3.8) is shown as a dashed line. The +points marked by red dots in Fig. 2 that delimit the part of the jump set satisfying (3.8) +will be called the Weierstrass points or W-points, for short. We have shown in [26] that the +solid portion of the jump set delimited by W-points is polyconvex for all sufficiently small +µ. As we show below, one can provide an almost explicit characterization of all values of µ +for which W-points are also points of polyconvexity assuming the quartic nonlinearity (3.2). +As discussed above, in order to prove the polyconvexity of W-points we need to establish +(2.5), where m is given by (2.6). This problem has been already analyzed in [24], where we +showed that (2.5) can be written as Φ(x, y) ≥ Φ(ε±, ε0) for all x, y, where +Φ(x, y) = µ +2(x2 + y2) − αx − βy − h(xy) − mxy, +9 + +α = 2√µR{{d}}, +β = µ2 + R4d+d− +R√µ +, +m = [[h′d]] +[[d]] , +R = +� +−[[h′]] +[[d]] . +According to the equations of the jump set (3.3) R = √µ/ε0. Hence, we also have +α = µ(ε+ + ε−), +β = µ +� +ε0 + ε+ε− +ε0 +� +, +m = {{h′}} − µ{{ε}} +ε0 +. +When we minimized Φ(x, y) over all (x, y), such that xy = d we have concluded that the +minimizer is (d/y, y), where y = y(d) is the largest root of +y4 − β0y3 + dα0y − d2 = 0, +α0 = ε+ + ε−, +β0 = ε0 + ε+ε− +ε0 +, +(3.9) +while the minimum of Φ(x, y) is achieved at a finite point corresponding to a critical point +of φ(d) = Φ(d/y(d), y(d)). +In the special case of W-points we have ε+ = ε0 and therefore α0 = β0 = ε− + ε0. In this +case equation (3.9) factors +(y2 − d)(y2 − α0y + d) = 0. +The largest root is y = 1 +2(α0 + +� +α2 +0 − 4d), provided 0 < d ≤ α2 +0/4. If d > α2 +0/4, then the +quartic has only two real roots y = ± +√ +d. Thus, +y(d) = +� +(α0 + +� +α2 +0 − 4d)/2, +d ≤ α2 +0/4, +√ +d, +d > α2 +0/4. +In [24] we have also computed +φ′(d) = µy(d)2 − β0y(d) +d ++ h′(d) − m. +In the case of W-points for which β0 = α0 we see that +y(d)2 − β0y(d) +d += −1 +when d ≤ α2 +0/4. Hence, any critical points of φ(d) in this regime would have to satisfy +h′(d) − µ − m = 0. +One of the solutions is d−, which always satisfies d− ≤ α2 +0/4. If this equation has 3 solutions, +the the middle one corresponds to a local maximum of φ(d), while the third d∗ > d+ always +fails to satisfy d∗ ≤ α2 +0/4 because d+ = ε2 +0 > (ε− + ε0)2/4. We conclude that the only critical +points of φ(d) that need to be checked are the ones that satisfy d > α2 +0/4, while +φ′(d) = µ +� +1 − α0 +√ +d +� ++ h′(d) − m. +10 + +Observe that φ′(d) > 0 when d ≥ max(α2 +0, �d+), where �d+ is the largest root of h′(d) − m. +Hence we only need to check for critical points in a specific bounded interval. In fact, if h(d) +is given by (3.2), then it is easy to see that φ′(d) > 0 for all d ≥ α2 +0. Hence, we only need to +check for critical points of φ(d) on (α2 +0/4, α2 +0). In addition, since {{h′}} = 0 for h(d), given by +(3.2) we have m = −µ{{ε}}/ε0 = −µα0/(2ε0). Thus, we obtain the following characterization +of polyconvexity of W-points. +Theorem 3.1. Let h(d) be given by (3.2), then W-points are polyconvex whenever +min +d∈ +� +α2 +0 +4 ,α2 +0 +� +� +h(d) + µ +� +d + α0d +2ε0 +− 2α0 +√ +d +�� += h(ε2 +0) − µε0 +�ε0 +2 + 3ε− +2 +� +. +(3.10) +where α0 = ε0 + ε−, with (ε0, ε−), (ε−, ε0), and (ε0, ε0) being the coordinates of W-points. +The right-hand side in (3.10) is just φ(ε2 +0), where φ(d) is the function being minimized +in (3.10). For quartic energy (3.2) we compute the coordinates of W-points by solving +−4d(d − d1)(d − d2) = µ. +Then ε2 +0 is the largest root, and +ε− = d1 + d2 − ε2 +0 +ε0 +. +We can compute the largest value of µ for which (3.10) holds by substituting µ = −4ε2 +0(ε2 +0 − +d1)(ε2 +0−d2) into (3.10) and regarding ε0 ≤ √d2 as a parameter. When ε0 = √d2, φ(d)−φ(d2) +is a positive polynomial in x = +√ +d. We then seek numerically the largest value of ε0 < +√d2 for which the polynomial P(x) = (φ(x2) − φ(ε2 +0))/(x − ε0)2 develops a double root. +Algebraically this means seeking the largest root ε0 < √d2 of the discriminant (computed in +Maple). This solution gives the largest value of µ below which the W-points are polyconvex. +For example, when d1 = 1, d2 = 3, we have polyconvexity of W-points for all µ < 6.35888. +In this paper we will be interested exclusively in the case when W-points are quasiconvex. +In Fig. 2 the W-points are polyconvex in the top right panel and unstable in the bottom left +panel. +Secondary jump set. The algebraic equations (2.7) describing the secondary jump set can +generally be solved only numerically. By contrast, when µ is small, the asymptotics of the +solutions can be computed explicitly, providing an excellent approximation to the computed +secondary jump set for µ < 3, with d1 = 1, d2 = 3. While the entire secondary jump set is +unstable [26], we will see that it provides an excellent (inside) bound for the binodal. +Suppose that F0 lies on the secondary jump set. Then there exists ε±, y and λ ∈ [0, 1], +such that the pair F0, F , where +F = +� +ε +0 +0 +ε0 +� +, +ε = λε+ + (1 − λ)ε−, +satisfies the jump set equations (2.7). We compute +P = λP+ + (1 − λ)P− = +� +µε + h′ε0 +0 +0 +µε0 + εh′ +� +. +11 + +We have +P0 = µF0 + h′(d0)cofF0 = µ +� +ε +0 +0 +ε0 +� ++ µb ⊗ m + h′(d0) +�� +ε0 +0 +0 +ε +� ++ b⊥ ⊗ m⊥ +� +. +Thus, the second and the third equations in the jump set system (2.7) become + + + + + + + + + + + + + + + + + + + + + (h′(d0) − h′)ε0 +0 +0 +h′(d0)ε − εh′ + + m = −µb, + + (h′(d0) − h′)ε0 +0 +0 +h′(d0)ε − εh′ + + b = −µ|b|2m. +These equations result in 3 possibilities +(a) (h′(d0) − h′)ε0 = h′(d0)ε − εh′ = −γ, µb = γm, m ∈ S1 +(b) (h′(d0) − h′)ε0 = −(h′(d0)ε − εh′) = −γ, µb = γI−m, I− = +� +1 +0 +0 +−1 +� +, m ∈ S1 +(c) (h′(d0) − h′)ε0 ̸= ±(h′(d0)ε − εh′) +Possibility (c) implies that F0 must be diagonal, and will be our main focus. In [26] we show +that possibilities (a) and (b) have no solutions. Let us therefore assume that F± is diagonal +and has the form +F± = +� +ε± +0 +0 +ε0 +� +. +This implies that F − F0 = βe2 ⊗ e2. In particular +F0 = +� +x0 +0 +0 +y0 +� +, +x0 = ε = λε+ + (1 − λ)ε−, +λ ∈ (0, 1). +Let us compute the diagonal matrices P± using equations (3.4) and (3.5). +P 11 +± = µε± + h′(d±)ε0 = µ{{ε}} = µ(d1 + d2) +2ε0 +, +P 22 +± = µε0 + h′(d±)ε± = µ +� +ε0 ∓ [[ε]]ε± +2ε0 +� +. +Let us compute the diagonal matrix P0. +P 11 +0 += µx0 + h′(d0)y0 = µε + h′(d0)d0 +ε , +P 22 +0 += µy0 + h′(d0)x0 = µd0 +ε ++ h′(d0)ε. +Traction continuity equation (P − P0)e2 = 0 then becomes +ε0 + [[ε]] +2ε0 +(ε− − 2λ{{ε}}) − d0 +ε − h′(d0) +µ +ε = 0. +12 + +It will be convenient to use ε as a variable in place of λ. Replacing λ above using ε = ε−+λ[[ε]] +we obtain +d0 +ε = ε0 + 1 +ε0 +(ε+ε− − {{ε}}ε) − h′(d0) +µ +ε. +(3.11) +Let us now compute all the terms in the last equation in (2.7). +W(F0) = µ +2 (ε2 + y2 +0) + h(d0) = µ +2 +� +ε2 + d2 +0 +ε2 +� ++ h(d0). +Next we compute +W = W− + λ[[W]] = W− + λµ[[ε]]{{ε}} = W− + µ(ε − ε−){{ε}}, +where [[h]] = −{{h′}}[[d]] = 0 has been used. We compute +h(d−) = [(d− − d1)(d− − d1)]2 = +µ2 +16ε4 +0 +, +according to (3.6). Therefore, +W− = µ +2 (ε2 +− + ε2 +0) + µ2 +16ε4 +0 +. +We then compute F0 − F = (y0 − ε0)e2 ⊗ e2. Therefore +⟨P , F0 − F ⟩ = µ +�d0 +ε − ε0 +� � +ε0 + 1 +ε0 +(ε+ε− − {{ε}}ε) +� +. +Finally, the Maxwell equation W(F0) − W = ⟨P , F0 − F ⟩ can be written as +1 +2 +� +ε2 + d2 +0 +ε2 +� ++h(d0) +µ +−(ε−ε−){{ε}}−1 +2(ε2 +−+ε2 +0)− µ +16ε4 +0 += +�d0 +ε − ε0 +� � +ε0 + 1 +ε0 +(ε+ε− − {{ε}}ε) +� +. +(3.12) +Next we replace in the above form of the Maxwell relation the expression d0/ε by its expres- +sion from (3.11) As a result of such a substitution the Maxwell relation will also become a +quadratic equation in ε. This permits us to eliminate this variable as a rational expression in +terms of ε0 and d0, while the Maxwell relation will also reduce to a rational relation between +d0 and ε0. This calculation can only be done with the aid of a computer algebra system, +since the remaining equation F(ε0, d0) = 0 is very long and complicated. Now for a given +choice of numerical values of µ, d1 and d2 we can solve F(ε0, d0) = 0 numerically and then +extract those solutions which satisfy λ ∈ [0, 1]. The result for d1 = 1, d2 = 2 and µ = µtop/3 +is shown as a green curve in Fig. 3. As we can see, it identifies all points between the green +curve and the dashed lines of the primary jump set as a part of the binodal region—an +improvement over the primary jump set bound. +While, it is not apparent from Fig. 3, the secondary jump set consists of two curves +related by symmetry with respect to the bisector of the first quadrant. Each of the curves +13 + +0.5 +1 +1.5 +2 +2.5 +1 +0.5 +1 +1.5 +2 +2.5 +2 + = 2.8168 +Figure 3: Secondary jump set computed by numerically solving equations (3.11) and (3.12). +are cut-off at their intersection with each other at the bisector. Each curve starts at a W- +point and ends at a point (not shown) on the dashed part of the jump set. The endpoints +of the secondary jump set correspond to the extreme values 0 and 1 of the volume fraction +λ in (2.8). The corresponding points on the secondary jump set must lie on the primary +jump set. There are two possibilities. Either F ̸= F or F = F at λ = 0 or 1. In the +former case the limiting position F+ of F is rank-one related to two different points on the +jump set: F− (layer normal e1) and F (layer normal e2). W-point F+ is the only one with +this property. All other points F+ on the jump set have a unique counterpart F−. In the +latter case a detailed asymptotic analysis shows that that the common limit point of F and +F must achieve equality in the “Legendre-Hadamard for phase boundaries” inequality from +[23]. This point lies on the dashed part of the jump set and is used in numerical calculations. +The details of the analysis will be spelled out in a forthcoming paper [20]. +Circular nucleus. In [26] it is shown that the secondary jump set (green curve in Fig. 3) +is unstable. That means that the corresponding bound on the binodal is not optimal. We +can improve the bound using another method of probing the binodal: nucleation of equilib- +rium energy-neutral inclusions. The theory justifying why such nucleation tests probe the +binodal was developed in [19]. In the case of the isotropic, objective energy (3.1) and a +hydrostatic loading it is natural that the shape of an optimal precipitate should be circular. +The deformation gradient inside the circular precipitate must be a constant hydrostatic field +F0 = εW +0 I2, since that field is rank-one connected to an infinite family of fields +FR = R +� +εW +− +0 +0 +εW +0 +� +RT, +R ∈ SO(2), +where (εW +− , εW +0 ) is a coordinate of one of the W-points. The deformation gradient outside of +the circular inclusion must solve an Euler-Lagrange equation for the energy (3.1) +µ∆y + (cof∇y)∇h′(det ∇y) = 0, +x ∈ R2 \ B(0, 1), +(3.13) +14 + +and agree with FR at the point Re1 on the boundary of the circular inclusion: +∇y(x) = εW +− n ⊗ n + εW +0 τ ⊗ τ, +x ∈ ∂B(0, 1). +(3.14) +In this case both equations (2.9) will be satisfied for the possibly marginally stable matrix +F∞ = lim +|x|→∞ ∇y(x) = ε∞I2. +We also know that that the values of ∇y(x) inside the circular inclusion and its trace on +the outside boundary of the inclusion are stable. Our results from [19] then say that either +F∞ lies on the binodal and all values ∇y(x) in the exterior of the inclusion are stable, or +F∞ lies in the interior of the binodal region B. +In our special radially symmetric case we look for a radially symmetric solution of (3.13) +y = η(r)ˆx, +|x| > 1. +The the unknown function η(r) must solve +� +η +r +d +drh′ � +ηη′ +r +� ++ µ +� +η′ + η +r +�′ = 0, +r > 1, +η′(1) = εW +− , +η(1) = εW +0 . +(3.15) +The nonlinear second order ODE (3.15) cannot be integrated explicitly, but can be solved +numerically. In order to do so, we need to convert the infinite range r > 1 into a finite one +by means of the change of the independent variable x = 1/r2. It will also be convenient to +change the dependent variable v = η/r, so that v(x) would have a finite limit, when x → 0. +Then v(x) solves +v′′ = − (v′)2vh′′(v2 − 2xvv′) +µ + v2h′′(v2 − 2xvv′), +x ∈ [0, 1], +v(1) = εW +0 , +v′(1) = εW +0 − εW +− +2 +. +(3.16) +The value ε∞ = v(0)I2 found numerically is shown as a blue dot in Fig. 4. It provides an +improved bound on the binodal compared to the secondary jump set (green line in Fig. 4) by +showing that hydrostatic strains between the blue dot and the green line are unstable. This +conclusion holds, provided the non-degeneracy condition (2.10) is verified. A calculation +shows that +� +Rd W ◦ +F (F0, ∇φ)dx = −I2 +� +R2 h′′(ε2 +∞)ε2 +∞ +� +η′(r) + η(r) +r +− 2ε∞ +� +dx. +Thus, +� +Rd W ◦ +F (F0, ∇φ)dx = −2πh′′(ε2 +∞)ε2 +∞I2 lim +r→∞(rη(r) − ε∞r2). +To see that the limit above exists and is non-zero, at least for small µ > 0, we simply solve +(3.15) for µ = 0, for which εW +0 = √d2, εW +− = +d1 +√d2. The solution is η(r) = √d1r2 + d2 − d1, +and we easily see that +lim +r→∞(rη(r) − ε∞r2) = d2 − d1 +2√d1 +. +15 + +Hence, the non-degeneracy condition (2.10) will hold, at least for sufficiently small µ > 0. +The non-degeneracy will also hold for all µ below the topological transition, because if we +write �η(r) = η(r) − ε∞r, then (assuming that �η′(r) → 0, as r → ∞) �η(r) will solve, when r +is large, the differential equation +ε∞h′′(ε2 +∞) +� +ε∞ +� +�η′ + �η +r +� ++ �η′�η +r +� ++ µ +� +�η′ + �η +r +� += 0. +This integrates to +ε∞h′′(ε2 +∞)(2ε∞r�η + �η2) + 2µr�η = 2c. +Since �η, satisfying �η′(r) → 0, as r → ∞, cannot be zero (it is the leading term of η(r)−ε∞r), +we conclude that the constant of integration c cannot be zero either. Hence, we obtain that +lim +r→∞(rη(r) − ε∞r2) = lim +r→∞ r�η(r) = +c +µ + ε2∞h′′(ε2∞) ̸= 0. +Polyconvexity limits along εI2. We now turn to the problem of proving polyconvexity at +points F = εI2. To succeed we need to find a constant m ∈ R, such that (2.5) holds. For +our energy we compute +W ◦(F , H) = µ +2|H|2 + h(ε2 + d + εθ) − h(ε2) − εh′(ε2)θ, +θ = Tr H, d = det H. +We also have +|H|2 = 1 +2|H − HT|2 − 2d + θ2. +Hence we need to find m ∈ R, such that +µθ2 +2 ++ h(ε2 + d + εθ) − h(ε2) − εh′(ε2)θ ≥ (m + µ)d, +∀d ≤ θ2 +4 . +(3.17) +In particular the inequality must hold for d = θ2/4. In that case we must have +m ≤ µ + 4 min +θ∈R +h(ε2 + θ2/4 + εθ) − h(ε2) − εh′(ε2)θ +θ2 += m∗. +(3.18) +The infimum of the smooth function +F(d, θ) = µθ2 +2 ++ h(ε2 + d + εθ) − h(ε2) − εh′(ε2)θ − (m + µ)d +must be attained either at a critical point or at infinity. Obviously, if along the minimizing +sequence the quantity δ = ε2 + d + εθ goes to infinity, then the values of F must also go to ++∞, which cannot happen along a minimizing sequence. Hence, (d, θ) go to infinity so that +δ stays bounded. Hence, we can switch variables and instead of the pair (d, θ) consider the +pair (δ, θ). In this case d = δ − εθ − ε2. Hence, +F(δ, θ) = µθ2 +2 ++ h(δ) − h(ε2) − εh′(ε2)θ − (m + µ)(δ − εθ − ε2), +16 + +where δ ≤ (θ/2 + ε)2. Minimizing F(δ, θ) with respect to θ we obtain +θ = −ε(m + µ − h′(ε2)) +µ +. +Hence we need to minimize +f(δ) = h(δ) − h(ε2) − (m + µ)(δ − ε2) − ε2(m + µ − h′(ε2))2 +2µ +, +over all δ satisfying +δ ≤ ε2(h′(ε2) + µ − m)2 +4µ2 +. +(3.19) +We remark that taking θ = −4ε in (3.18) we conclude that m∗ ≤ µ + h′(ε2). Thus, the +right-hand side of (3.19) is monotone decreasing in m, when m ≤ m∗. +Now, the minimum is achieved either at the boundary, where equality in (3.19) holds, or +at a critical point. It cannot be “achieved” at infinity, where f(δ) is +∞. If the minimum +is achieved at the boundary then f(δ) ≥ 0 for all δ, provided m ≤ m∗. If the minimum is +achieved at a critical point +h′(δ) = m + µ, +then several possibilities need to be considered. Let us first assume that the equation +h′(δ) = m∗ + µ +(3.20) +has a single root δ∗. If that root fails to satisfy (3.19) with m = m∗, then for m = m∗ the +function f(δ) has no critical points and polyconvexity holds. If that root satisfies (3.19), +then all values δ < δ∗ are admissible. We then substitute m = h′(δ) − µ, δ ≤ δ∗ into f(δ). +The resulting function +f(δ) = h(δ) − h(ε2) − h′(δ)(δ − ε2) − ε2(h′(δ) − h′(ε2))2 +2µ +(3.21) +can be plotted on (−∞, δ∗] versus m = h′(δ) − µ to see if there are values of m above which +all values of f(δ) are positive. +Yet another possibility is when equation (3.20) has 3 real roots. If even the smallest root +δ∗ fails to satisfy (3.19) with m = m∗, then there are no critical points and polyconvexity +holds. Otherwise, all values of δ ≤ δ∗ are admissible and we can prove failure of polyconvexity +by plotting (3.21) versus m(δ) = h′(δ)−µ on δ ≤ δ∗ and checking that it is negative. In fact, +we believe that polyconvexity fails in all cases when δ∗ satisfies (3.19). In order to exhibit +this failure we only need to produce a single value of admissible δ for which f(δ), given +by (3.21), is negative. Hence, if (3.21) is negative for all δ ≤ δ∗, then there is no need to +examine other intervals of admissible δ, since for any m ≤ m∗ there is always an admissible +δ ≤ δ∗, which makes f(δ) negative. However, if f(δ) has a region where it is positive, then +one needs to examine other areas of admissibility and check whether f(δ) is negative for the +same values of m. Thus, we obtain an algorithm that can prove polyconvexity or failure of +17 + +1.1 +1.15 +1.2 +1 +1.06 +1.08 +1.1 +1.12 +1.14 +1.16 +1.18 +1.2 +2 + = 2.8168 +nucleation bound +polyconvexity bound +secondary jump set +Figure 4: Bounds on the binodal from the inside and the outside of the binodal region along +hydrostatic strains. +it in many, but not all cases. Polyconvexity holds whenever +δ∗ > ε2(h′(ε2) + µ − m∗)2 +4µ2 +. +for all solutions δ∗ of h′(δ) = µ + m∗. Polyconvexity fails whenever f(δ) < 0 for all δ < δ∗, +provided +δ∗ ≤ ε2(h′(ε2) + µ − m∗)2 +4µ2 +. +If ε2 = d1, then the minimization problem (3.18) simplifies: +min +θ∈R +h(d1 + θ√d1 + θ2/4) +θ2 +. +We first observe that in general θ = 0 is not a minimizer. Then there are 3 minimizers: +θ = −4 +� +d1, +θ = ±2 +� +d2 − 2 +� +d1. +When ε = √d1 + x, then the minimizer θ(x) must be located near one of the above 3 +minimizers. We can then write θ = θ0 + y for the minimizer, where θ0 denotes one of the 3. +If we write the function under the minimum as H(ε, θ), then at the minimum we must have +∂H/∂θ = 0, which gives the equation +x ∂2H +∂θ∂ε + y∂2H +∂θ2 = 0. +After solving for y and substituting this solution back into H we obtain +H = x + + +∂H +∂ε − ∂H +∂θ +∂2H +∂θ∂ε +∂2H +∂θ2 + + + , +18 + +where derivatives are evaluated at (√d1, θ0). Maple calculation yields +H = + + + + + +x +2 +√d1h′′(d1), +θ0 = −4√d1, +xd1h′′(d1) +√d1+√d2 , +θ0 = −2√d2 − 2√d1, +xd1h′′(d1) +√d1−√d2 , +θ0 = 2√d2 − 2√d1. +This shows that θ = 2√d2 − 2√d1 + y is the minimizer, while +m∗ = µ − +4xd1 +√d2 − √d1 +h′′(d1). +In particular, the equation h′(δ) = m∗ + µ will have 3 real roots. The smallest one δ∗ will +be near d1: +δ∗ = d1 + µ + m∗ +h′′(d1) . +Finally, polyconvexity will hold if (3.19) fails when δ = δ∗ and m = m∗. In other words we +must have (asymptotically) +ε ≤ +� +d1 + +µ +h′′(d1)√d1 +√d2 − √d1 +√d2 + √d1 +. +(3.22) +Fig. 4 showing the right-hand side of (3.22) as a red dot implies that ε∞I2 fails to be +polyconvex, but by a very slim margin. The ordering of the bounds in Fig. 4 persists on +the entire range of µ. We see that the gap between established stability (along the bisector +below the red dot) and established instability (along the bisector above the blue dot) is very +small. +4 +Limiting case µ → 0 +In this section we derive explicit asymptotics of the secondary jump set and the nucleation +bound. +Secondary jump set. Expanding equation (3.7) to first order in µ we obtain +ε+ = d2 +ε0 +− +µ +4ε3 +0(d2 − d1) + O(µ2), +ε− = d1 +ε0 ++ +µ +4ε3 +0(d2 − d1) + O(µ2). +(4.1) +When d1 and d2 are fixed we think of ε± as functions of ε0 and µ, even if we suppress this +in the notation. Clearly, when µ → 0 we have ε+ → d2/ε0, ε− → d1/ε0. +The parametric equations (x0(ε0; µ), y0(ε0; µ)) of secondary jump set converge, when +µ → 0, to the hyperbola x0y0 = d1. In particular, d0(ε0, µ) → d1, as µ → 0. The volume +fraction λ of the rank-one laminate used in the second rank laminate is also a function of +ε0 and µ and must have a limit (at least along a subsequence) λ(ε0; µ) → λ0(ε0), as µ → 0. +Equation (3.11) shows that d0 = d1 + µδ + O(µ2), while δ satisfies +d +ε0 +� +ε0 + 1 +ε0 +�d1 +d2 +ε2 +0 − d1 + d2 +2ε2 +0 +d +�� +− d1 − 2δ(d2 − d1)2d +2 +ε2 +0 += 0, +(4.2) +19 + +where d = λd2 + (1 − λ)d1. Equation (4.2) was obtained simply by passing to the limit as +µ → 0 in equation (3.11). +When we pass to the limit as µ → 0 in (3.12) we obtain +(d − d1)2(ε4 +0 + d +2 − 2d2d) +2ε2 +0d +2 += 0. +(4.3) +The dependence of d on the volume fraction λ is essential and should not disappear in the +limit µ → 0. Therefore, the solution of (4.3) that we are after is +d = d2 − +� +d2 +2 − ε4 +0, +(4.4) +where the choice of the root was dictated by the requirement that d ≤ d2. Combining this +with the requirement that d ≥ d1 shows that +4� +d2 +2 − (d2 − d1)2 ≤ ε0 ≤ +� +d2. +(4.5) +Substituting (4.4) into (4.2) gives the explicit formula for δ: +δ = ε4 +0(d2 − d1) − 2(d2 +2 − ε4 +0)(d2 − +� +d2 +2 − ε4 +0) +4ε2 +0(d2 − d1)2(d2 − +� +d2 +2 − ε4 +0)2 +. +(4.6) +It seem that in order to obtain the correct asymptotics of the secondary jump set we need +to obtain the first order asymptotics of ε: +ε = d2 − +� +d2 +2 − ε4 +0 +ε0 ++ �εµ + O(µ2). +(4.7) +In fact, this is not necessary because the leading order asymptotics of d0 is a constant d1. +In that case, as far as the first order asymptotics as µ → 0 is concerned, using (4.7) simply +corresponds to reparametrizing the curve + + + + + + + +x0 = d2 − +� +d2 +2 − ε4 +0 +ε0 +, +y0 = d1 + µδ(ε0) +x0 +. +(4.8) +Indeed, if we change the curve parameter ε0 to ε0 + µ�ε/x′ +0(ε0), then +x0 +� +ε0 + +µ�ε +x′ +0(ε0) +� += x0(ε0) + µ�ε + O(µ2). +At the same time +y0 +� +ε0 + +µ�ε +x′ +0(ε0) +� += +d1 +x0(ε0) − +µd1�ε +x0(ε0)2 + µδ(ε0) +x0(ε0) + O(µ2) = d1 + µδ +x0 + µ�ε + O(µ2). +20 + +0 +1 +2 +3 +4 +5 +1 +1.05 +1.1 +1.15 +asymptotic +numerical +pcx +asymptotic +Figure 5: Comparison between the asymptotics (4.12) and ε∞ obtained from the numerical +solution of (3.15). +We conclude that equation (4.8) correctly describes the asymptotics of the secondary jump +set with O(µ2) error, where the parameter ε0 varies according to (4.5). When ε0 = √d2, +the secondary jump set enters one of the W-points, while when ε0 = +4� +d2 +2 − (d2 − d1)2 the +secondary jump set enters its other end at the “Legendre-Hadamard for phase boundaries” +bound that for small µ lies on the dashed part of the jump set in Fig. 3. +The plot of +(4.8) is in Fig. 3 and is indistinguishable from the numerically obtained curve using the full +(non-asymptotic) versions of secondary jump set equations. +Circular nucleus. In the near-liquid limit µ → 0 we can find the asymptotics of the +solution explicitly. +We know that in the limit µ → 0 the field d(x) = det ∇y(x) must +approach d1. Hence, +ηη′ +r += d1 + µδ(r) + O(µ2), +r > 1. +That implies +η(r) = +� +d1r2 + c0 + µ�η(r) + O(µ2), +(4.9) +and therefore, +δ(r) = 1 +r +� +�η(r) +� +d1r2 + c0 +�′ +. +Substituting this ansatz into (3.15) we obtain +µ +√d1r2 + c0 +r +h′′(d1)δ′(r) + µ +� +d1r +√d1r2 + c0 ++ +√d1r2 + c0 +r +�′ ++ O(µ2) = 0. +(4.10) +Initial conditions from (3.15) imply that +c0 = d2 − d1, +�η(1) = − +d2 − d1 +4d3/2 +2 h′′(d2) +, +�η′(1) = d1(d2 − d1) +2d3/2 +2 +� +1 +d1h′′(d1) + +1 +2d2h′′(d2) +� +. +21 + +Equation (4.10) is easy to integrate (observing that √d1r2 + c0/r is decreasing from √d2 to +√d1 and is therefore uniformly bounded away from zero and ∞). +h′′(d1)�η(r) = +c1r2 + c2 +√d1r2 + c0 +− +r2 +2√d1r2 + c0 +ln +√d1r2 + c0 +r +. +(4.11) +From initial conditions for �η(r) we obtain +c1 = 1 +2 ln +� +d2, +c2 = −(d2 − d1)h′′(d1) +4d2h′′(d2) +, +and hence +ε∞ = +�� +d1 + +µ +2h′′(d1)√d1 +ln +√d2 +√d1 +� +I2 + O(µ2). +(4.12) +Figure 5 shows the quality of the asymptotics for the entire range of shear moduli µ. The +numbers on the y-axis indicate that even for values of µ that are not particularly small the +asymptotics (4.12) gives a good approximation of the actual value of ε∞. For example, for +µ = 3 the relative discrepancy is only around 0.1%. +5 +A glimpse into the relaxed energy +Hypothetical bounds on the binodal. We have seen in the foregoing discussion that the energy +W(F ) is not polyconvex at F = ε∞I2. This is not very surprising, since polyconvexity is +usually strictly stronger that quasiconvexity and we expect and conjecture that F = ε∞I2 +lies on the binodal—at the very edge of quasiconvexity. Here we recall our observation that +if someone could prove that F = ε∞I2 is stable, then we would immediately conclude that +for every |x| > 1 +∇y(x) = η′(r)ˆx ⊗ ˆx + η(r) +r (I2 − ˆx ⊗ ˆx) +would be stable in the sense of Definition 2.2, providing a bound on the binodal from the +outside. For the entire range of µ for which W-points are polyconvex the union of the curves +� +ε1 = η(r) +r , +ε2 = η′(r), +and +� +ε1 = η′(r) +ε2 = η(r) +r , +r > 1 +(5.1) +are indistinguishable from the secondary jump set curves shown in green in Fig. 3. Fig. 6 +shows the same blown-up part of the strain space as in Fig. 4, where the curves (5.1) shown in +magenta are passing through the blue point from Fig. 4. Assuming the conjectured stability +of ε∞I2, the magenta curve must lie outside of binodal region, while secondary jump set +lies in its interior [26]. Thus, the binodal of the energy (3.1) would have to lie between the +green and the magenta curves. We will even go so far as to conjecture that the magenta +curve is in fact the actual binodal of the energy (3.1). Regardless, under the assumption of +stability of ε∞I2, the magenta line represents a rather tight outside bound on the binodal +region. Another byproduct of the assumed stability of ε∞I2 would be the formula for the +22 + +0.5 +1 +1.5 +2 +2.5 +1 +0.5 +1 +1.5 +2 +2.5 +2 + = 2.8168 +hypothetical binodal +known binodal region +1.1 +1.15 +1.2 +1 +1.06 +1.08 +1.1 +1.12 +1.14 +1.16 +1.18 +1.2 +2 + = 2.8168 +nucleation bound +polyconvexity bound +secondary jump set +outside bound +Figure 6: A hypothetical bound on the binodal region from the outside, assuming stability +of ε∞I2. +quasiconvex envelope QW(F ) for hydrostatic strains F . If F = ε∞I2 is stable, then our +radial solution ∇y(x) = η(r)ˆx of (3.15) is also a global minimizer in every finite ball B(0, R), +where it satisfies the affine boundary condition y(x) = (η(R)/R)x, x ∈ ∂B(0, R) [21]. The +energy of such configurations must necessarily be QW(η(R)I2/R)|B(0, R)|. This permits +us to compute QW(εI2) for all ε, as the energy of y(x) = η(r)ˆx in B(0, R). Using the +Clapeyron-type formula for the nonlinear elastic energy stored in an equilibrium stationary +configuration we obtain for F = η(R)I2/R: [21] +|B(0, R)|QW(F ) = 1 +2 +� +∂B(0,R) +{P (∇y)n · y + P ∗(∇y)n · x}dS. +(5.2) +Substituting n = ˆx, y = η(r)ˆx into (5.2) we obtain +QW +�η(R) +R I2 +� += 2(µ − h′(d))d − µη′(R)2 + (2h′(d) + µ)η(R)2 +R2 ++ 2h(d), +(5.3) +where +d = η′(R)η(R) +R +. +When µ is small we can use the explicit asymptotic formulas (4.9), (4.11) for η(r) to obtain +an explicit asymptotics for QW(εI2). The plot of QW(εI2), coming from the numerical +solution of (3.15), as well as its explicit asymptotic approximation, superposed on the plot +of W(εI2) is shown in Fig. 7. +6 +Conclusions +In this paper our far reaching goal was to solve analytically the relaxation problem for the +double well Hadamard energy (3.1) in two space dimensions when the rigidity measure µ +23 + +1 +1.2 +1.4 +1.6 +1.8 +3 +4 +5 +6 +7 +8 +9 +Energy +W( I2) +QW( I2) +QWasym( I2) +Figure 7: Quasiconvex envelope of W(F ) restricted to hydrostatic strains F = εI2. +is sufficiently small. An apparently more attainable target was to locate the corresponding +binodal region inside the strain space. The study of the limit µ → 0 was expected to show how +the ’cooperative’ , rigidity-controlled microstructures, dominating the quasiconvex envelope +at large µ, give rise to more arbitrary and less controlled microstructures characterizing first +order phase transitions in zero rigidity liquids. +We used some of our previously developed methods to pinpoint a substantial portion +of the binodal. +While our general methods apply for Hadamard materials in the entire +parameter range and are amenable to numerical implementation, here we were able to obtain +the explicit asymptotic formulas only in the ’near-liquid’ regime. In particular, we showed +that in an ’almost liquid’ limit, a subset of the jump set adjacent to the high strain phase +remains stable which ensures that simple lamination delivers the corresponding part of the +binodal. This means that even when the reference measure of rigidity µ is small, the high +strain phase maintains its tangential rigidity at the level which ensures solid-solid like nature +of the incipient phase transition. Instead, our analysis showed that the subset of the jump set +adjacent to the low strain and low rigidity phase is unstable in the µ → 0 limit. Moreover, the +secondary jump set is also unstable in this limit. This result suggests that laminates of any +finite rank are unstable near the corresponding subset of the binodal. As we’ve demonstrated +for hydrostatic strains, the reduced rigidity control in this range allows the incipient phase +transformation to proceed non-cooperatively through the formation of isolated nuclei of the +more rigid phase inside the matrix of the less rigid phase. Such transformation mechanism is +already very similar to the one believed to be operating in purely fluid-fluid phase transitions. +Whether the revealed asymmetry of the transformation mechanism between the direct +and reverse transformation is a peculiarity of the Hadamard material or whether this striking +phenomenon has a more general nature, remains to be established. It shows, however, the +intricate role of rigidity in structural transformations which, even if weak, can produce rather +complex structure of the relaxed energy. This complexity will then reflect a gradual transition +from geometrically ordered microstructures, controlled by long range elastic interactions, +24 + +to more ’fluid’ microstructures whose spatial organization is mostly affected by molecular +interactions operating at short range. In other words, in this limit the direct and reverse +solid-solid phase transitions can operate through different transformation mechanisms. The +fact that the ensuing complex structure of the relaxed energy at ’almost-liquid’ solid-solid +phase transitions is ultimately replaced by a simple energy convexification at fluid-fluid phase +transitions points to a singular nature of the limit µ → 0. +Acknowledgments. +YG was supported by the National Science Foundation under +Grant No. DMS-2005538. The work of LT was supported by the French grant ANR-10- +IDEX-0001-02 PSL. +References +[1] J. M. Ball. Progress and puzzles in nonlinear elasticity. In J¨org Schr¨oder and Patrizio +Neff, editors, Poly-, Quasi- and Rank-One Convexity in Applied Mechanics, pages 1–15. +Springer Vienna, Vienna, 2010. +[2] J. M. Ball and F. Murat. W 1,p-quasiconvexity and variational problems for multiple +integrals. J. Funct. Anal., 58(3):225–253, 1984. +[3] J.M. Ball and R.D. James. Incompatible sets of gradients and metastability. Archive +for Rational Mechanics and Analysis, 218(3):1363–1416, 2015. +[4] John M. Ball. 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Journal of Elasticity, 123(2):225–243, 2016. +[24] Yury Grabovsky and Lev Truskinovsky. +Explicit relaxation of a two-well hadamard +energy. Journal of Elasticity, 135(1-2):351–373, 2019. +[25] Yury Grabovsky and Lev Truskinovsky. When rank-one convexity meets polyconvexity: +An algebraic approach to elastic binodal. J. Nonlinear Sci., 28(1):229–253, 2019. +[26] Yury Grabovsky and Lev Truskinovsky. Ubiquity of infinite rank laminates. to besub- +mitted, In preparation. +[27] J. Hadamard. Le¸cons sur la propagation des ondes et les ´equations de l’hydrodynamique. +Hermann, Paris, 1903. +[28] Fritz John. Plane elastic waves of finite amplitude. hadamard materials and harmonic +materials. Communications on Pure and Applied Mathematics, 19(3):309–341, 1966. +26 + +[29] V. Kardonski and Roitburd. On the shape of coherent precipitates. Phys. Met. Metal- +lurg. USSR, 33:210–212, 1972. +[30] A. G. Khachaturyan. Theory of structural transformation in solids. Wiley, New York, +1983. +[31] L. B. Kublanov and A. B. Freidin. Nuclei of a solid phase in a deformable material. +Prikl. Mat. Mekh., 52(3):493–501, 1988. +[32] Lev Davidovich Landau and Evgenii Mikhailovich Lifshitz. Statistical Physics: Volume +5, volume 5. Elsevier, 2013. +[33] J. K. Lee, D. M. Barnett, and H. I. Aaronson. The elastic strain energy of coherent +ellipsoidal precipitates in anisotropic crystalline solids. Metall. Trans. A, 8A:963–970, +1977. +[34] J. C. Maxwell. On the dynamic evidence of the molecular composition of bodies. Nature, +11(279-280):357–359, 374–377, 1875. +[35] A. Pineau. Influence of uniaxial stress on the morphology of coherent precipitates during +coarsening — elastic energy considerations. Acta Metall., 24:559–564, 1976. +[36] J.D. van der Waals. The equilibrium between a solid body and a fluid phase, especially +in the neighbourhood of the critical state. In KNAW, Proceedings, volume 6, pages +1903–1904, 1903. +27 + diff --git a/1NAzT4oBgHgl3EQf8v6L/content/tmp_files/load_file.txt b/1NAzT4oBgHgl3EQf8v6L/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..158c5479fd06e853c17b23956c8bb81e79023dcc --- /dev/null +++ b/1NAzT4oBgHgl3EQf8v6L/content/tmp_files/load_file.txt @@ -0,0 +1,856 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf,len=855 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='01909v1 [math-ph] 5 Jan 2023 Solid-solid phase transitions in the ‘near-liquid’ limit Yury Grabovsky∗ Lev Truskinovsky† January 6, 2023 Abstract In this paper, dedicated to the memory of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Ericksen, we address the fundamental difference between solid-solid and liquid-liquid phase transitions while remaining within the Ericksen’s nonlinear elasticity paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' To this end we assume that rigidity is weak and explore the nature of solid-solid phase transitions in a ‘near-liquid’ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In the language of calculus of variations we probe limits of quasiconvexity in an ’almost liquid’ solid by comparing the thresholds for cooperative (laminate based) and non- cooperative (inclusion based) nucleation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We consider a 2D problem and work with a prototypical two-phase Hadamard material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Using these two types of nucleation tests we obtain for this material surprisingly tight two-sided bounds on the elastic binodal without computing the quasi-convex envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 1 Introduction In 1975 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Ericksen posed the problem of equilibrium for solids undergoing first order phase transitions in the framework of nonlinear elasticity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In this way he effectively reformu- lated the classical problem of physics into a problem of vectorial calculus of variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The contemporaneous physical theory viewed non-hydrostatically stressed solids as metastable and therefore did not distinguish between solid-solid and liquid-liquid phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Ericksen realized that at normal conditions the assumption of complete relaxation of non- hydrostatic stresses is impractical and his pioneering research program of studying materials with non-rank-one convex energies revolutionized elasticity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The goal of this paper is to elucidate the difference between solid-solid and liquid-liquid phase transitions within the Ericksen’s nonlinear elasticity paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' From the perspective of elasticity theory, the main difference between liquids and solids is that liquids do not resist shear [6, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' This degeneracy in the elastic constitutive structure of liquids is responsible for their peculiar behavior during first order phase transitions vis a vis the behavior of solids, characterized by finite rigidity [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' While in both cases reaching phase equilibrium usually leads to the formation of phase mixtures, in the case of solids ∗Department of Mathematics, Temple University, Philadelphia, PA 19122, USA †PMMH, CNRS – UMR 7636, ESPCI, PSL, 75005 Paris, France 1 the knowledge of phase fractions carries considerably more information about the geometry of the resulting microstructure than in the case of liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' More specifically, if the phase organization in liquid phase transitions is largely controlled by surface tension, in solid phase transitions the dominance of elastic long-range interactions leaves to surface tension only a minor role of a scale selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' First order phase transitions in liquids are well understood at both physical and mathe- matical level [32, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The reason is that the scalar problem confronted in the liquid case is fully solvable [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Instead, despite many dedicated efforts, largely inspired by the pioneering contributions of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Ericksen himself [12, 11, 13, 14, 15], the mathematical understanding of elastic phase transitions in solids is still far from being complete as the underlying nonconvex vectorial problems of the calculus of variations remain highly challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' To set the stage, we recall that in nonlinear elasticity the energy functional can be written in the form E[y] = � Ω W(F )dx, where F = ∇y and y : Ω → Rn is the deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' For the energy minimizing configurations the conventional physically informed energy density W(F ) can be replaced by a relaxed one QW(F ) = infφ∈C∞ 0 (D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='Rn) |D|−1 � D W(F + ∇φ)dx which is known as quasiconvexification of W(F ) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' To construct the function QW(F ) one must know the energy minimizing phase microstructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In the case of liquids the geometry of such microstructures is irrelevant and the construction of QW(F ) reduces to convexification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In solids the task of finding the equilibrium microstructures in a generic case is hardly tractable [4, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' With the aim of building a bridge between elastic phase transitions in liquids and solids, we consider a special limit of ‘near-liquid’ solids which are characterized by an arbitrarily weak resistance to shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' While we pose the general question of how in such a limit the tight control on the geometry of optimal microstructures by elastic interactions is progressively lost, we address a simpler problem of describing in this limit the boundary of the set of stable single-phase configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In the case of liquid-liquid phase transitions the incipient microstructures do not have any special features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The problem also simplifies in the case of ‘strongly-solid’ elastic phase transitions when the equilibrium microstructures are just simple laminates [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The goal of the present paper is to understand the opposite, ‘weakly- solid’ limit, when some of the simplest laminate-based microstructures are proved to be suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In the physics of phase transformations, the Maxwell-Gibbs critical/equilibrium condi- tions [34, 16], defining the incipient transitions in liquids, are designed to account for the possibility of phase nucleation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In other words, their role is to delimit the homogeneous configurations that are unstable to perturbations that are small only in extent and the set of such configurations is known in physics as the binodal region [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' From the perspective of the mathematical theory of elastic phase transitions the analog of the binodal region would incorporate the homogeneous states that fail to be strong minima of the energy functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Therefore, the binodal region is a subset in the configurational space of strain measures where the quasi convex envelope lays below the energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Locating the boundaries of the binodal region (known simply as a binodal) in the ’near-liquid’ limit constitutes the main task of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' While remaining nontrivial, this task appears, a priori as more tractable than the task of constructing the actual quasiconvex envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In our prior work we have developed a general method for identifying the subsets of the 2 d1 d2 d h(d) Figure 1: Double-well structure of the energy density h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' binodal supporting the laminate type energy minimizing configurations [18, 23, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Behind this method is the study of stability of the jump set—a codimension one variety in the phase space that has a dual nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' On the one hand it determines the set of pairs F± that could be the traces of the deformation gradient at the phase boundary in a stable configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' On the other, the jump set consists of points that are at most marginally stable in the sense that their every neighborhood contains points where quasiconvexity fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Therefore, if one can prove quasiconvexity at a point on the jump set, then this point must lie on the binodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In addition, we have also developed tools to constrain the location of the binodal by means of addressing nucleation phenomenon directly [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' As we show in this paper, combined together, these two types of approaches can produce in the ’near-liquid’ limit a rather good practical understanding of the whole structure of the binodal, and even allow one to obtain the exact formulas for the quasiconvex envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' To highlight ideas we focus here only on the simplest family of non-quasiconvex energy densities known as Hadamard materials [27, 28]: W(F ) = µ 2|F |2 + h(det F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Specifically, we’ll be interested in the case of two space dimensions and assume that the function h(d) describes a generic double-well potential modeling isotropic-to-isotropic phase transitions (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The main advantage of this class of elastic materials is that one can identify a single parameter µ, scaling the effective rigidity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' by varying this parameter we can study the entire range of intermediate rigidity responses from ’strong’ (µ ≫ 1) to ’weak’ (µ ≪ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' A notable feature of the Hadamard materials is that the phase with the larger value of det F (smaller density) is characterized by a larger tangential (effective) rigidity than the phase with the smaller value of det F (larger density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' As a result, the latter is more ’liquid-like’ than the former and therefore the incipient phase transformation induced by compression can be expected to be different from the incipient phase transformation induced by stretching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' As we show in what follows, this asymmetry leads to a coexistence of ’strongly-solid’ and ’weakly-solid’ responses inside a single material model as, even in the absence of hysteresis, the direct and reverse solid-solid phase transitions proceed according to morphologically different transformation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' While for an Hadamard material the double well energy structure is described by the simplest scalar potential h(d), the results of relaxation of W(F ) are nontrivial due to the inherent incompatibility of the energy wells [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We recall that W(F ) is quasiconvex if and only if h(d) is convex [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The relaxation of W(F ) with non-convex h(d) is known for 3 the ‘infinitely-weak’ solids (effectively fluids) with µ = 0, where QW(F ) = h∗∗(det F ) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Previously we explicitly constructed the quasiconvex envelope for W(F ) in the ‘strongly- solid’ limit assuming that the shear modulus µ is sufficiently large and the corresponding quadratic term dominates the double-well term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In this case the formula for QW(F ) couples |F | and det F and the relaxed energy is sandwiched between W(F ) above and U(F ) = µ 2|F |2 + h∗∗(det F ) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In this paper we show that the constraint on µ in [24] was not a technical limitation, and that, as µ decreases, our formula for QW(F ) ceases to be valid in the subsets of the binodal region close to the ’liquid-like’ phase with smaller rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In the limit of small µ, we show that the relaxation of W(F ) goes through a chain of structural transitions with simple lamination persisting only in the vicinity of the pure ’solid-like’ phase, being replaced by very different phase arrangements close to the ’liquid-like’ phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Our main technical approach is to generate bounds on the binodal surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The simplest bounds is obtained by probing the binodal by means of nucleating first rank laminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Their optimality is proved by establishing their polyconvexity (and therefore quasi-convexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In this setting this is an algebraic problem, because the supporting null- Lagrangians can be constructed explicitly, [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In contrast with the strongly solid regime of large µ analyzed in [24], in the near liquid regime of small µ, not all of the first rank laminate bounds are optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' These bounds are then improved by nucleating second rank laminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' However, as shown in [26], the second rank laminate bounds are not optimal either, and are further improved for hydrostatic strains by means of nucleating a bounded circular inclusion in the infinite plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We conjecture that this bound is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' If our conjecture is true, then the values of the deformation gradient in the exterior of the circular nucleus would provide a bound on the binodal from the outside of the binodal region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Another consequence of the assumed optimality of the inclusion-based nucleation bound is the explicit formula for the quasiconvex envelope QW(F ) at all hydrostatic strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' By juxtaposing the hypothetical bound provided by the study of bounded inclusions and unbounded second rank laminates we derive tight two-sided bounds on the binodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' As we demonstrate in [20], both bounds remain tight in the full range of parameters for which the bounds are meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Moreover, the hypothetical bound being in complete agreement with the numerically computed rank-one convex binodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In Section 2 we recall some general results from the calculus of variations for nonconvex vectorial problems, used in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In Section 3 we specialize these results for the Hadamard material and present the numerical illustrations of the obtained bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Analytical results for the limiting case µ → 0 are pre- sented in Section 4 where we also compare them with numerical computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In Section 5 we demonstrate the far reaching consequences of the assumed optimality of the nucleation bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The paper ends with a general discussion and conclusions in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 4 2 Preliminaries Binodal region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hyperelastic materials in a d-dimensional space have the following form of the energy stored in the deformed elastic body E[y] = � Ω W(∇y(x))dx, where Ω ⊂ Rd is the reference configuration, and y : Ω → Rd is the deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In order to understand the stable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' experimentally observable) configurations of the body it is often necessary to replace the energy density W(F ) with a relaxed one QW(F ), called quasiconvexification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Even though, there is a formula for QW(F ) [8]: QW(F ) = inf φ∈C∞ 0 (D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='Rn) 1 |D| � D W(F + ∇φ)dx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1) there is no systematic approaches to compute it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' A simpler, but just as useful an object, is the elastic binodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' An elastic binodal is the boundary of the binodal region B = {F : W(F ) < QW(F )}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2) Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We say that the matrix F is stable, if W(F ) = QW(F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Thus, the binodal is the boundary separating the binodal region from the set of stable points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Jump set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' While we acknowledge that there could be rank-one convex, non quasiconvex functions, most cases of practical interest in elastic phase transitions feature multiwell ener- gies that are not rank-one convex and possess a non-trivial jump set, stable points of which form a part of the binodal (or the entire binodal, if one is lucky).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The jump set is the set of solutions F = F− of the equations \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 F+ = F− + a ⊗ n, [[P ]]n = 0, [[P T ]]a = 0, [[W]] − ⟨{{P }}, [[F ]]⟩ = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='3) where a ̸= 0 and |n| = 1 are thought to be excluded from the above system resulting in a single scalar equation for F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We refer the reader to [26] for a discussion of the geometry of the solution set of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Here we used the standard notations P± = WF (F±), [[F ]] = F+ − F−, {{P }} = P+ + P− 2 , ⟨A, B⟩ = Tr (ABT), where WF indicates the matrix of partial derivatives Pij = ∂W/∂Fij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 5 The points on the jump set belong either to the binodal or to the binodal region B, [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hence, the jump set always represents a bound on the binodal region from within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' One of the easy ways to detect the unstable parts of the jump set is to use the Weierstrass condition, which is necessary for stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' W ◦(F , b ⊗ m) ≥ 0, ∀b ∈ Rn, |m| = 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='4) where W ◦(F , H) = W(F + H) − W(F ) − ⟨WF (F ), H⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We have proved in [22] that the pairs of points F± on the jump set are either both stable or both unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hence, a point F+ satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='4) can be still classified as unstable, if F− fails (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' While there are other conditions of stability that don’t follow from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='4) (see [23]) we will only make use of an easily verifiable corollary of(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='4) that restricts the rank-one test fields b ⊗ m to an infinitesimally small neighborhood of [[F ]] = a ⊗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Currently, the only general tool for establishing stability is polyconvexity, which is suf- ficient but rather far from necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In two dimensions it reduces to finding a constant m ∈ R, such that W ◦(F , H) − m det H ≥ 0, ∀H ∈ R2×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5) If (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5) holds, then F is stable in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' For points F± on the jump set, however, the only value of m that could possibly work is, as shown in [25], m = ⟨[[P ]], cof[[F ]]⟩ |[[F ]]|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='6) Secondary jump set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' An improved bound on the binodal is provided by the secondary jump set corresponding to the nucleation of a rank-two laminate in the infinite homoge- neously strained space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Thus, the secondary jump set is defined by the system of equations \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 F = F + b ⊗ m, P m = P m, P Tb = P Tb, W(F ) − W = P m · b, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='7) where the pair F±, to be determined, is assumed to satisfy the primary jump set equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='3), while W = λW(F+) + (1 − λ)W(F−), P = λP+ + (1 − λ)P−, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='8) for some λ ∈ [0, 1], which also plays the role of a variable to be solved for in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='7), along with F , b ̸= 0, and |m| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Once again, the secondary jump set represents a bound on the binodal region from within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Nucleation criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Let us now recall another method of probing the binodal: nucleation of inclusions either of a prescribed shape [5, 33, 31] or of an optimal inclusion, whose shape must be determined [29, 35, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The theory justifying why these tests probe the binodal 6 was developed in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In the case of “nucleation of a bounded inclusion”, the criterion for F0 to be “marginally stable”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' to lie in the closure of B, is the existence of a field φ ∈ S = {φ ∈ L2 loc(Rd) : ∇φ ∈ L2(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Rd)}, such that ∇ · P (F0 + ∇φ) = 0, ∇ · P ∗(F0 + ∇φ) = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='9) in the sense of distribution in Rd, where P (F ) = WF (F ), P ∗(F ) = W(F )Id − F TP (F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We also need to verify the non-degeneracy of the solution φ: � Rd W ◦ F (F0, ∇φ)dx ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='10) In the case of nucleation of an actual inclusion ω with smooth boundary the verification of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='9) consists in verifying that the field φ ∈ S solves ∇ · P (F0 + ∇φ) = 0 both inside and outside of ω, together with the condition that the traces F±(x) = F0 + ∇φ±(x) on the two sides of ∂ω form a corresponding pair on the jump set for each x ∈ ∂ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' If, in addition, we can somehow prove that F + ∇φ(x) is stable in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2, for each x ∈ Rd, then F0 must lie on the binodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Conversely, if it is known that that at some x0 ∈ Rd the matrix F0 + ∇φ(x0) is unstable, then F0 must lie in the interior of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 3 Hadamard material In this paper we focus our attention on a particularly simple, yet nontrivial energy W(F ) = µ 2|F |2 + h(d), F ∈ {F ∈ GL(n) : det F > 0}, d = det F , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1) where h(d) is a C2(0, +∞) function with a double-well shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In our explicit computations and illustrations we use the quartic double-well energy1 h(d) = (d − d1)2(d − d2)2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2) which affords certain simplification of general formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Jump set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We recall (see [24]) that in two dimensions the jump set of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1) consists of matrices F±, whose two singular values labelled ε0 and ε± satisfy the equations ε0[[h′]] + µ[[ε]] = 0, [[h]] − {{h′}}[[d]] = 0, d± = det F± = ε0ε±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='3) The notation reflects that for each pair F± on the jump set there is a frame in which both matrices are diagonal and share the same singular value ε0 with the same eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 1Formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2) only needs to hold in an arbitrary neighborhoodof [d1, d2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The potential h(d) can be modified outside of that neighborhood arbitrarily, as long as h∗∗(d) = h(d) there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In particular, the singular behavior of h(d) as d → 0+, required in nonlinear elasticity, can be easily assured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 7 Equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='3) can be used to derive the semi-explicit parametric equations of the jump set, where, say d+ = ε0ε+, can serve as a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Given d+ we can use the second equation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='3) to compute d− = D(d+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Then, multiplying the first equation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='3) by ε0 we obtain the parametric equations \uf8f1 \uf8f2 \uf8f3 ε0(d+) = � −µ[[d]] [[h′]] , ε+(d+) = d+ ε0(d+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In the case of potential (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2) we obtain [[h]] − {{h′}}[[d]] = [[d]]3(d1 + d2 − d+ − d−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hence, d− = d1 + d2 − d+ = D(d+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' It follows that {{h′}} = 0, ε+ + ε− = d1 + d2 ε0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='4) In particular, we can eliminate h′(d±) from our formulas by means of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='4): h′(d±) = {{h′}} ± 1 2[[h′]] = ∓µ 2 [[ε]] ε0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5) For quartic energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2) we can also write the equation of the jump set explicitly as ε± = ε±(ε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Indeed, ε± = d±/ε0, while d± solves (d± − d1)(d± − d2) = − µ 4ε2 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='6) The two roots of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='6) are the values of d±, where, by convention, we denote by d+ the larger root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='6) has exactly two real roots whenever ε0 > √µ/(d2 − d1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hence, explicitly, ε± = 1 2ε0 � d1 + d2 ± � (d2 − d1)2 − µ ε2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='7) In our calculations we will use equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5) to eliminate all occurrences of h′(d±) and equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='7) to eliminate ε±, since the pair ε± is uniquely determined by a single parameter ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Numerical illustrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' When µ is large we have shown in [24] that the jump set [18] comprises the entire binodal, each point of which corresponds to the nucleation of a simple laminate, leading to an explicit formula for the relaxation QW(F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' As the shear modulus µ decreases, parts of the jump set will become unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The jump set will then undergo a topological change at µ = µtop and in the limit µ → 0, which is the main focus of this paper, a specific portion of it will remain stable, as we will show using methods from [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 2 shows the jump sets and indicates their unstable parts for four different values of the shear modulus µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The values of µ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 2 are chosen to be µ = 0, µtop/3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='9µtop, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5µtop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Dotted lines indicate “convexification hyperbolas”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=', hyperbolas ε2 = d1/ε1 and ε2 = d2/ε1, where the interval [d1, d2] is the interval on which h(d) differs from its convex hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 8 1 2 3 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 3 2 = 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='8168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 2 2 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='6617 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='6757 1 2 3 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 3 2 Figure 2: Jump sets for h(d) given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2) with d1 = 1, d2 = 3, and different values of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' All points outside of the region bounded by the convexification hyperbolas are well-known to be stable (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' [9]), since they are obviously polyconvex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' W-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In [23] we have shown that the easily computable corollary of the Weierstrass condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='4) for the energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1) has the form ε0 ≥ ε±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='8) In [24] we have shown that this condition is always satisfied for large values of µ as is evident from the lower right panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 2, while it has an obvious geometric interpretation in the two panels in which the part of the jump set failing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='8) is shown as a dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The points marked by red dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 2 that delimit the part of the jump set satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='8) will be called the Weierstrass points or W-points, for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We have shown in [26] that the solid portion of the jump set delimited by W-points is polyconvex for all sufficiently small µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' As we show below, one can provide an almost explicit characterization of all values of µ for which W-points are also points of polyconvexity assuming the quartic nonlinearity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' As discussed above, in order to prove the polyconvexity of W-points we need to establish (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5), where m is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' This problem has been already analyzed in [24], where we showed that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5) can be written as Φ(x, y) ≥ Φ(ε±, ε0) for all x, y, where Φ(x, y) = µ 2(x2 + y2) − αx − βy − h(xy) − mxy, 9 α = 2√µR{{d}}, β = µ2 + R4d+d− R√µ , m = [[h′d]] [[d]] , R = � −[[h′]] [[d]] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' According to the equations of the jump set (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='3) R = √µ/ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hence, we also have α = µ(ε+ + ε−), β = µ � ε0 + ε+ε− ε0 � , m = {{h′}} − µ{{ε}} ε0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' When we minimized Φ(x, y) over all (x, y), such that xy = d we have concluded that the minimizer is (d/y, y), where y = y(d) is the largest root of y4 − β0y3 + dα0y − d2 = 0, α0 = ε+ + ε−, β0 = ε0 + ε+ε− ε0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='9) while the minimum of Φ(x, y) is achieved at a finite point corresponding to a critical point of φ(d) = Φ(d/y(d), y(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In the special case of W-points we have ε+ = ε0 and therefore α0 = β0 = ε− + ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In this case equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='9) factors (y2 − d)(y2 − α0y + d) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The largest root is y = 1 2(α0 + � α2 0 − 4d), provided 0 < d ≤ α2 0/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' If d > α2 0/4, then the quartic has only two real roots y = ± √ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Thus, y(d) = � (α0 + � α2 0 − 4d)/2, d ≤ α2 0/4, √ d, d > α2 0/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In [24] we have also computed φ′(d) = µy(d)2 − β0y(d) d + h′(d) − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In the case of W-points for which β0 = α0 we see that y(d)2 − β0y(d) d = −1 when d ≤ α2 0/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hence, any critical points of φ(d) in this regime would have to satisfy h′(d) − µ − m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' One of the solutions is d−, which always satisfies d− ≤ α2 0/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' If this equation has 3 solutions, the the middle one corresponds to a local maximum of φ(d), while the third d∗ > d+ always fails to satisfy d∗ ≤ α2 0/4 because d+ = ε2 0 > (ε− + ε0)2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We conclude that the only critical points of φ(d) that need to be checked are the ones that satisfy d > α2 0/4, while φ′(d) = µ � 1 − α0 √ d � + h′(d) − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 10 Observe that φ′(d) > 0 when d ≥ max(α2 0, �d+), where �d+ is the largest root of h′(d) − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hence we only need to check for critical points in a specific bounded interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In fact, if h(d) is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2), then it is easy to see that φ′(d) > 0 for all d ≥ α2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hence, we only need to check for critical points of φ(d) on (α2 0/4, α2 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In addition, since {{h′}} = 0 for h(d), given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2) we have m = −µ{{ε}}/ε0 = −µα0/(2ε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Thus, we obtain the following characterization of polyconvexity of W-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Let h(d) be given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2), then W-points are polyconvex whenever min d∈ � α2 0 4 ,α2 0 � � h(d) + µ � d + α0d 2ε0 − 2α0 √ d �� = h(ε2 0) − µε0 �ε0 2 + 3ε− 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='10) where α0 = ε0 + ε−, with (ε0, ε−), (ε−, ε0), and (ε0, ε0) being the coordinates of W-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The right-hand side in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='10) is just φ(ε2 0), where φ(d) is the function being minimized in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' For quartic energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2) we compute the coordinates of W-points by solving −4d(d − d1)(d − d2) = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Then ε2 0 is the largest root, and ε− = d1 + d2 − ε2 0 ε0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We can compute the largest value of µ for which (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='10) holds by substituting µ = −4ε2 0(ε2 0 − d1)(ε2 0−d2) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='10) and regarding ε0 ≤ √d2 as a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' When ε0 = √d2, φ(d)−φ(d2) is a positive polynomial in x = √ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We then seek numerically the largest value of ε0 < √d2 for which the polynomial P(x) = (φ(x2) − φ(ε2 0))/(x − ε0)2 develops a double root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Algebraically this means seeking the largest root ε0 < √d2 of the discriminant (computed in Maple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' This solution gives the largest value of µ below which the W-points are polyconvex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' For example, when d1 = 1, d2 = 3, we have polyconvexity of W-points for all µ < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='35888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In this paper we will be interested exclusively in the case when W-points are quasiconvex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 2 the W-points are polyconvex in the top right panel and unstable in the bottom left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Secondary jump set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The algebraic equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='7) describing the secondary jump set can generally be solved only numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' By contrast, when µ is small, the asymptotics of the solutions can be computed explicitly, providing an excellent approximation to the computed secondary jump set for µ < 3, with d1 = 1, d2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' While the entire secondary jump set is unstable [26], we will see that it provides an excellent (inside) bound for the binodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Suppose that F0 lies on the secondary jump set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Then there exists ε±, y and λ ∈ [0, 1], such that the pair F0, F , where F = � ε 0 0 ε0 � , ε = λε+ + (1 − λ)ε−, satisfies the jump set equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We compute P = λP+ + (1 − λ)P− = � µε + h′ε0 0 0 µε0 + εh′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 11 We have P0 = µF0 + h′(d0)cofF0 = µ � ε 0 0 ε0 � + µb ⊗ m + h′(d0) �� ε0 0 0 ε � + b⊥ ⊗ m⊥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Thus, the second and the third equations in the jump set system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='7) become \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 \uf8ee \uf8f0 (h′(d0) − h′)ε0 0 0 h′(d0)ε − εh′ \uf8f9 \uf8fb m = −µb, \uf8ee \uf8f0 (h′(d0) − h′)ε0 0 0 h′(d0)ε − εh′ \uf8f9 \uf8fb b = −µ|b|2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' These equations result in 3 possibilities (a) (h′(d0) − h′)ε0 = h′(d0)ε − εh′ = −γ, µb = γm, m ∈ S1 (b) (h′(d0) − h′)ε0 = −(h′(d0)ε − εh′) = −γ, µb = γI−m, I− = � 1 0 0 −1 � , m ∈ S1 (c) (h′(d0) − h′)ε0 ̸= ±(h′(d0)ε − εh′) Possibility (c) implies that F0 must be diagonal, and will be our main focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In [26] we show that possibilities (a) and (b) have no solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Let us therefore assume that F± is diagonal and has the form F± = � ε± 0 0 ε0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' This implies that F − F0 = βe2 ⊗ e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In particular F0 = � x0 0 0 y0 � , x0 = ε = λε+ + (1 − λ)ε−, λ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Let us compute the diagonal matrices P± using equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' P 11 ± = µε± + h′(d±)ε0 = µ{{ε}} = µ(d1 + d2) 2ε0 , P 22 ± = µε0 + h′(d±)ε± = µ � ε0 ∓ [[ε]]ε± 2ε0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Let us compute the diagonal matrix P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' P 11 0 = µx0 + h′(d0)y0 = µε + h′(d0)d0 ε , P 22 0 = µy0 + h′(d0)x0 = µd0 ε + h′(d0)ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Traction continuity equation (P − P0)e2 = 0 then becomes ε0 + [[ε]] 2ε0 (ε− − 2λ{{ε}}) − d0 ε − h′(d0) µ ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 12 It will be convenient to use ε as a variable in place of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Replacing λ above using ε = ε−+λ[[ε]] we obtain d0 ε = ε0 + 1 ε0 (ε+ε− − {{ε}}ε) − h′(d0) µ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='11) Let us now compute all the terms in the last equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' W(F0) = µ 2 (ε2 + y2 0) + h(d0) = µ 2 � ε2 + d2 0 ε2 � + h(d0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Next we compute W = W− + λ[[W]] = W− + λµ[[ε]]{{ε}} = W− + µ(ε − ε−){{ε}}, where [[h]] = −{{h′}}[[d]] = 0 has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We compute h(d−) = [(d− − d1)(d− − d1)]2 = µ2 16ε4 0 , according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Therefore, W− = µ 2 (ε2 − + ε2 0) + µ2 16ε4 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We then compute F0 − F = (y0 − ε0)e2 ⊗ e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Therefore ⟨P , F0 − F ⟩ = µ �d0 ε − ε0 � � ε0 + 1 ε0 (ε+ε− − {{ε}}ε) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Finally, the Maxwell equation W(F0) − W = ⟨P , F0 − F ⟩ can be written as 1 2 � ε2 + d2 0 ε2 � +h(d0) µ −(ε−ε−){{ε}}−1 2(ε2 −+ε2 0)− µ 16ε4 0 = �d0 ε − ε0 � � ε0 + 1 ε0 (ε+ε− − {{ε}}ε) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='12) Next we replace in the above form of the Maxwell relation the expression d0/ε by its expres- sion from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='11) As a result of such a substitution the Maxwell relation will also become a quadratic equation in ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' This permits us to eliminate this variable as a rational expression in terms of ε0 and d0, while the Maxwell relation will also reduce to a rational relation between d0 and ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' This calculation can only be done with the aid of a computer algebra system, since the remaining equation F(ε0, d0) = 0 is very long and complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Now for a given choice of numerical values of µ, d1 and d2 we can solve F(ε0, d0) = 0 numerically and then extract those solutions which satisfy λ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The result for d1 = 1, d2 = 2 and µ = µtop/3 is shown as a green curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' As we can see, it identifies all points between the green curve and the dashed lines of the primary jump set as a part of the binodal region—an improvement over the primary jump set bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' While, it is not apparent from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 3, the secondary jump set consists of two curves related by symmetry with respect to the bisector of the first quadrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Each of the curves 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='8168 Figure 3: Secondary jump set computed by numerically solving equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' are cut-off at their intersection with each other at the bisector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Each curve starts at a W- point and ends at a point (not shown) on the dashed part of the jump set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The endpoints of the secondary jump set correspond to the extreme values 0 and 1 of the volume fraction λ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The corresponding points on the secondary jump set must lie on the primary jump set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' There are two possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Either F ̸= F or F = F at λ = 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In the former case the limiting position F+ of F is rank-one related to two different points on the jump set: F− (layer normal e1) and F (layer normal e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' W-point F+ is the only one with this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' All other points F+ on the jump set have a unique counterpart F−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In the latter case a detailed asymptotic analysis shows that that the common limit point of F and F must achieve equality in the “Legendre-Hadamard for phase boundaries” inequality from [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' This point lies on the dashed part of the jump set and is used in numerical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The details of the analysis will be spelled out in a forthcoming paper [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Circular nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In [26] it is shown that the secondary jump set (green curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 3) is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' That means that the corresponding bound on the binodal is not optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We can improve the bound using another method of probing the binodal: nucleation of equilib- rium energy-neutral inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The theory justifying why such nucleation tests probe the binodal was developed in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In the case of the isotropic, objective energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1) and a hydrostatic loading it is natural that the shape of an optimal precipitate should be circular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The deformation gradient inside the circular precipitate must be a constant hydrostatic field F0 = εW 0 I2, since that field is rank-one connected to an infinite family of fields FR = R � εW − 0 0 εW 0 � RT, R ∈ SO(2), where (εW − , εW 0 ) is a coordinate of one of the W-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The deformation gradient outside of the circular inclusion must solve an Euler-Lagrange equation for the energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1) µ∆y + (cof∇y)∇h′(det ∇y) = 0, x ∈ R2 \\ B(0, 1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='13) 14 and agree with FR at the point Re1 on the boundary of the circular inclusion: ∇y(x) = εW − n ⊗ n + εW 0 τ ⊗ τ, x ∈ ∂B(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='14) In this case both equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='9) will be satisfied for the possibly marginally stable matrix F∞ = lim |x|→∞ ∇y(x) = ε∞I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We also know that that the values of ∇y(x) inside the circular inclusion and its trace on the outside boundary of the inclusion are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Our results from [19] then say that either F∞ lies on the binodal and all values ∇y(x) in the exterior of the inclusion are stable, or F∞ lies in the interior of the binodal region B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In our special radially symmetric case we look for a radially symmetric solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='13) y = η(r)ˆx, |x| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The the unknown function η(r) must solve � η r d drh′ � ηη′ r � + µ � η′ + η r �′ = 0, r > 1, η′(1) = εW − , η(1) = εW 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='15) The nonlinear second order ODE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='15) cannot be integrated explicitly, but can be solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In order to do so, we need to convert the infinite range r > 1 into a finite one by means of the change of the independent variable x = 1/r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' It will also be convenient to change the dependent variable v = η/r, so that v(x) would have a finite limit, when x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Then v(x) solves v′′ = − (v′)2vh′′(v2 − 2xvv′) µ + v2h′′(v2 − 2xvv′), x ∈ [0, 1], v(1) = εW 0 , v′(1) = εW 0 − εW − 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='16) The value ε∞ = v(0)I2 found numerically is shown as a blue dot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' It provides an improved bound on the binodal compared to the secondary jump set (green line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 4) by showing that hydrostatic strains between the blue dot and the green line are unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' This conclusion holds, provided the non-degeneracy condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='10) is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' A calculation shows that � Rd W ◦ F (F0, ∇φ)dx = −I2 � R2 h′′(ε2 ∞)ε2 ∞ � η′(r) + η(r) r − 2ε∞ � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Thus, � Rd W ◦ F (F0, ∇φ)dx = −2πh′′(ε2 ∞)ε2 ∞I2 lim r→∞(rη(r) − ε∞r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' To see that the limit above exists and is non-zero, at least for small µ > 0, we simply solve (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='15) for µ = 0, for which εW 0 = √d2, εW − = d1 √d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The solution is η(r) = √d1r2 + d2 − d1, and we easily see that lim r→∞(rη(r) − ε∞r2) = d2 − d1 2√d1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 15 Hence, the non-degeneracy condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='10) will hold, at least for sufficiently small µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The non-degeneracy will also hold for all µ below the topological transition, because if we write �η(r) = η(r) − ε∞r, then (assuming that �η′(r) → 0, as r → ∞) �η(r) will solve, when r is large, the differential equation ε∞h′′(ε2 ∞) � ε∞ � �η′ + �η r � + �η′�η r � + µ � �η′ + �η r � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' This integrates to ε∞h′′(ε2 ∞)(2ε∞r�η + �η2) + 2µr�η = 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Since �η, satisfying �η′(r) → 0, as r → ∞, cannot be zero (it is the leading term of η(r)−ε∞r), we conclude that the constant of integration c cannot be zero either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hence, we obtain that lim r→∞(rη(r) − ε∞r2) = lim r→∞ r�η(r) = c µ + ε2∞h′′(ε2∞) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Polyconvexity limits along εI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We now turn to the problem of proving polyconvexity at points F = εI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' To succeed we need to find a constant m ∈ R, such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' For our energy we compute W ◦(F , H) = µ 2|H|2 + h(ε2 + d + εθ) − h(ε2) − εh′(ε2)θ, θ = Tr H, d = det H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We also have |H|2 = 1 2|H − HT|2 − 2d + θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hence we need to find m ∈ R, such that µθ2 2 + h(ε2 + d + εθ) − h(ε2) − εh′(ε2)θ ≥ (m + µ)d, ∀d ≤ θ2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='17) In particular the inequality must hold for d = θ2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In that case we must have m ≤ µ + 4 min θ∈R h(ε2 + θ2/4 + εθ) − h(ε2) − εh′(ε2)θ θ2 = m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='18) The infimum of the smooth function F(d, θ) = µθ2 2 + h(ε2 + d + εθ) − h(ε2) − εh′(ε2)θ − (m + µ)d must be attained either at a critical point or at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Obviously, if along the minimizing sequence the quantity δ = ε2 + d + εθ goes to infinity, then the values of F must also go to +∞, which cannot happen along a minimizing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hence, (d, θ) go to infinity so that δ stays bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hence, we can switch variables and instead of the pair (d, θ) consider the pair (δ, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In this case d = δ − εθ − ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hence, F(δ, θ) = µθ2 2 + h(δ) − h(ε2) − εh′(ε2)θ − (m + µ)(δ − εθ − ε2), 16 where δ ≤ (θ/2 + ε)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Minimizing F(δ, θ) with respect to θ we obtain θ = −ε(m + µ − h′(ε2)) µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hence we need to minimize f(δ) = h(δ) − h(ε2) − (m + µ)(δ − ε2) − ε2(m + µ − h′(ε2))2 2µ , over all δ satisfying δ ≤ ε2(h′(ε2) + µ − m)2 4µ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='19) We remark that taking θ = −4ε in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='18) we conclude that m∗ ≤ µ + h′(ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Thus, the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='19) is monotone decreasing in m, when m ≤ m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Now, the minimum is achieved either at the boundary, where equality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='19) holds, or at a critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' It cannot be “achieved” at infinity, where f(δ) is +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' If the minimum is achieved at the boundary then f(δ) ≥ 0 for all δ, provided m ≤ m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' If the minimum is achieved at a critical point h′(δ) = m + µ, then several possibilities need to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Let us first assume that the equation h′(δ) = m∗ + µ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='20) has a single root δ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' If that root fails to satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='19) with m = m∗, then for m = m∗ the function f(δ) has no critical points and polyconvexity holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' If that root satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='19), then all values δ < δ∗ are admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We then substitute m = h′(δ) − µ, δ ≤ δ∗ into f(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The resulting function f(δ) = h(δ) − h(ε2) − h′(δ)(δ − ε2) − ε2(h′(δ) − h′(ε2))2 2µ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='21) can be plotted on (−∞, δ∗] versus m = h′(δ) − µ to see if there are values of m above which all values of f(δ) are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Yet another possibility is when equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='20) has 3 real roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' If even the smallest root δ∗ fails to satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='19) with m = m∗, then there are no critical points and polyconvexity holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Otherwise, all values of δ ≤ δ∗ are admissible and we can prove failure of polyconvexity by plotting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='21) versus m(δ) = h′(δ)−µ on δ ≤ δ∗ and checking that it is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In fact, we believe that polyconvexity fails in all cases when δ∗ satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In order to exhibit this failure we only need to produce a single value of admissible δ for which f(δ), given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='21), is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hence, if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='21) is negative for all δ ≤ δ∗, then there is no need to examine other intervals of admissible δ, since for any m ≤ m∗ there is always an admissible δ ≤ δ∗, which makes f(δ) negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' However, if f(δ) has a region where it is positive, then one needs to examine other areas of admissibility and check whether f(δ) is negative for the same values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Thus, we obtain an algorithm that can prove polyconvexity or failure of 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2 2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='8168 nucleation bound polyconvexity bound secondary jump set Figure 4: Bounds on the binodal from the inside and the outside of the binodal region along hydrostatic strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' it in many, but not all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Polyconvexity holds whenever δ∗ > ε2(h′(ε2) + µ − m∗)2 4µ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' for all solutions δ∗ of h′(δ) = µ + m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Polyconvexity fails whenever f(δ) < 0 for all δ < δ∗, provided δ∗ ≤ ε2(h′(ε2) + µ − m∗)2 4µ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' If ε2 = d1, then the minimization problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='18) simplifies: min θ∈R h(d1 + θ√d1 + θ2/4) θ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We first observe that in general θ = 0 is not a minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Then there are 3 minimizers: θ = −4 � d1, θ = ±2 � d2 − 2 � d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' When ε = √d1 + x, then the minimizer θ(x) must be located near one of the above 3 minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We can then write θ = θ0 + y for the minimizer, where θ0 denotes one of the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' If we write the function under the minimum as H(ε, θ), then at the minimum we must have ∂H/∂θ = 0, which gives the equation x ∂2H ∂θ∂ε + y∂2H ∂θ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' After solving for y and substituting this solution back into H we obtain H = x \uf8eb \uf8ec \uf8ed∂H ∂ε − ∂H ∂θ ∂2H ∂θ∂ε ∂2H ∂θ2 \uf8f6 \uf8f7 \uf8f8 , 18 where derivatives are evaluated at (√d1, θ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Maple calculation yields H = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 x 2 √d1h′′(d1), θ0 = −4√d1, xd1h′′(d1) √d1+√d2 , θ0 = −2√d2 − 2√d1, xd1h′′(d1) √d1−√d2 , θ0 = 2√d2 − 2√d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' This shows that θ = 2√d2 − 2√d1 + y is the minimizer, while m∗ = µ − 4xd1 √d2 − √d1 h′′(d1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In particular, the equation h′(δ) = m∗ + µ will have 3 real roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The smallest one δ∗ will be near d1: δ∗ = d1 + µ + m∗ h′′(d1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Finally, polyconvexity will hold if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='19) fails when δ = δ∗ and m = m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In other words we must have (asymptotically) ε ≤ � d1 + µ h′′(d1)√d1 √d2 − √d1 √d2 + √d1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='22) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 4 showing the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='22) as a red dot implies that ε∞I2 fails to be polyconvex, but by a very slim margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The ordering of the bounds in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 4 persists on the entire range of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We see that the gap between established stability (along the bisector below the red dot) and established instability (along the bisector above the blue dot) is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 4 Limiting case µ → 0 In this section we derive explicit asymptotics of the secondary jump set and the nucleation bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Secondary jump set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Expanding equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='7) to first order in µ we obtain ε+ = d2 ε0 − µ 4ε3 0(d2 − d1) + O(µ2), ε− = d1 ε0 + µ 4ε3 0(d2 − d1) + O(µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1) When d1 and d2 are fixed we think of ε± as functions of ε0 and µ, even if we suppress this in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Clearly, when µ → 0 we have ε+ → d2/ε0, ε− → d1/ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The parametric equations (x0(ε0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' µ), y0(ε0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' µ)) of secondary jump set converge, when µ → 0, to the hyperbola x0y0 = d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In particular, d0(ε0, µ) → d1, as µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The volume fraction λ of the rank-one laminate used in the second rank laminate is also a function of ε0 and µ and must have a limit (at least along a subsequence) λ(ε0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' µ) → λ0(ε0), as µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='11) shows that d0 = d1 + µδ + O(µ2), while δ satisfies d ε0 � ε0 + 1 ε0 �d1 d2 ε2 0 − d1 + d2 2ε2 0 d �� − d1 − 2δ(d2 − d1)2d 2 ε2 0 = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2) 19 where d = λd2 + (1 − λ)d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2) was obtained simply by passing to the limit as µ → 0 in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' When we pass to the limit as µ → 0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='12) we obtain (d − d1)2(ε4 0 + d 2 − 2d2d) 2ε2 0d 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='3) The dependence of d on the volume fraction λ is essential and should not disappear in the limit µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Therefore, the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='3) that we are after is d = d2 − � d2 2 − ε4 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='4) where the choice of the root was dictated by the requirement that d ≤ d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Combining this with the requirement that d ≥ d1 shows that 4� d2 2 − (d2 − d1)2 ≤ ε0 ≤ � d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5) Substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='4) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2) gives the explicit formula for δ: δ = ε4 0(d2 − d1) − 2(d2 2 − ε4 0)(d2 − � d2 2 − ε4 0) 4ε2 0(d2 − d1)2(d2 − � d2 2 − ε4 0)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='6) It seem that in order to obtain the correct asymptotics of the secondary jump set we need to obtain the first order asymptotics of ε: ε = d2 − � d2 2 − ε4 0 ε0 + �εµ + O(µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='7) In fact, this is not necessary because the leading order asymptotics of d0 is a constant d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In that case, as far as the first order asymptotics as µ → 0 is concerned, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='7) simply corresponds to reparametrizing the curve \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 x0 = d2 − � d2 2 − ε4 0 ε0 , y0 = d1 + µδ(ε0) x0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='8) Indeed, if we change the curve parameter ε0 to ε0 + µ�ε/x′ 0(ε0), then x0 � ε0 + µ�ε x′ 0(ε0) � = x0(ε0) + µ�ε + O(µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' At the same time y0 � ε0 + µ�ε x′ 0(ε0) � = d1 x0(ε0) − µd1�ε x0(ε0)2 + µδ(ε0) x0(ε0) + O(µ2) = d1 + µδ x0 + µ�ε + O(µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 20 0 1 2 3 4 5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='15 asymptotic numerical pcx asymptotic Figure 5: Comparison between the asymptotics (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='12) and ε∞ obtained from the numerical solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We conclude that equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='8) correctly describes the asymptotics of the secondary jump set with O(µ2) error, where the parameter ε0 varies according to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' When ε0 = √d2, the secondary jump set enters one of the W-points, while when ε0 = 4� d2 2 − (d2 − d1)2 the secondary jump set enters its other end at the “Legendre-Hadamard for phase boundaries” bound that for small µ lies on the dashed part of the jump set in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The plot of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='8) is in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 3 and is indistinguishable from the numerically obtained curve using the full (non-asymptotic) versions of secondary jump set equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Circular nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In the near-liquid limit µ → 0 we can find the asymptotics of the solution explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We know that in the limit µ → 0 the field d(x) = det ∇y(x) must approach d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Hence, ηη′ r = d1 + µδ(r) + O(µ2), r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' That implies η(r) = � d1r2 + c0 + µ�η(r) + O(µ2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='9) and therefore, δ(r) = 1 r � �η(r) � d1r2 + c0 �′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Substituting this ansatz into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='15) we obtain µ √d1r2 + c0 r h′′(d1)δ′(r) + µ � d1r √d1r2 + c0 + √d1r2 + c0 r �′ + O(µ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='10) Initial conditions from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='15) imply that c0 = d2 − d1, �η(1) = − d2 − d1 4d3/2 2 h′′(d2) , �η′(1) = d1(d2 − d1) 2d3/2 2 � 1 d1h′′(d1) + 1 2d2h′′(d2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 21 Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='10) is easy to integrate (observing that √d1r2 + c0/r is decreasing from √d2 to √d1 and is therefore uniformly bounded away from zero and ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' h′′(d1)�η(r) = c1r2 + c2 √d1r2 + c0 − r2 2√d1r2 + c0 ln √d1r2 + c0 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='11) From initial conditions for �η(r) we obtain c1 = 1 2 ln � d2, c2 = −(d2 − d1)h′′(d1) 4d2h′′(d2) , and hence ε∞ = �� d1 + µ 2h′′(d1)√d1 ln √d2 √d1 � I2 + O(µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='12) Figure 5 shows the quality of the asymptotics for the entire range of shear moduli µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The numbers on the y-axis indicate that even for values of µ that are not particularly small the asymptotics (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='12) gives a good approximation of the actual value of ε∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' For example, for µ = 3 the relative discrepancy is only around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 5 A glimpse into the relaxed energy Hypothetical bounds on the binodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We have seen in the foregoing discussion that the energy W(F ) is not polyconvex at F = ε∞I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' This is not very surprising, since polyconvexity is usually strictly stronger that quasiconvexity and we expect and conjecture that F = ε∞I2 lies on the binodal—at the very edge of quasiconvexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Here we recall our observation that if someone could prove that F = ε∞I2 is stable, then we would immediately conclude that for every |x| > 1 ∇y(x) = η′(r)ˆx ⊗ ˆx + η(r) r (I2 − ˆx ⊗ ˆx) would be stable in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2, providing a bound on the binodal from the outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' For the entire range of µ for which W-points are polyconvex the union of the curves � ε1 = η(r) r , ε2 = η′(r), and � ε1 = η′(r) ε2 = η(r) r , r > 1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1) are indistinguishable from the secondary jump set curves shown in green in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 6 shows the same blown-up part of the strain space as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 4, where the curves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1) shown in magenta are passing through the blue point from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Assuming the conjectured stability of ε∞I2, the magenta curve must lie outside of binodal region, while secondary jump set lies in its interior [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Thus, the binodal of the energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1) would have to lie between the green and the magenta curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We will even go so far as to conjecture that the magenta curve is in fact the actual binodal of the energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Regardless, under the assumption of stability of ε∞I2, the magenta line represents a rather tight outside bound on the binodal region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Another byproduct of the assumed stability of ε∞I2 would be the formula for the 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='5 2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='8168 hypothetical binodal known binodal region 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2 2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='8168 nucleation bound polyconvexity bound secondary jump set outside bound Figure 6: A hypothetical bound on the binodal region from the outside, assuming stability of ε∞I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' quasiconvex envelope QW(F ) for hydrostatic strains F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' If F = ε∞I2 is stable, then our radial solution ∇y(x) = η(r)ˆx of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='15) is also a global minimizer in every finite ball B(0, R), where it satisfies the affine boundary condition y(x) = (η(R)/R)x, x ∈ ∂B(0, R) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The energy of such configurations must necessarily be QW(η(R)I2/R)|B(0, R)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' This permits us to compute QW(εI2) for all ε, as the energy of y(x) = η(r)ˆx in B(0, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Using the Clapeyron-type formula for the nonlinear elastic energy stored in an equilibrium stationary configuration we obtain for F = η(R)I2/R: [21] |B(0, R)|QW(F ) = 1 2 � ∂B(0,R) {P (∇y)n · y + P ∗(∇y)n · x}dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2) Substituting n = ˆx, y = η(r)ˆx into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2) we obtain QW �η(R) R I2 � = 2(µ − h′(d))d − µη′(R)2 + (2h′(d) + µ)η(R)2 R2 + 2h(d), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='3) where d = η′(R)η(R) R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' When µ is small we can use the explicit asymptotic formulas (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='9), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='11) for η(r) to obtain an explicit asymptotics for QW(εI2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The plot of QW(εI2), coming from the numerical solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='15), as well as its explicit asymptotic approximation, superposed on the plot of W(εI2) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' 6 Conclusions In this paper our far reaching goal was to solve analytically the relaxation problem for the double well Hadamard energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='1) in two space dimensions when the rigidity measure µ 23 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content='8 3 4 5 6 7 8 9 Energy W( I2) QW( I2) QWasym( I2) Figure 7: Quasiconvex envelope of W(F ) restricted to hydrostatic strains F = εI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' An apparently more attainable target was to locate the corresponding binodal region inside the strain space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The study of the limit µ → 0 was expected to show how the ’cooperative’ , rigidity-controlled microstructures, dominating the quasiconvex envelope at large µ, give rise to more arbitrary and less controlled microstructures characterizing first order phase transitions in zero rigidity liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' We used some of our previously developed methods to pinpoint a substantial portion of the binodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' While our general methods apply for Hadamard materials in the entire parameter range and are amenable to numerical implementation, here we were able to obtain the explicit asymptotic formulas only in the ’near-liquid’ regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In particular, we showed that in an ’almost liquid’ limit, a subset of the jump set adjacent to the high strain phase remains stable which ensures that simple lamination delivers the corresponding part of the binodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' This means that even when the reference measure of rigidity µ is small, the high strain phase maintains its tangential rigidity at the level which ensures solid-solid like nature of the incipient phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Instead, our analysis showed that the subset of the jump set adjacent to the low strain and low rigidity phase is unstable in the µ → 0 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Moreover, the secondary jump set is also unstable in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' This result suggests that laminates of any finite rank are unstable near the corresponding subset of the binodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' As we’ve demonstrated for hydrostatic strains, the reduced rigidity control in this range allows the incipient phase transformation to proceed non-cooperatively through the formation of isolated nuclei of the more rigid phase inside the matrix of the less rigid phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Such transformation mechanism is already very similar to the one believed to be operating in purely fluid-fluid phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Whether the revealed asymmetry of the transformation mechanism between the direct and reverse transformation is a peculiarity of the Hadamard material or whether this striking phenomenon has a more general nature, remains to be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' It shows, however, the intricate role of rigidity in structural transformations which, even if weak, can produce rather complex structure of the relaxed energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' This complexity will then reflect a gradual transition from geometrically ordered microstructures, controlled by long range elastic interactions, 24 to more ’fluid’ microstructures whose spatial organization is mostly affected by molecular interactions operating at short range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' In other words, in this limit the direct and reverse solid-solid phase transitions can operate through different transformation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The fact that the ensuing complex structure of the relaxed energy at ’almost-liquid’ solid-solid phase transitions is ultimately replaced by a simple energy convexification at fluid-fluid phase transitions points to a singular nature of the limit µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' YG was supported by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' DMS-2005538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' The work of LT was supported by the French grant ANR-10- IDEX-0001-02 PSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} +page_content=' Ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQf8v6L/content/2301.01909v1.pdf'} 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a/69E4T4oBgHgl3EQfcgw0/content/tmp_files/2301.05083v1.pdf.txt b/69E4T4oBgHgl3EQfcgw0/content/tmp_files/2301.05083v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..96797936f8a855fb997ec172bfa2308e6cc40a53 --- /dev/null +++ b/69E4T4oBgHgl3EQfcgw0/content/tmp_files/2301.05083v1.pdf.txt @@ -0,0 +1,1509 @@ +Gravitational collapse of scalar and vector fields +Karim Mosani,∗ Koushiki,† Pankaj S. Joshi,‡ and Jay Verma Trivedi§ +International Centre for Space and Cosmology, School of Arts and Sciences, +Ahmedabad University, Ahmedabad-380009 (Guj), India. +Tapobroto Bhanja¶ +International Center for Cosmology, & PDPIAS, +Charotar University of Science and Technology, Anand- 388421 (Guj), India +(Dated: January 13, 2023) +We study here the unhindered gravitational collapse of spatially homogeneous (SH) scalar fields φ +with a potential Vs(φ), as well as vector fields ˜A with a potential Vv(B) where B = g( ˜A, ˜A) and g is +the metric tensor. We show that in both cases, classes of potentials exist that give rise to black holes +or naked singularities depending on the choice of the potential. The strength of the naked singular- +ity is examined, and they are seen to be strong, in the sense of Tipler, for a wide class of respective +potentials. We match the collapsing scalar/vector field with a generalized Vaidya spacetime outside. +We highlight that full generality is maintained within the domain of SH scalar or vector field collapse. +keywords: Gravitational collapse, singularity, scalar field, vector field, causal structure. +I. +INTRODUCTION +The contraction of a matter field under its gravita- +tional influence is called gravitational collapse. In 1939, +Oppenheimer and Snyder [1], and independently in 1938, +Datt [2] developed the first solution of Einstein’s field +equations (called the OSD model) depicting the gravita- +tional collapse of a massive star. They considered a very +specific case of spatially homogeneous (SH) dust collapse +(By spatial homogeneity, we mean homogeneous on a +three-dimensional spacelike orbit with a six-dimensional +isometry group G6 corresponding to the spacetime [3]). +Such a matter field undergoes gravitational collapse that +ends up in a singularity. Such a spacetime singularity +is hidden behind an event horizon, not visible to any ob- +server, and what we obtain is a black hole as the outcome +of continual collapse. +Extending the above special scenario, in 1969, Penrose +proposed what is now known as the cosmic censorship hy- +pothesis (CCH) [4]. The weaker version of the hypothesis +states that all singularities of gravitational collapse are +hidden within a black hole and hence, cannot be seen +by a distant observer (a globally naked singularity can- +not exist). The strong version of the hypothesis states +that no past inextendible nonspacelike geodesics can ex- +ist between the singularity and any point in the space- +time manifold. In other words, a causal geodesic with +a positive tangent “at” the singularity does not exist (a +locally naked singularity also cannot exist). +The sup- +porting argument for the validity of the strong CCH is +the desirability of the spacetime manifold to be globally +∗ kmosani2014@gmail.com +† koushiki.malda@gmail.com +‡ pankaj.joshi@ahduni.edu.in +§ jay.verma2210@gmail.com +¶ tapobroto.bhanja@gmail.com +hyperbolic. Global hyperbolicity implies the existence of +Cauchy surfaces embedded in the total manifold, thereby +making general relativity a deterministic theory [5–7]. +Now singularity theorems of Hawking and Penrose +[6, 11] do not imply that singularities are hidden from +an external observer under any possible circumstances. +In fact, singularity theorems take the causality condition +as one of the axioms to start with to prove the existence +of incomplete past (future) directed causal curves. Addi- +tionally, the OSD model that motivated cosmic censor- +ship is a special case. Joshi and Malafarina [12] showed +that any arbitrarily small neighbourhood of the initial +data giving rise to OSD collapse contains initial data cor- +responding to collapse evolution giving rise to a singular- +ity with the following property: one could trace outgoing +past singular causal geodesics. This means that the end +state of OSD collapse is unstable under small perturba- +tions in initial data. Moreover, one can show the forma- +tion of naked singularities (global and local) as an end +state of gravitational collapse from suitable, physically +reasonable initial data for various matter fields [13, 14]. +This implies that the initial conditions must be fine-tuned +for the cosmic censorship conjecture to hold. +In such a context, an important question one can ask +here is as follows: what will be the end state of an un- +hindered gravitational collapse of a fundamental matter +field, such as a scalar field or a vector field, derived from +an appropriate Lagrangian? +The answer to this question has been achieved up to +a certain extent. Scalar fields are fundamental matter +fields derived from suitable Lagrangian. +A real scalar +field is a map defined on a smooth manifold as φ : M → +R with a suitable continuity condition. +Christodoulou +showed that in the case of gravitational collapse of a +massless scalar field φ (the scalar field Lagrangian is +Lφ = (1/2)gµν∂µφ∂νφ), the set of initial data giving rise +to a naked singularity as an end state has positive codi- +mension in the entire initial data set [15, 16]. This means +arXiv:2301.05083v1 [gr-qc] 12 Jan 2023 + +2 +that the initial data set corresponding to naked singular- +ity has a zero measure in the total initial data set. In +other words, naked singularity in such cases is unstable +under arbitrarily small perturbations in the initial data. +One can have a massless scalar field with a poten- +tial function Vs(φ) that still be a fundamental matter +field. +A massive scalar field will then be a particular +case of a massless scalar field with a specific potential of +the form Vs(φ) = (1/2)µ2φ2, where µ is the mass term. +Goswami and Joshi [17] showed the example of the grav- +itational collapse of a massless SH scalar field with a cer- +tain potential Vs(φ) that ends up in a naked singularity. +Mosani, Dey, Bhattacharya, and Joshi [18] conducted a +similar investigation for a massless scalar field with a two- +dimensional analogue of the Mexican hat-shaped Higgs +field potential and found out that the end state of such +unhindered scalar field collapse is a naked singularity. +In addition to scalar fields as fundamental matter +fields, vector fields are also fundamental matter fields de- +rived from suitable matter Lagrangian. Geometrically, +vector fields on a smooth manifold M can be thought of +as sections on the tangent bundle π : TM → M, where +π is a continuous surjection. A section is a smooth map +σ : M → TM such that π ◦ σ is an identity map on +M. From a particle physics point of view, the funda- +mental nature of a vector field is different from that of a +scalar field. There are many aspects, but one of the most +important ones is that massive or massless vector fields +mediate most particle physics processes. These represent +the three fundamental interactions: quantum electrody- +namics and weak and strong processes. A massless vector +field with a potential function Vv(B) is again a funda- +mental matter field. A massive vector field will then be +a particular case of a massless vector field with a specific +potential of the form Vv(B) = (1/2)µ2B, where µ is the +mass term. Garfinkle, Mann, and Vuille [26] have studied +the collapse of a massive vector field and numerically ob- +tained the critical initial conditions. To our knowledge, +much analytical work has not been done in investigating +the causal structure of the end-state spacetime of the un- +hindered gravitational collapse of matter fields that are +vector fields. +In this paper, in both the massless SH scalar field as +well as vector field cases, we show that there are broad +classes of potentials for which the configuration collapses +and ends up in either a black hole or a naked singu- +larity depending on the potential function chosen. We +approach the causality investigation problem of scalar +field as well as vector field collapse in a unified way, so +to speak. +As far as general relativity is concerned, it +does not discriminate between whether a scalar field or +a vector field seeds the matter field. +The matter field +is entirely identified by a rank two tensor field that we +call the stress-energy tensor. As far as SH perfect fluid +is concerned, one can identify a given matter field by +the functional form of the equation of state parameter +ω(a), where a is the scale factor of the collapsing cloud. +We derive relevant equations of collapsing SH scalar field +φ(a) and vector field ˜A(a) in the sub-sections of section +II. The main body of section II contains discussions and +relevant relations regarding the gravitational collapse of +SH perfect fluids. In section III, we smoothly join the +interior collapsing perfect fluid with an external gener- +alized Vaidya spacetime. In section IV, we investigate +the causal structure of the spacetime (condition of ob- +taining a naked singularity) at the end of the collapse +of the interior perfect fluid that is either a scalar field φ +with potential Vs or a vector field ˜A with potential Vv. +We also depict a few examples of well-known scalar fields +and vector fields. In section V, we derive the criteria for +the singularity, thus obtained in the end, to be strong of +Tipler’s type. In the last section, we highlight the key +points of the investigation. Here we use the geometrized +units 8πG = c = 1 throughout. +II. +INTERIOR COLLAPSING MATTER FIELD +Consider a gravitational collapse of a SH perfect fluid. +The components of the stress-energy tensor in the coor- +dinate basis {dxµ � ∂ν|0 ≤ µ, ν ≤ 3} of the comoving +coordinates (t, x, y, z) are given by +T µ +ν = diag (−ρ, p, p, p) . +(1) +The spacetime geometry is governed by the flat (k = 0) +Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) metric +ds2 = −dt2 + a2dΣ2, +(2) +where dΣ2 = dx2 + dy2 + dz2. Here a = a(t) is the scale +factor such that a(0) = 1 and a(ts) = 0, where ts is the +time of formation of the singularity. R = R(t, r) is the +physical radius of the collapsing cloud and can be written +as +R(t, r) = ra(t), +(3) +where r is the radial spherical coordinate. For a FLRW +spacetime Eq.(2), we have +ρ = 3˙a2 +a2 , +(4) +and +p = −2¨a +a − ˙a2 +a2 . +(5) +The overhead dot denotes the partial time derivative of +a. Eq.(4) can be rewritten to obtain the dynamics of the +collapse as +˙a = − +� +ρ(a) +3 a. +(6) +Differentiating the above equation once again gives us +¨a = 1 +3a +�aρ,a +2 ++ ρ +� +. +(7) + +3 +Integrating Eq.(6), we obtain the time curve, which is +t(a) = +� 1 +a +�3 +ρ +da +a . +(8) +The dynamics of the scale factor a(t) is, thus, the inverse +of the LHS of the above equation. The time of formation +of the singularity ts = t(0) is +ts = +� 1 +0 +�3 +ρ +da +a . +(9) +Now, let us consider a particular matter field ˆT from a +set of all the possible SH perfect fluids. Choosing such +an element means choosing a specific functional form of +the equation of state parameter +ω(a) = p +ρ. +(10) +Using Eq.(4), Eq.(5), and Eq.(10), we can express the +density of the matter field with the equation of state +parameter ω as +ρ = ρ0 exp +�� 1 +a +3 (1 + ω(a)) +a +da +� +, +(11) +An SH perfect fluid is a fundamental matter field since +it can be derived by a fundamental matter Lagrangian. +In the following two subsections, we will describe two +distinct ways of obtaining such a matter field. +A. +Scalar field collapse +We prove that any SH perfect fluid is equivalent to a +SH scalar field φ(a) with a suitable potential Vs(a), as +far as the gravitational collapse is concerned. If φ(a) is +invertible, then the following statement holds: Any SH +perfect fluid is gravitationally equivalent to a SH scalar +field φ with a suitable potential Vs(φ). +Consider a real scalar field defined on the manifold M +as +φ : M → R. +(12) +The Lagrangian of a massless scalar field φ with potential +Vs(a) is given by +Lφ = 1 +2gµν∂µφ∂νφ − Vs(φ), +(13) +The stress-energy tensor is obtained from the Lagrangian +Lφ as +Tµν = − +2 +√−g +δ (√−gLφ) +δgµν +. +(14) +The density (ρs) and the isotropic pressure (ps) are sub- +sequently expressed in terms of the time derivative of the +scalar field and its potential as +ρs = 1 +2 +˙φ2 + Vs +(15) +and +ps = 1 +2 +˙φ2 − Vs. +(16) +The overhead dot denotes the time derivative of the func- +tions. +From Eq.(15) and Eq.(16), and from using the +chain rule ˙φ = φ,a ˙a, we get +ρs + ps = φ2 +,a ˙a2. +(17) +We now equate ρs = ρ and ps = p. Using Eq.(5) and +(17), along with replacing ˙a and ¨a using Eq.(6) and (7), +one obtains the expression of density as a function of a +as +ρs = ρ0 exp +�� 1 +a +aφ2 +,ada +� +. +(18) +From Eqs.(15) and (16), we get +ps = ρs − 2Vs. +(19) +Using Eq.(6) in Eq.(17), we get +ρs +� +1 − φ2 +,aa2 +3 +� ++ ps = 0. +(20) +Using Eqs.(19) and (20), we get +Vs(φ) = ρs +� +1 − φ2 +,aa2 +6 +� +. +(21) +Using Eq.(17), Eq.(6) in Eq.(5), one obtains +ρs,a +ρs += −φ,2 +a +a . +(22) +We have, using Eq.(10), Eq.(15) and Eq.(16), +Vs = ρs +2 (1 − ω) . +(23) +Now from Eq.(21) and Eq.(23), we have +φ(a),a = ± +� +3 (1 + ω(a)) +a +. +(24) +Integrating the above equation, one obtains +φ(a) = ± +� 1 +a +� +3 (1 + ω(a)) +a +da + c. +(25) +From Eq. (11) and Eq.(21) we have +Vs(a) = ρ0 +�1 − ω(a) +2 +� +exp +�� 1 +a +3 (1 + ω(a)) +a +da +� +. +(26) +Hence, we proved that given the functional form of + +4 +the equation of state parameter ω(a), one could obtain +the corresponding scalar field φ(a) given by Eq. +(25) +with potential Vs(a) given by Eq. (26). As long as φ(a) +is invertible (or, in other words, a bijective map from +(0, 1] → R), we obtain a(φ), at least in principle, using +which, we get Vs(φ). +Alternatively, given a scalar field φ(a), one can obtain +the corresponding perfect fluid ˆT (or the ω(a) by which +it is identified), using Eq.(24). +On the other hand, we can also start with a given scalar +field potential V (φ). One can use Eq.(18) and Eq.(21) to +obtain the ordinary nonlinear differential equation +H +� +a, φ, dφ +da , d2φ +da2 +� += 0, +(27) +that can be solved in principle, to obtain φ(a), and later +obtain ω(a) using Eq.(24). Hence, given a scalar field po- +tential Vs(φ), one can obtain the corresponding ˆT (iden- +tified by ω(a)) in the above manner. +B. +Vector field collapse +We prove that any SH perfect fluid is equivalent to a +SH vector field ˜A(a) with a suitable potential Vv(a), as far +as the gravitational collapse is concerned. If B(a) is in- +vertible, then the following statement holds: Any SH per- +fect fluid is gravitationally equivalent to a SH vector field +˜A with a suitable potential Vv(B) (where B = g( ˜A, ˜A)). +Consider a vector field +˜A : M → TM. +(28) +with potential V (B). For a fixed p ∈ M, ˜A(p) = Aµdxµ, +where Aµ = (A0, Ai), 1 < i < 3 (in the comoving carte- +sian coordinate basis). Here B = gαβAαAβ. We con- +sider a SH pure vector field: A0 = 0 and Ai = A ∈ R +∀i ∈ (1, 2, 3). For such a vector field, B = 3A2/a2. +The Lagrangian of a massless vector field ˜A with po- +tential Vv(B) is given by +L ˜ +A = −1 +4F µνFµν − Vv(B). +(29) +F is a two form called the field strength and can be writ- +ten in terms of wedge product as F = Fµνdxµ ∧ dxν. +The field strength is the exterior derivative of the vec- +tor field ˜A, i.e.F = d ˜A. The components are written as +Fµν = ∇µAν − ∇νAµ. +The stress-energy tensor is obtained from the La- +grangian L ˜ +A as +Tµν = − +2 +√−g +δ (√−gL ˜ +A) +δgµν +. +(30) +This gives us +Tµν = −1 +4FαβF αβgµν −Vv(B)gµν +FµαF +α +ν ++2V ′ +vAµAν. +(31) +The overhead prime denotes the ordinary derivative with +respect to B. The density and the isotropic pressure are +subsequently expressed in terms of the time derivative of +the vector field component and its potential as +ρv = 3 +2 +˙A2 +a2 + Vv(B), +(32) +and +pv = 1 +2 +˙A2 +a2 − Vv(B) + 2V ′ +v +A2 +a2 . +(33) +We now equate ρv = ρ and pv = p. +From Eq.(32) +and Eq.(4), we obtain +Vv = ρv +� +1 − 1 +2A,2 +a +� +. +(34) +Substituting for ρ(a) from Eq.(11), we obtain +Vv = ρ0 exp +�� 1 +a +3 (1 + ω(a)) +a +da +� � +1 − 1 +2A,2 +a +� +(35) +On differentiating Eq.(34) with respect to B we obtain, +V ′ +v = +ρv,a +� +1 − A,2 +a +2 +� +− ρvA,a A,aa +6A2 +a2 +� +A,a +A − 1 +a +� +(36) +Using Eq.(33), Eq.(4), and Eq.(5), we obtain +aρv,a +3 ++ ρv +� +1 + 1 +6A,2 +a +� += Vv − V ′ +v +A2 +a2 . +(37) +Substituting for Vv and V ′ +v from Eq.(34) and Eq.(36), +and also substituting for ρ,a (by differentiating Eq.(11)) +in Eq.(37), we obtain a second order nonlinear differential +equation +G +� +a, ω, A, dA +da , d2A +da2 +� += 0, +(38) +where G is +G =d2A +da2 − 4 +A +�dA +da +�2 ++ 1 +2a (5 − 3ω) dA +da + 6 +A (1 + ω) +− 3 (1 + ω) +a +�dA +da +�−1 +. +(39) +For a fixed ω(a), solving this differential equation with +two initial conditions gives us A(a), and consequently, +the vector field ˜A. +Hence, we proved that given the functional form of the +equation of state parameter ω(a), one could obtain the +corresponding vector field ˜A using Eq.(38), and conse- +quently, the vector field potential Vv(a) using Eq.(35). + +5 +SH Perfect Fluid +Characterised by the equation of state +parameter 𝜔(a) +SH Scalar Field +Characterised by 𝜙(a) +SH Vector Field +Characterised by Ã(a) +FIG. 1: A spatially homogeneous (SH) perfect fluid (governed by a flat FLRW spacetime metric) is completely +characterized by the equation of state parameter ω(a), Eq.(10) of the matter field. This matter field is obtained +from fundamental matter Lagrangian. Hence, the same matter field is also characterized by an SH scalar field φ(a), +Eq.(25) or its potential Vs(a), Eq.(26) (Vs(φ) if φ(a) is invertible). Similarly, it can also be characterized by an SH +vector field ˜A(a), Eq.(38) or its potential Vv(a), Eq.(35) (Vv(B) if B(a) is invertible). This schematic diagram +depicts the equivalence between the gravitational collapse of SH Perfect fluid, Scalar field and Vector field. By +spatial homogeneity, we mean homogeneous on a three-dimensional spacelike orbit with a six-dimensional isometry +group G6 corresponding to the spacetime [3]. +Now, from the functional form A(a), we obtain B(a). As +long as B(a) is invertible (or, in other words, a bijec- +tive map from (0, 1] → R), we obtain a(B), at least in +principle, using which, we get Vv(B). +Alternatively, given a vector field ˜A(a), one can obtain +the corresponding perfect fluid ˆT (or the ω(a) by which +it is identified), using Eq.(38). +On the other hand, we can also start with a given vec- +tor field potential Vv(B). One can differentiate Eq.(35), +and do some rearrangements to obtain +ω(a, A, dA +da , d2A +da2 ) +as +ω = 2AV ′ +aV +�A +a − dA +da +� +−a +3 +dA +da +d2A +da2 +� +1 − 1 +2 +�dA +da +�2�−1 +−1. +(40) +Substituting Eq.(40) in Eq.(38), we obtain +˜G +� +a, A, dA +da , d2A +da2 +� += 0. +(41) +In principle, this differential equation can be solved to +obtain A(a), which, when substituted in Eq.(40), gives +us ω(a). Hence, given a vector field potential V (B), one +can obtain the corresponding ˆT (identified by ω(a)) in +the above manner. +III. +EXTERIOR GENERALIZED VAIDYA +SPACETIME +The collapsing vector field spacetime (g− +µν) can be +joined smoothly with the exterior generalized Vaidya +spacetime (g+ +µν) so that their union forms a valid solution +of the Einstein’s field equations. The interior FLRW and +the exterior generalized Vaidya spacetime [19] are respec- +tively given as +ds2 +− = −dt2 + a(t)2dr2 + r2 +ba(t)2dΩ2, +(42) +and +ds2 ++ = − +� +1 − 2M(R, v) +R +� +dv2 − 2dvdR + R2dΩ2. (43) +Here, v is the retarded null coordinate, R is the general- +ized Vaidya radius, and rb is the value of the radial co- +ordinate r corresponding to the matching hypersurface, +or in other words, the radial coordinate of the outermost +shell of the collapsing scalar/vector field cloud. The mat- +ter field corresponding to the generalized Vaidya space- +time is a combination of Type I and type II, such that +the components of the stress-energy tensor written in the + +6 +orthonormal basis appear as +Tab = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +¯ϵ +2 + ϵ +¯ϵ +2 +0 +0 +¯ϵ +2 +¯ϵ +2 − ϵ 0 +0 +0 +0 +P +0 +0 +0 +0 P. +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(44) +ϵ = P = 0 and ¯ϵ ̸= 0 corresponds the usual Vaidya +spacetime as a special case. ¯ϵ = 0 and ϵ ̸= 0 corresponds +to a sub-class of Type I matter field. The generalized +Vaidya solution encompasses many known Einstein field +equations solutions. Matching the first and second fun- +damental forms for the interior and exterior metric on Σ +gives the following equations: +R(t) = R(t, rb) (= rba(t)) , +(45) +F(t, rb) = 2M(R, v), +(46) +�dv +dt +� +Σ += +1 + ˙R +1 − F (t,rb) +R +, +(47) +and +M(R, v),R = F(t, rb) +2R ++ R ¨R. +(48) +Here, F = R ˙R2 is the Misner-Sharp mass function of +the collapsing spherical SH perfect fluid. Using the rela- +tion (45), we can relate the generalized Vaidya mass with +the density of the interior collapsing SH spherical perfect +fluid cloud as +M = ρ +6R3. +(49) +Using Eq.(7), differentiation of Eq.(25) with respect to +a, and Eq.(49), in Eq.(48) we get +M,R = 3M +R +� +1 + (1 + ω(a))r2 +b +R2 +� +, +(50) +integrating which we obtain +M(R, v) = M1(v) exp +�� +3 +R +� +1 + (1 + ˜ω (R))r2 +b +R2 +� +dR +� +. +(51) +Here M1(v) is a constant of integration and is a function +of null coordinate v, and +˜ω(R) = ω +�R +rb +� +. +Eq.(51) gives us the expression of the generalized Vaidya +mass function of the exterior generalized Vaidya space- +time, in terms of interior collapsing perfect fluid equation +of state parameter ω, to ensure smooth matching at the +matching hypersurface. +For the exterior matter field to satisfy the weak energy +condition, ¯ϵ and ϵ should be non-negative [19]. +These +inequalities, in turn, put restrictions on the generalized +Vaidya mass function as +M,v ≤ 0, +and +M,R ≥ 0. +(52) +Using Eq.(50) and Eq.(51) in the above two relations, we +obtain +M1,v ≤ 0, +(53) +and +�1 + ω(a) +a2 +� +≥ 0. +(54) +The inequality (54) is always satisfied if the interior col- +lapsing matter field obeys the weak energy condition. +Hence, Eq.(53) is the only restriction on the generalized +Vaidya mass function for the exterior spacetime to obey +at least the weak energy condition. +Now, +we have a complete solution of Einstein’s +field equations consisting of an interior collapsing SH +scalar/vector field (with some potential) and the exte- +rior generalized Vaidya solution, matched smoothly at +the matching hypersurface. The free functions are cat- +egorically the potential function (Vs(φ) in case of scalar +field collapse, and Vv(B) in case of vector field collapse), +and the component of generalized Vaidya mass function +M1(v), the latter one restricted by the inequality (53). +It is evident that the choice of M1(v) does not affect the +causal structure of the spacetime obtained as an end- +state of unhindered gravitational collapse. +Of course, +instead of considering the potential function Vs(φ) (or +Vv(B)) as a free function, one could also consider any one +of the remaining functions: ω(a), ρ(a), φ(a) (or A(a)), +Vs(a) (or Vv(a)) as a free function, without any trouble. +In the next section, we study the end state of this class of +global dynamical spacetime identified by any one of the +free functions. +IV. +CAUSAL STRUCTURE AND STRENGTH +OF THE SINGULARITY +Once the singularity is formed as an end state of grav- +itational collapse of the interior scalar (vector) field with +potential Vs(φ) (Vv(B)), one can investigate whether or +not causal geodesics can escape the singularity. Addition- +ally, one can investigate whether or not such singularity +is gravitationally strong in the sense of Tipler. The fol- +lowing two subsections discuss these two properties. + +7 +Massless scalar field +Vs(φ) = 0 +φ(a) = c ± +√ +6 log a +strong +BH +Homogeneous dust (ω = 0) +Vs(φ) ∝ exp +�√ +3φ +� +φ(a) = c ± +√ +3 log a +strong +BH +Goswami/ Joshi [17] (ω = − 2 +3) (SF1) +Vs(φ) ∝ exp φ +φ(a) = c ± log a +strong +NS +Two dimensional analog of Mexican hat [18] +(SF2) +Vs(φ) = 1 +2µφ2 + λφ4 +φ(a) = ±2 +√ +2√c − log a +weak +NS +TABLE I: Four examples of spatially homogeneous scalar fields that collapse to form a singularity that is either +hidden (blackhole or BH) or (naked singularity or NS). In the fourth example, µ = − 16 +3 λ. The first three types end +up in a strong singularity in the sense of Tipler. +Massless vector field +Vv(B) = 0 +strong +BH +Massive vector field +Vv(B) = − 1 +2µ2B +strong +BH +VF1 +Vv(a) as in Fig.(3) +strong +NS +VF2 +Vv(a) as in Fig.(3) +weak +NS +TABLE II: Four examples of spatially homogeneous vector fields that collapse to form a singularity that is either +hidden within a black hole (BH) or is naked (NS). The ones mentioned in the third and the fourth row are newly +constructed vector fields from known scalar fields (mentioned in the third [17] and the fourth [18] row of Table 1, +respectively) by exploiting the gravitational equivalence depicted in Fig.(1). The corresponding vector field +component A(a) for each case is plotted in Fig.(2-3). The first three types end up in a gravitationally strong +singularity in the sense of Tipler. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-0.5 +0.0 +0.5 +1.0 +a +A +a +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +2 +4 +6 +a +V +FIG. 2: The dynamics of the vector field component A(a) in the case of the massive (µ = 1) vector field ˜A (Left +panel) and its potential Vv(a) (Right panel). First we obtain ω(a, A, dA +da , d2A +da2 ) by substituting Vv(B) = − 1 +2µ2B in +Eq.(40). Substituting for ω(a, A, dA +da , d2A +da2 ) in Eq.(41) and solving the differential equation with initial conditions +A(1) = 1 and A′(1) = 2, we obtain A(a). Consequently, substituting Vv(B) = − 1 +2µ2B, and the obtained A(a) in +Eq.(40), we obtain ω(a), Further substitution of ω(a) in Eq.(35), we obtain Vv(a). + +8 +0.2 +0.4 +0.6 +0.8 +1.0 +-3 +-2 +-1 +0 +1 +2 +3 +a +Vv +(a) +0.0 +0.5 +1.0 +1.5 +2.0 +-10 +-8 +-6 +-4 +-2 +0 +2 +4 +B +Vv +(b) +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +-300 +-200 +-100 +0 +100 +200 +a +Vv +(c) +0.0 +0.5 +1.0 +1.5 +2.0 +-800 +-600 +-400 +-200 +0 +200 +B +Vv +(d) +A1 +A2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +a +A +(e) +FIG. 3: (a) and (c): Vector field potentials Vv(a) corresponding to newly constructed vector fields VF1 (orange) +and VF2 (green), as mentioned in the third and fourth row of the Table (II), respectively. (b) and (d): The same +vector field potentials Vv(B) as function of B. (e): The vector field components A(a) in both of these cases. In the +latter example, µ = −8/3 and λ = 1. First, we obtain ωi(a) , using Eq.(25) (Here i ∈ 1, 2 corresponds to VF1 and +VF2 respectively). Then we obtain the vector field components Ai(a) by solving the differential Eq.(38) with initial +conditions Ai(1) = 1 and A′ +i(1) = 10. Further substitution of ωi(a) and the obtained Ai(a) in Eq.(35), we get +Vv(i)(a). Once Vv(i)(a) is obtained, we obtain Vv(i)(B). +A. +Causal structure of the singularity +We say that a singularity formed due to unhindered +gravitational collapse is naked if there exists a family of +outgoing causal curves whose past endpoint is the singu- +larity. In the future, these curves can either reach a far- +away observer or fall back to the singularity. The singu- +larities are then termed globally naked and locally naked, +respectively. Whether or not the singularity is naked es- +sentially depends on the geometry of trapped surfaces as +the collapse evolves. Trapped surfaces are two-surfaces in +the spacetime on which not only the ingoing congruence +but also the outgoing congruence necessarily converge. +Convergence or otherwise of the outgoing null geodesic +congruence is determined by the behaviour of its expan- +sion scalar, which we denote here as θl (t, r). It is ex- +pressed in terms of the metric coefficients, in comoving +spherical coordinates as, +θl = 2 +R +� +1 − +� +ρR2 +3 +� +. +(55) + +9 +EH +AH +0.0 +0.1 +0.2 +0.3 +0.4 +R +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +t +(a) Massless scalar field (Vs = 0) +EH +AH +0.0 +0.1 +0.2 +0.3 +0.4 +R +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +t +(b) Massless vector field (Vv = 0) +0.0 +0.1 +0.2 +0.3 +0.4 +R +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +t +(c) SF1/VF1 +0.0 +0.1 +0.2 +0.3 +0.4 +R +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +t +(d) SF2/VF2 +FIG. 4: Spacetime diagram of the examples of spatially homogeneous scalar fields and vector fields mentioned in +Tables I and II. The solid black curve in each of them represents the boundary of the collapsing cloud. Upper panel: +The singularity is not visible in both examples. Lower Panel: In the case of SF1/VF1, the singularity forms in a +finite comoving time and is globally visible because of the absence of the apparent and event horizons. In the case of +SF2/VF2, the singularity forms in an infinite comoving time. However, an ultra-high density region is obtained in +finite comoving time, which can be visible globally because of the absence of the apparent and event horizons. +The region in which θl < 0 is called the trapped region. +The boundary of the trapped region, given by θl = 0, is +called the apparent horizon. If the neighbourhood of the +singular center is surrounded by a trapped region since +before the time of formation of the singularity ts, then +it is covered, and we get a black hole. Hence, the nec- +essary condition for singular null geodesic congruence to +escape the singularity is the absence of a trapped region, +which is ensured by the condition θl(ts, r) > 0 for such +congruence. The absence of trapped region in the neigh- +bourhood of the singularity (t, r) = (ts, 0) is ensured by +the following inequality: +lim +t→ts +ρR2 +3 +≤ lim +a→0 +ρ(a)r2 +ba2 +3 +< 1, +(56) +The inequality (56) is definitely satisfied if +lim +a→0 ρ(a) < 1 +a2 . +(57) +For +lim +a→0 ρ(a) = k +a2 , +for some k ∈ R+, the inequality is satisfied only for +rb < +� +3/k. Rewriting the inequality (57) in terms of +the equation of state parameter ω(a) using Eq.(11), one +obtains +lim +a→0 ρ0a2 exp +�� 1 +a +3 (1 + ω(a)) +a +� +da < 1. +(58) +If a collapsing matter field with the equation of state +parameter ω(a) satisfies the inequality (58), then it will + +10 +up in a naked singularity [17]. In the case of otherwise, +the final outcome is a black hole. +Hence, as decided by the above inequality, we get a +class of SH matter fields that include scalar and vector +fields, identified by the functional form ω(a), that goes +to either the blackhole or naked singularity final state as +an end state of unhindered gravitational collapse. +In the case of scalar field collapse, the restriction (58) +on ω(a) gives us a restriction on the scalar field φ(a) us- +ing Eq.(25), and the scalar field potential function Vs(a) +using Eq.(26). Hence, obtaining a class of ω(a) is grav- +itationally equivalent to obtaining a class of scalar field +potentials Vs(a) that goes to the naked singularity as +an end state of unhindered gravitational collapse. More- +over, suppose φ(a) is a bijective map from (0, 1] → R. +In that case, obtaining a class of ω(a) is gravitationally +equivalent to obtaining a class of scalar field potentials +Vs(φ) = Vs(a(φ)) that goes to the naked singularity as +an end state of gravitational collapse. +Similarly, in the case of vector field collapse, the re- +striction (58) on ω(a) gives us a restriction on the vector +field ˜A (or more specifically, a restriction on the vec- +tor field component A(a)) obtained by solving the dif- +ferential Eq.(38), and the vector field potential function +Vv(a) obtained by substituting A(a) and ρ from Eq.(11), +in Eq.(34). +Hence, obtaining a class of ω(a) is gravi- +tationally equivalent to obtaining a class of vector field +potential Vv(a) that goes to the naked singularity as an +end state of unhindered gravitational collapse. +More- +over, suppose A(a) is a bijective map from (0, 1] → R. +In that case, obtaining a class of ω(a) is gravitationally +equivalent to obtaining a class of vector field potential +Vv(A) = Vv(a(A)) that goes to the naked singularity as +an end state of unhindered gravitational collapse. +In Table (I) and (II), we discuss examples of such +scalar field collapse and vector field collapse that ends +up in either a black hole or a naked singularity. +Ex- +ploiting the equivalence between SH perfect fluids, scalar +fields with potential Vs(a), and vector fields with poten- +tial Vv(a), we construct two examples of collapsing vector +fields with potential out-of-known examples of collapsing +scalar fields with potentials, giving rise to the naked sin- +gularity as an end state. +The first example of a collapsing vector field with po- +tential Vv(a) is constructed from the scalar field with po- +tential mentioned in the third row of Table (I) [17]. The +perfect fluid corresponding to such scalar field example +has an equation of state parameter ω(a) = − 2 +3. The con- +structed collapsing vector field ˜A = (0, A, A, A) (in the +comoving coordinate basis) has the property (dynamics +of A(a) and Vv(a)) as shown in Fig.(3). Refer to the third +row of Table (II). +The second example of a collapsing vector field with +potential Vv(a) is constructed from the scalar field with +potential mentioned in the fourth row of Table (I) [18]. +Such a scalar field has a two-dimensional analogue of +Mexican hat-shaped potential. The constructed collaps- +ing vector field ˜A = (0, A, A, A) (in the comoving co- +ordinate basis) has the property (dynamics of A(a) and +Vv(a)) as shown in Fig.(3). Refer to the fourth row of +Table (II). The spacetime diagrams of some of the ex- +amples in Table (I) and (II) are plotted in Fig.(4). +B. +Strength of the singularity +Generally, a singularity in the spacetime manifold is +identified by the existence of at least one past/future in- +complete geodesic. However, in the case of singularities +forming as the end state of a gravitational collapse, apart +from the existence of such incomplete geodesics, one ex- +pects an additional physical property as follows: An ob- +ject approaching such singularity should be crushed to +zero volume. We call such a singularity gravitationally +strong in the sense of Tipler [20]. A precise definition of +a strong singularity is as follows: +Consider a smooth spacetime manifold (M, g) and a +causal geodesic γ : [t0, 0) → M. Let λ be an affine pa- +rameter along this geodesic. Let ξ(i), (0 ≤ i ≤ 2 in the +case of null geodesic, 0 ≤ i ≤ 3 in the case of timelike +geodesic) be the independent Jacobi vector fields. The +wedge product of these Jacobi fields gives us the volume +form V = � ξ(i). We say that a singularity is gravita- +tionally strong in the sense of Tipler if this volume form +vanishes as λ → 0. +Clarke and Krolak [21] related the existence of a Tipler +strong singularity with the growth rate of the curvature +terms as follows: At least along one null geodesic with +affine parameter λ (such that λ → 0 as the singularity is +approached), the following inequality +lim +λ→0 λ2RijKiKj > 0 +(59) +should hold for the singularity to be strong in the sense of +Tipler. Here Ki = dxi +dλ are the tangents to the chosen null +geodesic, and xi is the coordinate system. This condition +puts a lower bound on the growth of the curvature scalar. +In the spherical coordinate system (t, r, θ, φ), the radial +null geodesic equation reads +dt +dr = a. +(60) +Hence, we have the relation between the tangents Kt and +Kr as +Kt = aKr, +(61) +and subsequently, in terms of the affine parameter, +Kt = R +λ , +and +Kr = r +λ. +(62) +The inequality (59) can then be written in terms of ω as +lim +a→0 +� +r2(1 + ω)ρ0 exp +�� 1 +a +3(1 + ω) +a +da +�� +> 0 +(63) + +11 +Hence, the singularity formed due to the gravitational +collapse of a scalar/vector field is strong in the sense of +Tipler if the following inequality holds (assuming that +the weak energy condition is respected): +lim +a→0 exp +�� 1 +a +3(1 + ω) +a +da +� +> 0. +(64) +Hence, (along with using the condition (58)) one can ob- +tain a naked singularity that is strong in the sense of +Tipler for that ω that satisfies the following constraint: +0 < lim +a→0 exp +�� 1 +a +3(1 + ω) +a +da +� +< O(a−2). +(65) +This constraint gives us the class of SH collapsing mat- +ter fields that we identify by ω(a), which ends up in +strong curvature naked singularity. Or in other words, +we have a class of scalar/vector field potentials corre- +sponding to the given scalar/vector field that collapses +to a strong naked singularity. +As an example, in Ta- +bles (I) and (II), we mention the causal property and +the strength of the singularity formed due to the gravi- +tational collapse of four different scalar/vector fields. +V. +CONCLUSIONS AND REMARKS +Following are the concluding remarks: +1. Unlike the singularity theorems that provide rig- +orous proof of the existence of incomplete causal +geodesics under rather generic conditions, one does +not currently have proof or disproof of the cosmic +censorship hypothesis. +In fact, we need a math- +ematically rigorous formulation of this conjecture, +which is not available currently, before we can prove +or disprove it. +Under the situation at present, we can only spec- +ulate its validity or otherwise. +Proposed coun- +terexamples, hence have great importance in under- +standing whether naked singularities, in fact, exist +or not in our universe. Through such analysis of +gravitational collapse models only, one could pos- +sibly hope to arrive at a suitable formulation of +cosmic censorship. The collapse of inhomogeneous +dust and the Vaidya null fluids were the first exam- +ples proposed to produce naked singularities. How- +ever, an important objection could be that, even if +astrophysically interesting, they are not fundamen- +tal forms of matter [7, 22]. +One could then ask +whether the collapse of matter configuration that +is obtained from a fundamental matter Lagrangian +ends up in a naked singularity. Scalar fields with +potential and vector fields with potential are fun- +damental matter fields in this sense. Here we show +that not just one particular choice of these fields +but an entire class of such types could collapse and +form a naked singularity as an end state. This basi- +cally divides the allowed class of potential functions +into classes that take the unhindered collapse to a +black hole or naked singularity. +2. To achieve this, we show equivalence between SH +(a) Perfect fluid: characterized by ω(a), +(b) Massless scalar field φ: characterized by φ(a) +or its potential Vs(a) or Vs(φ) (if φ(a) is in- +vertible), and +(c) Massless vector field +˜A: +characterized by +A(a), or its potential Vv(a), or Vv(B) (if B(a) +is invertible). +as far as the gravitational collapse is concerned. +This gravitational equivalence is described in sub- +sections of section (II) and depicted in Fig.(1). +Now, if the functional form of ω(a) satisfies the +inequality (58), then the singular null geodesic con- +gruence, if at all there exists, does not get trapped +as a → 0. +Hence, we have a class of functions +ω(a) corresponding to a naked singularity as an end +state of gravitational collapse. Now, because of the +above equivalence, in the case of an SH scalar field +collapse, one then has a class of scalar field func- +tion φ(a), or a class of scalar field potential Vs(a), +or a class of scalar field potential in terms of φ, +i.e. Vs(φ) (provided φ(a) is invertible), that corre- +sponds to the naked singularity as an end state. +Similarly, in the case of an SH vector field col- +lapse, one has a class of vector field component +function A(a), or a class of vector field potential +Vv(a), or a class of vector field potential in terms +of B = g( ˜A, ˜A), i.e. Vs(B) (provided B(a) is in- +vertible), that corresponds to the naked singularity +as an end state. +3. A naked singularity formed due to gravitational col- +lapse may or may not be relevant if they are not +gravitationally strong in the sense of Tipler [20]. +Here, we show a class of ω(a) that satisfies the in- +equalities (65) that corresponds to the formation +of a strong curvature naked singularity. Using ar- +guments similar to point no. 2 of this section, we +have equivalently shown a class of scalar field po- +tential (in case of scalar field collapse) and a class +of vector field potential (in case of vector field col- +lapse) that corresponds to a strong curvature naked +singularity. +4. For the sake of completion, we study the global +spacetime, consisting of the interior collapsing +scalar/vector field and the exterior generalized +Vaidya spacetime. The smooth matching demands +a restriction on the free function, that is, the gener- +alized Vaidya mass function, in terms of the prop- +erty of the interior collapsing scalar/vector field. + +12 +We have fulfilled this demand by deriving the ex- +pression of the generalized Vaidya mass in terms +of the equation of state parameter of the interior +collapsing field in Eq.(51). +[1] J. R. Oppenheimer and H. Snyder, Phys. Rev. Journals +Archive 56, 455 (1939). +[2] S. Datt, Zs. f. Phys. 108 314 (1938). +[3] G.F.R. Ellis, S.T.C. Siklos and J. Wainwrighit, in Dy- +namical systems in cosmology, Eds. J. Wainwright and +G.F.R. Ellis, (Cambridge University Press, Cambridge, +England, 1997). +[4] R. Penrose, Riv. Nuovo Cimento Soc. Ital. Fis. 1, 252 +(1969). +[5] R. Geroch, Journal of Mathematical Physics, 11, 2, 437- +449 (1970). +[6] S. W. Hawking and G. F. R. Ellis, The large scale struc- +ture of spacetime, Cambridge University Press (1973). +[7] P. S. Joshi, Global Aspects in Gravitation and Cosmology +(Clendron Press, Oxford, 1993). +[8] R. Geroch and G. Horowitz, ‘Global structure of space- +times’, in General Relativity: +An Einstein Centenary +Survey, eds S. W. Hawking and W. Israel, Cambridge: +Cambridge University Press (1979). +[9] S. W. Hawking and W.Israel, ‘An introductory survey’, in +General Relativity: An Einstein Centenary Survey, eds +S. W. Hawking and W. Israel. Cambridge: Cambridge +University Press (1979). +[10] R. Penrose, ‘Singularities and time asymmetry’, in Gen- +eral Relativity: An Einstein Centenary Survey, eds S. W. +Hawking and W. Israel. Cambridge: Cambridge Univer- +sity Press (1979). +[11] R. Penrose, Phys. Rev. Lett. 14, 57 (1965). +[12] P. S. Joshi and D. Malafarina, Phys. Rev. D 83, 024009 +(2011). +[13] P. S. Joshi, Gravitational Collapse, and Spacetime Singu- +larities, (Cambridge University Press, Cambridge, Eng- +land, 2007). +[14] K. Mosani, D. Dey, P. S. Joshi, Phys. Rev. D 102, +044037. +[15] D. Christodoulou, Annals of Mathematics Annals of +Mathematics, 140, 607 (1994). +[16] D. Christodoulou, Annals of Mathematics, 149, 183 +(1999). +[17] R. Goswami and P. S. Joshi, Modern Physics Letters A, +22, 01, pp. 65-74 (2007). +[18] Karim Mosani, Dipanjan Dey, Kaushik Bhattacharya +and Pankaj S. Joshi, Phys. Rev. D 105, 064048 (2022). +[19] A. Wang and Y. Wu, Gen. Relativ. Gravit. 31, 107 +(1999). +[20] F. J. Tipler, Phys. Lett. 64A, 8 (1977). +[21] C. J. S. Clarke and A. Krolak, J. Geom. Phys. 2, 127 +(1985). +[22] D. M. Eardley, in ’Gravitation in Astrophysics’, ed. B. +Carter and J. B. Hartle (Plenum, New York, 1987). +[23] Karim Mosani, Dipanjan Dey and Pankaj S. Joshi, Phys. +Rev. D 101, 044052 (2020). +[24] Demetrios Christodoulou, Commun. Math. Phys. 105, +337-361 (1986). +[25] K. S. Virbhadra, S. Jhingan and P. S. Joshi, International +Journal of Modern Physics D 06, 357-361 (1997). +[26] David Garfinkle, Robert Mann, and Chris Vuille Phys. +Rev. D 68, 064015 (2003). +[27] E. Poisson, “A Relativist’s Toolkit: The Mathematics +of Black-Hole Mechanics,” Cambridge University Press, +(2009). + diff --git a/69E4T4oBgHgl3EQfcgw0/content/tmp_files/load_file.txt b/69E4T4oBgHgl3EQfcgw0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b542277558301d3da97aa7dc643a3ccea17e849b --- /dev/null +++ b/69E4T4oBgHgl3EQfcgw0/content/tmp_files/load_file.txt @@ -0,0 +1,657 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf,len=656 +page_content='Gravitational collapse of scalar and vector fields Karim Mosani,∗ Koushiki,† Pankaj S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Joshi,‡ and Jay Verma Trivedi§ International Centre for Space and Cosmology, School of Arts and Sciences, Ahmedabad University, Ahmedabad-380009 (Guj), India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Tapobroto Bhanja¶ International Center for Cosmology, & PDPIAS, Charotar University of Science and Technology, Anand- 388421 (Guj), India (Dated: January 13, 2023) We study here the unhindered gravitational collapse of spatially homogeneous (SH) scalar fields φ with a potential Vs(φ), as well as vector fields ˜A with a potential Vv(B) where B = g( ˜A, ˜A) and g is the metric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' We show that in both cases, classes of potentials exist that give rise to black holes or naked singularities depending on the choice of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The strength of the naked singular- ity is examined, and they are seen to be strong, in the sense of Tipler, for a wide class of respective potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' We match the collapsing scalar/vector field with a generalized Vaidya spacetime outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' We highlight that full generality is maintained within the domain of SH scalar or vector field collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' keywords: Gravitational collapse, singularity, scalar field, vector field, causal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' INTRODUCTION The contraction of a matter field under its gravita- tional influence is called gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In 1939, Oppenheimer and Snyder [1], and independently in 1938, Datt [2] developed the first solution of Einstein’s field equations (called the OSD model) depicting the gravita- tional collapse of a massive star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' They considered a very specific case of spatially homogeneous (SH) dust collapse (By spatial homogeneity, we mean homogeneous on a three-dimensional spacelike orbit with a six-dimensional isometry group G6 corresponding to the spacetime [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Such a matter field undergoes gravitational collapse that ends up in a singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Such a spacetime singularity is hidden behind an event horizon, not visible to any ob- server, and what we obtain is a black hole as the outcome of continual collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Extending the above special scenario, in 1969, Penrose proposed what is now known as the cosmic censorship hy- pothesis (CCH) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The weaker version of the hypothesis states that all singularities of gravitational collapse are hidden within a black hole and hence, cannot be seen by a distant observer (a globally naked singularity can- not exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The strong version of the hypothesis states that no past inextendible nonspacelike geodesics can ex- ist between the singularity and any point in the space- time manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In other words, a causal geodesic with a positive tangent “at” the singularity does not exist (a locally naked singularity also cannot exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The sup- porting argument for the validity of the strong CCH is the desirability of the spacetime manifold to be globally ∗ kmosani2014@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='com † koushiki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='malda@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='com ‡ pankaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='joshi@ahduni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='in § jay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='verma2210@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='com ¶ tapobroto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='bhanja@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='com hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Global hyperbolicity implies the existence of Cauchy surfaces embedded in the total manifold, thereby making general relativity a deterministic theory [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Now singularity theorems of Hawking and Penrose [6, 11] do not imply that singularities are hidden from an external observer under any possible circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In fact, singularity theorems take the causality condition as one of the axioms to start with to prove the existence of incomplete past (future) directed causal curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Addi- tionally, the OSD model that motivated cosmic censor- ship is a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Joshi and Malafarina [12] showed that any arbitrarily small neighbourhood of the initial data giving rise to OSD collapse contains initial data cor- responding to collapse evolution giving rise to a singular- ity with the following property: one could trace outgoing past singular causal geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' This means that the end state of OSD collapse is unstable under small perturba- tions in initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Moreover, one can show the forma- tion of naked singularities (global and local) as an end state of gravitational collapse from suitable, physically reasonable initial data for various matter fields [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' This implies that the initial conditions must be fine-tuned for the cosmic censorship conjecture to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In such a context, an important question one can ask here is as follows: what will be the end state of an un- hindered gravitational collapse of a fundamental matter field, such as a scalar field or a vector field, derived from an appropriate Lagrangian?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The answer to this question has been achieved up to a certain extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Scalar fields are fundamental matter fields derived from suitable Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' A real scalar field is a map defined on a smooth manifold as φ : M → R with a suitable continuity condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Christodoulou showed that in the case of gravitational collapse of a massless scalar field φ (the scalar field Lagrangian is Lφ = (1/2)gµν∂µφ∂νφ), the set of initial data giving rise to a naked singularity as an end state has positive codi- mension in the entire initial data set [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' This means arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='05083v1 [gr-qc] 12 Jan 2023 2 that the initial data set corresponding to naked singular- ity has a zero measure in the total initial data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In other words, naked singularity in such cases is unstable under arbitrarily small perturbations in the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' One can have a massless scalar field with a poten- tial function Vs(φ) that still be a fundamental matter field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' A massive scalar field will then be a particular case of a massless scalar field with a specific potential of the form Vs(φ) = (1/2)µ2φ2, where µ is the mass term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Goswami and Joshi [17] showed the example of the grav- itational collapse of a massless SH scalar field with a cer- tain potential Vs(φ) that ends up in a naked singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Mosani, Dey, Bhattacharya, and Joshi [18] conducted a similar investigation for a massless scalar field with a two- dimensional analogue of the Mexican hat-shaped Higgs field potential and found out that the end state of such unhindered scalar field collapse is a naked singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In addition to scalar fields as fundamental matter fields, vector fields are also fundamental matter fields de- rived from suitable matter Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Geometrically, vector fields on a smooth manifold M can be thought of as sections on the tangent bundle π : TM → M, where π is a continuous surjection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' A section is a smooth map σ : M → TM such that π ◦ σ is an identity map on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' From a particle physics point of view, the funda- mental nature of a vector field is different from that of a scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' There are many aspects, but one of the most important ones is that massive or massless vector fields mediate most particle physics processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' These represent the three fundamental interactions: quantum electrody- namics and weak and strong processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' A massless vector field with a potential function Vv(B) is again a funda- mental matter field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' A massive vector field will then be a particular case of a massless vector field with a specific potential of the form Vv(B) = (1/2)µ2B, where µ is the mass term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Garfinkle, Mann, and Vuille [26] have studied the collapse of a massive vector field and numerically ob- tained the critical initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' To our knowledge, much analytical work has not been done in investigating the causal structure of the end-state spacetime of the un- hindered gravitational collapse of matter fields that are vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In this paper, in both the massless SH scalar field as well as vector field cases, we show that there are broad classes of potentials for which the configuration collapses and ends up in either a black hole or a naked singu- larity depending on the potential function chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' We approach the causality investigation problem of scalar field as well as vector field collapse in a unified way, so to speak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' As far as general relativity is concerned, it does not discriminate between whether a scalar field or a vector field seeds the matter field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The matter field is entirely identified by a rank two tensor field that we call the stress-energy tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' As far as SH perfect fluid is concerned, one can identify a given matter field by the functional form of the equation of state parameter ω(a), where a is the scale factor of the collapsing cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' We derive relevant equations of collapsing SH scalar field φ(a) and vector field ˜A(a) in the sub-sections of section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The main body of section II contains discussions and relevant relations regarding the gravitational collapse of SH perfect fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In section III, we smoothly join the interior collapsing perfect fluid with an external gener- alized Vaidya spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In section IV, we investigate the causal structure of the spacetime (condition of ob- taining a naked singularity) at the end of the collapse of the interior perfect fluid that is either a scalar field φ with potential Vs or a vector field ˜A with potential Vv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' We also depict a few examples of well-known scalar fields and vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In section V, we derive the criteria for the singularity, thus obtained in the end, to be strong of Tipler’s type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In the last section, we highlight the key points of the investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Here we use the geometrized units 8πG = c = 1 throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' INTERIOR COLLAPSING MATTER FIELD Consider a gravitational collapse of a SH perfect fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The components of the stress-energy tensor in the coor- dinate basis {dxµ � ∂ν|0 ≤ µ, ν ≤ 3} of the comoving coordinates (t, x, y, z) are given by T µ ν = diag (−ρ, p, p, p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (1) The spacetime geometry is governed by the flat (k = 0) Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) metric ds2 = −dt2 + a2dΣ2, (2) where dΣ2 = dx2 + dy2 + dz2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Here a = a(t) is the scale factor such that a(0) = 1 and a(ts) = 0, where ts is the time of formation of the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' R = R(t, r) is the physical radius of the collapsing cloud and can be written as R(t, r) = ra(t), (3) where r is the radial spherical coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' For a FLRW spacetime Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (2), we have ρ = 3˙a2 a2 , (4) and p = −2¨a a − ˙a2 a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (5) The overhead dot denotes the partial time derivative of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (4) can be rewritten to obtain the dynamics of the collapse as ˙a = − � ρ(a) 3 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (6) Differentiating the above equation once again gives us ¨a = 1 3a �aρ,a 2 + ρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (7) 3 Integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (6), we obtain the time curve, which is t(a) = � 1 a �3 ρ da a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (8) The dynamics of the scale factor a(t) is, thus, the inverse of the LHS of the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The time of formation of the singularity ts = t(0) is ts = � 1 0 �3 ρ da a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (9) Now, let us consider a particular matter field ˆT from a set of all the possible SH perfect fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Choosing such an element means choosing a specific functional form of the equation of state parameter ω(a) = p ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (10) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (4), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (5), and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (10), we can express the density of the matter field with the equation of state parameter ω as ρ = ρ0 exp �� 1 a 3 (1 + ω(a)) a da � , (11) An SH perfect fluid is a fundamental matter field since it can be derived by a fundamental matter Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In the following two subsections, we will describe two distinct ways of obtaining such a matter field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Scalar field collapse We prove that any SH perfect fluid is equivalent to a SH scalar field φ(a) with a suitable potential Vs(a), as far as the gravitational collapse is concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' If φ(a) is invertible, then the following statement holds: Any SH perfect fluid is gravitationally equivalent to a SH scalar field φ with a suitable potential Vs(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Consider a real scalar field defined on the manifold M as φ : M → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (12) The Lagrangian of a massless scalar field φ with potential Vs(a) is given by Lφ = 1 2gµν∂µφ∂νφ − Vs(φ), (13) The stress-energy tensor is obtained from the Lagrangian Lφ as Tµν = − 2 √−g δ (√−gLφ) δgµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (14) The density (ρs) and the isotropic pressure (ps) are sub- sequently expressed in terms of the time derivative of the scalar field and its potential as ρs = 1 2 ˙φ2 + Vs (15) and ps = 1 2 ˙φ2 − Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (16) The overhead dot denotes the time derivative of the func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (15) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (16), and from using the chain rule ˙φ = φ,a ˙a, we get ρs + ps = φ2 ,a ˙a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (17) We now equate ρs = ρ and ps = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (5) and (17), along with replacing ˙a and ¨a using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (6) and (7), one obtains the expression of density as a function of a as ρs = ρ0 exp �� 1 a aφ2 ,ada � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (18) From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (15) and (16), we get ps = ρs − 2Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (19) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (6) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (17), we get ρs � 1 − φ2 ,aa2 3 � + ps = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (20) Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (19) and (20), we get Vs(φ) = ρs � 1 − φ2 ,aa2 6 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (21) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (17), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (6) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (5), one obtains ρs,a ρs = −φ,2 a a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (22) We have, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (10), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (15) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (16), Vs = ρs 2 (1 − ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (23) Now from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (21) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (23), we have φ(a),a = ± � 3 (1 + ω(a)) a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (24) Integrating the above equation, one obtains φ(a) = ± � 1 a � 3 (1 + ω(a)) a da + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (25) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (11) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (21) we have Vs(a) = ρ0 �1 − ω(a) 2 � exp �� 1 a 3 (1 + ω(a)) a da � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (26) Hence, we proved that given the functional form of 4 the equation of state parameter ω(a), one could obtain the corresponding scalar field φ(a) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (25) with potential Vs(a) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' As long as φ(a) is invertible (or, in other words, a bijective map from (0, 1] → R), we obtain a(φ), at least in principle, using which, we get Vs(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Alternatively, given a scalar field φ(a), one can obtain the corresponding perfect fluid ˆT (or the ω(a) by which it is identified), using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' On the other hand, we can also start with a given scalar field potential V (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' One can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (18) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (21) to obtain the ordinary nonlinear differential equation H � a, φ, dφ da , d2φ da2 � = 0, (27) that can be solved in principle, to obtain φ(a), and later obtain ω(a) using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='(24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Hence, given a scalar field po- tential Vs(φ), one can obtain the corresponding ˆT (iden- tified by ω(a)) in the above manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Vector field collapse We prove that any SH perfect fluid is equivalent to a SH vector field ˜A(a) with a suitable potential Vv(a), as far as the gravitational collapse is concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' If B(a) is in- vertible, then the following statement holds: Any SH per- fect fluid is gravitationally equivalent to a SH vector field ˜A with a suitable potential Vv(B) (where B = g( ˜A, ˜A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Consider a vector field ˜A : M → TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (28) with potential V (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' For a fixed p ∈ M, ˜A(p) = Aµdxµ, where Aµ = (A0, Ai), 1 < i < 3 (in the comoving carte- sian coordinate basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Here B = gαβAαAβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' We con- sider a SH pure vector field: A0 = 0 and Ai = A ∈ R ∀i ∈ (1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' For such a vector field, B = 3A2/a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The Lagrangian of a massless vector field ˜A with po- tential Vv(B) is given by L ˜ A = −1 4F µνFµν − Vv(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (29) F is a two form called the field strength and can be writ- ten in terms of wedge product as F = Fµνdxµ ∧ dxν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The field strength is the exterior derivative of the vec- tor field ˜A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='F = d ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The components are written as Fµν = ∇µAν − ∇νAµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The stress-energy tensor is obtained from the La- grangian L ˜ A as Tµν = − 2 √−g δ (√−gL ˜ A) δgµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (30) This gives us Tµν = −1 4FαβF αβgµν −Vv(B)gµν +FµαF α ν +2V ′ vAµAν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (31) The overhead prime denotes the ordinary derivative with respect to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The density and the isotropic pressure are subsequently expressed in terms of the time derivative of the vector field component and its potential as ρv = 3 2 ˙A2 a2 + Vv(B), (32) and pv = 1 2 ˙A2 a2 − Vv(B) + 2V ′ v A2 a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (33) We now equate ρv = ρ and pv = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (32) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (4), we obtain Vv = ρv � 1 − 1 2A,2 a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (34) Substituting for ρ(a) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (11), we obtain Vv = ρ0 exp �� 1 a 3 (1 + ω(a)) a da � � 1 − 1 2A,2 a � (35) On differentiating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (34) with respect to B we obtain, V ′ v = ρv,a � 1 − A,2 a 2 � − ρvA,a A,aa 6A2 a2 � A,a A − 1 a � (36) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (33), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (4), and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (5), we obtain aρv,a 3 + ρv � 1 + 1 6A,2 a � = Vv − V ′ v A2 a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (37) Substituting for Vv and V ′ v from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (34) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (36), and also substituting for ρ,a (by differentiating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (11)) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (37), we obtain a second order nonlinear differential equation G � a, ω, A, dA da , d2A da2 � = 0, (38) where G is G =d2A da2 − 4 A �dA da �2 + 1 2a (5 − 3ω) dA da + 6 A (1 + ω) − 3 (1 + ω) a �dA da �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (39) For a fixed ω(a), solving this differential equation with two initial conditions gives us A(a), and consequently, the vector field ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Hence, we proved that given the functional form of the equation of state parameter ω(a), one could obtain the corresponding vector field ˜A using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (38), and conse- quently, the vector field potential Vv(a) using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' 5 SH Perfect Fluid Characterised by the equation of state parameter 𝜔(a) SH Scalar Field Characterised by 𝜙(a) SH Vector Field Characterised by Ã(a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' 1: A spatially homogeneous (SH) perfect fluid (governed by a flat FLRW spacetime metric) is completely characterized by the equation of state parameter ω(a), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (10) of the matter field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' This matter field is obtained from fundamental matter Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Hence, the same matter field is also characterized by an SH scalar field φ(a), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (25) or its potential Vs(a), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (26) (Vs(φ) if φ(a) is invertible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Similarly, it can also be characterized by an SH vector field ˜A(a), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (38) or its potential Vv(a), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (35) (Vv(B) if B(a) is invertible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' This schematic diagram depicts the equivalence between the gravitational collapse of SH Perfect fluid, Scalar field and Vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' By spatial homogeneity, we mean homogeneous on a three-dimensional spacelike orbit with a six-dimensional isometry group G6 corresponding to the spacetime [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Now, from the functional form A(a), we obtain B(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' As long as B(a) is invertible (or, in other words, a bijec- tive map from (0, 1] → R), we obtain a(B), at least in principle, using which, we get Vv(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Alternatively, given a vector field ˜A(a), one can obtain the corresponding perfect fluid ˆT (or the ω(a) by which it is identified), using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' On the other hand, we can also start with a given vec- tor field potential Vv(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' One can differentiate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (35), and do some rearrangements to obtain ω(a, A, dA da , d2A da2 ) as ω = 2AV ′ aV �A a − dA da � −a 3 dA da d2A da2 � 1 − 1 2 �dA da �2�−1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (40) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (40) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (38), we obtain ˜G � a, A, dA da , d2A da2 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (41) In principle, this differential equation can be solved to obtain A(a), which, when substituted in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (40), gives us ω(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Hence, given a vector field potential V (B), one can obtain the corresponding ˆT (identified by ω(a)) in the above manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' EXTERIOR GENERALIZED VAIDYA SPACETIME The collapsing vector field spacetime (g− µν) can be joined smoothly with the exterior generalized Vaidya spacetime (g+ µν) so that their union forms a valid solution of the Einstein’s field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The interior FLRW and the exterior generalized Vaidya spacetime [19] are respec- tively given as ds2 − = −dt2 + a(t)2dr2 + r2 ba(t)2dΩ2, (42) and ds2 + = − � 1 − 2M(R, v) R � dv2 − 2dvdR + R2dΩ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (43) Here, v is the retarded null coordinate, R is the general- ized Vaidya radius, and rb is the value of the radial co- ordinate r corresponding to the matching hypersurface, or in other words, the radial coordinate of the outermost shell of the collapsing scalar/vector field cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The mat- ter field corresponding to the generalized Vaidya space- time is a combination of Type I and type II, such that the components of the stress-energy tensor written in the 6 orthonormal basis appear as Tab = � � � � � � � � � � � � � � � ¯ϵ 2 + ϵ ¯ϵ 2 0 0 ¯ϵ 2 ¯ϵ 2 − ϵ 0 0 0 0 P 0 0 0 0 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' � � � � � � � � � � � � � � � (44) ϵ = P = 0 and ¯ϵ ̸= 0 corresponds the usual Vaidya spacetime as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' ¯ϵ = 0 and ϵ ̸= 0 corresponds to a sub-class of Type I matter field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The generalized Vaidya solution encompasses many known Einstein field equations solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Matching the first and second fun- damental forms for the interior and exterior metric on Σ gives the following equations: R(t) = R(t, rb) (= rba(t)) , (45) F(t, rb) = 2M(R, v), (46) �dv dt � Σ = 1 + ˙R 1 − F (t,rb) R , (47) and M(R, v),R = F(t, rb) 2R + R ¨R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (48) Here, F = R ˙R2 is the Misner-Sharp mass function of the collapsing spherical SH perfect fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Using the rela- tion (45), we can relate the generalized Vaidya mass with the density of the interior collapsing SH spherical perfect fluid cloud as M = ρ 6R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (49) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (7), differentiation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (25) with respect to a, and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (49), in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (48) we get M,R = 3M R � 1 + (1 + ω(a))r2 b R2 � , (50) integrating which we obtain M(R, v) = M1(v) exp �� 3 R � 1 + (1 + ˜ω (R))r2 b R2 � dR � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (51) Here M1(v) is a constant of integration and is a function of null coordinate v, and ˜ω(R) = ω �R rb � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (51) gives us the expression of the generalized Vaidya mass function of the exterior generalized Vaidya space- time, in terms of interior collapsing perfect fluid equation of state parameter ω, to ensure smooth matching at the matching hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' For the exterior matter field to satisfy the weak energy condition, ¯ϵ and ϵ should be non-negative [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' These inequalities, in turn, put restrictions on the generalized Vaidya mass function as M,v ≤ 0, and M,R ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (52) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (50) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (51) in the above two relations, we obtain M1,v ≤ 0, (53) and �1 + ω(a) a2 � ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (54) The inequality (54) is always satisfied if the interior col- lapsing matter field obeys the weak energy condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Hence, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (53) is the only restriction on the generalized Vaidya mass function for the exterior spacetime to obey at least the weak energy condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Now, we have a complete solution of Einstein’s field equations consisting of an interior collapsing SH scalar/vector field (with some potential) and the exte- rior generalized Vaidya solution, matched smoothly at the matching hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The free functions are cat- egorically the potential function (Vs(φ) in case of scalar field collapse, and Vv(B) in case of vector field collapse), and the component of generalized Vaidya mass function M1(v), the latter one restricted by the inequality (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' It is evident that the choice of M1(v) does not affect the causal structure of the spacetime obtained as an end- state of unhindered gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Of course, instead of considering the potential function Vs(φ) (or Vv(B)) as a free function, one could also consider any one of the remaining functions: ω(a), ρ(a), φ(a) (or A(a)), Vs(a) (or Vv(a)) as a free function, without any trouble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In the next section, we study the end state of this class of global dynamical spacetime identified by any one of the free functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' CAUSAL STRUCTURE AND STRENGTH OF THE SINGULARITY Once the singularity is formed as an end state of grav- itational collapse of the interior scalar (vector) field with potential Vs(φ) (Vv(B)), one can investigate whether or not causal geodesics can escape the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Addition- ally, one can investigate whether or not such singularity is gravitationally strong in the sense of Tipler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The fol- lowing two subsections discuss these two properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='Massless scalar field ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='Vs(φ) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='φ(a) = c ± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='6 log a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='strong ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='BH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='Homogeneous dust (ω = 0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='Vs(φ) ∝ exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='3φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='φ(a) = c ± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='3 log a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='strong ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='BH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='Goswami/ Joshi [17] (ω = − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='3) (SF1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='Vs(φ) ∝ exp φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='φ(a) = c ± log a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='strong ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='NS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='Two dimensional analog of Mexican hat [18] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='(SF2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='Vs(φ) = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='2µφ2 + λφ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='φ(a) = ±2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='2√c − log a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='weak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='NS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='TABLE I: Four examples of spatially homogeneous scalar fields that collapse to form a singularity that is either ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='hidden (blackhole or BH) or (naked singularity or NS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In the fourth example, µ = − 16 3 λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The first three types end up in a strong singularity in the sense of Tipler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Massless vector field Vv(B) = 0 strong BH Massive vector field Vv(B) = − 1 2µ2B strong BH VF1 Vv(a) as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (3) strong NS VF2 Vv(a) as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (3) weak NS TABLE II: Four examples of spatially homogeneous vector fields that collapse to form a singularity that is either hidden within a black hole (BH) or is naked (NS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The ones mentioned in the third and the fourth row are newly constructed vector fields from known scalar fields (mentioned in the third [17] and the fourth [18] row of Table 1, respectively) by exploiting the gravitational equivalence depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The corresponding vector field component A(a) for each case is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='(2-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The first three types end up in a gravitationally strong singularity in the sense of Tipler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 a A a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 0 2 4 6 a V FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' 2: The dynamics of the vector field component A(a) in the case of the massive (µ = 1) vector field ˜A (Left panel) and its potential Vv(a) (Right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' First we obtain ω(a, A, dA da , d2A da2 ) by substituting Vv(B) = − 1 2µ2B in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='(40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Substituting for ω(a, A, dA da , d2A da2 ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (41) and solving the differential equation with initial conditions A(1) = 1 and A′(1) = 2, we obtain A(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Consequently, substituting Vv(B) = − 1 2µ2B, and the obtained A(a) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (40), we obtain ω(a), Further substitution of ω(a) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (35), we obtain Vv(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 3 2 1 0 1 2 3 a Vv (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 10 8 6 4 2 0 2 4 B Vv (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 300 200 100 0 100 200 a Vv (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 800 600 400 200 0 200 B Vv (d) A1 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 a A (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' 3: (a) and (c): Vector field potentials Vv(a) corresponding to newly constructed vector fields VF1 (orange) and VF2 (green), as mentioned in the third and fourth row of the Table (II), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (b) and (d): The same vector field potentials Vv(B) as function of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (e): The vector field components A(a) in both of these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In the latter example, µ = −8/3 and λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' First, we obtain ωi(a) , using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (25) (Here i ∈ 1, 2 corresponds to VF1 and VF2 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Then we obtain the vector field components Ai(a) by solving the differential Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (38) with initial conditions Ai(1) = 1 and A′ i(1) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Further substitution of ωi(a) and the obtained Ai(a) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (35), we get Vv(i)(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Once Vv(i)(a) is obtained, we obtain Vv(i)(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Causal structure of the singularity We say that a singularity formed due to unhindered gravitational collapse is naked if there exists a family of outgoing causal curves whose past endpoint is the singu- larity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In the future, these curves can either reach a far- away observer or fall back to the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The singu- larities are then termed globally naked and locally naked, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Whether or not the singularity is naked es- sentially depends on the geometry of trapped surfaces as the collapse evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Trapped surfaces are two-surfaces in the spacetime on which not only the ingoing congruence but also the outgoing congruence necessarily converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Convergence or otherwise of the outgoing null geodesic congruence is determined by the behaviour of its expan- sion scalar, which we denote here as θl (t, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' It is ex- pressed in terms of the metric coefficients, in comoving spherical coordinates as, θl = 2 R � 1 − � ρR2 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (55) 9 EH AH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='4 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='5 t (a) Massless scalar field (Vs = 0) EH AH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='4 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='35 t (b) Massless vector field (Vv = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='4 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='5 t (c) SF1/VF1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='4 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='5 t (d) SF2/VF2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' 4: Spacetime diagram of the examples of spatially homogeneous scalar fields and vector fields mentioned in Tables I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The solid black curve in each of them represents the boundary of the collapsing cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Upper panel: The singularity is not visible in both examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Lower Panel: In the case of SF1/VF1, the singularity forms in a finite comoving time and is globally visible because of the absence of the apparent and event horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In the case of SF2/VF2, the singularity forms in an infinite comoving time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' However, an ultra-high density region is obtained in finite comoving time, which can be visible globally because of the absence of the apparent and event horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The region in which θl < 0 is called the trapped region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The boundary of the trapped region, given by θl = 0, is called the apparent horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' If the neighbourhood of the singular center is surrounded by a trapped region since before the time of formation of the singularity ts, then it is covered, and we get a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Hence, the nec- essary condition for singular null geodesic congruence to escape the singularity is the absence of a trapped region, which is ensured by the condition θl(ts, r) > 0 for such congruence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The absence of trapped region in the neigh- bourhood of the singularity (t, r) = (ts, 0) is ensured by the following inequality: lim t→ts ρR2 3 ≤ lim a→0 ρ(a)r2 ba2 3 < 1, (56) The inequality (56) is definitely satisfied if lim a→0 ρ(a) < 1 a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (57) For lim a→0 ρ(a) = k a2 , for some k ∈ R+, the inequality is satisfied only for rb < � 3/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Rewriting the inequality (57) in terms of the equation of state parameter ω(a) using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (11), one obtains lim a→0 ρ0a2 exp �� 1 a 3 (1 + ω(a)) a � da < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (58) If a collapsing matter field with the equation of state parameter ω(a) satisfies the inequality (58), then it will 10 up in a naked singularity [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In the case of otherwise, the final outcome is a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Hence, as decided by the above inequality, we get a class of SH matter fields that include scalar and vector fields, identified by the functional form ω(a), that goes to either the blackhole or naked singularity final state as an end state of unhindered gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In the case of scalar field collapse, the restriction (58) on ω(a) gives us a restriction on the scalar field φ(a) us- ing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (25), and the scalar field potential function Vs(a) using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='(26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Hence, obtaining a class of ω(a) is grav- itationally equivalent to obtaining a class of scalar field potentials Vs(a) that goes to the naked singularity as an end state of unhindered gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' More- over, suppose φ(a) is a bijective map from (0, 1] → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In that case, obtaining a class of ω(a) is gravitationally equivalent to obtaining a class of scalar field potentials Vs(φ) = Vs(a(φ)) that goes to the naked singularity as an end state of gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Similarly, in the case of vector field collapse, the re- striction (58) on ω(a) gives us a restriction on the vector field ˜A (or more specifically, a restriction on the vec- tor field component A(a)) obtained by solving the dif- ferential Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (38), and the vector field potential function Vv(a) obtained by substituting A(a) and ρ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (11), in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Hence, obtaining a class of ω(a) is gravi- tationally equivalent to obtaining a class of vector field potential Vv(a) that goes to the naked singularity as an end state of unhindered gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' More- over, suppose A(a) is a bijective map from (0, 1] → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In that case, obtaining a class of ω(a) is gravitationally equivalent to obtaining a class of vector field potential Vv(A) = Vv(a(A)) that goes to the naked singularity as an end state of unhindered gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In Table (I) and (II), we discuss examples of such scalar field collapse and vector field collapse that ends up in either a black hole or a naked singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Ex- ploiting the equivalence between SH perfect fluids, scalar fields with potential Vs(a), and vector fields with poten- tial Vv(a), we construct two examples of collapsing vector fields with potential out-of-known examples of collapsing scalar fields with potentials, giving rise to the naked sin- gularity as an end state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The first example of a collapsing vector field with po- tential Vv(a) is constructed from the scalar field with po- tential mentioned in the third row of Table (I) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The perfect fluid corresponding to such scalar field example has an equation of state parameter ω(a) = − 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The con- structed collapsing vector field ˜A = (0, A, A, A) (in the comoving coordinate basis) has the property (dynamics of A(a) and Vv(a)) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Refer to the third row of Table (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The second example of a collapsing vector field with potential Vv(a) is constructed from the scalar field with potential mentioned in the fourth row of Table (I) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Such a scalar field has a two-dimensional analogue of Mexican hat-shaped potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The constructed collaps- ing vector field ˜A = (0, A, A, A) (in the comoving co- ordinate basis) has the property (dynamics of A(a) and Vv(a)) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Refer to the fourth row of Table (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The spacetime diagrams of some of the ex- amples in Table (I) and (II) are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Strength of the singularity Generally, a singularity in the spacetime manifold is identified by the existence of at least one past/future in- complete geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' However, in the case of singularities forming as the end state of a gravitational collapse, apart from the existence of such incomplete geodesics, one ex- pects an additional physical property as follows: An ob- ject approaching such singularity should be crushed to zero volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' We call such a singularity gravitationally strong in the sense of Tipler [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' A precise definition of a strong singularity is as follows: Consider a smooth spacetime manifold (M, g) and a causal geodesic γ : [t0, 0) → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Let λ be an affine pa- rameter along this geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Let ξ(i), (0 ≤ i ≤ 2 in the case of null geodesic, 0 ≤ i ≤ 3 in the case of timelike geodesic) be the independent Jacobi vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The wedge product of these Jacobi fields gives us the volume form V = � ξ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' We say that a singularity is gravita- tionally strong in the sense of Tipler if this volume form vanishes as λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Clarke and Krolak [21] related the existence of a Tipler strong singularity with the growth rate of the curvature terms as follows: At least along one null geodesic with affine parameter λ (such that λ → 0 as the singularity is approached), the following inequality lim λ→0 λ2RijKiKj > 0 (59) should hold for the singularity to be strong in the sense of Tipler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Here Ki = dxi dλ are the tangents to the chosen null geodesic, and xi is the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' This condition puts a lower bound on the growth of the curvature scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In the spherical coordinate system (t, r, θ, φ), the radial null geodesic equation reads dt dr = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (60) Hence, we have the relation between the tangents Kt and Kr as Kt = aKr, (61) and subsequently, in terms of the affine parameter, Kt = R λ , and Kr = r λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (62) The inequality (59) can then be written in terms of ω as lim a→0 � r2(1 + ω)ρ0 exp �� 1 a 3(1 + ω) a da �� > 0 (63) 11 Hence, the singularity formed due to the gravitational collapse of a scalar/vector field is strong in the sense of Tipler if the following inequality holds (assuming that the weak energy condition is respected): lim a→0 exp �� 1 a 3(1 + ω) a da � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (64) Hence, (along with using the condition (58)) one can ob- tain a naked singularity that is strong in the sense of Tipler for that ω that satisfies the following constraint: 0 < lim a→0 exp �� 1 a 3(1 + ω) a da � < O(a−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (65) This constraint gives us the class of SH collapsing mat- ter fields that we identify by ω(a), which ends up in strong curvature naked singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Or in other words, we have a class of scalar/vector field potentials corre- sponding to the given scalar/vector field that collapses to a strong naked singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' As an example, in Ta- bles (I) and (II), we mention the causal property and the strength of the singularity formed due to the gravi- tational collapse of four different scalar/vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' CONCLUSIONS AND REMARKS Following are the concluding remarks: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Unlike the singularity theorems that provide rig- orous proof of the existence of incomplete causal geodesics under rather generic conditions, one does not currently have proof or disproof of the cosmic censorship hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' In fact, we need a math- ematically rigorous formulation of this conjecture, which is not available currently, before we can prove or disprove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Under the situation at present, we can only spec- ulate its validity or otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Proposed coun- terexamples, hence have great importance in under- standing whether naked singularities, in fact, exist or not in our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Through such analysis of gravitational collapse models only, one could pos- sibly hope to arrive at a suitable formulation of cosmic censorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The collapse of inhomogeneous dust and the Vaidya null fluids were the first exam- ples proposed to produce naked singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' How- ever, an important objection could be that, even if astrophysically interesting, they are not fundamen- tal forms of matter [7, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' One could then ask whether the collapse of matter configuration that is obtained from a fundamental matter Lagrangian ends up in a naked singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Scalar fields with potential and vector fields with potential are fun- damental matter fields in this sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Here we show that not just one particular choice of these fields but an entire class of such types could collapse and form a naked singularity as an end state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' This basi- cally divides the allowed class of potential functions into classes that take the unhindered collapse to a black hole or naked singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' To achieve this, we show equivalence between SH (a) Perfect fluid: characterized by ω(a), (b) Massless scalar field φ: characterized by φ(a) or its potential Vs(a) or Vs(φ) (if φ(a) is in- vertible), and (c) Massless vector field ˜A: characterized by A(a), or its potential Vv(a), or Vv(B) (if B(a) is invertible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' as far as the gravitational collapse is concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' This gravitational equivalence is described in sub- sections of section (II) and depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Now, if the functional form of ω(a) satisfies the inequality (58), then the singular null geodesic con- gruence, if at all there exists, does not get trapped as a → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Hence, we have a class of functions ω(a) corresponding to a naked singularity as an end state of gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Now, because of the above equivalence, in the case of an SH scalar field collapse, one then has a class of scalar field func- tion φ(a), or a class of scalar field potential Vs(a), or a class of scalar field potential in terms of φ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Vs(φ) (provided φ(a) is invertible), that corre- sponds to the naked singularity as an end state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Similarly, in the case of an SH vector field col- lapse, one has a class of vector field component function A(a), or a class of vector field potential Vv(a), or a class of vector field potential in terms of B = g( ˜A, ˜A), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Vs(B) (provided B(a) is in- vertible), that corresponds to the naked singularity as an end state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' A naked singularity formed due to gravitational col- lapse may or may not be relevant if they are not gravitationally strong in the sense of Tipler [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Here, we show a class of ω(a) that satisfies the in- equalities (65) that corresponds to the formation of a strong curvature naked singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' Using ar- guments similar to point no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' 2 of this section, we have equivalently shown a class of scalar field po- tential (in case of scalar field collapse) and a class of vector field potential (in case of vector field col- lapse) that corresponds to a strong curvature naked singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' For the sake of completion, we study the global spacetime, consisting of the interior collapsing scalar/vector field and the exterior generalized Vaidya spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' The smooth matching demands a restriction on the free function, that is, the gener- alized Vaidya mass function, in terms of the prop- erty of the interior collapsing scalar/vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' 12 We have fulfilled this demand by deriving the ex- pression of the generalized Vaidya mass in terms of the equation of state parameter of the interior collapsing field in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfcgw0/content/2301.05083v1.pdf'} +page_content=' [1] J.' metadata={'source': 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a/7NA0T4oBgHgl3EQfOP_L/content/tmp_files/2301.02159v1.pdf.txt b/7NA0T4oBgHgl3EQfOP_L/content/tmp_files/2301.02159v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1d08a7e34974fd630d0e2429f0657ff5ef7fa1b9 --- /dev/null +++ b/7NA0T4oBgHgl3EQfOP_L/content/tmp_files/2301.02159v1.pdf.txt @@ -0,0 +1,3207 @@ +arXiv:2301.02159v1 [math.NA] 5 Jan 2023 +Finite element approximation of scalar curvature in arbitrary +dimension +Evan S. Gawlik∗ +Michael Neunteufel† +Abstract +We analyze finite element discretizations of scalar curvature in dimension N ≥ 2. +Our +analysis focuses on piecewise polynomial interpolants of a smooth Riemannian metric g on a +simplicial triangulation of a polyhedral domain Ω ⊂ RN having maximum element diameter h. +We show that if such an interpolant gh has polynomial degree r ≥ 0 and possesses single-valued +tangential-tangential components on codimension-1 simplices, then it admits a natural notion +of (densitized) scalar curvature that converges in the H−2(Ω)-norm to the (densitized) scalar +curvature of g at a rate of O(hr+1) as h → 0, provided that either N = 2 or r ≥ 1. As a special +case, our result implies the convergence in H−2(Ω) of the widely used “angle defect” approxima- +tion of Gaussian curvature on two-dimensional triangulations, without stringent assumptions on +the interpolated metric gh. We present numerical experiments that indicate that our analytical +estimates are sharp. +1 +Introduction +Many partial differential equations that arise in mathematical physics and geometric analysis involve +the Riemann curvature tensor and its contractions. +The scalar curvature R, which is obtained +from two contractions of the Riemann curvature tensor, is particularly important; it serves as the +integrand in the Einstein-Hilbert functional from general relativity, and it appears in the governing +equation for two-dimensional Ricci flow. To approximate solutions to PDEs involving the scalar +curvature, it is necessary to discretize the nonlinear differential operator that sends a Riemannian +metric tensor to its scalar curvature. +The goal of this paper is to construct and analyze such +discretizations in arbitrary dimension N ≥ 2. +We are specifically interested in the setting where a smooth Riemannian metric tensor g on +a polyhedral domain Ω ⊂ RN is approximated by a piecewise polynomial Regge metric gh on a +simplicial triangulation T of Ω having maximum element diameter h. Here, a metric is called a +Regge metric on T if it is piecewise smooth and its tangential-tangential components are single- +valued on every codimension-1 simplex in T . When such a metric is piecewise polynomial, it belongs +to a finite element space called the Regge finite element space [11, 12, 21]. Regge metrics are not +classically differentiable, so our first task will be to assign meaning to the scalar curvature of gh. Our +definition, which is a natural generalization of one that is now well-established in dimension N = 2, +treats the scalar curvature of gh as a distribution and regards it as an approximation of the densitized +scalar curvature of g, i.e. the scalar curvature R times the volume form ω. For piecewise constant +Regge metrics, our definition reduces to the classical definition of the distributional densitized +∗Department of Mathematics, University of Hawai‘i at Manoa, Honolulu, HI, 96822, USA, egawlik@hawaii.edu +†Institute for Analysis and Scientific Computing, TU Wien, Wiedner Hauptstr. 8-10, 1040 Wien, Austria, +michael.neunteufel@tuwien.ac.at +1 + +curvature on piecewise flat spaces [8, 23]. It is a linear combination of Dirac delta distributions +supported on (N − 2)-simplices S, weighted by the angle defect at S: 2π minus the sum of the +dihedral angles incident at S. For piecewise polynomial Regge metrics of higher degree, it includes +additional contributions involving the scalar curvature in the interior of each N-simplex and the +jump in the mean curvature across each (N − 1)-simplex. +We study the convergence of the distributional densitized scalar curvature of gh to the densitized +scalar curvature of g under refinement of the triangulation. We show in Theorem 4.1 that in the +H−2(Ω)-norm, this convergence takes place at a rate of O(hr+1) when gh is an optimal-order +interpolant of g that is piecewise polynomial of degree r ≥ 0, provided that either N = 2 or r ≥ 1. +Our numerical experiments in Section 5 suggest that these estimates are sharp in general. +To put this convergence result into context, let us summarize some existing convergence results +in the literature on finite element approximation of the scalar curvature. We first need to assemble +some notation. +Notation. +In what follows, W s,p(Ω) denotes the Sobolev-Slobodeckij space of differentiability +index s ∈ [0, ∞) and integrability index p ∈ [1, ∞], and ∥ · ∥W s,p(Ω) and | · |W s,p(Ω) denote the +associated norm and semi-norm, which we always take with respect to the Euclidean metric. We +denote Lp(Ω) = W 0,p(Ω) and Hs(Ω) = W s,2(Ω). For k ∈ N, we denote H−k(Ω) = (Hk +0 (Ω))′, where +Hk +0 (Ω) denotes the space of functions in Hk(Ω) whose derivatives of order 0 through k − 1 have +vanishing trace on ∂Ω, and the prime denotes the dual space. Occasionally we use weighted Lp and +H−k spaces associated with a Riemannian metric g, which we denote by Lp(Ω, g) and H−k(Ω, g); +see Section 4 and [16, Equation 4.1] for details. +If g is a smooth Riemannian metric and gh is a Regge metric, then R(g) denotes the scalar +curvature of g, (Rω)(g) denotes the densitized scalar curvature of g, (Rω)dist(gh) denotes the +distributional densitized scalar curvature of gh (defined below in Definition 3.1), and R(q) +h (gh) +denotes the L2(Ω, gh)-projection of (Rω)dist(gh) onto the Lagrange finite element space of degree q. +We also use the terms optimal-order interpolant, canonical interpolant, and geodesic interpolant +below. The first of these is a catch-all term for any piecewise polynomial interpolant gh of g that +belongs to the Regge finite element space and enjoys error estimates of optimal order in W s,p(T)- +norms on N-simplices T; see Definition 4.2. The canonical interpolant is a specific interpolant +(which is optimal-order) detailed in [21, Chapter 2]. The geodesic interpolant of g is the unique +piecewise constant Regge metric gh with the property that the length of every edge in T , as +measured by gh, agrees with the geodesic distance between the corresponding vertices in T , as +measured by g. +Summary of existing results. +We can now summarize some existing results about the approx- +imation of g’s curvature by gh’s distributional curvature. Throughout what follows, the letter r +denotes the polynomial degree of gh. +1. Cheeger, M¨uller, and Schrader [8, Equation (5.7) and Theorem 5.1] proved that if r = 0 and +gh is the geodesic interpolant of g, then (Rω)dist(gh) converges to (Rω)(g) in the (setwise) +sense of measures at a rate of O(h) in dimension N = 2 and O(h1/2) in dimension N ≥ 3. +2. Gawlik [16, Theorem 4.1] proved that if r ≥ 1, N = 2, and gh is any optimal-order interpolant +of g, then R(q) +h (gh) converges to R(g) at a rate of O(hr) in the H−1(Ω, g)-norm and at a rate of +O(hr−k−1) in the broken Hk(Ω)-norm, k = 0, 1, 2, . . . , r − 2, provided that q ≥ max{1, r − 2}. +3. Berchenko-Kogan and Gawlik [4, Corollary 6.2] proved that if r ≥ 1, N = 2, and gh is any +optimal-order interpolant of g, then (Rω)dist(gh) converges to (Rω)(g) at a rate of O(hr) in +2 + +the norm ∥u∥V ′,h = supv∈V,v̸=0⟨u, v⟩V ′,V /∥v∥V,h, where +V = {v ∈ H1 +0(Ω) | v|T ∈ H2(T) ∀T ∈ T N} +(1) +and ∥v∥V,h = |v|H1(Ω) + +�� +T∈T N h2 +T |v|2 +H2(T) +�1/2 +. Here, hT denotes the diameter of T, and +T N denotes the set of N-simplices in T . +4. Gopalakrishnan, Neunteufel, Sch¨oberl, and Wardetzky [19, Theorem 6.5 and Corollary 6.6] +proved that if r ≥ 0, N = 2, and gh is the canonical interpolant of g, then R(r+1) +h +(gh) converges +to R(g) at a rate of O(hr+1) in the H−1(Ω, g)-norm and at a rate of O(hr−k) in the broken +Hk(Ω)-norm, k = 0, 1, 2, . . . , r − 1. +New results. +As one can see from above, our analysis in this paper covers two important cases +that have not yet been addressed in the literature: +1. We prove a convergence result in the case where N ≥ 3 and r ≥ 1. This opens the door +to the use of piecewise polynomial Regge metrics to approximate scalar curvature in high +dimensions. +2. We prove a convergence result in the case where N = 2, r = 0, and gh is an arbitrary +optimal-order interpolant of g. This has been a longstanding gap in the literature on Gaussian +curvature approximation. Previous efforts to address the case where N = 2 and r = 0 have +relied on subtle properties of the geodesic interpolant [8] and the canonical interpolant [19]. +Our results establish the validity of Gaussian curvature approximations involving the angle +defect without stringent assumptions on the interpolated metric tensor gh. +Note that our analysis predicts no convergence at all in the H−2(Ω)-norm when N ≥ 3 and r = 0. +Our numerical experiments suggest that this result is sharp for general optimal-order interpolants. +However, for the canonical interpolant, numerical experiments suggest that (Rω)dist(gh) converges +to (Rω)(g) in the H−2(Ω)-norm at a rate of O(h) when N ≥ 3 and r = 0. We intend to study this +superconvergence phenomenon exhibited by the canonical interpolant in future work. +Structure of the paper. +Our strategy for proving convergence of (Rω)dist(gh) to (Rω)(g) con- +sists of two steps. First, in Sections 2-3, we study the evolution of (Rω)dist(gh) under deformations +of the metric, leading to an integral formula for the error (Rω)dist(gh) − (Rω)(g) which reads +⟨(Rω)dist(gh) − (Rω)(g), v⟩V ′,V = +� 1 +0 +bh(�g(t); σ, v) − ah(�g(t); σ, v) dt, +∀v ∈ V. +(2) +Here, �g(t) = (1 − t)g + tgh, σ = +∂ +∂t�g(t) = gh − g, V is the space defined in (1), and bh(�g(t); ·, ·) +and ah(�g(t); ·, ·) are certain metric-dependent bilinear forms. In Section 4, we use techniques from +finite element theory to estimate the right-hand side of (2), leading to Theorem 4.1. +The approach above is similar to the one used in dimension N = 2 in [4, 16, 19], but there are +a few important differences. First, we work with an integral formula for the error (Rω)dist(gh) − +(Rω)(g) rather than an integral formula for the curvature itself. Previous analyses in [4, 16, 19] +hinged on formulas of the latter type. Loosely speaking, in this paper we compute the evolution of +the error along a one-parameter family of Regge metrics starting at g and ending at gh, whereas +the papers [4, 16, 19] compute the evolution of the curvature along a pair of one-parameter families +of metrics: one family that starts at the Euclidean metric δ and ends at gh, and one that starts at +3 + +δ and ends at g. The approach based on evolving the error appears to be better suited for proving +optimal error estimates. +Another key aspect of our analysis is our use of the H−2(Ω)-norm to measure the error. This +norm is weaker than the ones used in [4, 16, 19], and it appears to be more natural for measuring +the error in the curvature. For example, for piecewise constant Regge metrics in dimension N = 2, +we show that convergence of (Rω)dist(gh) to (Rω)(g) holds in the H−2(Ω)-norm for any optimal- +order interpolant of g, but numerical experiments suggest that it fails to hold in stronger norms +when gh is not the canonical interpolant of g. A key tool that we use to prove convergence in +H−2(Ω) is the near-equivalence of a certain pair of metric-dependent, mesh-dependent norms on +V ; see Proposition 4.5. This equivalence is similar to one that Walker [27, Theorems 4.1 and 4.3] +proved for an analogous family of mesh-dependent norms on triangulated surfaces. +Additional comments. +The formula (2) is not only useful for the error analysis, but it is also +interesting in its own right. It has a differential counterpart (see Theorem 3.6) that reads +d +dt⟨(Rω)dist(�g(t)), v⟩V ′,V = bh(�g(t); σ, v) − ah(�g(t); σ, v), +∀v ∈ V, +(3) +which mimics the formula +d +dt +� +Ω +Rvω = +� +Ω +(div div Sσ)vω − +� +Ω +⟨G, σ⟩vω, +∀v ∈ V +(4) +that holds for a family of smooth Riemannian metrics g(t) with densitized scalar curvature Rω and +Einstein tensor G = Ric − 1 +2Rg. Here, Sσ = σ−g Tr σ, and div is the covariant divergence operator; +see below for more notational details. +The correspondence between (3) and (4) becomes even more transparent when one inspects the +formulas for bh and ah (see Theorem 3.6). The bilinear form bh(�g; ·, ·) is (up to the appearance of +S) a non-Euclidean, N-dimensional generalization of a bilinear form that appears in the Hellan- +Herrmann-Johnson finite element method [1–3, 5–7, 9, 22]. +It can be regarded as the integral +of div div Sσ against v, where div div is interpreted in a distributional sense. This link with the +Hellan-Herrmann-Johnson method has previously been noted and used in dimension N = 2 [4, 16, +19]. +The bilinear form ah(�g; ·, ·), which is only nonzero in dimension N ≥ 3, appears to play the role +of +� +Ω⟨G, σ⟩vω, which is also only nonzero in dimension N ≥ 3. It gives rise to a natural way of +defining the Einstein tensor in a distributional sense for Regge metrics. We discuss this more in +Section 3.2. Among other things, we point out that the formula for ah contains a term involving +the jump in the trace-reversed second fundamental form across codimension-1 simplices; the same +quantity arises in studies of singular sources in general relativity, where it encodes the well-known +Israel junction conditions across a hypersurface on which stress-energy is concentrated [20]. +There are a few other connections between our calculations and ones that appear in the physics +literature. The variation of the Gibbons-Hawking-York boundary term in general relativity [17, 28] +is one example. It has many parallels to our calculations in Section 2.2, and one can undoubtedly +find formulas like (6) in the literature after reconciling notations. We still give a full derivation +of such formulas, not only to familiarize the reader with our notation, but also to provide careful +derivations that refrain from discarding total derivatives (which integrate to zero on manifolds +without boundary, but not in general) and minimize the use of local coordinate calculations where +possible. +4 + +2 +Evolution of geometric quantities +In this section, we consider an N-dimensional manifold M equipped with a smooth Riemannian +metric g, and we study the evolution of various geometric quantities under deformations of g. +We adopt the following notation in this section. The Levi-Civita connection associated with g +is denoted ∇. If σ is a (p, q)-tensor field, then its covariant derivative is the (p, q + 1)-tensor field +∇σ, and its covariant derivative in the direction of a vector field X is the (p, q)-tensor field ∇Xσ. +Its trace Tr σ is the contraction of σ along the first two indices, using g to raise or lower indices +as needed. We denote div σ = Tr ∇σ and ∆σ = div ∇σ. The g-inner product of two (p, q)-tensor +fields σ and ρ is denoted ⟨σ, ρ⟩. +The volume form associated with g is denoted ω. The Ricci tensor and the scalar curvature of g +are denoted Ric and R, respectively. When we wish to emphasize their dependence on g, we write +ω(g), Ric(g), R(g), etc. +If D is an embedded submanifold of M, then we denote by ωD the induced volume form on D. +If σ is a tensor field on M, then σ|D denotes the pullback of σ under the inclusion D ֒→ M. Later +we will introduce some additional notation related to embedded submanifolds of codimension 1, +like the mean curvature H and second fundamental form II; see Section 2.2. +We denote the exterior derivative of a differential form α by dα. If α is a one-form, then α♯ +denotes the vector field obtained by raising indices with g. If f is a scalar field, then we sometimes +interpret the one-form ∇f = df as the vector field (df)♯ without explicitly writing it. +Later, in Section 4, we will append a subscript g to many quantities like ∇ and ⟨·, ·⟩ to emphasize +their dependence on g. +In that section only, an absent subscript will generally signal that the +quantity in question is computed with respect to the Euclidean metric, which we denote by δ. We +say more about this notational shift in Section 4. +2.1 +Evolution of the densitized scalar curvature +First we study the evolution of the densitized scalar curvature Rω under deformations of the metric. +Proposition 2.1. Let g(t) be a family of smooth Riemannian metrics with time derivative ∂ +∂tg =: σ. +We have +∂ +∂t(Rω) = (div div Sσ)ω − ⟨G, σ⟩ω, +where G = Ric − 1 +2Rg denotes the Einstein tensor associated with g and +Sσ = σ − g Tr σ. +Proof. We compute +∂ +∂t(Rω) = ˙Rω + R ˙ω +and invoke the well-known formulas [15, Lemma 2] +˙R = div div σ − ∆ Tr σ − ⟨Ric, σ⟩ +and [10, Equation 2.4] +˙ω = 1 +2(Tr σ)ω. +Since ∆ Tr σ = div div(g Tr σ) and Tr σ = ⟨g, σ⟩, the result follows. +5 + +2.2 +Evolution of the mean curvature +Next we study the evolution of the mean curvature H of a hypersurface F. We assume that the tan- +gent bundle of F is trivial, so that there exists a smooth, g-orthonormal frame field τ1, τ2, . . . , τN−1 +on F. (If this is not the case, then one can simply fix a point p ∈ F and focus on a neighborhood of p +on which the tangent bundle is trivial.) We let n be the unit normal to F so that n, τ1, τ2, . . . , τN−1 +forms a right-handed g-orthonormal frame (in the ambient manifold) at each point on F. If the +metric g varies smoothly in time, then we assume that the vectors n, τ1, τ2, . . . , τN−1 also vary +smoothly in time and remain g-orthonormal at all times. +We use the notation +II(X, Y ) = g(∇Xn, Y ) = −g(n, ∇XY ) +for the second fundamental form on F. Our sign convention is such that Tr II = H, and H is +positive for a sphere with an outward normal vector. We also let ∇F and divF denote the surface +gradient and surface divergence operators on F, which have the following meanings. For a scalar +field v, +∇F v = ∇v − n∇nv = +N−1 +� +i=1 +τi∇τiv, +and for a one-form α, +divF α = Tr (∇α|F) = +N−1 +� +i=1 +(∇τiα)(τi). +Note that in the formula ∇F v = ∇v − n∇nv, we have regarded ∇v as a vector field rather than a +one-form. Recall that the surface divergence operator satisfies the identity +� +F +(divF α)ωF = +� +∂F +α(νF )ω∂F + +� +F +Hα(n)ωF , +(5) +where νF is the outward unit normal to ∂F and H is the mean curvature of F. +Proposition 2.2. Let g(t) be a family of smooth Riemannian metrics with time derivative ∂ +∂tg =: σ. +Let F be a time-independent hypersurface with mean curvature H and induced volume form ωF. +Then +∂ +∂t(HωF ) = −1 +2 +�� +II, σ|F +� ++ (div Sσ)(n) + divF (σ(n, ·)) − Hσ(n, n) +� +ωF, +(6) +where +II(X, Y ) = II(X, Y ) − Hg(X, Y ) +is the trace-reversed second fundamental form. +Remark 2.3. In dimension N = 2, the formula (6) simplifies considerably. +Letting τ and n +denote the unit tangent and unit normal to F, we have ∇τn = Hτ, −∇ττ = Hn, and II(τ, τ) = +g(∇τn, τ) − Hg(τ, τ) = H − H = 0, so II vanishes. In addition, +divF (σ(n, ·)) − Hσ(n, n) = ∇τ (σ(n, ·)) (τ) − Hσ(n, n) += ∇τ (σ(n, τ)) − σ(n, ∇ττ) − Hσ(n, n) += ∇τ (σ(n, τ)) . +Thus, in two dimensions, +∂ +∂t(HωF) = −1 +2 ((div Sσ)(n) + ∇τ (σ(n, τ))) ωF. +6 + +To prove Proposition 2.2, we write +˙H = − +N−1 +� +i=1 +∂ +∂tg(n, ∇τiτi) +(7) +and use the following lemmas. +Lemma 2.4. For any time-dependent vector fields X and Y , +∂ +∂t∇Y X = ∇ ˙Y X + ∇Y ˙X + 1 +2 ((∇Xσ)Y + (∇Y σ)X − (∇σ)(X, Y ))♯ , +where (∇σ)(X, Y ) denotes the one-form Z �→ (∇Zσ)(X, Y ), and (∇Xσ)Y denotes the one-form +Z �→ (∇Xσ)(Y, Z). +Proof. In coordinates, +(∇Y X)ℓ = Y j ∂Xℓ +∂xj + Γℓ +ijY jXi, +where Γℓ +ij denote the Christoffel symbols of the second kind associated with g. Thus, +∂ +∂t(∇Y X)ℓ = ˙Y j ∂Xℓ +∂xj + Γℓ +ij ˙Y jXi + Y j ∂ ˙Xℓ +∂xj + Γℓ +ijY j ˙Xi + ˙Γℓ +ijY jXi += (∇ ˙Y X)ℓ + (∇Y ˙X)ℓ + ˙Γℓ +ijY jXi. +Next, we recall the following formula for the rate of change of the Christoffel symbols under a +metric deformation [10, Equation 2.23]: +˙Γℓ +ij = 1 +2gℓm ((∇iσ)jm + (∇jσ)im − (∇mσ)ij) . +It follows that +˙Γℓ +ijY jXi = 1 +2gℓm � +(∇Xσ)jmY j + (∇Y σ)imXi − (∇mσ)ijY jXi� += 1 +2 [((∇Xσ)Y + (∇Y σ)X − (∇σ)(X, Y ))]ℓ . +Hence, +∂ +∂t(∇Y X)ℓ = (∇ ˙Y X)ℓ + (∇Y ˙X)ℓ + 1 +2 ((∇Xσ)Y + (∇Y σ)X − (∇σ)(X, Y ))ℓ . +Lemma 2.5. For any time-dependent vector field X, +∂ +∂tg(n, X) = 1 +2σ(n, n)g(n, X) + g(n, ˙X). +Proof. Writing X = ng(n, X) + �N−1 +i=1 τig(τi, X), we compute +∂ +∂tg(n, X) = σ(n, X) + g( ˙n, X) + g(n, ˙X) += σ(n, n)g(n, X) + +N−1 +� +i=1 +σ(n, τi)g(τi, X) + g( ˙n, n)g(n, X) + +N−1 +� +i=1 +g( ˙n, τi)g(τi, X) + g(n, ˙X) += (σ(n, n) + g( ˙n, n)) g(n, X) + +N−1 +� +i=1 +(σ(n, τi) + g( ˙n, τi)) g(τi, X) + g(n, ˙X). +7 + +For each i = 1, 2, . . . , N − 1, we have +0 = ∂ +∂tg(n, τi) = σ(n, τi) + g( ˙n, τi) + g(n, ˙τi) += σ(n, τi) + g( ˙n, τi) +since ˙τi is g-orthogonal to n. Likewise, +0 = ∂ +∂tg(n, n) = σ(n, n) + 2g(n, ˙n), +so the result follows. +We are now ready to compute the time derivative of the mean curvature H. By Lemma 2.5, we +have +˙H = − +N−1 +� +i=1 +∂ +∂tg(n, ∇τiτi) += − +N−1 +� +i=1 +�1 +2σ(n, n)g(n, ∇τiτi) + g +� +n, ∂ +∂t∇τiτi +�� += 1 +2Hσ(n, n) − +N−1 +� +i=1 +g +� +n, ∂ +∂t∇τiτi +� +. +(8) +Using Lemma 2.4 and the symmetry of the second fundamental form, we can write the second term +as +g +� +n, ∂ +∂t∇τiτi +� += g(n, ∇ ˙τiτi) + g(n, ∇τi ˙τi) + (∇τiσ)(n, τi) − 1 +2(∇nσ)(τi, τi) += 2g(n, ∇ ˙τiτi) + (∇τiσ)(n, τi) − 1 +2(∇nσ)(τi, τi). +The first term above, when summed over i, can be simplified as follows. We write ˙τi = �N−1 +j=1 τjg(τj, ˙τi) +and use the linearity of ∇XY in X to compute +2 +N−1 +� +i=1 +g(n, ∇ ˙τiτi) = 2 +N−1 +� +i=1 +N−1 +� +j=1 +g(n, ∇τjτi)g(τj, ˙τi) += +N−1 +� +i=1 +N−1 +� +j=1 +g(n, ∇τjτi) (g(τj, ˙τi) + g( ˙τj, τi)) += − +N−1 +� +i=1 +N−1 +� +j=1 +g(n, ∇τjτi)σ(τj, τi) += ⟨II, σ|F ⟩. +Above, we used the symmetry of the second fundamental form to pass from the first line to the +second, and we used the identity +0 = ∂ +∂tg(τj, τi) = σ(τj, τi) + g(τj, ˙τi) + g( ˙τj, τi) +8 + +to pass from the second line to the third. Inserting these results into (8), we get +˙H = 1 +2Hσ(n, n) − ⟨II, σ|F ⟩ + +N−1 +� +i=1 +�1 +2(∇nσ)(τi, τi) − (∇τiσ)(n, τi) +� +. +(9) +Lemma 2.6. We have +N−1 +� +i=1 +�1 +2(∇nσ)(τi, τi) − (∇τiσ)(n, τi) +� += 1 +2 (⟨II, σ|F⟩ − (div Sσ)(n) − divF (σ(n, ·))) . +(10) +Proof. The identity 0 = ∇τi (g(n, n)) = 2g(n, ∇τin) shows that ∇τin is in the span of {τj}N−1 +j=1 , so +the first term on the right-hand side of (10) satisfies +⟨II, σ|F ⟩ = +N−1 +� +i=1 +N−1 +� +j=1 +σ(τj, τi)g(τj, ∇τin) += +N−1 +� +i=1 +σ(∇τin, τi). +(11) +The second term on the right-hand side of (10) can be computed as follows. Recalling that Sσ = +σ − g Tr σ, we have +(div Sσ)(n) = ∇n(Sσ)(n, n) + +N−1 +� +i=1 +∇τi(Sσ)(n, τi) += (∇nσ)(n, n) − ∇n(g Tr σ)(n, n) + +N−1 +� +i=1 +[(∇τiσ)(n, τi) − ∇τi(g Tr σ)(n, τi)] += (∇nσ)(n, n) − g(n, n)∇n Tr σ + +N−1 +� +i=1 +[(∇τiσ)(n, τi) − g(n, τi)∇τi Tr σ] += (∇nσ)(n, n) − ∇n Tr σ + +N−1 +� +i=1 +(∇τiσ)(n, τi). +Since the trace commutes with covariant differentiation, +∇n Tr σ = Tr ∇nσ = (∇nσ)(n, n) + +N−1 +� +i=1 +(∇nσ)(τi, τi). +Thus, +(div Sσ)(n) = +N−1 +� +i=1 +[(∇τiσ)(n, τi) − (∇nσ)(τi, τi)] . +(12) +The third term on the right-hand side of (10) is given by +divF (σ(n, ·)) = +N−1 +� +i=1 +∇τi (σ(n, ·)) (τi) += +N−1 +� +i=1 +[∇τi (σ(n, τi)) − σ(n, ∇τiτi)] . +(13) +9 + +Combining (11), (12), and (13), we see that +1 +2 (⟨II, σ|F ⟩ − (div Sσ)(n) − divF (σ(n, ·))) += 1 +2 +N−1 +� +i=1 +[σ(∇τin, τi) − (∇τiσ)(n, τi) + (∇nσ)(τi, τi) − ∇τi (σ(n, τi)) + σ(n, ∇τiτi)] += 1 +2 +N−1 +� +i=1 +[(∇nσ)(τi, τi) − 2(∇τiσ)(n, τi)] . +Combining Lemma 2.6 with (9), we get +˙H = 1 +2 (−⟨II, σ|F ⟩ − (div Sσ)(n) − divF (σ(n, ·)) + Hσ(n, n)) . +(14) +Proposition 2.2 now follows from the identities +∂ +∂t(HωF) = ˙HωF + H ˙ωF = ˙HωF + 1 +2H Tr (σ|F ) ωF +and +⟨II, σ|F ⟩ − H Tr (σ|F) = ⟨II, σ|F⟩. +2.3 +Evolution of angles +Next we study the evolution of angles under deformations of the metric. +Lemma 2.7. Let g(t) be a family of smooth Riemannian metrics with time derivative +∂ +∂tg =: σ. +Let (¯n(t), ¯τ(t)) be a pair of g(t)-orthonormal vectors, and let (n(t), τ(t)) be another pair of g(t)- +orthonormal vectors lying in the span of (¯n(t), ¯τ(t)). Let θ(t) be the angle for which +τ = ¯τ cos θ + ¯n sin θ, +n = −¯τ sin θ + ¯n cos θ. +Assume that these vectors vary smoothly in time, and assume that n(t) (respectively, ¯n(t)) is at all +times g(t)-orthogonal to a time-independent hypersurface F (respectively, ¯F). Then, at all times +for which θ ∈ (0, π), we have +∂ +∂tθ = 1 +2σ(n, τ) − 1 +2σ(¯n, ¯τ). +(15) +Proof. Differentiating the relation cos θ = g(¯n, n) yields +− ˙θ sin θ = ∂ +∂t (g(¯n, n)) . +In particular, at any time s, we can write +− ˙θ(s) sin θ(s) = ∂ +∂t +���� +t=s +(g(t)(¯n(t), n(s))) + ∂ +∂t +���� +t=s +(g(t)(¯n(s), n(t))) − σ(s)(¯n(s), n(s)). +10 + +Using Lemma 2.5 and suppressing the evaluations at t = s, we get +− ˙θ sin θ = 1 +2σ(¯n, ¯n)g(¯n, n) + 1 +2σ(n, n)g(n, ¯n) − σ(¯n, n) += 1 +2σ(¯n, ¯n cos θ − n) + 1 +2σ(n cos θ − ¯n, n) += 1 +2σ(¯n, ¯τ sin θ) + 1 +2σ(−τ sin θ, n). +If θ ∈ (0, π) at time t = s, then we can divide by sin θ to get (15). +3 +Distributional densitized scalar curvature +Let T be a simplicial triangulation of a polyhedral domain Ω ⊂ RN. We use T k to denote the +set of all k-simplices in T . We also use ˚T k to denote the subset of T k consisting of k-simplices +that are not contained in the boundary of Ω. We call such simplices interior simplices. We call +(N − 1)-simplices faces. +Let g be a Regge metric on T . Recall that this means that g|T is a smooth Riemannian metric +on each T ∈ T N, and the induced metric g|F is single-valued on each F ∈ ˚ +T N−1 (and consequently +the induced metric is single-valued on all lower-dimensional simplices in T ). +On each T ∈ T N, we denote by RT the scalar curvature of g|T . On an interior face F ∈ ˚ +T N−1 +that lies on the boundary of two N-simplices T + and T −, the second fundamental form on F, as +measured by g|T +, generally differs from that measured by g|T −. We denote by �II�F the jump in +the second fundamental form across F. More precisely, +�II�F(X, Y ) = g|T + (∇Xn+, Y ) + g|T − (∇Xn−, Y ) +for any vectors X, Y tangent to F, where n± points outward from T ±, has unit length with respect +to g|T ±, and is g|T ±-orthogonal to F. We adopt similar notation for the jumps in other quantities +across F. For instance, �H�F denotes the jump in the mean curvature across F. We sometimes +drop the subscript F when there is no danger of confusion. If F is contained in ∂Ω, then we define +the jump in a scalar field v across F to be simply �v�F = v|F . +On each S ∈ ˚ +T N−2, the angle defect along S is +ΘS = 2π − +� +T∈T N +T⊃S +θST, +where θST denotes the dihedral angle formed by the two faces of T that contain S, as measured by +g|T . Generally this angle may vary along S. If F + and F − are the two faces of T that contain S, +and if n± denotes the unit normal to F ± with respect to g|T pointing outward from T, then +cos θST = − g|T (n+, n−). +Let +V = {v ∈ H1 +0(Ω) | ∀T ∈ T N, v|T ∈ H2(T)}. +Note that if v ∈ V , then v admits a single-valued trace on every simplex in T of dimension ≥ N −3. +Definition 3.1. Let g be a Regge metric. The distributional densitized scalar curvature of g is the +linear functional (Rω)dist(g) ∈ V ′ defined by +⟨(Rω)dist(g), v⟩V ′,V = +� +T∈T N +� +T +RT vωT + 2 +� +F ∈˚ +T N−1 +� +F +�H�FvωF + 2 +� +S∈˚ +T N−2 +� +S +ΘSvωS, +∀v ∈ V. +(16) +11 + +This definition generalizes Definition 3.1 of [4], where the distributional curvature two-form (i.e. +the Gaussian curvature times the volume form) is defined for Regge metrics in dimension N = 2. +Note that the factors of 2 appearing in all but the first term in (16) are consistent with the fact +that in dimension N = 2, the scalar curvature R is twice the Gaussian curvature. +One can heuristically motivate Definition 3.1 in much the same way that one motivates its +two-dimensional counterpart. When g is piecewise constant, Definition 3.1 recovers the classical +notion [23] that the distributional densitized scalar curvature is a linear combination of Dirac delta +distributions supported on (N − 2)-simplices, with weights given by angle defects. When g is not +piecewise constant, additional terms appear which account for the nonzero (classically defined) +curvature of g in the interior of each N-simplex T and the jump in the mean curvature across each +interior face F. The jump in the mean curvature across F can be understood by recalling that the +scalar curvature R at a point p ∈ F can be expressed as (two times) a sum of sectional curvatures +of N(N − 1)/2 tangent planes that are mutually g-orthogonal at p, (N − 1)(N − 2)/2 of which +are tangent to F at p and N − 1 of which are g-orthogonal to F at p. The sectional curvatures +corresponding to planes tangent to F are nonsingular, owing to the tangential-tangential continuity +of g. The remaining N − 1 sectional curvatures are singular, and by considering an N-dimensional +region that encloses a portion of F and has small thickness in the direction that is g-orthogonal of F, +one can use the Gauss-Bonnet theorem (along two-dimensional slices) to approximate the (volume- +)integrated sum of these sectional curvatures by the (surface-)integrated jump in the mean curvature +across F. +(In this calculation, one must bear in mind that sectional curvatures and Gaussian +curvatures are related via the Gauss-Codazzi equations.) See the discussion after Definition 3.1 +in [4], as well as [26], for more insight in dimension N = 2. See also [13] for a justification of +Definition 3.1 in the case where g is piecewise constant and N ≥ 2. +In the sequel, we will consistently use the letters T, F, and S to refer to simplices of dimension +N, N − 1, and N − 2, respectively. We will therefore write � +T , � +F , and � +S in place of � +T∈T N , +� +F ∈T N−1, and � +S∈T N−2, respectively. When we wish to sum over interior simplices of a given +dimension, we put a ring on top of the summation symbol. Thus, for example, ˚ +� +F is shorthand +for � +F ∈˚ +T N−1. +3.1 +Evolution of the distributional scalar curvature +We are interested in how (16) changes under deformations of the metric. To this end, consider a +one-parameter family of Regge metrics g(t) with time derivative +σ = ∂ +∂tg. +Our goal will be to compute +d +dt⟨(Rω)dist(g(t)), v⟩V ′,V +with v ∈ V arbitrary. +According to Propositions 2.1 and 2.2, the derivatives of the first two terms on the right-hand +side of (16) satisfy +d +dt +� +T +RT vωT = +� +T +(div div Sσ − ⟨G, σ⟩) vωT +and +2 d +dt +� +F +�H�FvωF = − +� +F +�� +II, σ|F +� ++ (div Sσ)(n) + divF (σ(n, ·)) − Hσ(n, n) +� +vωF . +(17) +For the third term on the right-hand side of (16), we use the following lemma. +12 + +Lemma 3.2. Along any interior (N − 2)-simplex S, we have +∂ +∂t(ΘSωS) = 1 +2 +�� +F ⊃S +�σ(n, τ)�F + ΘS Tr(σ|S) +� +ωS, +where the sum is over all (N −1)-simplices F that contain S, n is the unit normal to F with respect +to g, and τ is the unit vector with respect to g that points into F from S and is g-orthogonal to +both S and n. Here, our convention is that if F is shared by two N-simplices T + and T −, then +�σ(n, τ)�F = σ+(n+, τ) + σ−(n−, τ), +where σ± = σ|T ± and n± points outward from T ±. +Remark 3.3. Note that n generally differs on either side of F, whereas τ does not, because g has +single-valued tangential-tangential components along F. +Proof. We compute +˙ΘS = − +� +T⊃S +˙θST +and use Lemma 2.7 to differentiate each angle θST. +The resulting expression for ˙ΘS involves +differences between σ(n, τ) evaluated on consecutive pairs of faces F emanating from S. This sum +can be rearranged to give +˙ΘS = 1 +2 +� +F ⊃S +�σ(n, τ)�F. +(18) +We thus get +∂ +∂t(ΘSωS) = ˙ΘSωS + ΘS ˙ωS += 1 +2 +� +F ⊃S +�σ(n, τ)�F ωS + 1 +2ΘS Tr (σ|S) ωS. +It follows from the above lemma that +2 d +dt +� +S +ΘSvωS = +� +S +� +F ⊃S +�σ(n, τ)�F vωS + +� +S +ΘS Tr(σ|S)vωS += +� +S +� +F ⊃S +�σ(n, τ)�F vωS + +� +S +⟨ΘSg|S, σ|S⟩ vωS. +Collecting our results, we obtain +d +dt⟨(Rω)dist(g(t)), v⟩V ′,V = +� +T +� +T +(div div Sσ)vωT +− ˚ +� +F +� +F +�(div Sσ)(n) + divF (σ(n, ·)) − Hσ(n, n)�F vωF + ˚ +� +S +� +S +� +F ⊃S +�σ(n, τ)�F vωS +(19) +− +� +T +� +T +⟨G, σ⟩vωT − ˚ +� +F +� +F +� +�II�F, σ|F +� +vωF + ˚ +� +S +� +S +⟨ΘSg|S, σ|S⟩ vωS. +We will now use integration by parts to rewrite the first three terms in a way that involves no +derivatives of σ. +13 + +Lemma 3.4. For any v ∈ V , we have +� +T +� +T +(div div Sσ)vωT − ˚ +� +F +� +F +�(div Sσ)(n) + divF (σ(n, ·)) − Hσ(n, n)�F vωF ++ ˚ +� +S +� +S +� +F ⊃S +�σ(n, τ)�FvωS = +� +T +� +T +⟨Sσ, ∇∇v⟩ω − +� +F +� +F +Sσ(n, n)�∇nv�ωF. +Proof. We have +� +T +� +T +⟨Sσ, ∇∇v⟩ω − +� +F +� +F +Sσ(n, n)�∇nv�ωF +(20) += +� +T +� � +T +⟨Sσ, ∇∇v⟩ω − +� +∂T +Sσ(n, n)∇nv ω∂T +� += +� +T +� � +∂T +Sσ(n, ∇v)ω∂T − +� +T +(div Sσ)(∇v)ω − +� +∂T +Sσ(n, n)∇nv ω∂T +� += +� +T +� � +∂T +Sσ(n, ∇v)ω∂T − +� +∂T +(div Sσ)(n)vω∂T + +� +T +(div div Sσ)vω +− +� +∂T +Sσ(n, n)∇nv ω∂T +� +. +(21) +Note that here we are regarding ∇v as a vector field rather than a one-form. On each N-simplex +T, we can write +� +∂T Sσ(n, ∇v)ω∂T − +� +∂T Sσ(n, n)∇nv ω∂T as a sum of integrals over faces F ⊂ ∂T: +� +∂T +Sσ(n, ∇v)ω∂T − +� +∂T +Sσ(n, n)∇nv ω∂T = +� +F ⊂∂T +� +F +Sσ(n, ∇v − n∇nv)ωF += +� +F ⊂∂T +� +F +Sσ(n, ∇F v)ωF += +� +F ⊂∂T +� +F +σ(n, ∇F v)ωF . +In the last line above, we used the fact that ∇F v is g-orthogonal to n, so +Sσ(n, ∇Fv) = σ(n, ∇F v) − g(n, ∇F v) Tr σ = σ(n, ∇F v). +Each integral over F can be integrated by parts as follows. We have +σ(n, ∇F v) = divF (σ(n, ·)v) − divF (σ(n, ·)) v, +so the identity (5) applied to α = σ(n, ·)v implies that +� +F +σ(n, ∇F v)ωF = +� +∂F +σ(n, νF )vω∂F − +� +F +(divF (σ(n, ·)) − Hσ(n, n)) vωF. +Now we insert this result into (21) to get +� +T +� +T +⟨Sσ, ∇∇v⟩ω − +� +F +� +F +Sσ(n, n)�∇nv�ωF += +� +T +� � +F ⊂∂T +� +∂F +σ(n, νF )vω∂F − +� +F ⊂∂T +� +F +(divF (σ(n, ·)) − Hσ(n, n)) vωF +− +� +∂T +(div Sσ)(n)vω∂T + +� +T +(div div Sσ)vω +� +. +14 + +The first term can be re-expressed as a sum over interior (N − 2)-simplices S using our notation +from Lemma 3.2, and the next two terms can be re-expressed in terms of jumps across interior +faces F. (Integrals over (N − 2)-simplices S ⊂ ∂Ω and (N − 1)-simplices F ⊂ ∂Ω vanish because +v = 0 on ∂Ω.) The result is +� +T +� +T +⟨Sσ, ∇∇v⟩ω − +� +F +� +F +Sσ(n, n)�∇nv�ωF = ˚ +� +S +� +S +� +F ⊃S +�σ(n, τ)�FvωS +− ˚ +� +F +� +F +�divF (σ(n, ·)) − Hσ(n, n) + (div Sσ)(n)� vωF + +� +T +� +T +(div div Sσ)vω. +Remark 3.5. Many of the above calculations are similar to the ones in [4, Proposition 4.2], except +that here we are in dimension N rather than 2. +We can now state the main result of this subsection. +Theorem 3.6. Let g(t) be a family of Regge metrics with time derivative +∂ +∂tg =: σ. For every +v ∈ V , we have +d +dt⟨(Rω)dist(g(t)), v⟩V ′,V = bh(g; σ, v) − ah(g; σ, v), +(22) +where +bh(g; σ, v) = +� +T +� +T +⟨Sσ, ∇∇v⟩ω − +� +F +� +F +Sσ(n, n)�∇nv�FωF, +ah(g; σ, v) = +� +T +� +T +⟨G, σ⟩vωT + ˚ +� +F +� +F +� +�II�F, σ|F +� +vωF − ˚ +� +S +� +S +⟨ΘSg|S, σ|S⟩ vωS. +Proof. Combine (19) with Lemma 3.4. +3.2 +Distributional densitized Einstein tensor +We now pause to make a few remarks about the bilinear forms ah(g; ·, ·) and bh(g; ·, ·) appearing +in Theorem 3.6. These remarks will play no role in our analysis, but they help to elucidate the +content of Theorem 3.6. The reader can safely skip ahead to Section 4 if desired. +Numerical analysts will likely recognize the bilinear form bh(g; ·, ·) appearing in Theorem 3.6. As +we mentioned in Section 1, it is (up to the appearance of S) a non-Euclidean, N-dimensional gener- +alization of a bilinear form that appears in the Hellan-Herrmann-Johnson finite element method [1– +3, 5–7, 9, 22]. It can be regarded as the integral of div div Sσ against v, where div div is interpreted +in a distributional sense. +The bilinear form ah(g; ·, ·) can be understood by comparing Theorem 3.6 with Proposition 2.1, +which, when integrated against a continuous function v, states that for a family of smooth Rieman- +nian metrics g(t) with scalar curvature R, +d +dt +� +Ω +Rvω = +� +Ω +(div div Sσ)vω − +� +Ω +⟨G, σ⟩vω, +(23) +where σ = ∂ +∂tg and G = Ric − 1 +2Rg is the Einstein tensor associated with g. A comparison of (23) +with (22) suggests that for a Regge metric g, the bilinear form ah(g; σ, v) should be regarded as a +distributional counterpart of +� +Ω⟨G, σ⟩vω. +15 + +This motivates the following definition. Fix a number s > 1, and let Σ denote the space of +square-integrable symmetric (0, 2)-tensor fields σ with the following properties: the restriction of +σ to each T ∈ T N belongs to Hs(T), and the tangential-tangential components of σ along any +face F ∈ ˚ +T N−1 are single-valued. Note that these conditions imply that the tangential-tangential +components of σ along any S ∈ ˚ +T N−2 are well-defined and single-valued as well. +Definition 3.7. Let g be a Regge metric. The distributional densitized Einstein tensor associated +with g is the linear functional (Gω)dist(g) ∈ Σ′ defined by +⟨(Gω)dist(g), σ⟩Σ′,Σ = +� +T +� +T +⟨G, σ⟩ωT + ˚ +� +F +� +F +� +�II�F, σ|F +� +ωF − ˚ +� +S +� +S +⟨ΘSg|S, σ|S⟩ ωS, +∀σ ∈ Σ. +Remark 3.8. In dimension N = 2, we have (Gω)dist(g) = 0 for any Regge metric g, because +G vanishes within each triangle, ¯II vanishes on each edge, and the restriction of σ to each vertex +vanishes. +Remark 3.9. The appearance of the trace-reversed second fundamental form II in Definition 3.7 +is quite natural. The same quantity arises in studies of singular sources in general relativity, with +the jump in II encoding the well-known Israel junction conditions across a hypersurface on which +stress-energy is concentrated [20]. +Remark 3.10. If we define a map (div div S)dist : Σ → V ′ by +⟨(div div S)distσ, v⟩V ′,V = bh(g; σ, v), +∀v ∈ V, +then, by construction, we have +d +dt +���� +t=0 +⟨(Rω)dist(g + tσ), v⟩V ′,V = ⟨(div div S)distσ, v⟩V ′,V − ⟨(Gω)dist(g), vσ⟩Σ′,Σ +for every piecewise smooth σ ∈ Σ and every smooth function v with compact support in Ω. In +particular, suppose that Ω has no boundary (e.g., suppose that Ω is an N-dimensional cube and +we identify its opposing faces). Then bh(g; σ, 1) = 0 and +d +dt +���� +t=0 +⟨(Rω)dist(g + tσ), 1⟩V ′,V = −⟨(Gω)dist(g), σ⟩Σ′,Σ +for every piecewise smooth σ ∈ Σ. This implies that a Regge metric g is a stationary point of +⟨(Rω)dist(g), 1⟩Σ′,Σ if its distributional densitized Einstein tensor vanishes: (Gω)dist(g) = 0. +The functional ⟨(Rω)dist(g), 1⟩Σ′,Σ is a counterpart of the Einstein-Hilbert functional +� +Ω Rω from +general relativity, whose stationary points are solutions to the (vacuum) Einstein field equations +G = 0. It reduces to the Regge action from Regge calculus when g is piecewise constant. That is, +⟨(Rω)dist(g), 1⟩Σ′,Σ = 2 ˚ +� +S +ΘSVS, +if g is piecewise constant, +where VS = +� +S ωS denotes the volume of S. If g varies with t and remains piecewise constant for +all t, then +d +dt2 ˚ +� +S +ΘSVS = 2 ˚ +� +S +˙ΘSVS + 2 ˚ +� +S +ΘS ˙VS, +16 + +and one checks that (on a domain without boundary) +2 ˚ +� +S +˙ΘSVS = bh(g; σ, 1) = 0 +and +2 ˚ +� +S +ΘS ˙VS = −ah(g; σ, 1) = −⟨(Gω)dist(g), σ⟩Σ′,Σ, +where σ = +∂ +∂tg. The fact that ˚ +� +S ˙ΘSVS = 0 for any piecewise constant Regge metric g (on a +domain without boundary) was proved in Regge’s classic paper [23] using very different techniques. +Remark 3.11. If g is a Regge metric and σ = gv for some smooth function v with compact support +in Ω, then: +1. On each N-simplex T, we have +⟨G, σ⟩ = ⟨G, g⟩v = (Tr G)v = − +�N − 2 +2 +� +Rv. +2. On either side of each interior (N − 1)-simplex F, we have: +� +II, σ|F +� += ⟨II, g|F ⟩ v − ⟨g|F , g|F⟩ Hv += Hv − (N − 1)Hv += −(N − 2)Hv. +3. On each interior (N − 2)-simplex S, we have +⟨ΘSg|S, σ|S⟩ = ΘSv Tr(g|S) = (N − 2)ΘSv. +This shows that +⟨(Gω)dist(g), gv⟩Σ′,Σ = − +�N − 2 +2 +� �� +T +� +T +RT vωT + 2 ˚ +� +F +� +F +�H�FvωF + 2 ˚ +� +S +� +S +ΘSvωS +� += − +�N − 2 +2 +� +⟨(Rω)dist(g), v⟩V ′,V +for every smooth function v with compact support in Ω. One can interpret this as saying that the +trace of (Gω)dist(g) is − +� N−2 +2 +� +(Rω)dist(g). +Remark 3.12. If g is a piecewise constant Regge metric and σ ∈ Σ is piecewise constant, then +⟨(Gω)dist(g), σ⟩Σ′,Σ = − ˚ +� +S +� +S +ΘS Tr(σ|S)ωS. +If we linearize around the Euclidean metric g = δ, then we see from (18) that +d +dt +���� +t=0 +⟨(Gω)dist(δ + tρ), σ⟩Σ′,Σ = − ˚ +� +S +� +S +˙ΘS Tr(σ|S)ωS += −1 +2 +˚ +� +S +� +S +� +F ⊃S +�ρ(n, τ)�F Tr(σ|S)ωS +17 + +for every piecewise constant ρ, σ ∈ Σ. (Note that there are no additional terms on the right-hand +side because ΘS = 0 at t = 0.) Hence, if Ω has no boundary, then +d2 +dt2 +���� +t=0 +⟨(Rω)dist(δ + tσ), 1⟩V ′,V = − d +dt +���� +t=0 +⟨(Gω)dist(δ + tσ), σ⟩Σ′,Σ += 1 +2 +˚ +� +S +� +S +� +F ⊃S +�σ(n, τ)�F Tr(σ|S)ωS +for every piecewise constant σ ∈ Σ. This is equivalent to Christiansen’s formula [12, Theorem 2 +and Equations (25-26)] for the second variation of the Regge action around the Euclidean metric +in dimension N = 3. (There, the Regge action is taken to be 1 +2⟨(Rω)dist(g), 1⟩V ′,V rather than +⟨(Rω)dist(g), 1⟩V ′,V .) +4 +Convergence +In this section, we prove a convergence result for the distributional densitized scalar curvature in +the norm +∥u∥H−2(Ω) = +sup +v∈H2 +0(Ω), +v̸=0 +⟨u, v⟩H−2(Ω),H2 +0(Ω) +∥v∥H2(Ω) +. +(24) +Our convergence result will be applicable to a family {gh}h>0 of Regge metrics defined on a shape- +regular family {Th}h>0 of triangulations of Ω parametrized by h = maxT∈T N +h hT , where hT = +diam(T). Shape-regularity means that there exists a constant C0 independent of h such that +max +T∈T N +h +hT +ρT +≤ C0 +for all h > 0, where ρT denotes the inradius of T. +Theorem 4.1. Let Ω ⊂ RN be a polyhedral domain equipped with a smooth Riemannian metric g. +Let {gh}h>0 be a family of Regge metrics defined on a shape-regular family {Th}h>0 of triangulations +of Ω. Assume that limh→0 ∥gh − g∥L∞(Ω) = 0 and C1 := suph>0 maxT∈T N +h ∥gh∥W 1,∞(T) < ∞. The +following statements hold: +(i) If N = 2, then there exist positive constants C and h0 such that +∥(Rω)dist(gh) − (Rω)(g)∥H−2(Ω) ≤ C +� +1 + max +T +h−1 +T ∥gh − g∥L∞(T) + max +T +|gh − g|W 1,∞(T) +� +× +� +∥gh − g∥2 +L2(Ω) + +� +T +h2 +T |gh − g|2 +H1(T) +�1/2 +(25) +for all h ≤ h0. The constants C and h0 depend on ∥g∥W 1,∞(Ω), ∥g−1∥L∞(Ω), C0, and C1. +18 + +(ii) If N ≥ 3, assume additionally that C2 := suph>0 maxT∈T N +h |gh|W 2,∞(T) < ∞. Then there exist +positive constants C and h0 such that +∥(Rω)dist(gh) − (Rω)(g)∥H−2(Ω) ≤ C +� +1 + max +T +h−2 +T ∥gh − g∥L∞(T) + max +T +h−1 +T |gh − g|W 1,∞(T) +� +× +� +∥gh − g∥2 +L2(Ω) + +� +T +h2 +T |gh − g|2 +H1(T) + +� +T +h4 +T |gh − g|2 +H2(T) +�1/2 +(26) +for all h ≤ h0. The constants C and h0 depend on N, ∥g∥W 1,∞(Ω), ∥g−1∥L∞(Ω), C0, C1, and +C2. +The above theorem leads immediately to error estimates of optimal order for piecewise poly- +nomial interpolants of g having degree r ≥ 0, provided that either N = 2 or r ≥ 1. To make this +statement precise, we introduce a definition. Recall that the Regge finite element space of degree +r ≥ 0 consists of symmetric (0, 2)-tensor fields on Ω that are piecewise polynomial of degree at +most r and possess single-valued tangential-tangential components on interior (N − 1)-simplices. +Definition 4.2. Let Ih be a map that sends smooth symmetric (0, 2)-tensor fields on Ω to the Regge +finite element space of degree r ≥ 0. We say that Ih is an optimal-order interpolation operator of +degree r if there exists a number m ∈ {0, 1, . . . , N} and a constant C3 = C3(N, r, hT /ρT , t, s) such +that for every p ∈ [1, ∞], every s ∈ (m/p, r + 1], every t ∈ [0, s], and every symmetric (0, 2)-tensor +field g possessing W s,p(Ω)-regularity, Ihg exists (upon continuously extending Ih) and satisfies +|Ihg − g|W t,p(T) ≤ C3hs−t +T +|g|W s,p(T) +(27) +for every T ∈ T N +h . +We call the number m the codimension index of Ih. +A Regge metric gh +is called an optimal-order interpolant of g having degree r and codimension index m if it is the +image of a Riemannian metric g under an optimal-order interpolation operator having degree r and +codimension index m. +An example of an optimal-order interpolation operator is the canonical interpolation operator +onto the degree-r Regge finite element space introduced in [21, Chapter 2]. Its degrees of freedom +involve integrals over simplices of codimension at most N − 1, so its action on a tensor field g is +well-defined so long as g admits traces on simplices of codimension at most N − 1, i.e. g possesses +W s,p(Ω)-regularity with s > (N − 1)/p. Correspondingly, its codimension index is m = N − 1. +Corollary 4.3. Let Ω, g, and {Th}h>0 be as in Theorem 4.1. Let {gh}h>0 be a family of optimal- +order interpolants of g having degree r ≥ 0 and codimension index m. If N ≥ 3, assume that r ≥ 1. +Then there exist positive constants C and h0 such that +∥(Rω)dist(gh) − (Rω)(g)∥H−2(Ω) ≤ C +�� +T +hp(r+1) +T +|g|p +W r+1,p(T) +�1/p +for all h ≤ h0 and all p ∈ [2, ∞] satisfying p > +m +r+1. +(We interpret the right-hand side as +C maxT hr+1 +T +|g|W r+1,∞(T) if p = ∞.) The constants C and h0 depend on the same quantities listed +in (i) (if N = 2) and (ii) (if N ≥ 3), as well as on Ω, r, and (if N ≥ 3) |g|W 2,∞(Ω). +19 + +Remark 4.4. The corollary above continues to hold if we allow slightly more general interpolants +in Definition 4.2. For example, it holds if (27) is replaced by +|Ihg − g|W t,p(T) ≤ C3hs−t +T +� +T ′:T ′∩T̸=∅ +|g|W s,p(T ′), +(28) +where the sum is over all T ′ ∈ T N +h +that share a subsimplex with T. +In what follows, we reuse the letter C to denote a positive constant that may change at each +occurrence and may depend on N, ∥g∥W 1,∞(Ω), ∥g−1∥L∞(Ω), C0, and C1. Beginning in Lemma 4.8, +we allow C to also depend on C2. +Our strategy for proving Theorem 4.1 will be to consider an evolving metric +�g(t) = (1 − t)g + tgh +with time derivative +σ = ∂ +∂t�g(t) = gh − g. +Note that �g(t), being piecewise smooth and tangential-tangential continuous, is a Regge metric for +all t ∈ [0, 1], and it happens to be a (globally) smooth Riemannian metric at t = 0. Since �g(0) = g +and �g(1) = gh, Theorem 3.6 implies that +⟨(Rω)dist(gh) − (Rω)(g), v⟩V ′,V = +� 1 +0 +bh(�g(t); σ, v) − ah(�g(t); σ, v) dt, +∀v ∈ V. +Thus, we can estimate (Rω)dist(gh) − (Rω)(g) by estimating the bilinear forms bh(�g(t); ·, ·) and +ah(�g(t); ·, ·). +To do this, we introduce some notation. Given any Regge metric g, we let ∇g and ∇ denote +the covariant derivatives with respect to g and δ, respectively. Similarly, we append a subscript +g to other operators like Tr, S, and div when they are taken with respect to g, and we omit the +subscript when they are taken with respect to δ. On the boundary of any N-simplex T, we let ng +and n denote the outward unit normal vectors with respect to g|T and δ, respectively. These two +vectors are related to one another in coordinates via +ng = +1 +� +nT g−1n +g−1n, +(29) +where we are thinking of g as a matrix and n and ng as column vectors. We write ⟨·, ·⟩g for the +g-inner product of two tensor fields. If D is a submanifold of Ω on which the induced metric g|D +is well-defined, and if ρ is a tensor field on D, then we denote +∥ρ∥Lp(D,g) = +��� +D |ρ|p +g ωD(g) +�1/p , +if 1 ≤ p < ∞, +supD |ρ|g, +if p = ∞, +where ωD(g) is the induced volume form on D and |ρ|g = ⟨ρ, ρ⟩1/2 +g +. We abbreviate ∥ · ∥Lp(D) = +∥ · ∥Lp(D,δ) and | · | = | · |δ. +We introduce two metric-dependent, mesh-dependent norms. For v ∈ V , we set +∥v∥2 +2,h,g = +� +T +∥∇g∇gv∥2 +L2(T,g) + +� +F +h−1 +F ∥�dv(ng)�∥2 +L2(F,g) . +20 + +If σ is a symmetric (0, 2)-tensor field with the property that σ(ng, ng) is well-defined and single- +valued on every F ∈ T N−1 +h +, then we set +∥σ∥2 +0,h,g = +� +T +∥σ∥2 +L2(T,g) + +� +F +hF ∥σ(ng, ng)∥2 +L2(F,g), +where hF is the Euclidean diameter of F. Note that the image under Sg of any symmetric (0, 2)- +tensor field possessing single-valued tangential-tangential components along faces automatically +possesses single-valued normal-normal components along faces, because +Sgσ(ng, ng) = σ(ng, ng) − g(ng, ng) Trg σ = − Trg (σ|F ) . +Now we return to the setting of Theorem 4.1 and the discussion thereafter: g is a smooth +Riemannian metric, gh is a Regge metric, �g(t) = (1 − t)g + tgh, and σ = gh − g. We assume +throughout what follows that limh→0 ∥gh − g∥L∞(Ω) = 0 and suph>0 maxT∈T N +h ∥gh∥W 1,∞(T) < ∞. +These assumptions have some elementary consequences that we record here for reference (see [16] +for a derivation). For every h sufficiently small, every t ∈ [0, 1], and every vector w with unit +Euclidean length, +∥�g∥L∞(Ω) + ∥�g−1∥L∞(Ω) ≤ C, +(30) +max +T +|�g|W 1,∞(T) ≤ C, +(31) +C−1 ≤ inf +Ω (wT �gw) ≤ sup +Ω +(wT �gw) ≤ C, +(32) +where we are thinking of �g as a matrix and w as a column vector in the last line. Note that the +last line implies the existence of positive lower and upper bounds on wT �g−1w as well: +C−1 ≤ inf +Ω (wT �g−1w) ≤ sup +Ω +(wT �g−1w) ≤ C. +(33) +In addition, the inequalities ∥�g∥L∞(Ω) ≤ C and ∥�g−1∥L∞(Ω) ≤ C imply that +C−1∥ρ∥Lp(D,�g(t2)) ≤ ∥ρ∥Lp(D,�g(t1)) ≤ C∥ρ∥Lp(D,�g(t2)) +(34) +and +C−1∥ρ∥Lp(D) ≤ ∥ρ∥Lp(D,�g(t1)) ≤ C∥ρ∥Lp(D) +(35) +for every t1, t2 ∈ [0, 1], every admissible submanifold D, every p ∈ [1, ∞], every tensor field ρ having +finite Lp(D)-norm, and every h sufficiently small. We select h0 > 0 so that (30-35) hold for all +h ≤ h0, and we tacitly use these inequalities throughout our analysis. +We will show the following near-equivalence of the norms ∥ · ∥2,h,�g and ∥ · ∥2,h,g. +Proposition 4.5. For every v ∈ V , every h ≤ h0, and every t ∈ [0, 1], +∥v∥2 +2,h,�g ≤ C +� +∥v∥2 +2,h,g + +� +max +T +h−2 +T ∥gh − g∥2 +L∞(T) + max +T +|gh − g|2 +W 1,∞(T) +� +× +� +T +� +∥dv∥2 +L2(T) + h2 +T |dv|2 +H1(T) +� � +. +The proof of Proposition 4.5 relies on the following lemma. +21 + +Lemma 4.6. Let g1 and g2 be two symmetric positive definite matrices, and let n be a unit vector. +Let +ngi = +1 +� +nTg−1 +i +n +g−1 +i +n, +i = 1, 2. +Then there exists a constant c depending on |g1|, |g2|, |g−1 +1 |, |g−1 +2 | such that +|ng1 − ng2| ≤ c|g1 − g2|. +Proof. Using the identity +1 +� +nT g−1 +1 n +− +1 +� +nTg−1 +2 n += +nT(g−1 +2 +− g−1 +1 )n +nTg−1 +1 n +� +nTg−1 +2 n + nT g−1 +2 n +� +nT g−1 +1 n +, +(36) +we can write +ng1 − ng2 = +nT (g−1 +2 +− g−1 +1 )n +nT g−1 +1 n +� +nT g−1 +2 n + nTg−1 +2 n +� +nTg−1 +1 n +g−1 +1 n + +1 +� +nT g−1 +2 n +(g−1 +1 +− g−1 +2 )n. +Since g−1 +1 +− g−1 +2 += g−1 +1 (g2 − g1)g−1 +2 , the bound follows easily. +Notice that in view of (29), Lemma 4.6 implies that +∥n�g − ng∥L∞(F ) ≤ C∥�g − g∥L∞(F ) +(37) +on either side of any face F. +Now we are ready to begin proving Proposition 4.5. Consider the term � +F h−1 +F +���dv(n�g)� +��2 +L2(F,�g) +that appears in the definition of ∥v∥2 +2,h,�g. Notice that +dv(n�g) = dv(ng) + dv(n�g − ng), +and we can use the bound (37) to estimate +∥dv(n�g − ng)∥L2(F,�g) ≤ C∥dv(n�g − ng)∥L2(F ) +≤ C∥dv∥L2(F )∥n�g − ng∥L∞(F ) +≤ C∥dv∥L2(F )∥�g − g∥L∞(F ) +≤ C∥dv∥L2(F )∥gh − g∥L∞(F ) +on either side of F. Using the trace inequality +∥dv∥2 +L2(F ) ≤ C +� +h−1 +T ∥dv∥2 +L2(T) + hT |dv|2 +H1(T) +� +, +F ⊂ T ∈ T N +h , +(38) +it follows that +� +F +h−1 +F ∥�dv(n�g)�∥2 +L2(F,�g) +≤ C +�� +F +h−1 +F ∥�dv(ng)�∥2 +L2(F,g) + +� +T +h−1 +T +� +h−1 +T ∥dv∥2 +L2(T) + hT |dv|2 +H1(T) +� +∥gh − g∥2 +L∞(T) +� += C +�� +F +h−1 +F ∥�dv(ng)�∥2 +L2(F,g) + +� +T +� +h−2 +T ∥gh − g∥2 +L∞(T)∥dv∥2 +L2(T) + ∥gh − g∥2 +L∞(T)|dv|2 +H1(T) +�� +, +22 + +where we have used (34), (38), and the bound hT ≤ ChF, which follows from the shape-regularity +of Th. +Next, consider the term � +T ∥∇�g∇�gv∥2 +L2(T,�g) that appears in the definition of ∥v∥2 +2,h,�g. Notice +that +� +∇�g∇�gv +� +ij = (∇g∇gv)ij + (Γk +ij − �Γk +ij) ∂v +∂xk , +where Γk +ij and �Γk +ij are the Christoffel symbols of the second kind associated with g and �g, respec- +tively. We have +∥Γk +ij − �Γk +ij∥L∞(T) ≤ C∥�g − g∥W 1,∞(T) ≤ C∥gh − g∥W 1,∞(T), +so +∥∇�g∇�gv∥L2(T,�g) ≤ C∥∇�g∇�gv∥L2(T) +≤ C +� +∥∇g∇gv∥L2(T) + ∥gh − g∥W 1,∞(T)∥dv∥L2(T) +� +≤ C +� +∥∇g∇gv∥L2(T,g) + ∥gh − g∥W 1,∞(T)∥dv∥L2(T) +� +. +It follows that +∥v∥2 +2,h,�g ≤ C +� +∥v∥2 +2,h,g + +� +max +T +h−2 +T ∥gh − g∥2 +L∞(T) + max +T +|gh − g|2 +W 1,∞(T) +� +× +� +T +� +∥dv∥2 +L2(T) + h2 +T |dv|2 +H1(T) +� � +. +This completes the proof of Proposition 4.5. +Our next step will be to estimate the bilinear form bh(�g; ·, ·). +Proposition 4.7. For every h ≤ h0, every t ∈ [0, 1], and every v ∈ H2 +0(Ω), we have (with +σ = gh − g) +|bh(�g; σ, v)| ≤ C +� +∥gh − g∥2 +L2(Ω) + +� +T +h2 +T |gh − g|2 +H1(T) +�1/2 +× +� +1 + max +T +h−1 +T ∥gh − g∥L∞(T) + max +T +|gh − g|W 1,∞(T) +� +∥v∥H2(Ω). +Proof. In view of the definitions of ∥ · ∥0,h,�g and ∥ · ∥2,h,�g, we have +|bh(�g; σ, v)| ≤ ∥S�gσ∥0,h,�g∥v∥2,h,�g. +(39) +Recalling that +∥S�gσ∥2 +0,h,�g = +� +T +∥S�gσ∥2 +L2(T,�g) + +� +F +hF ∥S�gσ(n�g, n�g)∥2 +L2(F,�g), +we compute +⟨S�gσ, S�gσ⟩�g = +� +σ − �g⟨�g, σ⟩�g, σ − �g⟨�g, σ⟩�g +� +�g += ⟨σ, σ⟩�g − 2⟨�g, σ⟩2 +�g + ⟨�g, �g⟩�g⟨�g, σ⟩2 +�g += ⟨σ, σ⟩�g + (N − 2)⟨�g, σ⟩2 +�g, +23 + +which leads to the bound +∥S�gσ∥L2(T,�g) ≤ C∥σ∥L2(T,�g) ≤ C∥σ∥L2(T). +Also, by the trace inequality, +∥S�gσ(n�g, n�g)∥2 +L2(∂T,�g) ≤ C∥S�gσ∥2 +L2(∂T,�g) +≤ C∥σ∥2 +L2(∂T) +≤ C +� +h−1 +T ∥σ∥2 +L2(T) + hT |σ|2 +H1(T) +� +. +(Here we are measuring the L2(∂T, �g)-norm of the full tensor S�gσ rather than its restriction to the +tangent bundle of ∂T.) Thus, +∥S�gσ∥2 +0,h,�g ≤ C +� +∥σ∥2 +L2(Ω) + +� +T +h2 +T |σ|2 +H1(T) +� += C +� +∥gh − g∥2 +L2(Ω) + +� +T +h2 +T |gh − g|2 +H1(T) +� +. +(40) +Consider now the term ∥v∥2,h,�g in (39). Proposition 4.5 implies that +∥v∥2,h,�g ≤ C +� +∥v∥2,h,g + +� +max +T +h−1 +T ∥gh − g∥L∞(T) + max +T +|gh − g|W 1,∞(T) +� +∥v∥H2(Ω) +� +since v ∈ H2 +0(Ω). +Furthermore, since g is smooth and v ∈ H2 +0(Ω), we have �dv(ng)� = 0 on +every interior face F and �dv(ng)� = dv(ng) = 0 on every face F ⊂ ∂Ω. +Thus, ∥v∥2 +2,h,g = +� +T ∥∇g∇gv∥2 +L2(T,g) = ∥∇g∇gv∥2 +L2(Ω,g). Since +(∇g∇gv)ij = (∇∇v)ij − Γk +ij +∂v +∂xk , +we see that +∥v∥2,h,g = ∥∇g∇gv∥L2(Ω) ≤ C(|v|H2(Ω) + |v|H1(Ω)) ≤ C∥v∥H2(Ω). +Thus, +∥v∥2,h,�g ≤ C +� +1 + max +T +h−1 +T ∥gh − g∥L∞(T) + max +T +|gh − g|W 1,∞(T) +� +∥v∥H2(Ω). +(41) +Combining (39), (40), and (41) completes the proof. +At this point, we have finished proving part (i) of Theorem 4.1. Indeed, in dimension N = 2, +ah vanishes, so we can write +��⟨(Rω)dist(gh) − (Rω)(g), v⟩V ′,V +�� ≤ +� 1 +0 +|bh(�g(t); σ, v)| dt +and apply Proposition 4.7 to deduce (25). +To prove part (ii) of Theorem 4.1, we suppose that N ≥ 3 and that suph>0 maxT∈T N +h |gh|W 2,∞(T) < +∞, and we proceed as follows. Recall that +ah(�g; σ, v) = +� +T +� +T +⟨G(�g), σ⟩�gvωT (�g)+ ˚ +� +F +� +F +� +�II(�g)�F, σ|F +� +�g vωF(�g)− ˚ +� +S +� +S +⟨ΘS(�g)�g|S, σ|S⟩�gvωS(�g), +(42) +24 + +where have made all dependencies on the metric explicit in the notation. We will bound each of +the three terms above, beginning with the first. Throughout what follows, we continue to denote +σ = gh − g, and we let v be an arbitrary member of V . +Lemma 4.8. We have +����� +� +T +� +T +⟨G(�g), σ⟩�g vωT (�g) +����� ≤ C∥gh − g∥L2(Ω)∥v∥L2(Ω). +Proof. Since we are now assuming that suph>0 maxT∈T N +h ∥gh∥W 2,∞(T) < ∞, the Einstein tensor +associated with �g satisfies +∥G(�g)∥L∞(T) ≤ C +for every h ≤ h0, every t ∈ [0, 1], and every T ∈ T N +h . It follows that +���� +� +T +⟨G(�g), σ⟩�g vωT (�g) +���� ≤ ∥G(�g)∥L∞(T,�g)∥σ∥L2(T,�g)∥v∥L2(T,�g) +≤ C∥G(�g)∥L∞(T)∥σ∥L2(T)∥v∥L2(T) +≤ C∥σ∥L2(T)∥v∥L2(T) += C∥gh − g∥L2(T)∥v∥L2(T). +Summing over all T ∈ T N +h +completes the proof. +Lemma 4.9. We have +����� +˚ +� +F +� +F +� +�II(�g)�F, σ|F +� +�g vωF(�g) +����� ≤ C max +T +� +h−1 +T ∥gh − g∥W 1,∞(T) +� +× +�� +T +∥gh − g∥2 +L2(T) + h2 +T |gh − g|2 +H1(T) +�1/2 �� +T +∥v∥2 +L2(T) + h2 +T |v|2 +H1(T) +�1/2 +. +Proof. Consider an interior (N − 1)-simplex F. By applying a Euclidean rotation and translation +to the coordinates, we may assume without loss of generality that F lies in the plane xN = 0. In +these coordinates, the second fundamental form associated with �g is given by +IIij(�g) = −�g(n�g, ∇�g,eiej) += −�g(n�g, �Γk +ijek) += −nℓ +�g�gℓk�Γk +ij, +i, j = 1, 2, . . . , N − 1, +where e1, e2, . . . , eN are the Euclidean coordinate basis vectors. Since n�g = �g−1n/ +� +nT �g−1n and n +points in the xN direction, we get +IIij(�g) = − +1 +� +nT �g−1n +�ΓN +ij . +The jump in this quantity across F can be computed using the identity �ab� = �a�{b} + {a}�b�, +where {·} denotes the average across F, giving +−�IIij(�g)� = +� +1 +� +nT �g−1n +� � +�ΓN +ij +� ++ +� +1 +� +nT �g−1n +� � +�ΓN +ij +� +. +25 + +In view of (36), we have +����� +� +1 +� +nT �g−1n +������ +L∞(F ) +≤ C ∥��g�∥L∞(F ) +≤ C ∥�gh − g�∥L∞(F ) +≤ C +� +∥gh − g∥L∞(T1) + ∥gh − g∥L∞(T2) +� +, +where T1 and T2 are the two N-simplices that share the face F. +Here, we used the fact that +�g = g + t(gh − g) and g is smooth. Similarly, we have +��� +� +�ΓN +ij +���� +L∞(F ) ≤ C∥��g�∥W 1,∞(F ) +≤ C∥�gh − g�∥W 1,∞(F ) +≤ C +� +∥gh − g∥W 1,∞(T1) + ∥gh − g∥W 1,∞(T2) +� +. +(43) +Thus, +∥�II(�g)�∥L∞(F ) ≤ C +� +∥gh − g∥W 1,∞(T1) + ∥gh − g∥W 1,∞(T2) +� +. +From this it follows easily that the same bound holds, possibly with a larger constant C, for the +trace-reversed tensor II(�g) = II(�g) − H(�g)�g: +∥�II(�g)�∥L∞(F ) ≤ C +� +∥gh − g∥W 1,∞(T1) + ∥gh − g∥W 1,∞(T2) +� +. +It follows that +���� +� +F +� +�II(�g)�F , σ|F +� +�g vωF (�g) +���� +≤ ∥�II(�g)�∥L∞(F,�g)∥σ|F ∥L2(F,�g)∥v∥L2(F,�g) +≤ C∥�II(�g)�∥L∞(F )∥σ|F ∥L2(F )∥v∥L2(F ) +≤ C +� 2 +� +i=1 +∥gh − g∥W 1,∞(Ti) +� � +h−1 +T1 ∥σ∥2 +L2(T1) + hT1|σ|2 +H1(T1) +�1/2 � +h−1 +T1 ∥v∥2 +L2(T1) + hT1|v|2 +H1(T1) +�1/2 +. +By the shape-regularity of Th, we have C−1 ≤ hT1/hT2 ≤ C for some constant C independent of h +and F, so +����� +˚ +� +F +� +F +� +�II(�g)�F, σ|F +� +�g vωF(�g) +����� ≤ C max +T +� +h−1 +T ∥gh − g∥W 1,∞(T) +� +× +�� +T +∥gh − g∥2 +L2(T) + h2 +T |gh − g|2 +H1(T) +�1/2 �� +T +∥v∥2 +L2(T) + h2 +T |v|2 +H1(T) +�1/2 +. +Remark 4.10. If gh is piecewise constant, then in (43) we have the sharper bound +∥�gh − g�∥W 1,∞(F ) = ∥�gh − g�∥L∞(F ) ≤ C +� +∥gh − g∥L∞(T1) + ∥gh − g∥L∞(T2) +� +26 + +because ∂gh +∂xi = 0 and +∂g +∂xi is continuous for each i. This implies that for piecewise constant gh, we +can replace ∥gh − g∥W 1,∞(T) by ∥gh − g∥L∞(T) in Lemma 4.9, yielding +����� +˚ +� +F +� +F +� +�II(�g)�F, σ|F +� +�g vωF(�g) +����� ≤ C max +T +� +h−1 +T ∥gh − g∥L∞(T) +� +× +�� +T +∥gh − g∥2 +L2(T) + h2 +T |gh − g|2 +H1(T) +�1/2 �� +T +∥v∥2 +L2(T) + h2 +T |v|2 +H1(T) +�1/2 +. +Now we turn our attention toward the third integral in (42). In preparation for this, we will +first use the shape-regularity assumption to show that the dihedral angles of every N-simplex in +Th (measured in the Euclidean metric) are uniformly bounded above and below. +Lemma 4.11. There exist constants θmin, θmax ∈ (0, π) such that for every h > 0 and every +T ∈ T N +h , the dihedral angles in T (measured in the Euclidean metric) all lie between θmin and θmax. +Proof. This fact is proved in dimension N = 3 in [18, Lemma 3.6]. +We generalize their proof +to dimension N ≥ 3 as follows. Given N + 1 points x1, x2, . . . , xN+1 in general position in RN, +let T = [x1, x2, . . . , xN+1] denote the N-simplex with vertices x1, x2, . . . , xN+1. +Consider two +faces F1 = [x1, x3, x4, . . . , xN+1] and F2 = [x2, x3, x4, . . . , xN+1] that intersect along the (N − 2)- +dimensional subsimplex S = [x3, x4, . . . , xN+1]. Throughout what follows, we work in the Euclidean +metric. Let A be the orthogonal projection of x1 onto the (N−1)-dimensional hyperplane containing +F2, and let B be the orthogonal projection of x1 onto the (N −2)-dimensional hyperplane containing +S. Observe that both [x1, A] and [x1, B] are orthogonal to S, since S ⊂ F2. Thus, the triangle +[x1, A, B] is orthogonal to S. This triangle is a right triangle with hypotenuse [x1, B], so the dihedral +angle θST along S satisfies +sin θST = |[x1, A]| +|[x1, B]|, +where | · | denotes the Euclidean volume (i.e. length in this case). Obviously, |[x1, B]| is bounded +above by hT , the diameter of T. In addition, |[x1, A]| is bounded from below by 2 times ρT , the +inradius of T. To see why, we generalize the argument in [18, Proposition 2.3], bearing in mind +that our definition of ρT differs from theirs by a factor of 2. Consider the inscribed (N − 1)-sphere +in T, whose center C lies at a distance ρT from F2. Let D be the point where this inscribed sphere +touches F2, and let E be the point diametrically opposite to D on this sphere. The line segment +[D, E] is orthogonal to F2, so the volume of the N-simplex T ′ = [E, x2, x3, x4, . . . , xN+1] satisfies +|T ′| = 1 +N |[D, E]||F2| = 2ρT +N |F2|. +Since T ′ ⊂ T, we have +|T ′| ≤ |T| = 1 +N |[x1, A]||F2|, +so +2ρT ≤ |[x1, A]|. +Thus, +sin θST ≥ 2ρT +hT +. +The result follows from this bound and the shape-regularity of Th. +27 + +Next we show that Lemma 4.11 remains valid when one measures angles with g rather than the +Euclidean metric δ. +Lemma 4.12. Upon reducing the value of h0 if necessary, there exist constants θmin,g, θmax,g ∈ (0, π) +such that for every h ≤ h0, every T ∈ T N +h , every (N − 2)-simplex S ⊂ ∂T, and every point p ∈ S, +the dihedral angle in T at p (measured by g) lies between θmin,g and θmax,g. +Proof. If there were no such lower bound θmin,g > 0, then there would exist a sequence of N- +simplices T1 ∈ Th1, T2 ∈ Th2, . . . with faces F (1) +1 +, F (2) +1 +⊂ T1, F (1) +2 , F (2) +2 +⊂ T2, . . . and points +p1 ∈ F (1) +1 +∩ F (2) +1 , p2 ∈ F (1) +2 +∩ F (2) +2 +, . . . such that +∠ g|Ti(pi)(F (1) +i +, F (2) +i +) → 0 +as i → ∞, where ∠g(X, Y ) denotes the angle between X and Y as measured by g. Using the +compactness of the Grassmannian, this implies that, after extracting a subsequence which we do +not relabel, +∠δ(F (1) +i +, F (2) +i +) → 0, +where ∠δ(X, Y ) denotes the angle between X and Y as measured by the Euclidean metric δ. This +contradicts the assumed positive lower bound on the Euclidean dihedral angles. The existence of +an upper bound θmax,g < π is proved similarly. +Now we are ready to estimate the third integral in (42). +Lemma 4.13. We have +����� +˚ +� +S +� +S +⟨ΘS(�g) �g|S , σ|S⟩�g vωS(�g) +����� +≤ C +� +max +T +h−2 +T ∥gh − g∥L∞(T) +� �� +T +∥gh − g∥2 +L2(T) + h2 +T |gh − g|2 +H1(T) + h4 +T |gh − g|2 +H2(T) +�1/2 +× +�� +T +∥v∥2 +L2(T) + h2 +T |v|2 +H1(T) + h4 +T |v|2 +H2(T) +�1/2 +. +Proof. Fix an interior (N − 2)-simplex S and an N-simplex T containing S. At any point p along +S, we have +cos θST(g) − cos θST (�g) = �g(n(1) +�g , n(2) +�g ) − g(n(1) +g , n(2) +g ) += �g(n(1) +�g +− n(1) +g , n(2) +�g +− n(2) +g ) + �g(n(1) +�g +− n(1) +g , n(2) +g ) + �g(n(1) +g , n(2) +�g +− n(2) +g ) ++ �g(n(1) +g , n(2) +g ) − g(n(1) +g , n(2) +g ), +where n(1) +g +and n(2) +g +are suitably oriented unit normal vectors (with respect to g|T ) to the two faces +of T containing S, and similarly for n(1) +�g +and n(2) +�g . Using Lemma 4.6, we see that at the point p, +| cos θST(�g) − cos θST(g)| ≤ C|�g − g| ≤ C|gh − g| +for all h sufficiently small. +Since there are constants θmin,g, θmax,g ∈ (0, π) such that θmin,g ≤ +θST(g) ≤ θmax,g, we get +|θST (�g) − θST(g)| ≤ C|gh − g| ≤ C∥gh − g∥L∞(T). +28 + +Summing over T ⊃ S and noting that � +T⊃S θST (g) = 2π, we get +|ΘS(�g)| = |ΘS(�g) − ΘS(g)| ≤ +� +T⊃S +|θST(�g) − θST(g)| ≤ C +� +T⊃S +∥gh − g∥L∞(T). +(44) +Now we are almost ready to estimate the integral +� +S ⟨ΘS(�g) �g|S , σ|S⟩�g vωS(�g). We first note that +∥v∥2 +L2(S) ≤ C +� +h−2 +T ∥v∥2 +L2(T) + |v|2 +H1(T) + h2 +T |v|2 +H2(T) +� +, +which can be proved using a codimension-2 trace inequality and a scaling argument, or by applying +the codimension-1 trace inequality (38) twice (to v rather than dv). +If T1, T2, . . . , Tm are the +N-simplices that share the (N − 2)-simplex S, then we have +���� +� +S +⟨ΘS(�g) �g|S , σ|S⟩�g vωS(�g) +���� +≤ C∥ΘS(�g)∥L∞(S,�g)∥σ|S∥L2(S,�g)∥v∥L2(S,�g) +≤ C∥ΘS(�g)∥L∞(S)∥σ|S∥L2(S)∥v∥L2(S) +≤ C +� m +� +i=1 +∥gh − g∥L∞(Ti) +� � +h−2 +T1 ∥σ∥2 +L2(T1) + |σ|2 +H1(T1) + h2 +T1|σ|2 +H2(T1) +�1/2 +× +� +h−2 +T1 ∥v∥2 +L2(T1) + |v|2 +H1(T1) + h2 +T1|v|2 +H2(T1) +�1/2 +. +The proof is completed by summing over all interior (N − 2)-simplices S and substituting σ = +gh − g. +Collecting our results, we can state a bound on the bilinear form ah(�g; ·, ·). +Proposition 4.14. For every h ≤ h0, every t ∈ [0, 1], and every v ∈ V , we have (with σ = gh −g), +|ah(�g; σ, v)| ≤ C +� +1 + max +T +h−2 +T ∥gh − g∥L∞(T) + max +T +h−1 +T |gh − g|W 1,∞(T) +� +× +�� +T +∥gh − g∥2 +L2(T) + h2 +T |gh − g|2 +H1(T) + h4 +T |gh − g|2 +H2(T) +�1/2 +× +�� +T +∥v∥2 +L2(T) + h2 +T |v|2 +H1(T) + h4 +T |v|2 +H2(T) +�1/2 +. +Proof. Combine Lemmas 4.8, 4.9, and 4.13. +Upon combining Proposition 4.7 with Proposition 4.14, we see that +∥(Rω)dist(gh) − (Rω)(g)∥H−2(Ω) ≤ C +� +1 + max +T +h−2 +T ∥gh − g∥L∞(T) + max +T +h−1 +T |gh − g|W 1,∞(T) +� +× +� +∥gh − g∥2 +L2(Ω) + +� +T +h2 +T |gh − g|2 +H1(T) + +� +T +h4 +T |gh − g|2 +H2(T) +�1/2 +. +29 + +This completes the proof of Theorem 4.1. Corollary 4.3 then follows from (27) and the bounds +∥gh − g∥L2(Ω) ≤ |Ω|1/2−1/p∥gh − g∥Lp(Ω), +�� +T +h2 +T |gh − g|2 +H1(T) +�1/2 +≤ |Ω|1/2−1/p +�� +T +hp +T |gh − g|p +W 1,p(T) +�1/p +, +�� +T +h4 +T |gh − g|2 +H2(T) +�1/2 +≤ |Ω|1/2−1/p +�� +T +h2p +T |gh − g|p +W 2,p(T) +�1/p +, +which hold for all p ∈ [2, ∞] (with the obvious modifications for p = ∞). +Remark 4.15. Notice that the analysis above yields +|bh(�g; σ, v)| = O(hr+1), +(by Proposition 4.7), +(45) +����� +� +T +� +T +⟨G(�g), σ⟩�g vωT (�g) +����� = O(hr+1), +(by Lemma 4.8), +(46) +����� +˚ +� +F +� +F +� +�II(�g)�F , σ|F +� +�g vωF (�g) +����� = +� +O(h), +if r = 0, +O(h2r), +if r ≥ 1, +(by Remark 4.10), +(by Lemma 4.9), +(47) +����� +˚ +� +S +� +S +⟨ΘS(�g) �g|S , σ|S⟩�g vωS(�g) +����� = O(h2r), +(by Lemma 4.13) +(48) +for any optimal-order interpolant gh of g having degree r ≥ 0. Bearing in mind that (46-48) vanish +when N = 2, we see that the above estimates lead to an optimal error estimate ∥(Rω)dist(gh) − +(Rω)(g)∥H−2(Ω) = O(hr+1) in all cases except when N ≥ 3 and r = 0, where we obtain ∥(Rω)dist(gh)− +(Rω)(g)∥H−2(Ω) = O(1) because of (48). Numerical experiments suggest that these analytical re- +sults are sharp for a general optimal-order interpolant, whereas for the canonical interpolant the +estimate (48) improves to O(h2(r+1)), yielding ∥(Rω)dist(gh) − (Rω)(g)∥H−2(Ω) = O(h) when r = 0; +cf. Figure 2. +5 +Numerical examples +In this section we present numerical experiments in dimension N = 2, 3 to illustrate the predicted +convergence rates. The examples were performed in the open source finite element library NGSolve1 +[24, 25], where the Regge finite elements are available for arbitrary polynomial order. We construct +an optimal-order interpolant gh of a given metric tensor g as follows. On each element T, the local +L2 best-approximation ¯gh|T of g|T is computed. Then the tangential-tangential degrees of freedom +shared by two or more neighboring elements are averaged to obtain a globally tangential-tangential +continuous interpolant gh. +We verify in Appendix A that this interpolant is an optimal-order +interpolant in the sense of Remark 4.4 on shape-regular, quasi-uniform triangulations. +To compute the H−2(Ω)-norm of the error f := (Rω)dist(gh) − (Rω)(g) we make use of the fact +that ∥f∥H−2(Ω) is equivalent to ∥u∥H2(Ω), where u ∈ H2 +0(Ω) solves the biharmonic equation ∆2u = f. +This equation will be solved numerically using the (Euclidean) Hellan–Herrmann–Johnson method. +To prevent the discretization error from spoiling the real error, we use for uh two polynomial orders +more than for gh. +1www.ngsolve.org +30 + +We consider in dimension N = 2 the numerical example proposed in [16], where on the square +Ω = (−1, 1)2 the smooth Riemannian metric tensor +g(x, y) := +� +1 + (∂f +∂x)2 +∂f +∂x +∂f +∂y +∂f +∂x +∂f +∂y +1 + (∂f +∂y )2 +� +with f(x, y) := 1 +2(x2 + y2) − 1 +12(x4 + y4) is defined. This metric corresponds to the surface induced +by the embedding +� +x, y +� +�→ +� +x, y, f(x, y) +� +, and its exact scalar curvature is given by +R(g)(x, y) = +162(1 − x2)(1 − y2) +(9 + x2(x2 − 3)2 + y2(y2 − 3)2)2 . +For a three-dimensional example we consider the cube Ω = (−1, 1)3 and the Riemannian metric +tensor induced by the embedding +� +x, y, z +� +�→ +� +x, y, z, f(x, y, z) +� +, where f(x, y, z) := 1 +2(x2 + y2 + +z2) − 1 +12(x4 + y4 + z4). The scalar curvature is +R(g)(x, y, z) = 18 +� +(1 − x2)(1 − y2)(9 + q(z)) + (1 − y2)(1 − z2)(9 + q(x)) + (1 − z2)(1 − x2)(9 + q(y)) +� +(9 + q(x) + q(y) + q(z))2 +, +where q(x) = x2(x2 − 3)2. +We start with a structured mesh consisting of 2·22k triangles and 6·23k tetrahedra, respectively, +in two and three dimensions with ˜h = maxT hT = +√ +N 21−k (and minimal edge length 21−k) +for k = 0, 1, . . . . To avoid possible superconvergence due to mesh symmetries, we perturb each +component of the inner mesh vertices by a random number drawn from a uniform distribution in +the range [−˜h 2−(2N+1)/2, ˜h 2−(2N+1)/2]. As depicted in Figure 1 (left) and listed in Table 1, linear +convergence is observed when N = 2 and gh has polynomial degree r = 0. This is consistent with +Theorem 4.1(i). For r = 1 and r = 2, higher convergence rates are obtained as expected. +In the three-dimensional case, the same convergence rates as for N = 2 are obtained, cf. Figure 1 +(right) and Table 2. This indicates that Theorem 4.1(ii) is sharp for r ≥ 1. For r = 0 we observe +numerically linear convergence, which is better than predicted by Theorem 4.1(ii). However, further +investigation suggests that the observed linear convergence for r = 0 is pre-asymptotic. Indeed, to +test if (48) is sharp, we compute the H−2(Ω)-norm of the linear functional +v �→ +� 1 +0 +˚ +� +S +� +S +⟨ΘS(�g(t)) �g(t)|S , σ|S⟩�g(t) vωS(�g(t)) dt, +(49) +where we approximate the parameter integral by a Gauss quadrature of order seven. As depicted +in Figure 2, the norm of this functional for the optimal-order interpolant gh with r = 0 stagnates at +about 4·10−4, which is below the overall error of 4.296·10−3 for the finest grid; cf. Table 2. There- +fore, the lack of convergence predicted by Theorem 4.1(ii) is not yet visible in Figure 1. For r = 1, 2 +the proven rate of O(h2r) for (49) (see (48)) is clearly obtained. Interestingly, using the canonical +interpolant appears to increase the convergence rate of (49) to O(h2(r+1)) (i.e. an increase of two +orders), as observed in Figure 2. Thus, it appears that the canonical interpolant achieves conver- +gence in the lowest-order case. We intend to study this superconvergence phenomenon exhibited +by the canonical interpolant in future work. +Acknowledgments +We thank Yasha Berchenko-Kogan for many helpful discussions, especially about the mean curva- +ture term in Definition 3.1. We also thank Snorre Christiansen for pointing out the link with the +31 + +101 +102 +103 +104 +105 +106 +10−8 +10−6 +10−4 +10−2 +100 +ndof +error +r = 0 +r = 1 +r = 2 +O(h) +O(h2) +O(h3) +101 +102 +103 +104 +105 +106 +107 +108 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +ndof +error +r = 0 +r = 1 +r = 2 +O(h) +O(h2) +O(h3) +Figure 1: Convergence of the distributional scalar curvature in the H−2(Ω)-norm for N = 2 (left) +and N = 3 (right) with respect to the number of degrees of freedom (ndof) of gh for r = 0, 1, 2. +101 +102 +103 +104 +105 +106 +107 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +ndof +H−2(Ω)-norm of (49) +r = 0 +r = 1 +r = 2 +r = 0 c.i. +r = 1 c.i. +r = 2 c.i. +O(h2) +O(h4) +O(h6) +Figure 2: Convergence of (49) in the H−2(Ω)-norm with respect to number of degrees of freedom +(ndof) for an optimal-order interpolant and the canonical interpolant (c.i.) +for r = 0, 1, 2 in +dimension N = 3. +r = 0 +r = 1 +r = 2 +h +Error +Order +Error +Order +Error +Order +2.828 · 10−0 +1.534 · 10−0 +8.584 · 10−1 +4.609 · 10−1 +2.417 · 10−1 +1.251 · 10−1 +6.260 · 10−2 +3.198 · 10−2 +2.237 · 10−1 +1.945 · 10−1 +0.23 +6.220 · 10−2 +1.96 +2.336 · 10−2 +1.57 +9.434 · 10−3 +1.41 +4.457 · 10−3 +1.14 +2.181 · 10−3 +1.03 +1.067 · 10−3 +1.06 +8.613 · 10−2 +8.448 · 10−2 +0.03 +4.565 · 10−2 +1.06 +1.335 · 10−2 +1.98 +3.689 · 10−3 +1.99 +9.205 · 10−4 +2.11 +2.280 · 10−4 +2.02 +5.777 · 10−5 +2.04 +2.720 · 10−2 +1.364 · 10−2 +1.13 +2.213 · 10−3 +3.13 +3.615 · 10−4 +2.91 +4.189 · 10−5 +3.34 +5.504 · 10−6 +3.08 +7.028 · 10−7 +2.97 +8.784 · 10−8 +3.1 +Table 1: Same as Figure 1 (left), but in tabular form. +32 + +r = 0 +r = 1 +r = 2 +h +Error +Order +Error +Order +Error +Order +3.464 · 10−0 +1.850 · 10−0 +9.709 · 10−1 +4.999 · 10−1 +2.753 · 10−1 +1.358 · 10−1 +6.878 · 10−2 +7.869 · 10−2 +3.215 · 10−1 +-2.24 +1.132 · 10−1 +1.62 +4.152 · 10−2 +1.51 +1.838 · 10−2 +1.37 +8.733 · 10−3 +1.05 +4.296 · 10−3 +1.04 +1.359 · 10−1 +6.613 · 10−2 +1.15 +2.912 · 10−2 +1.27 +8.633 · 10−3 +1.83 +2.391 · 10−3 +2.15 +6.194 · 10−4 +1.91 +1.579 · 10−4 +2.01 +1.871 · 10−2 +4.133 · 10−2 +-1.26 +5.286 · 10−3 +3.19 +7.342 · 10−4 +2.97 +9.753 · 10−5 +3.38 +1.261 · 10−5 +2.89 +1.604 · 10−6 +3.03 +Table 2: Same as Figure 1 (right), but in tabular form. +Israel formalism mentioned in Remark 3.9. EG was supported by NSF grant DMS-2012427. MN +acknowledges support by the Austrian Science Fund (FWF) project F 65. +A +Optimal-order interpolation via averaging +Below we verify that the interpolant described in Section 5 is an optimal-order interpolant in the +sense of Remark 4.4, assuming that {Th}h>0 is shape-regular and quasi-uniform. Recall that quasi- +uniformity means that maxT∈T N +h h/hT is bounded above by a constant independent of h. In what +follows, the letter C may depend on this constant as well as on the parameters N, hT /ρT , r, s, and +t appearing below. +Let ℓ(1), ℓ(2), . . . , ℓ(M) denote the canonical degrees of freedom for the Regge finite element space +of degree r ≥ 0 on Th [21, Equation (2.4b)]. Each linear functional ℓ(i) is associated with a simplex +D ∈ T k +h of dimension k ≥ 1 in the following sense: ℓ(i) sends a symmetric (0, 2)-tensor field g to +the integral of g|D against a (symmetric tensor-valued) polynomial of degree ≤ r − k + 1 over D. +We enumerate these degrees of freedom with a local numbering system as follows. +On a +given N-simplex T ∈ T N +h , the degrees of freedom associated with subsimplices of T are denoted +ℓT +1 , ℓT +2 , . . . , ℓT +MT . If T, T ′ ∈ T N +h +are two N-simplices with nonempty intersection, then it may happen +that ℓT +i and ℓT ′ +j +coincide for some and i and j. We let S(i, T) denote the set of all pairs (j, T ′) for +which ℓT +i and ℓT ′ +j +coincide. +With the above local numbering system, let ψT +1 , ψT +2 , . . . , ψT +MT denote the basis for the degree-r +Regge finite element space that is dual to the above degrees of freedom. That is, +ℓT +i (ψT ′ +j ) = +� +1, +if (j, T ′) ∈ S(i, T), +0, +otherwise. +Let us assume that the degrees of freedom and basis functions above are first defined on a reference +simplex and then transported to T via an affine transformation. A scaling argument shows that [21, +Lemma 2.11] +∥ψT +i ∥Lp(T) ≤ ChN/p−2 +T +(50) +and +|ℓT +i (g)| ≤ Ch−N/p+2 +T +∥g∥Lp(T) +(51) +for all g in the domain of ℓT +i . Note that the −2 and the +2 appearing in the exponents above +arise because of the way that pullbacks of (0, 2)-tensor fields behave under affine transformations; +see [21, Lemma 2.11]. +33 + +Let g be a symmetric (0, 2)-tensor field possessing W s,p(Ω)-regularity for every p ∈ [1, ∞] and +every s > (N − 1)/p. The canonical interpolation operator Jh onto the Regge finite element space +is defined elementwise by +Jhg|T = J T +h (g|T ) = +MT +� +i=1 +ℓT +i (g)ψT +i . +Let ¯gh denote the elementwise L2-projection of g onto the space of discontinuous piecewise +polynomial symmetric (0, 2)-tensor fields of degree at most r. Since Jh is a projector, we have +¯gh|T = J T +h ( ¯gh|T ) = +MT +� +i=1 +ℓT +i (¯gh)ψT +i . +The interpolant discussed in Section 5 is defined by +gh|T = +MT +� +i=1 + + +1 +|S(i, T)| +� +(j,T ′)∈S(i,T) +ℓT ′ +j (¯gh) + + ψT +i , +where |S(i, T)| denotes the cardinality of S(i, T). +To analyze the error gh − g, let p ∈ [1, ∞], s ∈ ((N − 1)/p, r + 1], and t ∈ [0, s]. We have +|gh − g|W t,p(T) ≤ |gh − Jhg|W t,p(T) + |Jhg − g|W t,p(T). +The second term satisfies [21, Theorem 2.5] +|Jhg − g|W t,p(T) ≤ Chs−t +T +|g|W s,p(T). +(52) +To bound the first term, we use the fact that +ℓT +i (g) = +1 +|S(i, T)| +� +(j,T ′)∈S(i,T) +ℓT ′ +j (g) +to write +(gh − Jhg)|T = +MT +� +i=1 +1 +|S(i, T)| +� +(j,T ′)∈S(i,T) +ℓT ′ +j (¯gh − g)ψT +i . +Using an inverse estimate, (50), (51), and a standard error estimate [14, Proposition 1.135] for the +elementwise L2-projector, we obtain +|gh − Jhg|W t,p(T) ≤ Ch−t +T ∥gh − Jhg∥Lp(T) +≤ Ch−t +T +� +T ′:T ′∩T̸=∅ +h−N/p+2 +T ′ +∥¯gh − g∥Lp(T ′)hN/p−2 +T +≤ Ch−t +T +� +T ′:T ′∩T̸=∅ +∥¯gh − g∥Lp(T ′) +≤ Ch−t +T +� +T ′:T ′∩T̸=∅ +hs +T ′|g|W s,p(T ′) +≤ Chs−t +T +� +T ′:T ′∩T̸=∅ +|g|W s,p(T ′). +(53) +Here, we have repeatedly used the fact that the ratio hT /hT ′ is bounded uniformly above and below +by positive constants. Combining (52) and (53) shows that the error gh − g satisfies (28). +34 + +References +[1] +D. N. 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In: Physical +Review Letters 28.16 (1972), pp. 1082–1085. +36 + diff --git a/7NA0T4oBgHgl3EQfOP_L/content/tmp_files/load_file.txt b/7NA0T4oBgHgl3EQfOP_L/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..45cdd275a79595337ac8eea98384dfd6a8b144cd --- /dev/null +++ b/7NA0T4oBgHgl3EQfOP_L/content/tmp_files/load_file.txt @@ -0,0 +1,1181 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf,len=1180 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='02159v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='NA] 5 Jan 2023 Finite element approximation of scalar curvature in arbitrary dimension Evan S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Gawlik∗ Michael Neunteufel† Abstract We analyze finite element discretizations of scalar curvature in dimension N ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Our analysis focuses on piecewise polynomial interpolants of a smooth Riemannian metric g on a simplicial triangulation of a polyhedral domain Ω ⊂ RN having maximum element diameter h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We show that if such an interpolant gh has polynomial degree r ≥ 0 and possesses single-valued tangential-tangential components on codimension-1 simplices, then it admits a natural notion of (densitized) scalar curvature that converges in the H−2(Ω)-norm to the (densitized) scalar curvature of g at a rate of O(hr+1) as h → 0, provided that either N = 2 or r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' As a special case, our result implies the convergence in H−2(Ω) of the widely used “angle defect” approxima- tion of Gaussian curvature on two-dimensional triangulations, without stringent assumptions on the interpolated metric gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We present numerical experiments that indicate that our analytical estimates are sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 1 Introduction Many partial differential equations that arise in mathematical physics and geometric analysis involve the Riemann curvature tensor and its contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The scalar curvature R, which is obtained from two contractions of the Riemann curvature tensor, is particularly important;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' it serves as the integrand in the Einstein-Hilbert functional from general relativity, and it appears in the governing equation for two-dimensional Ricci flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' To approximate solutions to PDEs involving the scalar curvature, it is necessary to discretize the nonlinear differential operator that sends a Riemannian metric tensor to its scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The goal of this paper is to construct and analyze such discretizations in arbitrary dimension N ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We are specifically interested in the setting where a smooth Riemannian metric tensor g on a polyhedral domain Ω ⊂ RN is approximated by a piecewise polynomial Regge metric gh on a simplicial triangulation T of Ω having maximum element diameter h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Here, a metric is called a Regge metric on T if it is piecewise smooth and its tangential-tangential components are single- valued on every codimension-1 simplex in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' When such a metric is piecewise polynomial, it belongs to a finite element space called the Regge finite element space [11, 12, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Regge metrics are not classically differentiable, so our first task will be to assign meaning to the scalar curvature of gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Our definition, which is a natural generalization of one that is now well-established in dimension N = 2, treats the scalar curvature of gh as a distribution and regards it as an approximation of the densitized scalar curvature of g, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' the scalar curvature R times the volume form ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For piecewise constant Regge metrics, our definition reduces to the classical definition of the distributional densitized ∗Department of Mathematics, University of Hawai‘i at Manoa, Honolulu, HI, 96822, USA, egawlik@hawaii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='edu †Institute for Analysis and Scientific Computing, TU Wien, Wiedner Hauptstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 8-10, 1040 Wien, Austria, michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='neunteufel@tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='at 1 curvature on piecewise flat spaces [8, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' It is a linear combination of Dirac delta distributions supported on (N − 2)-simplices S, weighted by the angle defect at S: 2π minus the sum of the dihedral angles incident at S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For piecewise polynomial Regge metrics of higher degree, it includes additional contributions involving the scalar curvature in the interior of each N-simplex and the jump in the mean curvature across each (N − 1)-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We study the convergence of the distributional densitized scalar curvature of gh to the densitized scalar curvature of g under refinement of the triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We show in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1 that in the H−2(Ω)-norm, this convergence takes place at a rate of O(hr+1) when gh is an optimal-order interpolant of g that is piecewise polynomial of degree r ≥ 0, provided that either N = 2 or r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Our numerical experiments in Section 5 suggest that these estimates are sharp in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' To put this convergence result into context, let us summarize some existing convergence results in the literature on finite element approximation of the scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We first need to assemble some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In what follows, W s,p(Ω) denotes the Sobolev-Slobodeckij space of differentiability index s ∈ [0, ∞) and integrability index p ∈ [1, ∞], and ∥ · ∥W s,p(Ω) and | · |W s,p(Ω) denote the associated norm and semi-norm, which we always take with respect to the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We denote Lp(Ω) = W 0,p(Ω) and Hs(Ω) = W s,2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For k ∈ N, we denote H−k(Ω) = (Hk 0 (Ω))′, where Hk 0 (Ω) denotes the space of functions in Hk(Ω) whose derivatives of order 0 through k − 1 have vanishing trace on ∂Ω, and the prime denotes the dual space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Occasionally we use weighted Lp and H−k spaces associated with a Riemannian metric g, which we denote by Lp(Ω, g) and H−k(Ω, g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' see Section 4 and [16, Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If g is a smooth Riemannian metric and gh is a Regge metric, then R(g) denotes the scalar curvature of g, (Rω)(g) denotes the densitized scalar curvature of g, (Rω)dist(gh) denotes the distributional densitized scalar curvature of gh (defined below in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1), and R(q) h (gh) denotes the L2(Ω, gh)-projection of (Rω)dist(gh) onto the Lagrange finite element space of degree q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We also use the terms optimal-order interpolant, canonical interpolant, and geodesic interpolant below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The first of these is a catch-all term for any piecewise polynomial interpolant gh of g that belongs to the Regge finite element space and enjoys error estimates of optimal order in W s,p(T)- norms on N-simplices T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The canonical interpolant is a specific interpolant (which is optimal-order) detailed in [21, Chapter 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The geodesic interpolant of g is the unique piecewise constant Regge metric gh with the property that the length of every edge in T , as measured by gh, agrees with the geodesic distance between the corresponding vertices in T , as measured by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Summary of existing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We can now summarize some existing results about the approx- imation of g’s curvature by gh’s distributional curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Throughout what follows, the letter r denotes the polynomial degree of gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Cheeger, M¨uller, and Schrader [8, Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='7) and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1] proved that if r = 0 and gh is the geodesic interpolant of g, then (Rω)dist(gh) converges to (Rω)(g) in the (setwise) sense of measures at a rate of O(h) in dimension N = 2 and O(h1/2) in dimension N ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Gawlik [16, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1] proved that if r ≥ 1, N = 2, and gh is any optimal-order interpolant of g, then R(q) h (gh) converges to R(g) at a rate of O(hr) in the H−1(Ω, g)-norm and at a rate of O(hr−k−1) in the broken Hk(Ω)-norm, k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , r − 2, provided that q ≥ max{1, r − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Berchenko-Kogan and Gawlik [4, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='2] proved that if r ≥ 1, N = 2, and gh is any optimal-order interpolant of g, then (Rω)dist(gh) converges to (Rω)(g) at a rate of O(hr) in 2 the norm ∥u∥V ′,h = supv∈V,v̸=0⟨u, v⟩V ′,V /∥v∥V,h, where V = {v ∈ H1 0(Ω) | v|T ∈ H2(T) ∀T ∈ T N} (1) and ∥v∥V,h = |v|H1(Ω) + �� T∈T N h2 T |v|2 H2(T) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Here, hT denotes the diameter of T, and T N denotes the set of N-simplices in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Gopalakrishnan, Neunteufel, Sch¨oberl, and Wardetzky [19, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='5 and Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='6] proved that if r ≥ 0, N = 2, and gh is the canonical interpolant of g, then R(r+1) h (gh) converges to R(g) at a rate of O(hr+1) in the H−1(Ω, g)-norm and at a rate of O(hr−k) in the broken Hk(Ω)-norm, k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' New results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' As one can see from above, our analysis in this paper covers two important cases that have not yet been addressed in the literature: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We prove a convergence result in the case where N ≥ 3 and r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This opens the door to the use of piecewise polynomial Regge metrics to approximate scalar curvature in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We prove a convergence result in the case where N = 2, r = 0, and gh is an arbitrary optimal-order interpolant of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This has been a longstanding gap in the literature on Gaussian curvature approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Previous efforts to address the case where N = 2 and r = 0 have relied on subtle properties of the geodesic interpolant [8] and the canonical interpolant [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Our results establish the validity of Gaussian curvature approximations involving the angle defect without stringent assumptions on the interpolated metric tensor gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Note that our analysis predicts no convergence at all in the H−2(Ω)-norm when N ≥ 3 and r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Our numerical experiments suggest that this result is sharp for general optimal-order interpolants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' However, for the canonical interpolant, numerical experiments suggest that (Rω)dist(gh) converges to (Rω)(g) in the H−2(Ω)-norm at a rate of O(h) when N ≥ 3 and r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We intend to study this superconvergence phenomenon exhibited by the canonical interpolant in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Structure of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Our strategy for proving convergence of (Rω)dist(gh) to (Rω)(g) con- sists of two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' First, in Sections 2-3, we study the evolution of (Rω)dist(gh) under deformations of the metric, leading to an integral formula for the error (Rω)dist(gh) − (Rω)(g) which reads ⟨(Rω)dist(gh) − (Rω)(g), v⟩V ′,V = � 1 0 bh(�g(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v) − ah(�g(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v) dt, ∀v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (2) Here, �g(t) = (1 − t)g + tgh, σ = ∂ ∂t�g(t) = gh − g, V is the space defined in (1), and bh(�g(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' ·, ·) and ah(�g(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' ·, ·) are certain metric-dependent bilinear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In Section 4, we use techniques from finite element theory to estimate the right-hand side of (2), leading to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The approach above is similar to the one used in dimension N = 2 in [4, 16, 19], but there are a few important differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' First, we work with an integral formula for the error (Rω)dist(gh) − (Rω)(g) rather than an integral formula for the curvature itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Previous analyses in [4, 16, 19] hinged on formulas of the latter type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Loosely speaking, in this paper we compute the evolution of the error along a one-parameter family of Regge metrics starting at g and ending at gh, whereas the papers [4, 16, 19] compute the evolution of the curvature along a pair of one-parameter families of metrics: one family that starts at the Euclidean metric δ and ends at gh, and one that starts at 3 δ and ends at g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The approach based on evolving the error appears to be better suited for proving optimal error estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Another key aspect of our analysis is our use of the H−2(Ω)-norm to measure the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This norm is weaker than the ones used in [4, 16, 19], and it appears to be more natural for measuring the error in the curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For example, for piecewise constant Regge metrics in dimension N = 2, we show that convergence of (Rω)dist(gh) to (Rω)(g) holds in the H−2(Ω)-norm for any optimal- order interpolant of g, but numerical experiments suggest that it fails to hold in stronger norms when gh is not the canonical interpolant of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' A key tool that we use to prove convergence in H−2(Ω) is the near-equivalence of a certain pair of metric-dependent, mesh-dependent norms on V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This equivalence is similar to one that Walker [27, Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='3] proved for an analogous family of mesh-dependent norms on triangulated surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Additional comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The formula (2) is not only useful for the error analysis, but it is also interesting in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' It has a differential counterpart (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='6) that reads d dt⟨(Rω)dist(�g(t)), v⟩V ′,V = bh(�g(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v) − ah(�g(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v), ∀v ∈ V, (3) which mimics the formula d dt � Ω Rvω = � Ω (div div Sσ)vω − � Ω ⟨G, σ⟩vω, ∀v ∈ V (4) that holds for a family of smooth Riemannian metrics g(t) with densitized scalar curvature Rω and Einstein tensor G = Ric − 1 2Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Here, Sσ = σ−g Tr σ, and div is the covariant divergence operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' see below for more notational details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The correspondence between (3) and (4) becomes even more transparent when one inspects the formulas for bh and ah (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The bilinear form bh(�g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' ·, ·) is (up to the appearance of S) a non-Euclidean, N-dimensional generalization of a bilinear form that appears in the Hellan- Herrmann-Johnson finite element method [1–3, 5–7, 9, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' It can be regarded as the integral of div div Sσ against v, where div div is interpreted in a distributional sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This link with the Hellan-Herrmann-Johnson method has previously been noted and used in dimension N = 2 [4, 16, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The bilinear form ah(�g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' ·, ·), which is only nonzero in dimension N ≥ 3, appears to play the role of � Ω⟨G, σ⟩vω, which is also only nonzero in dimension N ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' It gives rise to a natural way of defining the Einstein tensor in a distributional sense for Regge metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We discuss this more in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Among other things, we point out that the formula for ah contains a term involving the jump in the trace-reversed second fundamental form across codimension-1 simplices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' the same quantity arises in studies of singular sources in general relativity, where it encodes the well-known Israel junction conditions across a hypersurface on which stress-energy is concentrated [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' There are a few other connections between our calculations and ones that appear in the physics literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The variation of the Gibbons-Hawking-York boundary term in general relativity [17, 28] is one example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' It has many parallels to our calculations in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='2, and one can undoubtedly find formulas like (6) in the literature after reconciling notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We still give a full derivation of such formulas, not only to familiarize the reader with our notation, but also to provide careful derivations that refrain from discarding total derivatives (which integrate to zero on manifolds without boundary, but not in general) and minimize the use of local coordinate calculations where possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 4 2 Evolution of geometric quantities In this section, we consider an N-dimensional manifold M equipped with a smooth Riemannian metric g, and we study the evolution of various geometric quantities under deformations of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We adopt the following notation in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The Levi-Civita connection associated with g is denoted ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If σ is a (p, q)-tensor field, then its covariant derivative is the (p, q + 1)-tensor field ∇σ, and its covariant derivative in the direction of a vector field X is the (p, q)-tensor field ∇Xσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Its trace Tr σ is the contraction of σ along the first two indices, using g to raise or lower indices as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We denote div σ = Tr ∇σ and ∆σ = div ∇σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The g-inner product of two (p, q)-tensor fields σ and ρ is denoted ⟨σ, ρ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The volume form associated with g is denoted ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The Ricci tensor and the scalar curvature of g are denoted Ric and R, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' When we wish to emphasize their dependence on g, we write ω(g), Ric(g), R(g), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If D is an embedded submanifold of M, then we denote by ωD the induced volume form on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If σ is a tensor field on M, then σ|D denotes the pullback of σ under the inclusion D ֒→ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Later we will introduce some additional notation related to embedded submanifolds of codimension 1, like the mean curvature H and second fundamental form II;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We denote the exterior derivative of a differential form α by dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If α is a one-form, then α♯ denotes the vector field obtained by raising indices with g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If f is a scalar field, then we sometimes interpret the one-form ∇f = df as the vector field (df)♯ without explicitly writing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Later, in Section 4, we will append a subscript g to many quantities like ∇ and ⟨·, ·⟩ to emphasize their dependence on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In that section only, an absent subscript will generally signal that the quantity in question is computed with respect to the Euclidean metric, which we denote by δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We say more about this notational shift in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1 Evolution of the densitized scalar curvature First we study the evolution of the densitized scalar curvature Rω under deformations of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let g(t) be a family of smooth Riemannian metrics with time derivative ∂ ∂tg =: σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We have ∂ ∂t(Rω) = (div div Sσ)ω − ⟨G, σ⟩ω, where G = Ric − 1 2Rg denotes the Einstein tensor associated with g and Sσ = σ − g Tr σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We compute ∂ ∂t(Rω) = ˙Rω + R ˙ω and invoke the well-known formulas [15, Lemma 2] ˙R = div div σ − ∆ Tr σ − ⟨Ric, σ⟩ and [10, Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='4] ˙ω = 1 2(Tr σ)ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Since ∆ Tr σ = div div(g Tr σ) and Tr σ = ⟨g, σ⟩, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='2 Evolution of the mean curvature Next we study the evolution of the mean curvature H of a hypersurface F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We assume that the tan- gent bundle of F is trivial, so that there exists a smooth, g-orthonormal frame field τ1, τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , τN−1 on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (If this is not the case, then one can simply fix a point p ∈ F and focus on a neighborhood of p on which the tangent bundle is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=') We let n be the unit normal to F so that n, τ1, τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , τN−1 forms a right-handed g-orthonormal frame (in the ambient manifold) at each point on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If the metric g varies smoothly in time, then we assume that the vectors n, τ1, τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , τN−1 also vary smoothly in time and remain g-orthonormal at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We use the notation II(X, Y ) = g(∇Xn, Y ) = −g(n, ∇XY ) for the second fundamental form on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Our sign convention is such that Tr II = H, and H is positive for a sphere with an outward normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We also let ∇F and divF denote the surface gradient and surface divergence operators on F, which have the following meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For a scalar field v, ∇F v = ∇v − n∇nv = N−1 � i=1 τi∇τiv, and for a one-form α, divF α = Tr (∇α|F) = N−1 � i=1 (∇τiα)(τi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Note that in the formula ∇F v = ∇v − n∇nv, we have regarded ∇v as a vector field rather than a one-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Recall that the surface divergence operator satisfies the identity � F (divF α)ωF = � ∂F α(νF )ω∂F + � F Hα(n)ωF , (5) where νF is the outward unit normal to ∂F and H is the mean curvature of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let g(t) be a family of smooth Riemannian metrics with time derivative ∂ ∂tg =: σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let F be a time-independent hypersurface with mean curvature H and induced volume form ωF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Then ∂ ∂t(HωF ) = −1 2 �� II, σ|F � + (div Sσ)(n) + divF (σ(n, ·)) − Hσ(n, n) � ωF, (6) where II(X, Y ) = II(X, Y ) − Hg(X, Y ) is the trace-reversed second fundamental form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In dimension N = 2, the formula (6) simplifies considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Letting τ and n denote the unit tangent and unit normal to F, we have ∇τn = Hτ, −∇ττ = Hn, and II(τ, τ) = g(∇τn, τ) − Hg(τ, τ) = H − H = 0, so II vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In addition, divF (σ(n, ·)) − Hσ(n, n) = ∇τ (σ(n, ·)) (τ) − Hσ(n, n) = ∇τ (σ(n, τ)) − σ(n, ∇ττ) − Hσ(n, n) = ∇τ (σ(n, τ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Thus, in two dimensions, ∂ ∂t(HωF) = −1 2 ((div Sσ)(n) + ∇τ (σ(n, τ))) ωF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 6 To prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='2, we write ˙H = − N−1 � i=1 ∂ ∂tg(n, ∇τiτi) (7) and use the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For any time-dependent vector fields X and Y , ∂ ∂t∇Y X = ∇ ˙Y X + ∇Y ˙X + 1 2 ((∇Xσ)Y + (∇Y σ)X − (∇σ)(X, Y ))♯ , where (∇σ)(X, Y ) denotes the one-form Z �→ (∇Zσ)(X, Y ), and (∇Xσ)Y denotes the one-form Z �→ (∇Xσ)(Y, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In coordinates, (∇Y X)ℓ = Y j ∂Xℓ ∂xj + Γℓ ijY jXi, where Γℓ ij denote the Christoffel symbols of the second kind associated with g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Thus, ∂ ∂t(∇Y X)ℓ = ˙Y j ∂Xℓ ∂xj + Γℓ ij ˙Y jXi + Y j ∂ ˙Xℓ ∂xj + Γℓ ijY j ˙Xi + ˙Γℓ ijY jXi = (∇ ˙Y X)ℓ + (∇Y ˙X)ℓ + ˙Γℓ ijY jXi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Next, we recall the following formula for the rate of change of the Christoffel symbols under a metric deformation [10, Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='23]: ˙Γℓ ij = 1 2gℓm ((∇iσ)jm + (∇jσ)im − (∇mσ)ij) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' It follows that ˙Γℓ ijY jXi = 1 2gℓm � (∇Xσ)jmY j + (∇Y σ)imXi − (∇mσ)ijY jXi� = 1 2 [((∇Xσ)Y + (∇Y σ)X − (∇σ)(X, Y ))]ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Hence, ∂ ∂t(∇Y X)ℓ = (∇ ˙Y X)ℓ + (∇Y ˙X)ℓ + 1 2 ((∇Xσ)Y + (∇Y σ)X − (∇σ)(X, Y ))ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For any time-dependent vector field X, ∂ ∂tg(n, X) = 1 2σ(n, n)g(n, X) + g(n, ˙X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Writing X = ng(n, X) + �N−1 i=1 τig(τi, X), we compute ∂ ∂tg(n, X) = σ(n, X) + g( ˙n, X) + g(n, ˙X) = σ(n, n)g(n, X) + N−1 � i=1 σ(n, τi)g(τi, X) + g( ˙n, n)g(n, X) + N−1 � i=1 g( ˙n, τi)g(τi, X) + g(n, ˙X) = (σ(n, n) + g( ˙n, n)) g(n, X) + N−1 � i=1 (σ(n, τi) + g( ˙n, τi)) g(τi, X) + g(n, ˙X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 7 For each i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , N − 1, we have 0 = ∂ ∂tg(n, τi) = σ(n, τi) + g( ˙n, τi) + g(n, ˙τi) = σ(n, τi) + g( ˙n, τi) since ˙τi is g-orthogonal to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Likewise, 0 = ∂ ∂tg(n, n) = σ(n, n) + 2g(n, ˙n), so the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We are now ready to compute the time derivative of the mean curvature H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='5, we have ˙H = − N−1 � i=1 ∂ ∂tg(n, ∇τiτi) = − N−1 � i=1 �1 2σ(n, n)g(n, ∇τiτi) + g � n, ∂ ∂t∇τiτi �� = 1 2Hσ(n, n) − N−1 � i=1 g � n, ∂ ∂t∇τiτi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (8) Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='4 and the symmetry of the second fundamental form, we can write the second term as g � n, ∂ ∂t∇τiτi � = g(n, ∇ ˙τiτi) + g(n, ∇τi ˙τi) + (∇τiσ)(n, τi) − 1 2(∇nσ)(τi, τi) = 2g(n, ∇ ˙τiτi) + (∇τiσ)(n, τi) − 1 2(∇nσ)(τi, τi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The first term above, when summed over i, can be simplified as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We write ˙τi = �N−1 j=1 τjg(τj, ˙τi) and use the linearity of ∇XY in X to compute 2 N−1 � i=1 g(n, ∇ ˙τiτi) = 2 N−1 � i=1 N−1 � j=1 g(n, ∇τjτi)g(τj, ˙τi) = N−1 � i=1 N−1 � j=1 g(n, ∇τjτi) (g(τj, ˙τi) + g( ˙τj, τi)) = − N−1 � i=1 N−1 � j=1 g(n, ∇τjτi)σ(τj, τi) = ⟨II, σ|F ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Above, we used the symmetry of the second fundamental form to pass from the first line to the second, and we used the identity 0 = ∂ ∂tg(τj, τi) = σ(τj, τi) + g(τj, ˙τi) + g( ˙τj, τi) 8 to pass from the second line to the third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Inserting these results into (8), we get ˙H = 1 2Hσ(n, n) − ⟨II, σ|F ⟩ + N−1 � i=1 �1 2(∇nσ)(τi, τi) − (∇τiσ)(n, τi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (9) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We have N−1 � i=1 �1 2(∇nσ)(τi, τi) − (∇τiσ)(n, τi) � = 1 2 (⟨II, σ|F⟩ − (div Sσ)(n) − divF (σ(n, ·))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The identity 0 = ∇τi (g(n, n)) = 2g(n, ∇τin) shows that ∇τin is in the span of {τj}N−1 j=1 , so the first term on the right-hand side of (10) satisfies ⟨II, σ|F ⟩ = N−1 � i=1 N−1 � j=1 σ(τj, τi)g(τj, ∇τin) = N−1 � i=1 σ(∇τin, τi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (11) The second term on the right-hand side of (10) can be computed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Recalling that Sσ = σ − g Tr σ, we have (div Sσ)(n) = ∇n(Sσ)(n, n) + N−1 � i=1 ∇τi(Sσ)(n, τi) = (∇nσ)(n, n) − ∇n(g Tr σ)(n, n) + N−1 � i=1 [(∇τiσ)(n, τi) − ∇τi(g Tr σ)(n, τi)] = (∇nσ)(n, n) − g(n, n)∇n Tr σ + N−1 � i=1 [(∇τiσ)(n, τi) − g(n, τi)∇τi Tr σ] = (∇nσ)(n, n) − ∇n Tr σ + N−1 � i=1 (∇τiσ)(n, τi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Since the trace commutes with covariant differentiation, ∇n Tr σ = Tr ∇nσ = (∇nσ)(n, n) + N−1 � i=1 (∇nσ)(τi, τi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Thus, (div Sσ)(n) = N−1 � i=1 [(∇τiσ)(n, τi) − (∇nσ)(τi, τi)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (12) The third term on the right-hand side of (10) is given by divF (σ(n, ·)) = N−1 � i=1 ∇τi (σ(n, ·)) (τi) = N−1 � i=1 [∇τi (σ(n, τi)) − σ(n, ∇τiτi)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (13) 9 Combining (11), (12), and (13), we see that 1 2 (⟨II, σ|F ⟩ − (div Sσ)(n) − divF (σ(n, ·))) = 1 2 N−1 � i=1 [σ(∇τin, τi) − (∇τiσ)(n, τi) + (∇nσ)(τi, τi) − ∇τi (σ(n, τi)) + σ(n, ∇τiτi)] = 1 2 N−1 � i=1 [(∇nσ)(τi, τi) − 2(∇τiσ)(n, τi)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Combining Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='6 with (9), we get ˙H = 1 2 (−⟨II, σ|F ⟩ − (div Sσ)(n) − divF (σ(n, ·)) + Hσ(n, n)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (14) Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='2 now follows from the identities ∂ ∂t(HωF) = ˙HωF + H ˙ωF = ˙HωF + 1 2H Tr (σ|F ) ωF and ⟨II, σ|F ⟩ − H Tr (σ|F) = ⟨II, σ|F⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='3 Evolution of angles Next we study the evolution of angles under deformations of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let g(t) be a family of smooth Riemannian metrics with time derivative ∂ ∂tg =: σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let (¯n(t), ¯τ(t)) be a pair of g(t)-orthonormal vectors, and let (n(t), τ(t)) be another pair of g(t)- orthonormal vectors lying in the span of (¯n(t), ¯τ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let θ(t) be the angle for which τ = ¯τ cos θ + ¯n sin θ, n = −¯τ sin θ + ¯n cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Assume that these vectors vary smoothly in time, and assume that n(t) (respectively, ¯n(t)) is at all times g(t)-orthogonal to a time-independent hypersurface F (respectively, ¯F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Then, at all times for which θ ∈ (0, π), we have ∂ ∂tθ = 1 2σ(n, τ) − 1 2σ(¯n, ¯τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Differentiating the relation cos θ = g(¯n, n) yields − ˙θ sin θ = ∂ ∂t (g(¯n, n)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In particular, at any time s, we can write − ˙θ(s) sin θ(s) = ∂ ∂t ���� t=s (g(t)(¯n(t), n(s))) + ∂ ∂t ���� t=s (g(t)(¯n(s), n(t))) − σ(s)(¯n(s), n(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 10 Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='5 and suppressing the evaluations at t = s, we get − ˙θ sin θ = 1 2σ(¯n, ¯n)g(¯n, n) + 1 2σ(n, n)g(n, ¯n) − σ(¯n, n) = 1 2σ(¯n, ¯n cos θ − n) + 1 2σ(n cos θ − ¯n, n) = 1 2σ(¯n, ¯τ sin θ) + 1 2σ(−τ sin θ, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If θ ∈ (0, π) at time t = s, then we can divide by sin θ to get (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 3 Distributional densitized scalar curvature Let T be a simplicial triangulation of a polyhedral domain Ω ⊂ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We use T k to denote the set of all k-simplices in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We also use ˚T k to denote the subset of T k consisting of k-simplices that are not contained in the boundary of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We call such simplices interior simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We call (N − 1)-simplices faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let g be a Regge metric on T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Recall that this means that g|T is a smooth Riemannian metric on each T ∈ T N, and the induced metric g|F is single-valued on each F ∈ ˚ T N−1 (and consequently the induced metric is single-valued on all lower-dimensional simplices in T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' On each T ∈ T N, we denote by RT the scalar curvature of g|T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' On an interior face F ∈ ˚ T N−1 that lies on the boundary of two N-simplices T + and T −, the second fundamental form on F, as measured by g|T +, generally differs from that measured by g|T −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We denote by �II�F the jump in the second fundamental form across F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' More precisely, �II�F(X, Y ) = g|T + (∇Xn+, Y ) + g|T − (∇Xn−, Y ) for any vectors X, Y tangent to F, where n± points outward from T ±, has unit length with respect to g|T ±, and is g|T ±-orthogonal to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We adopt similar notation for the jumps in other quantities across F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For instance, �H�F denotes the jump in the mean curvature across F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We sometimes drop the subscript F when there is no danger of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If F is contained in ∂Ω, then we define the jump in a scalar field v across F to be simply �v�F = v|F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' On each S ∈ ˚ T N−2, the angle defect along S is ΘS = 2π − � T∈T N T⊃S θST, where θST denotes the dihedral angle formed by the two faces of T that contain S, as measured by g|T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Generally this angle may vary along S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If F + and F − are the two faces of T that contain S, and if n± denotes the unit normal to F ± with respect to g|T pointing outward from T, then cos θST = − g|T (n+, n−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let V = {v ∈ H1 0(Ω) | ∀T ∈ T N, v|T ∈ H2(T)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Note that if v ∈ V , then v admits a single-valued trace on every simplex in T of dimension ≥ N −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let g be a Regge metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The distributional densitized scalar curvature of g is the linear functional (Rω)dist(g) ∈ V ′ defined by ⟨(Rω)dist(g), v⟩V ′,V = � T∈T N � T RT vωT + 2 � F ∈˚ T N−1 � F �H�FvωF + 2 � S∈˚ T N−2 � S ΘSvωS, ∀v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (16) 11 This definition generalizes Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1 of [4], where the distributional curvature two-form (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' the Gaussian curvature times the volume form) is defined for Regge metrics in dimension N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Note that the factors of 2 appearing in all but the first term in (16) are consistent with the fact that in dimension N = 2, the scalar curvature R is twice the Gaussian curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' One can heuristically motivate Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1 in much the same way that one motivates its two-dimensional counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' When g is piecewise constant, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1 recovers the classical notion [23] that the distributional densitized scalar curvature is a linear combination of Dirac delta distributions supported on (N − 2)-simplices, with weights given by angle defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' When g is not piecewise constant, additional terms appear which account for the nonzero (classically defined) curvature of g in the interior of each N-simplex T and the jump in the mean curvature across each interior face F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The jump in the mean curvature across F can be understood by recalling that the scalar curvature R at a point p ∈ F can be expressed as (two times) a sum of sectional curvatures of N(N − 1)/2 tangent planes that are mutually g-orthogonal at p, (N − 1)(N − 2)/2 of which are tangent to F at p and N − 1 of which are g-orthogonal to F at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The sectional curvatures corresponding to planes tangent to F are nonsingular, owing to the tangential-tangential continuity of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The remaining N − 1 sectional curvatures are singular, and by considering an N-dimensional region that encloses a portion of F and has small thickness in the direction that is g-orthogonal of F, one can use the Gauss-Bonnet theorem (along two-dimensional slices) to approximate the (volume- )integrated sum of these sectional curvatures by the (surface-)integrated jump in the mean curvature across F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (In this calculation, one must bear in mind that sectional curvatures and Gaussian curvatures are related via the Gauss-Codazzi equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=') See the discussion after Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1 in [4], as well as [26], for more insight in dimension N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' See also [13] for a justification of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1 in the case where g is piecewise constant and N ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In the sequel, we will consistently use the letters T, F, and S to refer to simplices of dimension N, N − 1, and N − 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We will therefore write � T , � F , and � S in place of � T∈T N , � F ∈T N−1, and � S∈T N−2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' When we wish to sum over interior simplices of a given dimension, we put a ring on top of the summation symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Thus, for example, ˚ � F is shorthand for � F ∈˚ T N−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1 Evolution of the distributional scalar curvature We are interested in how (16) changes under deformations of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' To this end, consider a one-parameter family of Regge metrics g(t) with time derivative σ = ∂ ∂tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Our goal will be to compute d dt⟨(Rω)dist(g(t)), v⟩V ′,V with v ∈ V arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' According to Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='2, the derivatives of the first two terms on the right-hand side of (16) satisfy d dt � T RT vωT = � T (div div Sσ − ⟨G, σ⟩) vωT and 2 d dt � F �H�FvωF = − � F �� II, σ|F � + (div Sσ)(n) + divF (σ(n, ·)) − Hσ(n, n) � vωF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (17) For the third term on the right-hand side of (16), we use the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 12 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Along any interior (N − 2)-simplex S, we have ∂ ∂t(ΘSωS) = 1 2 �� F ⊃S �σ(n, τ)�F + ΘS Tr(σ|S) � ωS, where the sum is over all (N −1)-simplices F that contain S, n is the unit normal to F with respect to g, and τ is the unit vector with respect to g that points into F from S and is g-orthogonal to both S and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Here, our convention is that if F is shared by two N-simplices T + and T −, then �σ(n, τ)�F = σ+(n+, τ) + σ−(n−, τ), where σ± = σ|T ± and n± points outward from T ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Note that n generally differs on either side of F, whereas τ does not, because g has single-valued tangential-tangential components along F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We compute ˙ΘS = − � T⊃S ˙θST and use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='7 to differentiate each angle θST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The resulting expression for ˙ΘS involves differences between σ(n, τ) evaluated on consecutive pairs of faces F emanating from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This sum can be rearranged to give ˙ΘS = 1 2 � F ⊃S �σ(n, τ)�F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (18) We thus get ∂ ∂t(ΘSωS) = ˙ΘSωS + ΘS ˙ωS = 1 2 � F ⊃S �σ(n, τ)�F ωS + 1 2ΘS Tr (σ|S) ωS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' It follows from the above lemma that 2 d dt � S ΘSvωS = � S � F ⊃S �σ(n, τ)�F vωS + � S ΘS Tr(σ|S)vωS = � S � F ⊃S �σ(n, τ)�F vωS + � S ⟨ΘSg|S, σ|S⟩ vωS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Collecting our results, we obtain d dt⟨(Rω)dist(g(t)), v⟩V ′,V = � T � T (div div Sσ)vωT − ˚ � F � F �(div Sσ)(n) + divF (σ(n, ·)) − Hσ(n, n)�F vωF + ˚ � S � S � F ⊃S �σ(n, τ)�F vωS (19) − � T � T ⟨G, σ⟩vωT − ˚ � F � F � �II�F, σ|F � vωF + ˚ � S � S ⟨ΘSg|S, σ|S⟩ vωS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We will now use integration by parts to rewrite the first three terms in a way that involves no derivatives of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 13 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For any v ∈ V , we have � T � T (div div Sσ)vωT − ˚ � F � F �(div Sσ)(n) + divF (σ(n, ·)) − Hσ(n, n)�F vωF + ˚ � S � S � F ⊃S �σ(n, τ)�FvωS = � T � T ⟨Sσ, ∇∇v⟩ω − � F � F Sσ(n, n)�∇nv�ωF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We have � T � T ⟨Sσ, ∇∇v⟩ω − � F � F Sσ(n, n)�∇nv�ωF (20) = � T � � T ⟨Sσ, ∇∇v⟩ω − � ∂T Sσ(n, n)∇nv ω∂T � = � T � � ∂T Sσ(n, ∇v)ω∂T − � T (div Sσ)(∇v)ω − � ∂T Sσ(n, n)∇nv ω∂T � = � T � � ∂T Sσ(n, ∇v)ω∂T − � ∂T (div Sσ)(n)vω∂T + � T (div div Sσ)vω − � ∂T Sσ(n, n)∇nv ω∂T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (21) Note that here we are regarding ∇v as a vector field rather than a one-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' On each N-simplex T, we can write � ∂T Sσ(n, ∇v)ω∂T − � ∂T Sσ(n, n)∇nv ω∂T as a sum of integrals over faces F ⊂ ∂T: � ∂T Sσ(n, ∇v)ω∂T − � ∂T Sσ(n, n)∇nv ω∂T = � F ⊂∂T � F Sσ(n, ∇v − n∇nv)ωF = � F ⊂∂T � F Sσ(n, ∇F v)ωF = � F ⊂∂T � F σ(n, ∇F v)ωF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In the last line above, we used the fact that ∇F v is g-orthogonal to n, so Sσ(n, ∇Fv) = σ(n, ∇F v) − g(n, ∇F v) Tr σ = σ(n, ∇F v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Each integral over F can be integrated by parts as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We have σ(n, ∇F v) = divF (σ(n, ·)v) − divF (σ(n, ·)) v, so the identity (5) applied to α = σ(n, ·)v implies that � F σ(n, ∇F v)ωF = � ∂F σ(n, νF )vω∂F − � F (divF (σ(n, ·)) − Hσ(n, n)) vωF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Now we insert this result into (21) to get � T � T ⟨Sσ, ∇∇v⟩ω − � F � F Sσ(n, n)�∇nv�ωF = � T � � F ⊂∂T � ∂F σ(n, νF )vω∂F − � F ⊂∂T � F (divF (σ(n, ·)) − Hσ(n, n)) vωF − � ∂T (div Sσ)(n)vω∂T + � T (div div Sσ)vω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 14 The first term can be re-expressed as a sum over interior (N − 2)-simplices S using our notation from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='2, and the next two terms can be re-expressed in terms of jumps across interior faces F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (Integrals over (N − 2)-simplices S ⊂ ∂Ω and (N − 1)-simplices F ⊂ ∂Ω vanish because v = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=') The result is � T � T ⟨Sσ, ∇∇v⟩ω − � F � F Sσ(n, n)�∇nv�ωF = ˚ � S � S � F ⊃S �σ(n, τ)�FvωS − ˚ � F � F �divF (σ(n, ·)) − Hσ(n, n) + (div Sσ)(n)� vωF + � T � T (div div Sσ)vω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Many of the above calculations are similar to the ones in [4, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='2], except that here we are in dimension N rather than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We can now state the main result of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let g(t) be a family of Regge metrics with time derivative ∂ ∂tg =: σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For every v ∈ V , we have d dt⟨(Rω)dist(g(t)), v⟩V ′,V = bh(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v) − ah(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v), (22) where bh(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v) = � T � T ⟨Sσ, ∇∇v⟩ω − � F � F Sσ(n, n)�∇nv�FωF, ah(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v) = � T � T ⟨G, σ⟩vωT + ˚ � F � F � �II�F, σ|F � vωF − ˚ � S � S ⟨ΘSg|S, σ|S⟩ vωS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Combine (19) with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='2 Distributional densitized Einstein tensor We now pause to make a few remarks about the bilinear forms ah(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' ·, ·) and bh(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' ·, ·) appearing in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' These remarks will play no role in our analysis, but they help to elucidate the content of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The reader can safely skip ahead to Section 4 if desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Numerical analysts will likely recognize the bilinear form bh(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' ·, ·) appearing in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' As we mentioned in Section 1, it is (up to the appearance of S) a non-Euclidean, N-dimensional gener- alization of a bilinear form that appears in the Hellan-Herrmann-Johnson finite element method [1– 3, 5–7, 9, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' It can be regarded as the integral of div div Sσ against v, where div div is interpreted in a distributional sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The bilinear form ah(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' ·, ·) can be understood by comparing Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='6 with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1, which, when integrated against a continuous function v, states that for a family of smooth Rieman- nian metrics g(t) with scalar curvature R, d dt � Ω Rvω = � Ω (div div Sσ)vω − � Ω ⟨G, σ⟩vω, (23) where σ = ∂ ∂tg and G = Ric − 1 2Rg is the Einstein tensor associated with g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' A comparison of (23) with (22) suggests that for a Regge metric g, the bilinear form ah(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v) should be regarded as a distributional counterpart of � Ω⟨G, σ⟩vω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 15 This motivates the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Fix a number s > 1, and let Σ denote the space of square-integrable symmetric (0, 2)-tensor fields σ with the following properties: the restriction of σ to each T ∈ T N belongs to Hs(T), and the tangential-tangential components of σ along any face F ∈ ˚ T N−1 are single-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Note that these conditions imply that the tangential-tangential components of σ along any S ∈ ˚ T N−2 are well-defined and single-valued as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let g be a Regge metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The distributional densitized Einstein tensor associated with g is the linear functional (Gω)dist(g) ∈ Σ′ defined by ⟨(Gω)dist(g), σ⟩Σ′,Σ = � T � T ⟨G, σ⟩ωT + ˚ � F � F � �II�F, σ|F � ωF − ˚ � S � S ⟨ΘSg|S, σ|S⟩ ωS, ∀σ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In dimension N = 2, we have (Gω)dist(g) = 0 for any Regge metric g, because G vanishes within each triangle, ¯II vanishes on each edge, and the restriction of σ to each vertex vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The appearance of the trace-reversed second fundamental form II in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='7 is quite natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The same quantity arises in studies of singular sources in general relativity, with the jump in II encoding the well-known Israel junction conditions across a hypersurface on which stress-energy is concentrated [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If we define a map (div div S)dist : Σ → V ′ by ⟨(div div S)distσ, v⟩V ′,V = bh(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v), ∀v ∈ V, then, by construction, we have d dt ���� t=0 ⟨(Rω)dist(g + tσ), v⟩V ′,V = ⟨(div div S)distσ, v⟩V ′,V − ⟨(Gω)dist(g), vσ⟩Σ′,Σ for every piecewise smooth σ ∈ Σ and every smooth function v with compact support in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In particular, suppose that Ω has no boundary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=', suppose that Ω is an N-dimensional cube and we identify its opposing faces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Then bh(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, 1) = 0 and d dt ���� t=0 ⟨(Rω)dist(g + tσ), 1⟩V ′,V = −⟨(Gω)dist(g), σ⟩Σ′,Σ for every piecewise smooth σ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This implies that a Regge metric g is a stationary point of ⟨(Rω)dist(g), 1⟩Σ′,Σ if its distributional densitized Einstein tensor vanishes: (Gω)dist(g) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The functional ⟨(Rω)dist(g), 1⟩Σ′,Σ is a counterpart of the Einstein-Hilbert functional � Ω Rω from general relativity, whose stationary points are solutions to the (vacuum) Einstein field equations G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' It reduces to the Regge action from Regge calculus when g is piecewise constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' That is, ⟨(Rω)dist(g), 1⟩Σ′,Σ = 2 ˚ � S ΘSVS, if g is piecewise constant, where VS = � S ωS denotes the volume of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If g varies with t and remains piecewise constant for all t, then d dt2 ˚ � S ΘSVS = 2 ˚ � S ˙ΘSVS + 2 ˚ � S ΘS ˙VS, 16 and one checks that (on a domain without boundary) 2 ˚ � S ˙ΘSVS = bh(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, 1) = 0 and 2 ˚ � S ΘS ˙VS = −ah(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, 1) = −⟨(Gω)dist(g), σ⟩Σ′,Σ, where σ = ∂ ∂tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The fact that ˚ � S ˙ΘSVS = 0 for any piecewise constant Regge metric g (on a domain without boundary) was proved in Regge’s classic paper [23] using very different techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If g is a Regge metric and σ = gv for some smooth function v with compact support in Ω, then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' On each N-simplex T, we have ⟨G, σ⟩ = ⟨G, g⟩v = (Tr G)v = − �N − 2 2 � Rv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' On either side of each interior (N − 1)-simplex F, we have: � II, σ|F � = ⟨II, g|F ⟩ v − ⟨g|F , g|F⟩ Hv = Hv − (N − 1)Hv = −(N − 2)Hv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' On each interior (N − 2)-simplex S, we have ⟨ΘSg|S, σ|S⟩ = ΘSv Tr(g|S) = (N − 2)ΘSv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This shows that ⟨(Gω)dist(g), gv⟩Σ′,Σ = − �N − 2 2 � �� T � T RT vωT + 2 ˚ � F � F �H�FvωF + 2 ˚ � S � S ΘSvωS � = − �N − 2 2 � ⟨(Rω)dist(g), v⟩V ′,V for every smooth function v with compact support in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' One can interpret this as saying that the trace of (Gω)dist(g) is − � N−2 2 � (Rω)dist(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If g is a piecewise constant Regge metric and σ ∈ Σ is piecewise constant, then ⟨(Gω)dist(g), σ⟩Σ′,Σ = − ˚ � S � S ΘS Tr(σ|S)ωS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If we linearize around the Euclidean metric g = δ, then we see from (18) that d dt ���� t=0 ⟨(Gω)dist(δ + tρ), σ⟩Σ′,Σ = − ˚ � S � S ˙ΘS Tr(σ|S)ωS = −1 2 ˚ � S � S � F ⊃S �ρ(n, τ)�F Tr(σ|S)ωS 17 for every piecewise constant ρ, σ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (Note that there are no additional terms on the right-hand side because ΘS = 0 at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=') Hence, if Ω has no boundary, then d2 dt2 ���� t=0 ⟨(Rω)dist(δ + tσ), 1⟩V ′,V = − d dt ���� t=0 ⟨(Gω)dist(δ + tσ), σ⟩Σ′,Σ = 1 2 ˚ � S � S � F ⊃S �σ(n, τ)�F Tr(σ|S)ωS for every piecewise constant σ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This is equivalent to Christiansen’s formula [12, Theorem 2 and Equations (25-26)] for the second variation of the Regge action around the Euclidean metric in dimension N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (There, the Regge action is taken to be 1 2⟨(Rω)dist(g), 1⟩V ′,V rather than ⟨(Rω)dist(g), 1⟩V ′,V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=') 4 Convergence In this section, we prove a convergence result for the distributional densitized scalar curvature in the norm ∥u∥H−2(Ω) = sup v∈H2 0(Ω), v̸=0 ⟨u, v⟩H−2(Ω),H2 0(Ω) ∥v∥H2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (24) Our convergence result will be applicable to a family {gh}h>0 of Regge metrics defined on a shape- regular family {Th}h>0 of triangulations of Ω parametrized by h = maxT∈T N h hT , where hT = diam(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Shape-regularity means that there exists a constant C0 independent of h such that max T∈T N h hT ρT ≤ C0 for all h > 0, where ρT denotes the inradius of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let Ω ⊂ RN be a polyhedral domain equipped with a smooth Riemannian metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let {gh}h>0 be a family of Regge metrics defined on a shape-regular family {Th}h>0 of triangulations of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Assume that limh→0 ∥gh − g∥L∞(Ω) = 0 and C1 := suph>0 maxT∈T N h ∥gh∥W 1,∞(T) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The following statements hold: (i) If N = 2, then there exist positive constants C and h0 such that ∥(Rω)dist(gh) − (Rω)(g)∥H−2(Ω) ≤ C � 1 + max T h−1 T ∥gh − g∥L∞(T) + max T |gh − g|W 1,∞(T) � × � ∥gh − g∥2 L2(Ω) + � T h2 T |gh − g|2 H1(T) �1/2 (25) for all h ≤ h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The constants C and h0 depend on ∥g∥W 1,∞(Ω), ∥g−1∥L∞(Ω), C0, and C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 18 (ii) If N ≥ 3, assume additionally that C2 := suph>0 maxT∈T N h |gh|W 2,∞(T) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Then there exist positive constants C and h0 such that ∥(Rω)dist(gh) − (Rω)(g)∥H−2(Ω) ≤ C � 1 + max T h−2 T ∥gh − g∥L∞(T) + max T h−1 T |gh − g|W 1,∞(T) � × � ∥gh − g∥2 L2(Ω) + � T h2 T |gh − g|2 H1(T) + � T h4 T |gh − g|2 H2(T) �1/2 (26) for all h ≤ h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The constants C and h0 depend on N, ∥g∥W 1,∞(Ω), ∥g−1∥L∞(Ω), C0, C1, and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The above theorem leads immediately to error estimates of optimal order for piecewise poly- nomial interpolants of g having degree r ≥ 0, provided that either N = 2 or r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' To make this statement precise, we introduce a definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Recall that the Regge finite element space of degree r ≥ 0 consists of symmetric (0, 2)-tensor fields on Ω that are piecewise polynomial of degree at most r and possess single-valued tangential-tangential components on interior (N − 1)-simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let Ih be a map that sends smooth symmetric (0, 2)-tensor fields on Ω to the Regge finite element space of degree r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We say that Ih is an optimal-order interpolation operator of degree r if there exists a number m ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , N} and a constant C3 = C3(N, r, hT /ρT , t, s) such that for every p ∈ [1, ∞], every s ∈ (m/p, r + 1], every t ∈ [0, s], and every symmetric (0, 2)-tensor field g possessing W s,p(Ω)-regularity, Ihg exists (upon continuously extending Ih) and satisfies |Ihg − g|W t,p(T) ≤ C3hs−t T |g|W s,p(T) (27) for every T ∈ T N h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We call the number m the codimension index of Ih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' A Regge metric gh is called an optimal-order interpolant of g having degree r and codimension index m if it is the image of a Riemannian metric g under an optimal-order interpolation operator having degree r and codimension index m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' An example of an optimal-order interpolation operator is the canonical interpolation operator onto the degree-r Regge finite element space introduced in [21, Chapter 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Its degrees of freedom involve integrals over simplices of codimension at most N − 1, so its action on a tensor field g is well-defined so long as g admits traces on simplices of codimension at most N − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' g possesses W s,p(Ω)-regularity with s > (N − 1)/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Correspondingly, its codimension index is m = N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let Ω, g, and {Th}h>0 be as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let {gh}h>0 be a family of optimal- order interpolants of g having degree r ≥ 0 and codimension index m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If N ≥ 3, assume that r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Then there exist positive constants C and h0 such that ∥(Rω)dist(gh) − (Rω)(g)∥H−2(Ω) ≤ C �� T hp(r+1) T |g|p W r+1,p(T) �1/p for all h ≤ h0 and all p ∈ [2, ∞] satisfying p > m r+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (We interpret the right-hand side as C maxT hr+1 T |g|W r+1,∞(T) if p = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=') The constants C and h0 depend on the same quantities listed in (i) (if N = 2) and (ii) (if N ≥ 3), as well as on Ω, r, and (if N ≥ 3) |g|W 2,∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 19 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The corollary above continues to hold if we allow slightly more general interpolants in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For example, it holds if (27) is replaced by |Ihg − g|W t,p(T) ≤ C3hs−t T � T ′:T ′∩T̸=∅ |g|W s,p(T ′), (28) where the sum is over all T ′ ∈ T N h that share a subsimplex with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In what follows, we reuse the letter C to denote a positive constant that may change at each occurrence and may depend on N, ∥g∥W 1,∞(Ω), ∥g−1∥L∞(Ω), C0, and C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Beginning in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='8, we allow C to also depend on C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Our strategy for proving Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1 will be to consider an evolving metric �g(t) = (1 − t)g + tgh with time derivative σ = ∂ ∂t�g(t) = gh − g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Note that �g(t), being piecewise smooth and tangential-tangential continuous, is a Regge metric for all t ∈ [0, 1], and it happens to be a (globally) smooth Riemannian metric at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Since �g(0) = g and �g(1) = gh, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='6 implies that ⟨(Rω)dist(gh) − (Rω)(g), v⟩V ′,V = � 1 0 bh(�g(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v) − ah(�g(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v) dt, ∀v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Thus, we can estimate (Rω)dist(gh) − (Rω)(g) by estimating the bilinear forms bh(�g(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' ·, ·) and ah(�g(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' ·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' To do this, we introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Given any Regge metric g, we let ∇g and ∇ denote the covariant derivatives with respect to g and δ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Similarly, we append a subscript g to other operators like Tr, S, and div when they are taken with respect to g, and we omit the subscript when they are taken with respect to δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' On the boundary of any N-simplex T, we let ng and n denote the outward unit normal vectors with respect to g|T and δ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' These two vectors are related to one another in coordinates via ng = 1 � nT g−1n g−1n, (29) where we are thinking of g as a matrix and n and ng as column vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We write ⟨·, ·⟩g for the g-inner product of two tensor fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If D is a submanifold of Ω on which the induced metric g|D is well-defined, and if ρ is a tensor field on D, then we denote ∥ρ∥Lp(D,g) = ��� D |ρ|p g ωD(g) �1/p , if 1 ≤ p < ∞, supD |ρ|g, if p = ∞, where ωD(g) is the induced volume form on D and |ρ|g = ⟨ρ, ρ⟩1/2 g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We abbreviate ∥ · ∥Lp(D) = ∥ · ∥Lp(D,δ) and | · | = | · |δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We introduce two metric-dependent, mesh-dependent norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For v ∈ V , we set ∥v∥2 2,h,g = � T ∥∇g∇gv∥2 L2(T,g) + � F h−1 F ∥�dv(ng)�∥2 L2(F,g) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 20 If σ is a symmetric (0, 2)-tensor field with the property that σ(ng, ng) is well-defined and single- valued on every F ∈ T N−1 h , then we set ∥σ∥2 0,h,g = � T ∥σ∥2 L2(T,g) + � F hF ∥σ(ng, ng)∥2 L2(F,g), where hF is the Euclidean diameter of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Note that the image under Sg of any symmetric (0, 2)- tensor field possessing single-valued tangential-tangential components along faces automatically possesses single-valued normal-normal components along faces, because Sgσ(ng, ng) = σ(ng, ng) − g(ng, ng) Trg σ = − Trg (σ|F ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Now we return to the setting of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1 and the discussion thereafter: g is a smooth Riemannian metric, gh is a Regge metric, �g(t) = (1 − t)g + tgh, and σ = gh − g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We assume throughout what follows that limh→0 ∥gh − g∥L∞(Ω) = 0 and suph>0 maxT∈T N h ∥gh∥W 1,∞(T) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' These assumptions have some elementary consequences that we record here for reference (see [16] for a derivation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For every h sufficiently small, every t ∈ [0, 1], and every vector w with unit Euclidean length, ∥�g∥L∞(Ω) + ∥�g−1∥L∞(Ω) ≤ C, (30) max T |�g|W 1,∞(T) ≤ C, (31) C−1 ≤ inf Ω (wT �gw) ≤ sup Ω (wT �gw) ≤ C, (32) where we are thinking of �g as a matrix and w as a column vector in the last line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Note that the last line implies the existence of positive lower and upper bounds on wT �g−1w as well: C−1 ≤ inf Ω (wT �g−1w) ≤ sup Ω (wT �g−1w) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (33) In addition, the inequalities ∥�g∥L∞(Ω) ≤ C and ∥�g−1∥L∞(Ω) ≤ C imply that C−1∥ρ∥Lp(D,�g(t2)) ≤ ∥ρ∥Lp(D,�g(t1)) ≤ C∥ρ∥Lp(D,�g(t2)) (34) and C−1∥ρ∥Lp(D) ≤ ∥ρ∥Lp(D,�g(t1)) ≤ C∥ρ∥Lp(D) (35) for every t1, t2 ∈ [0, 1], every admissible submanifold D, every p ∈ [1, ∞], every tensor field ρ having finite Lp(D)-norm, and every h sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We select h0 > 0 so that (30-35) hold for all h ≤ h0, and we tacitly use these inequalities throughout our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We will show the following near-equivalence of the norms ∥ · ∥2,h,�g and ∥ · ∥2,h,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For every v ∈ V , every h ≤ h0, and every t ∈ [0, 1], ∥v∥2 2,h,�g ≤ C � ∥v∥2 2,h,g + � max T h−2 T ∥gh − g∥2 L∞(T) + max T |gh − g|2 W 1,∞(T) � × � T � ∥dv∥2 L2(T) + h2 T |dv|2 H1(T) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='5 relies on the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 21 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let g1 and g2 be two symmetric positive definite matrices, and let n be a unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let ngi = 1 � nTg−1 i n g−1 i n, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Then there exists a constant c depending on |g1|, |g2|, |g−1 1 |, |g−1 2 | such that |ng1 − ng2| ≤ c|g1 − g2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Using the identity 1 � nT g−1 1 n − 1 � nTg−1 2 n = nT(g−1 2 − g−1 1 )n nTg−1 1 n � nTg−1 2 n + nT g−1 2 n � nT g−1 1 n , (36) we can write ng1 − ng2 = nT (g−1 2 − g−1 1 )n nT g−1 1 n � nT g−1 2 n + nTg−1 2 n � nTg−1 1 n g−1 1 n + 1 � nT g−1 2 n (g−1 1 − g−1 2 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Since g−1 1 − g−1 2 = g−1 1 (g2 − g1)g−1 2 , the bound follows easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Notice that in view of (29), Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='6 implies that ∥n�g − ng∥L∞(F ) ≤ C∥�g − g∥L∞(F ) (37) on either side of any face F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Now we are ready to begin proving Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Consider the term � F h−1 F ���dv(n�g)� ��2 L2(F,�g) that appears in the definition of ∥v∥2 2,h,�g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Notice that dv(n�g) = dv(ng) + dv(n�g − ng), and we can use the bound (37) to estimate ∥dv(n�g − ng)∥L2(F,�g) ≤ C∥dv(n�g − ng)∥L2(F ) ≤ C∥dv∥L2(F )∥n�g − ng∥L∞(F ) ≤ C∥dv∥L2(F )∥�g − g∥L∞(F ) ≤ C∥dv∥L2(F )∥gh − g∥L∞(F ) on either side of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Using the trace inequality ∥dv∥2 L2(F ) ≤ C � h−1 T ∥dv∥2 L2(T) + hT |dv|2 H1(T) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' F ⊂ T ∈ T N h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (38) it follows that � F h−1 F ∥�dv(n�g)�∥2 L2(F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='�g) ≤ C �� F h−1 F ∥�dv(ng)�∥2 L2(F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='g) + � T h−1 T � h−1 T ∥dv∥2 L2(T) + hT |dv|2 H1(T) � ∥gh − g∥2 L∞(T) � = C �� F h−1 F ∥�dv(ng)�∥2 L2(F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='g) + � T � h−2 T ∥gh − g∥2 L∞(T)∥dv∥2 L2(T) + ∥gh − g∥2 L∞(T)|dv|2 H1(T) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 22 where we have used (34),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (38),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' and the bound hT ≤ ChF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' which follows from the shape-regularity of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Next, consider the term � T ∥∇�g∇�gv∥2 L2(T,�g) that appears in the definition of ∥v∥2 2,h,�g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Notice that � ∇�g∇�gv � ij = (∇g∇gv)ij + (Γk ij − �Γk ij) ∂v ∂xk , where Γk ij and �Γk ij are the Christoffel symbols of the second kind associated with g and �g, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We have ∥Γk ij − �Γk ij∥L∞(T) ≤ C∥�g − g∥W 1,∞(T) ≤ C∥gh − g∥W 1,∞(T), so ∥∇�g∇�gv∥L2(T,�g) ≤ C∥∇�g∇�gv∥L2(T) ≤ C � ∥∇g∇gv∥L2(T) + ∥gh − g∥W 1,∞(T)∥dv∥L2(T) � ≤ C � ∥∇g∇gv∥L2(T,g) + ∥gh − g∥W 1,∞(T)∥dv∥L2(T) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' It follows that ∥v∥2 2,h,�g ≤ C � ∥v∥2 2,h,g + � max T h−2 T ∥gh − g∥2 L∞(T) + max T |gh − g|2 W 1,∞(T) � × � T � ∥dv∥2 L2(T) + h2 T |dv|2 H1(T) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This completes the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Our next step will be to estimate the bilinear form bh(�g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' ·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For every h ≤ h0, every t ∈ [0, 1], and every v ∈ H2 0(Ω), we have (with σ = gh − g) |bh(�g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v)| ≤ C � ∥gh − g∥2 L2(Ω) + � T h2 T |gh − g|2 H1(T) �1/2 × � 1 + max T h−1 T ∥gh − g∥L∞(T) + max T |gh − g|W 1,∞(T) � ∥v∥H2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In view of the definitions of ∥ · ∥0,h,�g and ∥ · ∥2,h,�g, we have |bh(�g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v)| ≤ ∥S�gσ∥0,h,�g∥v∥2,h,�g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (39) Recalling that ∥S�gσ∥2 0,h,�g = � T ∥S�gσ∥2 L2(T,�g) + � F hF ∥S�gσ(n�g, n�g)∥2 L2(F,�g), we compute ⟨S�gσ, S�gσ⟩�g = � σ − �g⟨�g, σ⟩�g, σ − �g⟨�g, σ⟩�g � �g = ⟨σ, σ⟩�g − 2⟨�g, σ⟩2 �g + ⟨�g, �g⟩�g⟨�g, σ⟩2 �g = ⟨σ, σ⟩�g + (N − 2)⟨�g, σ⟩2 �g, 23 which leads to the bound ∥S�gσ∥L2(T,�g) ≤ C∥σ∥L2(T,�g) ≤ C∥σ∥L2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Also, by the trace inequality, ∥S�gσ(n�g, n�g)∥2 L2(∂T,�g) ≤ C∥S�gσ∥2 L2(∂T,�g) ≤ C∥σ∥2 L2(∂T) ≤ C � h−1 T ∥σ∥2 L2(T) + hT |σ|2 H1(T) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (Here we are measuring the L2(∂T, �g)-norm of the full tensor S�gσ rather than its restriction to the tangent bundle of ∂T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=') Thus, ∥S�gσ∥2 0,h,�g ≤ C � ∥σ∥2 L2(Ω) + � T h2 T |σ|2 H1(T) � = C � ∥gh − g∥2 L2(Ω) + � T h2 T |gh − g|2 H1(T) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (40) Consider now the term ∥v∥2,h,�g in (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='5 implies that ∥v∥2,h,�g ≤ C � ∥v∥2,h,g + � max T h−1 T ∥gh − g∥L∞(T) + max T |gh − g|W 1,∞(T) � ∥v∥H2(Ω) � since v ∈ H2 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Furthermore, since g is smooth and v ∈ H2 0(Ω), we have �dv(ng)� = 0 on every interior face F and �dv(ng)� = dv(ng) = 0 on every face F ⊂ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Thus, ∥v∥2 2,h,g = � T ∥∇g∇gv∥2 L2(T,g) = ∥∇g∇gv∥2 L2(Ω,g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Since (∇g∇gv)ij = (∇∇v)ij − Γk ij ∂v ∂xk , we see that ∥v∥2,h,g = ∥∇g∇gv∥L2(Ω) ≤ C(|v|H2(Ω) + |v|H1(Ω)) ≤ C∥v∥H2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Thus, ∥v∥2,h,�g ≤ C � 1 + max T h−1 T ∥gh − g∥L∞(T) + max T |gh − g|W 1,∞(T) � ∥v∥H2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (41) Combining (39), (40), and (41) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' At this point, we have finished proving part (i) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Indeed, in dimension N = 2, ah vanishes, so we can write ��⟨(Rω)dist(gh) − (Rω)(g), v⟩V ′,V �� ≤ � 1 0 |bh(�g(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v)| dt and apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='7 to deduce (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' To prove part (ii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1, we suppose that N ≥ 3 and that suph>0 maxT∈T N h |gh|W 2,∞(T) < ∞, and we proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Recall that ah(�g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v) = � T � T ⟨G(�g), σ⟩�gvωT (�g)+ ˚ � F � F � �II(�g)�F, σ|F � �g vωF(�g)− ˚ � S � S ⟨ΘS(�g)�g|S, σ|S⟩�gvωS(�g), (42) 24 where have made all dependencies on the metric explicit in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We will bound each of the three terms above, beginning with the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Throughout what follows, we continue to denote σ = gh − g, and we let v be an arbitrary member of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We have ����� � T � T ⟨G(�g), σ⟩�g vωT (�g) ����� ≤ C∥gh − g∥L2(Ω)∥v∥L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Since we are now assuming that suph>0 maxT∈T N h ∥gh∥W 2,∞(T) < ∞, the Einstein tensor associated with �g satisfies ∥G(�g)∥L∞(T) ≤ C for every h ≤ h0, every t ∈ [0, 1], and every T ∈ T N h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' It follows that ���� � T ⟨G(�g), σ⟩�g vωT (�g) ���� ≤ ∥G(�g)∥L∞(T,�g)∥σ∥L2(T,�g)∥v∥L2(T,�g) ≤ C∥G(�g)∥L∞(T)∥σ∥L2(T)∥v∥L2(T) ≤ C∥σ∥L2(T)∥v∥L2(T) = C∥gh − g∥L2(T)∥v∥L2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Summing over all T ∈ T N h completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We have ����� ˚ � F � F � �II(�g)�F, σ|F � �g vωF(�g) ����� ≤ C max T � h−1 T ∥gh − g∥W 1,∞(T) � × �� T ∥gh − g∥2 L2(T) + h2 T |gh − g|2 H1(T) �1/2 �� T ∥v∥2 L2(T) + h2 T |v|2 H1(T) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Consider an interior (N − 1)-simplex F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' By applying a Euclidean rotation and translation to the coordinates, we may assume without loss of generality that F lies in the plane xN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In these coordinates, the second fundamental form associated with �g is given by IIij(�g) = −�g(n�g, ∇�g,eiej) = −�g(n�g, �Γk ijek) = −nℓ �g�gℓk�Γk ij, i, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , N − 1, where e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , eN are the Euclidean coordinate basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Since n�g = �g−1n/ � nT �g−1n and n points in the xN direction, we get IIij(�g) = − 1 � nT �g−1n �ΓN ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The jump in this quantity across F can be computed using the identity �ab� = �a�{b} + {a}�b�, where {·} denotes the average across F, giving −�IIij(�g)� = � 1 � nT �g−1n � � �ΓN ij � + � 1 � nT �g−1n � � �ΓN ij � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 25 In view of (36), we have ����� � 1 � nT �g−1n ������ L∞(F ) ≤ C ∥��g�∥L∞(F ) ≤ C ∥�gh − g�∥L∞(F ) ≤ C � ∥gh − g∥L∞(T1) + ∥gh − g∥L∞(T2) � , where T1 and T2 are the two N-simplices that share the face F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Here, we used the fact that �g = g + t(gh − g) and g is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Similarly, we have ��� � �ΓN ij ���� L∞(F ) ≤ C∥��g�∥W 1,∞(F ) ≤ C∥�gh − g�∥W 1,∞(F ) ≤ C � ∥gh − g∥W 1,∞(T1) + ∥gh − g∥W 1,∞(T2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (43) Thus, ∥�II(�g)�∥L∞(F ) ≤ C � ∥gh − g∥W 1,∞(T1) + ∥gh − g∥W 1,∞(T2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' From this it follows easily that the same bound holds, possibly with a larger constant C, for the trace-reversed tensor II(�g) = II(�g) − H(�g)�g: ∥�II(�g)�∥L∞(F ) ≤ C � ∥gh − g∥W 1,∞(T1) + ∥gh − g∥W 1,∞(T2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' It follows that ���� � F � �II(�g)�F , σ|F � �g vωF (�g) ���� ≤ ∥�II(�g)�∥L∞(F,�g)∥σ|F ∥L2(F,�g)∥v∥L2(F,�g) ≤ C∥�II(�g)�∥L∞(F )∥σ|F ∥L2(F )∥v∥L2(F ) ≤ C � 2 � i=1 ∥gh − g∥W 1,∞(Ti) � � h−1 T1 ∥σ∥2 L2(T1) + hT1|σ|2 H1(T1) �1/2 � h−1 T1 ∥v∥2 L2(T1) + hT1|v|2 H1(T1) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' By the shape-regularity of Th, we have C−1 ≤ hT1/hT2 ≤ C for some constant C independent of h and F, so ����� ˚ � F � F � �II(�g)�F, σ|F � �g vωF(�g) ����� ≤ C max T � h−1 T ∥gh − g∥W 1,∞(T) � × �� T ∥gh − g∥2 L2(T) + h2 T |gh − g|2 H1(T) �1/2 �� T ∥v∥2 L2(T) + h2 T |v|2 H1(T) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If gh is piecewise constant, then in (43) we have the sharper bound ∥�gh − g�∥W 1,∞(F ) = ∥�gh − g�∥L∞(F ) ≤ C � ∥gh − g∥L∞(T1) + ∥gh − g∥L∞(T2) � 26 because ∂gh ∂xi = 0 and ∂g ∂xi is continuous for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This implies that for piecewise constant gh, we can replace ∥gh − g∥W 1,∞(T) by ∥gh − g∥L∞(T) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='9, yielding ����� ˚ � F � F � �II(�g)�F, σ|F � �g vωF(�g) ����� ≤ C max T � h−1 T ∥gh − g∥L∞(T) � × �� T ∥gh − g∥2 L2(T) + h2 T |gh − g|2 H1(T) �1/2 �� T ∥v∥2 L2(T) + h2 T |v|2 H1(T) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Now we turn our attention toward the third integral in (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In preparation for this, we will first use the shape-regularity assumption to show that the dihedral angles of every N-simplex in Th (measured in the Euclidean metric) are uniformly bounded above and below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' There exist constants θmin, θmax ∈ (0, π) such that for every h > 0 and every T ∈ T N h , the dihedral angles in T (measured in the Euclidean metric) all lie between θmin and θmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This fact is proved in dimension N = 3 in [18, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We generalize their proof to dimension N ≥ 3 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Given N + 1 points x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , xN+1 in general position in RN, let T = [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , xN+1] denote the N-simplex with vertices x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , xN+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Consider two faces F1 = [x1, x3, x4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , xN+1] and F2 = [x2, x3, x4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , xN+1] that intersect along the (N − 2)- dimensional subsimplex S = [x3, x4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , xN+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Throughout what follows, we work in the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let A be the orthogonal projection of x1 onto the (N−1)-dimensional hyperplane containing F2, and let B be the orthogonal projection of x1 onto the (N −2)-dimensional hyperplane containing S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Observe that both [x1, A] and [x1, B] are orthogonal to S, since S ⊂ F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Thus, the triangle [x1, A, B] is orthogonal to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This triangle is a right triangle with hypotenuse [x1, B], so the dihedral angle θST along S satisfies sin θST = |[x1, A]| |[x1, B]|, where | · | denotes the Euclidean volume (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' length in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Obviously, |[x1, B]| is bounded above by hT , the diameter of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In addition, |[x1, A]| is bounded from below by 2 times ρT , the inradius of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' To see why, we generalize the argument in [18, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='3], bearing in mind that our definition of ρT differs from theirs by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Consider the inscribed (N − 1)-sphere in T, whose center C lies at a distance ρT from F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let D be the point where this inscribed sphere touches F2, and let E be the point diametrically opposite to D on this sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The line segment [D, E] is orthogonal to F2, so the volume of the N-simplex T ′ = [E, x2, x3, x4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , xN+1] satisfies |T ′| = 1 N |[D, E]||F2| = 2ρT N |F2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Since T ′ ⊂ T, we have |T ′| ≤ |T| = 1 N |[x1, A]||F2|, so 2ρT ≤ |[x1, A]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Thus, sin θST ≥ 2ρT hT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The result follows from this bound and the shape-regularity of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 27 Next we show that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='11 remains valid when one measures angles with g rather than the Euclidean metric δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Upon reducing the value of h0 if necessary, there exist constants θmin,g, θmax,g ∈ (0, π) such that for every h ≤ h0, every T ∈ T N h , every (N − 2)-simplex S ⊂ ∂T, and every point p ∈ S, the dihedral angle in T at p (measured by g) lies between θmin,g and θmax,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If there were no such lower bound θmin,g > 0, then there would exist a sequence of N- simplices T1 ∈ Th1, T2 ∈ Th2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' with faces F (1) 1 , F (2) 1 ⊂ T1, F (1) 2 , F (2) 2 ⊂ T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' and points p1 ∈ F (1) 1 ∩ F (2) 1 , p2 ∈ F (1) 2 ∩ F (2) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' such that ∠ g|Ti(pi)(F (1) i , F (2) i ) → 0 as i → ∞, where ∠g(X, Y ) denotes the angle between X and Y as measured by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Using the compactness of the Grassmannian, this implies that, after extracting a subsequence which we do not relabel, ∠δ(F (1) i , F (2) i ) → 0, where ∠δ(X, Y ) denotes the angle between X and Y as measured by the Euclidean metric δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This contradicts the assumed positive lower bound on the Euclidean dihedral angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The existence of an upper bound θmax,g < π is proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Now we are ready to estimate the third integral in (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We have ����� ˚ � S � S ⟨ΘS(�g) �g|S , σ|S⟩�g vωS(�g) ����� ≤ C � max T h−2 T ∥gh − g∥L∞(T) � �� T ∥gh − g∥2 L2(T) + h2 T |gh − g|2 H1(T) + h4 T |gh − g|2 H2(T) �1/2 × �� T ∥v∥2 L2(T) + h2 T |v|2 H1(T) + h4 T |v|2 H2(T) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Fix an interior (N − 2)-simplex S and an N-simplex T containing S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' At any point p along S, we have cos θST(g) − cos θST (�g) = �g(n(1) �g , n(2) �g ) − g(n(1) g , n(2) g ) = �g(n(1) �g − n(1) g , n(2) �g − n(2) g ) + �g(n(1) �g − n(1) g , n(2) g ) + �g(n(1) g , n(2) �g − n(2) g ) + �g(n(1) g , n(2) g ) − g(n(1) g , n(2) g ), where n(1) g and n(2) g are suitably oriented unit normal vectors (with respect to g|T ) to the two faces of T containing S, and similarly for n(1) �g and n(2) �g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='6, we see that at the point p, | cos θST(�g) − cos θST(g)| ≤ C|�g − g| ≤ C|gh − g| for all h sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Since there are constants θmin,g, θmax,g ∈ (0, π) such that θmin,g ≤ θST(g) ≤ θmax,g, we get |θST (�g) − θST(g)| ≤ C|gh − g| ≤ C∥gh − g∥L∞(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 28 Summing over T ⊃ S and noting that � T⊃S θST (g) = 2π, we get |ΘS(�g)| = |ΘS(�g) − ΘS(g)| ≤ � T⊃S |θST(�g) − θST(g)| ≤ C � T⊃S ∥gh − g∥L∞(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (44) Now we are almost ready to estimate the integral � S ⟨ΘS(�g) �g|S , σ|S⟩�g vωS(�g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We first note that ∥v∥2 L2(S) ≤ C � h−2 T ∥v∥2 L2(T) + |v|2 H1(T) + h2 T |v|2 H2(T) � , which can be proved using a codimension-2 trace inequality and a scaling argument, or by applying the codimension-1 trace inequality (38) twice (to v rather than dv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , Tm are the N-simplices that share the (N − 2)-simplex S, then we have ���� � S ⟨ΘS(�g) �g|S , σ|S⟩�g vωS(�g) ���� ≤ C∥ΘS(�g)∥L∞(S,�g)∥σ|S∥L2(S,�g)∥v∥L2(S,�g) ≤ C∥ΘS(�g)∥L∞(S)∥σ|S∥L2(S)∥v∥L2(S) ≤ C � m � i=1 ∥gh − g∥L∞(Ti) � � h−2 T1 ∥σ∥2 L2(T1) + |σ|2 H1(T1) + h2 T1|σ|2 H2(T1) �1/2 × � h−2 T1 ∥v∥2 L2(T1) + |v|2 H1(T1) + h2 T1|v|2 H2(T1) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The proof is completed by summing over all interior (N − 2)-simplices S and substituting σ = gh − g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Collecting our results, we can state a bound on the bilinear form ah(�g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' ·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For every h ≤ h0, every t ∈ [0, 1], and every v ∈ V , we have (with σ = gh −g), |ah(�g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v)| ≤ C � 1 + max T h−2 T ∥gh − g∥L∞(T) + max T h−1 T |gh − g|W 1,∞(T) � × �� T ∥gh − g∥2 L2(T) + h2 T |gh − g|2 H1(T) + h4 T |gh − g|2 H2(T) �1/2 × �� T ∥v∥2 L2(T) + h2 T |v|2 H1(T) + h4 T |v|2 H2(T) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Combine Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='8, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='9, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Upon combining Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='7 with Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='14, we see that ∥(Rω)dist(gh) − (Rω)(g)∥H−2(Ω) ≤ C � 1 + max T h−2 T ∥gh − g∥L∞(T) + max T h−1 T |gh − g|W 1,∞(T) � × � ∥gh − g∥2 L2(Ω) + � T h2 T |gh − g|2 H1(T) + � T h4 T |gh − g|2 H2(T) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 29 This completes the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='3 then follows from (27) and the bounds ∥gh − g∥L2(Ω) ≤ |Ω|1/2−1/p∥gh − g∥Lp(Ω), �� T h2 T |gh − g|2 H1(T) �1/2 ≤ |Ω|1/2−1/p �� T hp T |gh − g|p W 1,p(T) �1/p , �� T h4 T |gh − g|2 H2(T) �1/2 ≤ |Ω|1/2−1/p �� T h2p T |gh − g|p W 2,p(T) �1/p , which hold for all p ∈ [2, ∞] (with the obvious modifications for p = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Notice that the analysis above yields |bh(�g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' σ, v)| = O(hr+1), (by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='7), (45) ����� � T � T ⟨G(�g), σ⟩�g vωT (�g) ����� = O(hr+1), (by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='8), (46) ����� ˚ � F � F � �II(�g)�F , σ|F � �g vωF (�g) ����� = � O(h), if r = 0, O(h2r), if r ≥ 1, (by Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='10), (by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='9), (47) ����� ˚ � S � S ⟨ΘS(�g) �g|S , σ|S⟩�g vωS(�g) ����� = O(h2r), (by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='13) (48) for any optimal-order interpolant gh of g having degree r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Bearing in mind that (46-48) vanish when N = 2, we see that the above estimates lead to an optimal error estimate ∥(Rω)dist(gh) − (Rω)(g)∥H−2(Ω) = O(hr+1) in all cases except when N ≥ 3 and r = 0, where we obtain ∥(Rω)dist(gh)− (Rω)(g)∥H−2(Ω) = O(1) because of (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Numerical experiments suggest that these analytical re- sults are sharp for a general optimal-order interpolant, whereas for the canonical interpolant the estimate (48) improves to O(h2(r+1)), yielding ∥(Rω)dist(gh) − (Rω)(g)∥H−2(Ω) = O(h) when r = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 5 Numerical examples In this section we present numerical experiments in dimension N = 2, 3 to illustrate the predicted convergence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The examples were performed in the open source finite element library NGSolve1 [24, 25], where the Regge finite elements are available for arbitrary polynomial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We construct an optimal-order interpolant gh of a given metric tensor g as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' On each element T, the local L2 best-approximation ¯gh|T of g|T is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Then the tangential-tangential degrees of freedom shared by two or more neighboring elements are averaged to obtain a globally tangential-tangential continuous interpolant gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We verify in Appendix A that this interpolant is an optimal-order interpolant in the sense of Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='4 on shape-regular, quasi-uniform triangulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' To compute the H−2(Ω)-norm of the error f := (Rω)dist(gh) − (Rω)(g) we make use of the fact that ∥f∥H−2(Ω) is equivalent to ∥u∥H2(Ω), where u ∈ H2 0(Ω) solves the biharmonic equation ∆2u = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This equation will be solved numerically using the (Euclidean) Hellan–Herrmann–Johnson method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' To prevent the discretization error from spoiling the real error, we use for uh two polynomial orders more than for gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 1www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='ngsolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='org 30 We consider in dimension N = 2 the numerical example proposed in [16], where on the square Ω = (−1, 1)2 the smooth Riemannian metric tensor g(x, y) := � 1 + (∂f ∂x)2 ∂f ∂x ∂f ∂y ∂f ∂x ∂f ∂y 1 + (∂f ∂y )2 � with f(x, y) := 1 2(x2 + y2) − 1 12(x4 + y4) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This metric corresponds to the surface induced by the embedding � x, y � �→ � x, y, f(x, y) � , and its exact scalar curvature is given by R(g)(x, y) = 162(1 − x2)(1 − y2) (9 + x2(x2 − 3)2 + y2(y2 − 3)2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For a three-dimensional example we consider the cube Ω = (−1, 1)3 and the Riemannian metric tensor induced by the embedding � x, y, z � �→ � x, y, z, f(x, y, z) � , where f(x, y, z) := 1 2(x2 + y2 + z2) − 1 12(x4 + y4 + z4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The scalar curvature is R(g)(x, y, z) = 18 � (1 − x2)(1 − y2)(9 + q(z)) + (1 − y2)(1 − z2)(9 + q(x)) + (1 − z2)(1 − x2)(9 + q(y)) � (9 + q(x) + q(y) + q(z))2 , where q(x) = x2(x2 − 3)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We start with a structured mesh consisting of 2·22k triangles and 6·23k tetrahedra, respectively, in two and three dimensions with ˜h = maxT hT = √ N 21−k (and minimal edge length 21−k) for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' To avoid possible superconvergence due to mesh symmetries, we perturb each component of the inner mesh vertices by a random number drawn from a uniform distribution in the range [−˜h 2−(2N+1)/2, ˜h 2−(2N+1)/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' As depicted in Figure 1 (left) and listed in Table 1, linear convergence is observed when N = 2 and gh has polynomial degree r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This is consistent with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For r = 1 and r = 2, higher convergence rates are obtained as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In the three-dimensional case, the same convergence rates as for N = 2 are obtained, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Figure 1 (right) and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' This indicates that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1(ii) is sharp for r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For r = 0 we observe numerically linear convergence, which is better than predicted by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' However, further investigation suggests that the observed linear convergence for r = 0 is pre-asymptotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Indeed, to test if (48) is sharp, we compute the H−2(Ω)-norm of the linear functional v �→ � 1 0 ˚ � S � S ⟨ΘS(�g(t)) �g(t)|S , σ|S⟩�g(t) vωS(�g(t)) dt, (49) where we approximate the parameter integral by a Gauss quadrature of order seven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' As depicted in Figure 2, the norm of this functional for the optimal-order interpolant gh with r = 0 stagnates at about 4·10−4, which is below the overall error of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='296·10−3 for the finest grid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' There- fore, the lack of convergence predicted by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1(ii) is not yet visible in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' For r = 1, 2 the proven rate of O(h2r) for (49) (see (48)) is clearly obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Interestingly, using the canonical interpolant appears to increase the convergence rate of (49) to O(h2(r+1)) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' an increase of two orders), as observed in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Thus, it appears that the canonical interpolant achieves conver- gence in the lowest-order case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We intend to study this superconvergence phenomenon exhibited by the canonical interpolant in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Acknowledgments We thank Yasha Berchenko-Kogan for many helpful discussions, especially about the mean curva- ture term in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We also thank Snorre Christiansen for pointing out the link with the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='10−8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='10−6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='ndof ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='r = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='r = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='r = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='O(h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='O(h2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='O(h3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='107 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='108 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='10−6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='10−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='ndof ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='r = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='r = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='r = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='O(h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='O(h2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='O(h3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='Figure 1: Convergence of the distributional scalar curvature in the H−2(Ω)-norm for N = 2 (left) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='and N = 3 (right) with respect to the number of degrees of freedom (ndof) of gh for r = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 101 102 103 104 105 106 107 10−12 10−10 10−8 10−6 10−4 10−2 ndof H−2(Ω)-norm of (49) r = 0 r = 1 r = 2 r = 0 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' r = 1 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' r = 2 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' O(h2) O(h4) O(h6) Figure 2: Convergence of (49) in the H−2(Ω)-norm with respect to number of degrees of freedom (ndof) for an optimal-order interpolant and the canonical interpolant (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=') for r = 0, 1, 2 in dimension N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' r = 0 r = 1 r = 2 h Error Order Error Order Error Order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='828 · 10−0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='534 · 10−0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='584 · 10−1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='609 · 10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='417 · 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='251 · 10−1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='260 · 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='198 · 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='237 · 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='945 · 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='23 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='220 · 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='336 · 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='57 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='434 · 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='41 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='457 · 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='181 · 10−3 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='335 · 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='98 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='689 · 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='99 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='205 · 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='280 · 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='777 · 10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='720 · 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='364 · 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='213 · 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='615 · 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='91 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='189 · 10−5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='504 · 10−6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='08 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='028 · 10−7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='97 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='784 · 10−8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='1 Table 1: Same as Figure 1 (left), but in tabular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 32 r = 0 r = 1 r = 2 h Error Order Error Order Error Order 3.' metadata={'source': 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10−2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='869 · 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='215 · 10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='132 · 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='62 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='152 · 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='51 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='838 · 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='37 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='733 · 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='296 · 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='359 · 10−1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='613 · 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='912 · 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='27 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='633 · 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='83 2.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='133 · 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='286 · 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='19 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='342 · 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='97 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='753 · 10−5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='261 · 10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='604 · 10−6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='03 Table 2: Same as Figure 1 (right), but in tabular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Israel formalism mentioned in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' EG was supported by NSF grant DMS-2012427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' MN acknowledges support by the Austrian Science Fund (FWF) project F 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' A Optimal-order interpolation via averaging Below we verify that the interpolant described in Section 5 is an optimal-order interpolant in the sense of Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='4, assuming that {Th}h>0 is shape-regular and quasi-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Recall that quasi- uniformity means that maxT∈T N h h/hT is bounded above by a constant independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In what follows, the letter C may depend on this constant as well as on the parameters N, hT /ρT , r, s, and t appearing below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let ℓ(1), ℓ(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , ℓ(M) denote the canonical degrees of freedom for the Regge finite element space of degree r ≥ 0 on Th [21, Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='4b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Each linear functional ℓ(i) is associated with a simplex D ∈ T k h of dimension k ≥ 1 in the following sense: ℓ(i) sends a symmetric (0, 2)-tensor field g to the integral of g|D against a (symmetric tensor-valued) polynomial of degree ≤ r − k + 1 over D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We enumerate these degrees of freedom with a local numbering system as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' On a given N-simplex T ∈ T N h , the degrees of freedom associated with subsimplices of T are denoted ℓT 1 , ℓT 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , ℓT MT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' If T, T ′ ∈ T N h are two N-simplices with nonempty intersection, then it may happen that ℓT i and ℓT ′ j coincide for some and i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We let S(i, T) denote the set of all pairs (j, T ′) for which ℓT i and ℓT ′ j coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' With the above local numbering system, let ψT 1 , ψT 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' , ψT MT denote the basis for the degree-r Regge finite element space that is dual to the above degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' That is, ℓT i (ψT ′ j ) = � 1, if (j, T ′) ∈ S(i, T), 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let us assume that the degrees of freedom and basis functions above are first defined on a reference simplex and then transported to T via an affine transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' A scaling argument shows that [21, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='11] ∥ψT i ∥Lp(T) ≤ ChN/p−2 T (50) and |ℓT i (g)| ≤ Ch−N/p+2 T ∥g∥Lp(T) (51) for all g in the domain of ℓT i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Note that the −2 and the +2 appearing in the exponents above arise because of the way that pullbacks of (0, 2)-tensor fields behave under affine transformations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' see [21, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 33 Let g be a symmetric (0, 2)-tensor field possessing W s,p(Ω)-regularity for every p ∈ [1, ∞] and every s > (N − 1)/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The canonical interpolation operator Jh onto the Regge finite element space is defined elementwise by Jhg|T = J T h (g|T ) = MT � i=1 ℓT i (g)ψT i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Let ¯gh denote the elementwise L2-projection of g onto the space of discontinuous piecewise polynomial symmetric (0, 2)-tensor fields of degree at most r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Since Jh is a projector, we have ¯gh|T = J T h ( ¯gh|T ) = MT � i=1 ℓT i (¯gh)ψT i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The interpolant discussed in Section 5 is defined by gh|T = MT � i=1 \uf8eb \uf8ed 1 |S(i, T)| � (j,T ′)∈S(i,T) ℓT ′ j (¯gh) \uf8f6 \uf8f8 ψT i , where |S(i, T)| denotes the cardinality of S(i, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' To analyze the error gh − g, let p ∈ [1, ∞], s ∈ ((N − 1)/p, r + 1], and t ∈ [0, s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' We have |gh − g|W t,p(T) ≤ |gh − Jhg|W t,p(T) + |Jhg − g|W t,p(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' The second term satisfies [21, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='5] |Jhg − g|W t,p(T) ≤ Chs−t T |g|W s,p(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (52) To bound the first term, we use the fact that ℓT i (g) = 1 |S(i, T)| � (j,T ′)∈S(i,T) ℓT ′ j (g) to write (gh − Jhg)|T = MT � i=1 1 |S(i, T)| � (j,T ′)∈S(i,T) ℓT ′ j (¯gh − g)ψT i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Using an inverse estimate, (50), (51), and a standard error estimate [14, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='135] for the elementwise L2-projector, we obtain |gh − Jhg|W t,p(T) ≤ Ch−t T ∥gh − Jhg∥Lp(T) ≤ Ch−t T � T ′:T ′∩T̸=∅ h−N/p+2 T ′ ∥¯gh − g∥Lp(T ′)hN/p−2 T ≤ Ch−t T � T ′:T ′∩T̸=∅ ∥¯gh − g∥Lp(T ′) ≤ Ch−t T � T ′:T ′∩T̸=∅ hs T ′|g|W s,p(T ′) ≤ Chs−t T � T ′:T ′∩T̸=∅ |g|W s,p(T ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' (53) Here, we have repeatedly used the fact that the ratio hT /hT ′ is bounded uniformly above and below by positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Combining (52) and (53) shows that the error gh − g satisfies (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 34 References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Arnold and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' Brezzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' “Mixed and nonconforming finite element methods: implementa- tion, postprocessing and error 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 227– 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' York Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' “Role of conformal three-geometry in the dynamics of gravitation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' In: Physical Review Letters 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content='16 (1972), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 1082–1085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} +page_content=' 36' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NA0T4oBgHgl3EQfOP_L/content/2301.02159v1.pdf'} diff --git a/7NE1T4oBgHgl3EQfBgKg/content/tmp_files/2301.02853v1.pdf.txt b/7NE1T4oBgHgl3EQfBgKg/content/tmp_files/2301.02853v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..456c1ed256faad0999ff06de6061667e1cdf0d48 --- /dev/null +++ b/7NE1T4oBgHgl3EQfBgKg/content/tmp_files/2301.02853v1.pdf.txt @@ -0,0 +1,1502 @@ +Using a Penalized Likelihood to Detect Mortality +Deceleration +Silvio C. Patricio*1 and Trifon I. Missov1 +1The Interdisciplinary Centre on Population Dynamics, University of Southern Denmark +Abstract +We propose a novel method to detect deceleration in mortality patterns. For a gamma- +Gompertz frailty model, we suggest maximizing a penalized likelihood in a Bayesian setting +as an alternative to traditional likelihood inference and hypothesis testing. We compare the +performance of the two methods on simulated and real mortality data. +Keywords: Gompertz model; gamma-Gompertz model, mortality deceleration; penalized +likelihood function; maximum a posteriori probability. +1 +Introduction +Human death-rate patterns are astoundingly log-linear over a wide range of adult ages. The +Gompertz distribution (Gompertz, 1825) with an exponentially increasing hazard function cap- +tures this accurately. The theory of unobserved heterogeneity and the associated frailty model +(Vaupel et al., 1979) predicts a downward deviation at the oldest ages, to which only the most +robust individuals in the population survive. Detecting such a deceleration in real data is not +always successful (Gavrilova and Gavrilov, 2015; Newman, 2018), even though the vast major- +ity of studies indicate that death rates at older ages increase at lower rates and can even level +off (Curtsinger et al., 1992; Fukui et al., 1993, 1996; Carey et al., 1995; Khazaeli et al., 1998; +Gampe, 2010, 2021; Rootz´en and Zholud, 2017; Alvarez et al., 2021; Camarda, 2022; Belzile +et al., 2022). In a frailty model setting, testing for mortality deceleration is equivalent to testing +whether the non-negative frailty parameter is strictly positive. +Formally, denote by X a non-negative continuous random variable that describes individual +human lifespans (complete or after a given adult age). If X ∼ Gompertz(a, b), where a is the +mortality level at the initial age and b is the rate of aging, the associated hazard function (force +of mortality) at time x +µ(x) = lim +ε↓0 P(x ≤ X < x + ε|X ≥ x). +*silca@sam.sdu.dk +1 +arXiv:2301.02853v1 [stat.ME] 7 Jan 2023 + +is µ(x) = aebx . Vaupel et al. (1979) introduce a positive continuous random variable Z, called +frailty, that acts multiplicatively on µ(x) and captures one’s unobserved susceptibility to death. +The force of mortality for an individual with frailty Z = z is +µ(x | Z = z) = z µ(x) . +For a gamma-distributed frailty with E(Z) = 1 and VAR(Z) = σ2, the force of mortality of the +population, i.e., the marginal hazard is +¯µ(x) = +aebx +1 + σ2 a +b (ebx − 1) +(1) +(see Vaupel et al. (1979) and Vaupel and Missov (2014) for all technicalities). Note that the +variance of Z is often denoted by γ (e.g., in Vaupel and Missov, 2014) because it is also equal +to the squared coefficient of variation of the distribution of frailty among survivors to any age x. +If σ2 > 0, the force of mortality for the population ¯µ(x) starts deviating from the exponential +pattern with increasing x and reaches an asymptote b/σ2. When σ2 = 0, i.e., when there is +no unobserved heterogeneity, the model for the population reduces to the (Gompertz) model for +individuals with an exponentially increasing hazard function µ(x) = aebx. +Testing for mortality deceleration in this setting reduces to statistical testing whether σ2 = 0 +given the alternative σ2 > 0. The frailty parameter σ2 can take a value on the boundary of +the parameter space (σ2 = 0). This violates the standard underlying assumptions about the +asymptotic properties of likelihood-based inference and statistical hypothesis testing (see, for +example, B¨ohnstedt and Gampe, 2019). As a result, the asymptotic distribution of the maximum +likelihood estimator may not be Gaussian. +In this paper, we treat the problem of identifying whether σ2 > 0 or σ2 = 0 as a model +misspecification problem, i.e., we consider the gamma-Gompertz model when it is the Gompertz +model that actually holds. In this setting, we suggest subtracting a penalty from the log-likelihood +function. This penalty will be responsible for shrinking σ2 to zero when there is no heterogeneity, +as well as for adding a small bias to the Maximum Likelihood Estimator (MLE) when the effect +of unobserved heterogeneity is non-negligible. We carry out Monte Carlo simulation experiments +to evaluate the accuracy and precision of the estimates obtained by maximizing the likelihood +function, on the one hand, and the penalized likelihood function, on the other. +In Section 2, we formulate the model misspecification problem and introduce inference +methodology taking advantage of the maximum a posteriori probability (MAP). Then we carry +out a Monte Carlo simulation study to compare the performance of maximizing a standard and a +penalized likelihood. In Section 3, we compare the latter on mortality data for France, Japan and +the USA. Section 4 discusses the advantages and drawbacks of applying our method to detect +heterogeneity (deceleration) in mortality patterns. +2 +Methodology +Suppose X is a random sample with a cumulative distribution function G(x), and we fit the +incorrect family of densities {f(x; θ), θ ∈ Θ} to the data using MLE. The misspecified log- +likelihood is +2 + +ℓ(θ; X) = +n +� +i=1 +log f(Xi; θ). +Applying the law of large numbers, we get in the limit what the misspecified log-likelihood +function ℓ(θ; X) looks like for each θ ∈ Θ (see the right-hand side below): +1 +nℓ(θ; X) = 1 +n +n +� +i=1 +log f(Xi; θ) +a.s +−→ Eg (log f(X1; θ)) = +� +Im(X1) +log f(x; θ)dG(x) . +(2) +Assume there is no heterogeneity in the data (σ2 = 0), and we fit a gamma-Gompertz model. +In other words, we observe an exponential death-rate increase in the data, but we estimate a +model that implies a downward deviation from the exponential at the oldest ages. As shown in +(2), we will estimate σ2 close to but never equal to zero. +In this model setting, the standard technique is to estimate both the Gompertz and the gamma- +Gompertz models and compare their goodness of fit. However, minor changes in the data can +result in different models being selected, which can reduce prediction accuracy and lead to mis- +interpretations about the mortality deceleration and the mortality plateau. B¨ohnstedt and Gampe +(2019) derive the asymptotic distribution of the likelihood ratio test statistic to detect heterogene- +ity. Here, we would like to suggest an alternative that does not involve hypothesis testing. Using +the latter has been widely discussed and rethought in the Statistics community (Berk et al., 2010; +Head et al., 2015; Vidgen and Yasseri, 2016; Bruns and Ioannidis, 2016), especially in relation +to the arbitrary choice of the α-level (most often 0.1, 0.05, or 0.01) and sample size issues. +Maximum likelihood estimators, obtained by maximizing the log-likelihood function, often +have low bias and large variance. Estimation accuracy can sometimes be improved by shrinking +some parameters to zero (Tibshirani, 1996). The associated shrinkage estimator improves the +overall prediction accuracy at the expense of introducing a small bias to reduce the variance of +the parameters. This class of estimators is implicit in Bayesian inference and penalized likelihood +inference. Using shrinkage estimators is applied as an alternative to hypothesis testing. Lasso, +Ridge and Stein-type estimators are the most widely used examples of penalizing methods (see, +for example, Hastie et al., 2009). +2.1 +Inference +Let Dx be the number of deaths in a given age interval [x, x + 1) for x = 0, . . . , m, and Ex +denote the number of person-years lived in the same interval (see, for example Brillinger, 1986; +Macdonald et al., 2018). Define D = (D0, D1, . . . , Dm)⊤ and E = (E0, E1, . . . , Em)⊤. In +addition, let θ = (a, b, σ2)⊤ ∈ Θ be the parameter vector that characterizes the force of mortality +at age x of the gamma-Gompertz model given by (1). +Assume Dx are Poisson-distributed with E(Dx) = VAR(Dx) = µ(x; θ)Ex for x = 0, . . . , m +(Brillinger, 1986). Under this assumption, the log-likelihood function for θ = (a, b, σ2)⊤ is +given by +ℓ(θ) = ℓ(θ|D, E) = +m +� +x=0 +[Dx ln µ(x; θ)) − Ex µ(x; θ)] . +(3) +3 + +Maximizing ℓ(θ) with respect to θ = (a, b, σ2)⊤ yields the maximum-likelihood (ML) estimate +ˆθ. +Let us now define a penalized log-likelihood function as +ℓp(θ) = ℓ(θ) − p(σ2) , +(4) +where ℓ(θ) is the standard log-likelihood (3), while p(σ2) is a penalty function. The penalized +maximum-likelihood estimate is obtained by maximizing ℓp(θ) with respect to θ = (a, b, σ2)⊤. +For the problem addressed in this paper, the penalty function p(σ2) must be a non-decreasing +monotonic continuous function and lim +σ2↓0 p(σ2) > p(σ2) for all σ2 > 0. +In a Bayesian framework, maximizing (4) is equivalent to maximizing a posterior distribution +in a setting, in which e−p(σ2)/Cp, Cp := +� +Θ e−p(σ2)∇θ < ∞, is taken as a prior distribution of +θ. This procedure yields the maximum a posteriori probability (MAP) estimator. MAP is the +only Bayesian estimator that minimizes the expected canonical loss (Pereyra, 2019) and is widely +used in image and video processing (Greig et al., 1989; Afonso et al., 2010; Belekos et al., 2010). +As σ2 describes the variance of frailty at the starting age of analysis, the standard approach +would be to specify an inverse gamma prior distribution for it (Gelman et al., 1995). The inverse +gamma distribution is heavy-tailed and keeps probability mass further from zero than the gamma +distribution. In addition, while the inverse-gamma mode is always positive, the gamma mode +can also be zero (Llera and Beckmann, 2016). As we aim to test whether σ2 = 0 or σ2 > 0, we +will use the log-kernel of the gamma distribution to define the penalty function as +p(σ2) = λ +� +σ2 + ln σ2� +(5) +for some non-negative λ. When λ < 1, using (5) is equivalent to specifying a gamma prior +distribution for σ2 with parameters α = 1 − λ and β = λ. +When m → ∞, the effect of the penalty diminishes regardless of the size of λ. For human +life table data m is finite, thus λ ≥ 0 is a constant that controls the relative impact of the penalty +function on the estimates. When λ = 0, the penalty term has no effect, and maximizing the +penalized likelihood will produce the standard maximum likelihood estimates (MLE). However, +as λ → ∞, the impact of the penalty grows, and the maximum penalized likelihood estimates +for σ2 will approach zero, providing high precision, but low accuracy. +Choosing λ is sensible in a wide range of applications (Li et al., 2009; Bhattacharya and +McNicholas, 2014). Therefore, in accordance with the recommendations in Li et al. (2009), +we carry out a pilot simulation study, in which we find that choosing λ = 1 +2 provides similar +precision to the one by MLE when σ2 > 0, but better accuracy and precision when σ2 = 0 +(simulation results are presented in the next subsection). As a result, the final expression for the +penalized log-likelihood we propose is +ℓp(θ) = +m +� +x=0 +[Dx ln µ(x; θ) − Ex µ(x; θ)] − 1 +2 +� +ln σ2 + σ2� +. +(6) +From a Bayesian perspective, choosing λ = 1 +2 provides an informative prior distribution for +σ2. As for human populations we are likely to estimate σ2 < 1 (Missov, 2013), the specified +prior will provide for σ2 a distribution with a mode equal to zero, a median equal to 0.4549, and +4 + +0.00000 +0.00010 +0.00020 +0.00030 +−600 +−400 +−200 +0 +Parameter a +a +log−likelihood +MLE +MAP +a = 1e−04 +b = 0.1 +σ2 = 0.1 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +−8000 +−4000 +0 +Parameter b +b +log−likelihood +MLE +MAP +a = 1e−04 +b = 0.1 +σ2 = 0.1 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +−5.5 +−4.5 +−3.5 +Parameter σ2 +σ2 +log−likelihood +MLE +MAP +a = 1e−04 +b = 0.1 +σ2 = 0.1 +0.00000 +0.00010 +0.00020 +0.00030 +−800 +−600 +−400 +−200 +0 +Parameter a +a +log−likelihood +MLE +MAP +a = 1e−04 +b = 0.1 +σ2 = 0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +−8000 +−4000 +0 +Parameter b +b +log−likelihood +MLE +MAP +a = 1e−04 +b = 0.1 +σ2 = 0 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +−3.55 +−3.45 +−3.35 +Parameter σ2 +σ2 +log−likelihood +MLE +MAP +a = 1e−04 +b = 0.1 +σ2 = 0 +Figure 1: Plots of the profile log-likelihood and penalized log-likelihood functions of the param- +eters. In the first row we used synthetic data from a gamma-Gompertz model with parameters +a = 0.0001, b = 0.1 and σ2 = 0.1, in the second row we from a Gompertz model with parameters +a = 0.0001 and b = 0.1. +mean equal to 1. Furthermore, the prior provides a probability mass of 0.6826 in the interval +(0, 1]. +Figure 1 shows the log-likelihood and penalized log-likelihood functions for all parameters +when σ2 > 0 (first row) and σ2 = 0 (second row). When σ2 > 0, the penalty function affects +neither the shape of the log-likelihood, nor the location of its maximum. However, when σ2 = 0, +adding a penalty yields a higher maximum at 0. Moreover, when σ2 = 0, the first and second +derivatives of the penalized log-likelihood are higher than their respective counterparts of the +log-likelihood. As a result, derivative-based optimization methods may reach the maximum +point faster, and the estimator ˆσ2 may have a smaller variance. +2.2 +Monte Carlo simulations +We carry out Monte Carlo simulations to explore the performance of the MAP and ML methods +in estimating the gamma-Gompertz model parameters. We use the R software (Team et al., +2022) to maximize the log-likelihood and the penalized log-likelihood functions via the optim +function applying as a pre-step differential evolution (Storn and Price, 1997; Ardia et al., 2011). +The performance of the ML and MAP estimators are evaluated by calculating two measures: the +bias and the standard deviation. +We generate 10,000 random samples from this model for some parameter values (scenarios +with sample sizes of 2,000 and 5,000 were also considered, and are presented in the appendix). +From these samples, we generate life tables and use them to estimate model parameters via the +5 + +MAP and MLE methods. This process was repeated 2,000 times. In the presence of unobserved +heterogeneity, the true parameter values are a1 = 0.0001 and a2 = 0.00001 for a, b1 = 0.1 and +b2 = 0.15 for b, and σ2 +1 = 0.2 and σ2 +2 = 0.8 for σ2. When there is no heterogeneity (σ2 = 0), +the true parameter values are a1 = 0.0001, a2 = 0.0003 and a3 = 0.0005 for a, and b1 = 0.09, +b2 = 0.10 and b3 = 0.11 for b. +Table 1: Simulation results: gamma-Gompertz model and sample size 10,000. +There is heterogeneity +MLE estimator +MAP estimator +Bias +Standard deviation +Bias +Standard deviation +Parameter +a +b +σ2 +a +b +σ2 +a +b +σ2 +a +b +σ2 +(a1, b1, σ2 +1) +0.000053 +-0.000051 +-0.000223 +0.000051 +0.001499 +0.020721 +0.000055 +-0.000134 +-0.001626 +0.000052 +0.001502 +0.020791 +(a1, b1, σ2 +2) +0.000060 +-0.000292 +-0.007787 +0.000056 +0.001784 +0.035822 +0.000061 +-0.000354 +-0.009229 +0.000056 +0.001783 +0.035795 +(a1, b2, σ2 +1) +0.000077 +0.000131 +0.004431 +0.000056 +0.002181 +0.020569 +0.000080 +0.000015 +0.003096 +0.000057 +0.002186 +0.020631 +(a1, b2, σ2 +2) +0.000085 +-0.000262 +-0.003547 +0.000061 +0.002557 +0.034714 +0.000087 +-0.000348 +-0.004920 +0.000061 +0.002556 +0.034687 +(a2, b1, σ2 +1) +0.000007 +-0.000349 +-0.003349 +0.000008 +0.001315 +0.019464 +0.000008 +-0.000417 +-0.004597 +0.000008 +0.001318 +0.019523 +(a2, b1, σ2 +2) +0.000009 +-0.000635 +-0.013592 +0.000008 +0.001515 +0.032551 +0.000009 +-0.000683 +-0.014810 +0.000008 +0.001514 +0.032528 +(a2, b2, σ2 +1) +0.000009 +-0.000170 +0.002295 +0.000008 +0.001963 +0.019377 +0.000009 +-0.000268 +0.001078 +0.000008 +0.001966 +0.019430 +(a2, b2, σ2 +2) +0.000011 +-0.000650 +-0.007809 +0.000009 +0.002273 +0.032534 +0.000011 +-0.000721 +-0.009014 +0.000009 +0.002272 +0.032510 +There is no heterogeneity +MLE estimator +MAP estimator +Bias +Standard deviation +Bias +Standard deviation +Parameter +a +b +σ2 +a +b +σ2 +a +b +σ2(10−16) +a +b +σ2(10−15) +(a1, b1, σ2) +0.000005 +-0.000055 +0.000181 +0.000006 +0.000875 +0.008460 +0.000007 +-0.000239 +0.125942 +0.000006 +0.000791 +0.6308937 +(a1, b2, σ2) +0.000006 +-0.000050 +0.000222 +0.000006 +0.000968 +0.008907 +0.000007 +-0.000407 +0.060433 +0.000006 +0.000870 +0.1460452 +(a1, b3, σ2) +0.000006 +-0.000057 +0.000428 +0.000006 +0.001082 +0.009726 +0.000008 +-0.000305 +0.089035 +0.000006 +0.000978 +8.212895 +(a2, b1, σ2) +0.000011 +0.000138 +0.001864 +0.000016 +0.000913 +0.009010 +0.000016 +-0.000198 +0.004470 +0.000015 +0.000811 +0.294978 +(a2, b2, σ2) +0.000014 +0.000117 +0.001094 +0.000016 +0.001042 +0.009873 +0.000019 +-0.000264 +0.003661 +0.000015 +0.000859 +4.922260 +(a2, b3, σ2) +0.000015 +0.000164 +0.002204 +0.000016 +0.001150 +0.010369 +0.000022 +-0.000216 +0.007167 +0.000016 +0.001017 +3.551245 +(a3, b1, σ2) +0.000018 +0.000143 +0.001062 +0.000025 +0.000979 +0.009712 +0.000027 +-0.000138 +0.001124 +0.000025 +0.000876 +1.898573 +(a3, b2, σ2) +0.000024 +0.000076 +0.000505 +0.000025 +0.001057 +0.009721 +0.000030 +-0.000112 +0.001177 +0.000025 +0.000942 +8.114498 +(a3, b3, σ2) +0.000025 +0.000117 +0.001427 +0.000025 +0.001178 +0.009999 +0.000033 +-0.000171 +0.001415 +0.000024 +0.000990 +0.000025 +The simulation results are presented in Table 1. In the presence of unobserved heterogeneity, +both methods underestimate b and σ2. They also introduce a small positive bias to a, the one pro- +vided by ML estimator being slightly smaller. However, in general the ML and MAP estimators +perform equally well, with a similar bias and standard deviation. +In the absence of unobserved heterogeneity, the ML estimator provides again a smaller bias +for a and b than the MAP estimator. However, in this case, the MAP method estimates more +precisely the frailty parameter σ2, with a bias and a standard deviation close to zero (∝ 10−15). +The MAP estimator also provides a slight reduction in the standard deviation of parameter b. +By the Monte Carlo simulation we also calculate the proportion of trials in which MAP +estimates σ2 > 0 when the true values is σ2 = 0 (error type I), as well as the proportion of trials +in which MAP estimates σ2 = 0 when the true values is σ2 > 0 (error type II). Based on our +simulations, the type I errro equals 0.001502, while the type II error is 0.001126. +The Monte Carlo simulations show that using a penalizing likelihood function (6) is an alter- +native to hypothesis testing, the latter being dependent on the asymptotic distribution of the ML +estimator, sample size and the arbitrary choice of the α-level (B¨ohnstedt and Gampe, 2019). +3 +Performance of MAP and ML estimators on HMD data +In this section, we estimate the gamma-Gompertz model via ML and MAP using mortality data +from the Human Mortality Database (HMD, 2022). We take exposures and raw death counts for +6 + +the female population of France, Japan and the USA in the years 1960, 1980, 2000, and 2020, +after age 70. We apply again R (Team et al., 2022) to compute the ML and MAP estimates of +θ = (a, b, σ2)′ by using differential evolution. We use the mean squared error given by +MSE = 1 +n +m +� +x=0 +� +ln mx − ln ¯µ(x; ˆθ) +�2 +, +to assess the goodness of fit. +Table 2: Life expectancy: gamma Gompertz–Makeham model and ML estimates. +ML Estimates +MAP Estimates +Country +Year +a +b +σ2 +MSE +a +b +σ2 +MSE +France +1960 +0.003582 +0.107599 +0.016726 +0.100588 +0.003593 +0.107483 +0.015902 +0.100540 +1980 +0.002210 +0.112032 +0.003393 +0.045776 +0.002220 +0.111729 +0.003117 +0.046747 +2000 +0.001247 +0.117749 +0.000001 +0.073648 +0.001250 +0.117689 +0 +0.073262 +2020 +0.000957 +0.119494 +0.000002 +0.094243 +0.000960 +0.119416 +0 +0.093697 +Japan +1960 +0.004782 +0.105858 +0.039989 +0.011366 +0.004782 +0.105845 +0.039593 +0.011493 +1980 +0.002009 +0.117886 +0.015251 +0.053944 +0.002009 +0.117941 +0.015942 +0.053328 +2000 +0.001140 +0.115268 +0.000015 +0.064604 +0.001142 +0.115118 +0 +0.063728 +2020 +0.000575 +0.125814 +0.000233 +0.104944 +0.000574 +0.125870 +0.000225 +0.105597 +USA +1960 +0.004701 +0.095797 +0.032146 +0.111711 +0.004699 +0.095814 +0.032802 +0.110740 +1980 +0.003612 +0.093688 +0.000001 +0.054483 +0.003609 +0.093720 +0 +0.054652 +2000 +0.002712 +0.100566 +0.000003 +0.030967 +0.002714 +0.100540 +0 +0.030873 +2020 +0.002473 +0.101652 +0.000001 +0.022735 +0.002476 +0.101612 +0 +0.022618 +Table 2 shows the results of applying ML and MAP methods to the datasets described above. +The MAP estimator provides lower MSEs in 8 of the 12 datasets. When the standard ML method +estimates σ2 < 10−4, our novel method estimates σ2 = 0 and provides a smaller MSE. This +suggests that the MAP provides a slightly better fit to the data. Overall, MAP performs better +than ML when unobserved heterogeneity is not detected, and while for estimates of ˆσ2 > 0 ML +has a slight advantage. +The results from the real-data application back up the results from the Monte Carlo simula- +tions in Section 2. In the presence of unobserved heterogeneity, the MLE method provides the +most precise and accurate estimates. The MAP method, though, has just slightly lower precision. +On the other hand, in the absence of unobserved heterogeneity, the MAP provides smaller bias +and variance in its estimates compared to MLE. +3.1 +Examples when MAP and ML estimators yield different outcomes +Using MAP and ML estimators does not always lead to the same statistical inference. One of +them can detect heterogeneity in cases when the other does not. We will illustrate this on HMD +data for the Japanese female population in 2009 and the French female population born in 1848, +ages 70+. To assess the goodness of fit, we will use again MSE. +For Japanese females in 2009, ML yields estimates ˆθMLE = (0.006359, 0.133805, 0.070513)′ +with standard errors SE(a) = 0.000188, SE(b) = 0.002263 and SE(σ2) = 0.021156. The 95% +confidence interval for σ2 is (0.029047, 0.111978) indicating stasitically significant unobserved +7 + +heterogeneity, i.e., the existence of mortality deceleration. On the other hand, the MAP method +estimates ˆθMAP = (0.006966, 0.125440, 0)′, indicating the absence of unobserved heterogeneity. +Comparing the goodness of fit of both methods speaks in favor of the MAP outcome: MAP’s +MSE is by 37% lower than ML’s LSE (0.018691 for MAP vs 0.029958 for ML). It indicates that +unobserved heterogeneity is negligible and that the gamma-Gompertz model is misspecified. +70 +80 +90 +100 +110 +−5 +−4 +−3 +−2 +−1 +0 +Age +log−force of mortality +Mortality rate +MLE +MAP +Japan +70 +75 +80 +85 +90 +95 +100 +−3.0 +−2.0 +−1.0 +0.0 +Age +log−force of mortality +Mortality rate +MLE +MAP +France +Figure 2: MAP and MLE estimates of the force of mortality for the Japanese population in 2009 +and the Swedish population born in 1881, after age 70 +The left panel of Figure 2 shows that both methods estimate a similar logarithmic force of +mortality at most ages. However, after age 100, the MLE deviates downward from the observed +logarithmic death rates. +The MAP also provides a better fit and different conclusion for the cohort of French females +born in 1848. While ML estimates ˆθMLE = (0.053748, 0.090552, 0.008604)′ with SE(a) = +0.000317, SE(b) = 0.001273, SE(σ2) = 0.007562 and provides an MSE equal 0.046222, +MAP estimates ˆθMAP = (0.053113, 0.094921, 0.036466)′ and provides MSE = 0.034226, i.e., +MAP’s MSE is by 26% smaller than ML’s MSE. +Furthermore, while the MAP estimate of σ2 suggests that there is non-negligible unobserved +heterogeneity, the ML estimate and standard error for σ2 indicates the opposite: the amount +of unobserved heterogeneity is not statistically significant. The right panel of Figure 2 shows +the difference between these estimates. MAP’s estimate shows a leveling-off in the force of +mortality, while the MLE shows a log-linear increase in the hazard function. +4 +Concluding remarks +B¨ohnstedt and Gampe (2019) introduced a formal procedure to identify whether σ2 > 0 or +σ2 = 0 in a hypothesis testing setting: they studied the asymptotic properties of the maximum +likelihood estimator and the likelihood ratio test (LRT) for H0 : σ2 = 0 vs. H1 : σ2 = 0 for +8 + +the gamma-Gompertz model. However, LRTs are based on the asymptotic distribution of the +maximum likelihood estimator, hence its convergence depends on the sample size. Moreover, +conclusions drawn from hypothesis tests are dependent on the arbitrary choice of the significance +level or p-value. +We suggest an alternative method by considering the problem as model misspecification. We +add a penalty function to the likelihood so that we make sure that ˆσ2 = 0 when there is no het- +erogeneity. We also present a Bayesian interpretation (MAP) to our method. We take advantage +of robust Monte Carlo simulations to measure the bias and standard deviation of the ML and +MAP methods in scenarios with and without unobserved heterogeneity. We also compare the +performance of both methods for estimating the gamma-Gompertz model parameters using ac- +tual mortality data from the Human Mortality Database. The two methods work almost equally +well, the ML having a slight advantage, in the presence of unobserved heterogeneity. However, +in the absence of the latter, the MAP method provides an estimate closer to 0 (ˆσ2 ≈ 10−20) and a +better fit to the model in comparison to ML. As a result, the method we propose here can be used +as an alternative to likelihood ratio testing for the gamma-Gompertz model with H0 : σ2 = 0 vs. +H1 : σ2 > 0. 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Frontiers in Physics, +4:6. +Appendices +Table 3: Simulation results: gamma-Gompertz model and sample size 2,000. +There is heterogeneity +MLE estimator +MAP estimator +Bias +Standard deviation +Bias +Standard deviation +Parameter +a +b +σ2 +a +b +σ2 +a +b +σ2 +a +b +σ2 +(a1, b1, σ2 +1) +0.000089 +-0.001172 +-0.020361 +0.000111 +0.003286 +0.047684 +0.000104 +-0.001709 +-0.029261 +0.000117 +0.003465 +0.051082 +(a1, b1, σ2 +2) +0.000110 +-0.002075 +-0.048634 +0.000128 +0.003972 +0.078621 +0.000118 +-0.002397 +-0.055984 +0.000129 +0.003970 +0.078467 +(a1, b2, σ2 +1) +0.000104 +-0.000906 +-0.008883 +0.000124 +0.004826 +0.047186 +0.000120 +-0.001603 +-0.016820 +0.000130 +0.005032 +0.049798 +(a1, b2, σ2 +2) +0.000124 +-0.001978 +-0.029154 +0.000139 +0.005738 +0.077069 +0.000133 +-0.002420 +-0.036104 +0.000140 +0.005733 +0.076901 +(a2, b1, σ2 +1) +0.000030 +-0.003810 +-0.046694 +0.000021 +0.003024 +0.045092 +0.000034 +-0.004407 +-0.057147 +0.000024 +0.003388 +0.052275 +(a2, b1, σ2 +2) +0.000038 +-0.005056 +-0.097144 +0.000023 +0.003338 +0.070146 +0.000039 +-0.005316 +-0.103429 +0.000024 +0.003333 +0.069973 +(a2, b2, σ2 +1) +0.000027 +-0.003998 +-0.030152 +0.000021 +0.004259 +0.043531 +0.000030 +-0.004673 +-0.038167 +0.000022 +0.004481 +0.046813 +(a2, b2, σ2 +2) +0.000032 +-0.005191 +-0.065676 +0.000025 +0.005158 +0.075375 +0.000034 +-0.005565 +-0.071836 +0.000025 +0.005146 +0.075141 +There is no heterogeneity +MLE estimator +MAP estimator +Bias +Standard deviation +Bias +Standard deviation +Parameter +a +b +σ2 +a +b +σ2 +a +b +σ2(10−12) +a +b +σ2(10−12) +(a1, b1, σ2) +0.000015 +-0.001225 +0.000002 +0.000013 +0.001615 +0.006546 +0.000016 +-0.001341 +0.021800 +0.000014 +0.001693 +0.000287 +(a1, b2, σ2) +0.000014 +-0.001224 +0.000002 +0.000013 +0.001833 +0.007389 +0.000016 +-0.001312 +0.018541 +0.000014 +0.001858 +0.004010 +(a1, b3, σ2) +0.000014 +-0.001323 +0.000002 +0.000013 +0.001998 +0.008857 +0.000015 +-0.001324 +0.024195 +0.000015 +0.002098 +0.001130 +(a2, b1, σ2) +0.000028 +-0.000591 +0.000003 +0.000031 +0.001703 +0.009242 +0.000031 +-0.000983 +0.000923 +0.000033 +0.001666 +0.000003 +(a2, b2, σ2) +0.000029 +-0.000837 +0.000003 +0.000031 +0.001857 +0.010440 +0.000031 +-0.000842 +0.000731 +0.000031 +0.001804 +0.000009 +(a2, b3, σ2) +0.000030 +-0.000810 +0.000004 +0.000031 +0.002067 +0.011604 +0.000035 +-0.001196 +0.000701 +0.000034 +0.002086 +0.000008 +(a3, b1, σ2) +0.000034 +-0.000398 +0.000003 +0.000046 +0.001739 +0.011618 +0.000037 +-0.000606 +0.000220 +0.000048 +0.001681 +0.000001 +(a3, b2, σ2) +0.000036 +-0.000503 +0.000005 +0.000049 +0.002062 +0.013431 +0.000040 +-0.000670 +0.000151 +0.000048 +0.001901 +0.000002 +(a3, b3, σ2) +0.000040 +-0.000238 +0.000008 +0.000050 +0.002247 +0.014168 +0.000045 +-0.000722 +0.000108 +0.000051 +0.002061 +0.000008 +Table 4: Simulation results: gamma-Gompertz model and sample size 5,000. +There is heterogeneity +MLE estimator +MAP estimator +Bias +Standard deviation +Bias +Standard deviation +Parameter +a +b +σ2 +a +b +σ2 +a +b +σ2 +a +b +σ2 +(a1, b1, σ2 +1) +0.000067 +-0.000379 +-0.004470 +0.000071 +0.002078 +0.029046 +0.000071 +-0.000552 +-0.007370 +0.000072 +0.002090 +0.029281 +(a1, b1, σ2 +2) +0.000070 +-0.000582 +-0.015178 +0.000077 +0.002506 +0.049888 +0.000074 +-0.000708 +-0.018078 +0.000077 +0.002504 +0.049808 +(a1, b2, σ2 +1) +0.000090 +-0.000273 +0.001254 +0.000077 +0.002969 +0.028061 +0.000095 +-0.000509 +-0.001483 +0.000078 +0.002982 +0.028253 +(a1, b2, σ2 +2) +0.000094 +-0.000514 +-0.007569 +0.000083 +0.003583 +0.048230 +0.000097 +-0.000687 +-0.010324 +0.000084 +0.003579 +0.048152 +(a2, b1, σ2 +1) +0.000014 +-0.001320 +-0.014568 +0.000011 +0.001786 +0.026808 +0.000014 +-0.001465 +-0.017208 +0.000011 +0.001795 +0.027016 +(a2, b1, σ2 +2) +0.000015 +-0.001650 +-0.033574 +0.000013 +0.002344 +0.049087 +0.000016 +-0.001748 +-0.036030 +0.000013 +0.002342 +0.049016 +(a2, b2, σ2 +1) +0.000014 +-0.001299 +-0.006177 +0.000012 +0.002706 +0.026681 +0.000015 +-0.001505 +-0.008713 +0.000012 +0.002717 +0.026863 +(a2, b2, σ2 +2) +0.000015 +-0.001571 +-0.019479 +0.000014 +0.003352 +0.046735 +0.000016 +-0.001714 +-0.021904 +0.000014 +0.003349 +0.046666 +There is no heterogeneity +MLE estimator +MAP estimator +Bias +Standard deviation +Bias +Standard deviation +Parameter +a +b +σ2 +a +b +σ2 +a +b +σ2(10−12) +a +b +σ2(10−12) +(a1, b1, σ2) +0.000008 +-0.000312 +0.000009 +0.000008 +0.001139 +0.006564 +0.000008 +-0.000441 +0.024410 +0.000009 +0.001142 +0.000058 +(a1, b2, σ2) +0.000008 +-0.000339 +0.000011 +0.000009 +0.001290 +0.007315 +0.000009 +-0.000497 +0.024494 +0.000009 +0.001259 +0.000179 +(a1, b3, σ2) +0.000008 +-0.000237 +0.000011 +0.000009 +0.001398 +0.007896 +0.000009 +-0.000483 +0.037635 +0.000009 +0.001417 +0.000985 +(a2, b1, σ2) +0.000015 +0.000055 +0.000823 +0.000021 +0.001212 +0.008106 +0.000016 +-0.000102 +0.001137 +0.000022 +0.001184 +0.000012 +(a2, b2, σ2) +0.000017 +0.000028 +0.000732 +0.000022 +0.001363 +0.009211 +0.000020 +-0.000228 +0.001954 +0.000023 +0.001319 +0.001225 +(a2, b3, σ2) +0.000019 +0.000122 +0.000182 +0.000022 +0.001473 +0.009371 +0.000021 +-0.000024 +0.002634 +0.000022 +0.001387 +0.000014 +(a3, b1, σ2) +0.000023 +0.000212 +0.001357 +0.000033 +0.001265 +0.009088 +0.000023 +-0.000068 +0.000266 +0.000033 +0.001200 +0.000001 +(a3, b2, σ2) +0.000025 +0.000198 +0.000845 +0.000034 +0.001391 +0.009982 +0.000030 +-0.000154 +0.000558 +0.000035 +0.001332 +0.000048 +(a3, b3, σ2) +0.000026 +0.000127 +0.000722 +0.000035 +0.001590 +0.011029 +0.000034 +-0.000151 +0.001303 +0.000035 +0.001464 +0.003727 +12 + diff --git a/7NE1T4oBgHgl3EQfBgKg/content/tmp_files/load_file.txt b/7NE1T4oBgHgl3EQfBgKg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..539face44d79fcd6cb73e45b059689b7c73a5d0c --- /dev/null +++ b/7NE1T4oBgHgl3EQfBgKg/content/tmp_files/load_file.txt @@ -0,0 +1,1298 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf,len=1297 +page_content='Using a Penalized Likelihood to Detect Mortality Deceleration Silvio C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Patricio*1 and Trifon I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Missov1 1The Interdisciplinary Centre on Population Dynamics, University of Southern Denmark Abstract We propose a novel method to detect deceleration in mortality patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' For a gamma- Gompertz frailty model, we suggest maximizing a penalized likelihood in a Bayesian setting as an alternative to traditional likelihood inference and hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' We compare the performance of the two methods on simulated and real mortality data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Keywords: Gompertz model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' gamma-Gompertz model, mortality deceleration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' penalized likelihood function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' maximum a posteriori probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' 1 Introduction Human death-rate patterns are astoundingly log-linear over a wide range of adult ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The Gompertz distribution (Gompertz, 1825) with an exponentially increasing hazard function cap- tures this accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The theory of unobserved heterogeneity and the associated frailty model (Vaupel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 1979) predicts a downward deviation at the oldest ages, to which only the most robust individuals in the population survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Detecting such a deceleration in real data is not always successful (Gavrilova and Gavrilov, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Newman, 2018), even though the vast major- ity of studies indicate that death rates at older ages increase at lower rates and can even level off (Curtsinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Fukui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 1993, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Carey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Khazaeli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Gampe, 2010, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Rootz´en and Zholud, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Alvarez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Camarda, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Belzile et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' In a frailty model setting, testing for mortality deceleration is equivalent to testing whether the non-negative frailty parameter is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Formally, denote by X a non-negative continuous random variable that describes individual human lifespans (complete or after a given adult age).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' If X ∼ Gompertz(a, b), where a is the mortality level at the initial age and b is the rate of aging, the associated hazard function (force of mortality) at time x µ(x) = lim ε↓0 P(x ≤ X < x + ε|X ≥ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' silca@sam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='dk 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='02853v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='ME] 7 Jan 2023 is µ(x) = aebx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Vaupel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' (1979) introduce a positive continuous random variable Z, called frailty, that acts multiplicatively on µ(x) and captures one’s unobserved susceptibility to death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The force of mortality for an individual with frailty Z = z is µ(x | Z = z) = z µ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' For a gamma-distributed frailty with E(Z) = 1 and VAR(Z) = σ2, the force of mortality of the population, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', the marginal hazard is ¯µ(x) = aebx 1 + σ2 a b (ebx − 1) (1) (see Vaupel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' (1979) and Vaupel and Missov (2014) for all technicalities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Note that the variance of Z is often denoted by γ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', in Vaupel and Missov, 2014) because it is also equal to the squared coefficient of variation of the distribution of frailty among survivors to any age x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' If σ2 > 0, the force of mortality for the population ¯µ(x) starts deviating from the exponential pattern with increasing x and reaches an asymptote b/σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' When σ2 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', when there is no unobserved heterogeneity, the model for the population reduces to the (Gompertz) model for individuals with an exponentially increasing hazard function µ(x) = aebx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Testing for mortality deceleration in this setting reduces to statistical testing whether σ2 = 0 given the alternative σ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The frailty parameter σ2 can take a value on the boundary of the parameter space (σ2 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' This violates the standard underlying assumptions about the asymptotic properties of likelihood-based inference and statistical hypothesis testing (see, for example, B¨ohnstedt and Gampe, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' As a result, the asymptotic distribution of the maximum likelihood estimator may not be Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' In this paper, we treat the problem of identifying whether σ2 > 0 or σ2 = 0 as a model misspecification problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', we consider the gamma-Gompertz model when it is the Gompertz model that actually holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' In this setting, we suggest subtracting a penalty from the log-likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' This penalty will be responsible for shrinking σ2 to zero when there is no heterogeneity, as well as for adding a small bias to the Maximum Likelihood Estimator (MLE) when the effect of unobserved heterogeneity is non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' We carry out Monte Carlo simulation experiments to evaluate the accuracy and precision of the estimates obtained by maximizing the likelihood function, on the one hand, and the penalized likelihood function, on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' In Section 2, we formulate the model misspecification problem and introduce inference methodology taking advantage of the maximum a posteriori probability (MAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Then we carry out a Monte Carlo simulation study to compare the performance of maximizing a standard and a penalized likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' In Section 3, we compare the latter on mortality data for France, Japan and the USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Section 4 discusses the advantages and drawbacks of applying our method to detect heterogeneity (deceleration) in mortality patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' 2 Methodology Suppose X is a random sample with a cumulative distribution function G(x), and we fit the incorrect family of densities {f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' θ), θ ∈ Θ} to the data using MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The misspecified log- likelihood is 2 ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' X) = n � i=1 log f(Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Applying the law of large numbers, we get in the limit what the misspecified log-likelihood function ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' X) looks like for each θ ∈ Θ (see the right-hand side below): 1 nℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' X) = 1 n n � i=1 log f(Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' θ) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='s −→ Eg (log f(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' θ)) = � Im(X1) log f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' θ)dG(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' (2) Assume there is no heterogeneity in the data (σ2 = 0), and we fit a gamma-Gompertz model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' In other words, we observe an exponential death-rate increase in the data, but we estimate a model that implies a downward deviation from the exponential at the oldest ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' As shown in (2), we will estimate σ2 close to but never equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' In this model setting, the standard technique is to estimate both the Gompertz and the gamma- Gompertz models and compare their goodness of fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' However, minor changes in the data can result in different models being selected, which can reduce prediction accuracy and lead to mis- interpretations about the mortality deceleration and the mortality plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' B¨ohnstedt and Gampe (2019) derive the asymptotic distribution of the likelihood ratio test statistic to detect heterogene- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Here, we would like to suggest an alternative that does not involve hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Using the latter has been widely discussed and rethought in the Statistics community (Berk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Head et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Vidgen and Yasseri, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Bruns and Ioannidis, 2016), especially in relation to the arbitrary choice of the α-level (most often 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='05, or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='01) and sample size issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Maximum likelihood estimators, obtained by maximizing the log-likelihood function, often have low bias and large variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Estimation accuracy can sometimes be improved by shrinking some parameters to zero (Tibshirani, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The associated shrinkage estimator improves the overall prediction accuracy at the expense of introducing a small bias to reduce the variance of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' This class of estimators is implicit in Bayesian inference and penalized likelihood inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Using shrinkage estimators is applied as an alternative to hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Lasso, Ridge and Stein-type estimators are the most widely used examples of penalizing methods (see, for example, Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='1 Inference Let Dx be the number of deaths in a given age interval [x, x + 1) for x = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' , m, and Ex denote the number of person-years lived in the same interval (see, for example Brillinger, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Macdonald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Define D = (D0, D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' , Dm)⊤ and E = (E0, E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' , Em)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' In addition, let θ = (a, b, σ2)⊤ ∈ Θ be the parameter vector that characterizes the force of mortality at age x of the gamma-Gompertz model given by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Assume Dx are Poisson-distributed with E(Dx) = VAR(Dx) = µ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' θ)Ex for x = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' , m (Brillinger, 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Under this assumption, the log-likelihood function for θ = (a, b, σ2)⊤ is given by ℓ(θ) = ℓ(θ|D, E) = m � x=0 [Dx ln µ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' θ)) − Ex µ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' θ)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' (3) 3 Maximizing ℓ(θ) with respect to θ = (a, b, σ2)⊤ yields the maximum-likelihood (ML) estimate ˆθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Let us now define a penalized log-likelihood function as ℓp(θ) = ℓ(θ) − p(σ2) , (4) where ℓ(θ) is the standard log-likelihood (3), while p(σ2) is a penalty function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The penalized maximum-likelihood estimate is obtained by maximizing ℓp(θ) with respect to θ = (a, b, σ2)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' For the problem addressed in this paper, the penalty function p(σ2) must be a non-decreasing monotonic continuous function and lim σ2↓0 p(σ2) > p(σ2) for all σ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' In a Bayesian framework, maximizing (4) is equivalent to maximizing a posterior distribution in a setting, in which e−p(σ2)/Cp, Cp := � Θ e−p(σ2)∇θ < ∞, is taken as a prior distribution of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' This procedure yields the maximum a posteriori probability (MAP) estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' MAP is the only Bayesian estimator that minimizes the expected canonical loss (Pereyra, 2019) and is widely used in image and video processing (Greig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Afonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Belekos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' As σ2 describes the variance of frailty at the starting age of analysis, the standard approach would be to specify an inverse gamma prior distribution for it (Gelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The inverse gamma distribution is heavy-tailed and keeps probability mass further from zero than the gamma distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' In addition, while the inverse-gamma mode is always positive, the gamma mode can also be zero (Llera and Beckmann, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' As we aim to test whether σ2 = 0 or σ2 > 0, we will use the log-kernel of the gamma distribution to define the penalty function as p(σ2) = λ � σ2 + ln σ2� (5) for some non-negative λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' When λ < 1, using (5) is equivalent to specifying a gamma prior distribution for σ2 with parameters α = 1 − λ and β = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' When m → ∞, the effect of the penalty diminishes regardless of the size of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' For human life table data m is finite, thus λ ≥ 0 is a constant that controls the relative impact of the penalty function on the estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' When λ = 0, the penalty term has no effect, and maximizing the penalized likelihood will produce the standard maximum likelihood estimates (MLE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' However, as λ → ∞, the impact of the penalty grows, and the maximum penalized likelihood estimates for σ2 will approach zero, providing high precision, but low accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Choosing λ is sensible in a wide range of applications (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Bhattacharya and McNicholas, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Therefore, in accordance with the recommendations in Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' (2009), we carry out a pilot simulation study, in which we find that choosing λ = 1 2 provides similar precision to the one by MLE when σ2 > 0, but better accuracy and precision when σ2 = 0 (simulation results are presented in the next subsection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' As a result, the final expression for the penalized log-likelihood we propose is ℓp(θ) = m � x=0 [Dx ln µ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' θ) − Ex µ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' θ)] − 1 2 � ln σ2 + σ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' (6) From a Bayesian perspective, choosing λ = 1 2 provides an informative prior distribution for σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' As for human populations we are likely to estimate σ2 < 1 (Missov, 2013), the specified prior will provide for σ2 a distribution with a mode equal to zero, a median equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='4549, and 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='00010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='00020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='00030 −600 −400 −200 0 Parameter a a log−likelihood MLE MAP a = 1e−04 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='1 σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='30 −8000 −4000 0 Parameter b b log−likelihood MLE MAP a = 1e−04 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='1 σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='30 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='5 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='5 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='5 Parameter σ2 σ2 log−likelihood MLE MAP a = 1e−04 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='1 σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='00010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='00020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='00030 −800 −600 −400 −200 0 Parameter a a log−likelihood MLE MAP a = 1e−04 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='1 σ2 = 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='30 −8000 −4000 0 Parameter b b log−likelihood MLE MAP a = 1e−04 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='1 σ2 = 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='030 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='55 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='45 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='35 Parameter σ2 σ2 log−likelihood MLE MAP a = 1e−04 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='1 σ2 = 0 Figure 1: Plots of the profile log-likelihood and penalized log-likelihood functions of the param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' In the first row we used synthetic data from a gamma-Gompertz model with parameters a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='0001, b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='1 and σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='1, in the second row we from a Gompertz model with parameters a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='0001 and b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' mean equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Furthermore, the prior provides a probability mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='6826 in the interval (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Figure 1 shows the log-likelihood and penalized log-likelihood functions for all parameters when σ2 > 0 (first row) and σ2 = 0 (second row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' When σ2 > 0, the penalty function affects neither the shape of the log-likelihood, nor the location of its maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' However, when σ2 = 0, adding a penalty yields a higher maximum at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Moreover, when σ2 = 0, the first and second derivatives of the penalized log-likelihood are higher than their respective counterparts of the log-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' As a result, derivative-based optimization methods may reach the maximum point faster, and the estimator ˆσ2 may have a smaller variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='2 Monte Carlo simulations We carry out Monte Carlo simulations to explore the performance of the MAP and ML methods in estimating the gamma-Gompertz model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' We use the R software (Team et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 2022) to maximize the log-likelihood and the penalized log-likelihood functions via the optim function applying as a pre-step differential evolution (Storn and Price, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Ardia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The performance of the ML and MAP estimators are evaluated by calculating two measures: the bias and the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' We generate 10,000 random samples from this model for some parameter values (scenarios with sample sizes of 2,000 and 5,000 were also considered, and are presented in the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' From these samples, we generate life tables and use them to estimate model parameters via the 5 MAP and MLE methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' This process was repeated 2,000 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' In the presence of unobserved heterogeneity, the true parameter values are a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='0001 and a2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='00001 for a, b1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='1 and b2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='15 for b, and σ2 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='2 and σ2 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='8 for σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' When there is no heterogeneity (σ2 = 0), the true parameter values are a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='0001, a2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='0003 and a3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='0005 for a, and b1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='09, b2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='10 and b3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='11 for b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Table 1: Simulation results: gamma-Gompertz model and sample size 10,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' There is heterogeneity MLE estimator MAP estimator Bias Standard deviation Bias Standard deviation Parameter a b σ2 a b σ2 a b σ2 a b σ2 (a1, b1, σ2 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000053 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='001626 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='001502 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='020791 (a1, b1, σ2 2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000292 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='007787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='001784 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='035822 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000354 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='009229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='001783 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='035795 (a1, b2, σ2 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='004431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='002181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='020569 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='003096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='002186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' In the presence of unobserved heterogeneity, both methods underestimate b and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' They also introduce a small positive bias to a, the one pro- vided by ML estimator being slightly smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' However, in general the ML and MAP estimators perform equally well, with a similar bias and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' In the absence of unobserved heterogeneity, the ML estimator provides again a smaller bias for a and b than the MAP estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' However, in this case, the MAP method estimates more precisely the frailty parameter σ2, with a bias and a standard deviation close to zero (∝ 10−15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The MAP estimator also provides a slight reduction in the standard deviation of parameter b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' By the Monte Carlo simulation we also calculate the proportion of trials in which MAP estimates σ2 > 0 when the true values is σ2 = 0 (error type I), as well as the proportion of trials in which MAP estimates σ2 = 0 when the true values is σ2 > 0 (error type II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Based on our simulations, the type I errro equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='001502, while the type II error is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='001126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The Monte Carlo simulations show that using a penalizing likelihood function (6) is an alter- native to hypothesis testing, the latter being dependent on the asymptotic distribution of the ML estimator, sample size and the arbitrary choice of the α-level (B¨ohnstedt and Gampe, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' 3 Performance of MAP and ML estimators on HMD data In this section, we estimate the gamma-Gompertz model via ML and MAP using mortality data from the Human Mortality Database (HMD, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' We take exposures and raw death counts for 6 the female population of France, Japan and the USA in the years 1960, 1980, 2000, and 2020, after age 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' We apply again R (Team et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', 2022) to compute the ML and MAP estimates of θ = (a, b, σ2)′ by using differential evolution.' metadata={'source': 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+page_content='101612 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='022618 Table 2 shows the results of applying ML and MAP methods to the datasets described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The MAP estimator provides lower MSEs in 8 of the 12 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' When the standard ML method estimates σ2 < 10−4, our novel method estimates σ2 = 0 and provides a smaller MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' This suggests that the MAP provides a slightly better fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Overall, MAP performs better than ML when unobserved heterogeneity is not detected, and while for estimates of ˆσ2 > 0 ML has a slight advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The results from the real-data application back up the results from the Monte Carlo simula- tions in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' In the presence of unobserved heterogeneity, the MLE method provides the most precise and accurate estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The MAP method, though, has just slightly lower precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' On the other hand, in the absence of unobserved heterogeneity, the MAP provides smaller bias and variance in its estimates compared to MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='1 Examples when MAP and ML estimators yield different outcomes Using MAP and ML estimators does not always lead to the same statistical inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' One of them can detect heterogeneity in cases when the other does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' We will illustrate this on HMD data for the Japanese female population in 2009 and the French female population born in 1848, ages 70+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' To assess the goodness of fit, we will use again MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' For Japanese females in 2009, ML yields estimates ˆθMLE = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='006359, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='133805, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='070513)′ with standard errors SE(a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000188, SE(b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='002263 and SE(σ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='021156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The 95% confidence interval for σ2 is (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='029047, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='111978) indicating stasitically significant unobserved 7 heterogeneity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', the existence of mortality deceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' On the other hand, the MAP method estimates ˆθMAP = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='006966, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='125440, 0)′, indicating the absence of unobserved heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Comparing the goodness of fit of both methods speaks in favor of the MAP outcome: MAP’s MSE is by 37% lower than ML’s LSE (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='018691 for MAP vs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='029958 for ML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' It indicates that unobserved heterogeneity is negligible and that the gamma-Gompertz model is misspecified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' 70 80 90 100 110 −5 −4 −3 −2 −1 0 Age log−force of mortality Mortality rate MLE MAP Japan 70 75 80 85 90 95 100 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='0 Age log−force of mortality Mortality rate MLE MAP France Figure 2: MAP and MLE estimates of the force of mortality for the Japanese population in 2009 and the Swedish population born in 1881, after age 70 The left panel of Figure 2 shows that both methods estimate a similar logarithmic force of mortality at most ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' However, after age 100, the MLE deviates downward from the observed logarithmic death rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The MAP also provides a better fit and different conclusion for the cohort of French females born in 1848.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' While ML estimates ˆθMLE = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='053748, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='090552, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='008604)′ with SE(a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='000317, SE(b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='001273, SE(σ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='007562 and provides an MSE equal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='046222, MAP estimates ˆθMAP = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='053113, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='094921, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='036466)′ and provides MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='034226, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=', MAP’s MSE is by 26% smaller than ML’s MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Furthermore, while the MAP estimate of σ2 suggests that there is non-negligible unobserved heterogeneity, the ML estimate and standard error for σ2 indicates the opposite: the amount of unobserved heterogeneity is not statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The right panel of Figure 2 shows the difference between these estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' MAP’s estimate shows a leveling-off in the force of mortality, while the MLE shows a log-linear increase in the hazard function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' 4 Concluding remarks B¨ohnstedt and Gampe (2019) introduced a formal procedure to identify whether σ2 > 0 or σ2 = 0 in a hypothesis testing setting: they studied the asymptotic properties of the maximum likelihood estimator and the likelihood ratio test (LRT) for H0 : σ2 = 0 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' H1 : σ2 = 0 for 8 the gamma-Gompertz model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' However, LRTs are based on the asymptotic distribution of the maximum likelihood estimator, hence its convergence depends on the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Moreover, conclusions drawn from hypothesis tests are dependent on the arbitrary choice of the significance level or p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' We suggest an alternative method by considering the problem as model misspecification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' We add a penalty function to the likelihood so that we make sure that ˆσ2 = 0 when there is no het- erogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' We also present a Bayesian interpretation (MAP) to our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' We take advantage of robust Monte Carlo simulations to measure the bias and standard deviation of the ML and MAP methods in scenarios with and without unobserved heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' We also compare the performance of both methods for estimating the gamma-Gompertz model parameters using ac- tual mortality data from the Human Mortality Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' The two methods work almost equally well, the ML having a slight advantage, in the presence of unobserved heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' However, in the absence of the latter, the MAP method provides an estimate closer to 0 (ˆσ2 ≈ 10−20) and a better fit to the model in comparison to ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' As a result, the method we propose here can be used as an alternative to likelihood ratio testing for the gamma-Gompertz model with H0 : σ2 = 0 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' H1 : σ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' On the one hand, the MAP method does not depend on any asymptomatic distribu- tion, its performance is not strongly affected by sample size, and it also does not depend on the arbitrary choice of the significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' On the other hand, MAP provides similar estimates to the ones by ML when σ2 > 0 and more accurate estimates when σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Acknowledgments The research leading to this publication is a part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' 884328 – Unequal Lifespans).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Extremes, 20(4):713– 728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Storn, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' and Price, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Journal of Global Optimization, 11:341–359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Team, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' R: A language and environment for statistical computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Tibshirani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Regression shrinkage and selection via the lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B (Methodological), 58(1):267–288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' Vaupel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' and Missov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfBgKg/content/2301.02853v1.pdf'} +page_content=' (2014).' metadata={'source': 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a/99E4T4oBgHgl3EQf3g3K/content/tmp_files/2301.05307v1.pdf.txt b/99E4T4oBgHgl3EQf3g3K/content/tmp_files/2301.05307v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dbd92543ba5dad01c18ebe2981db468b22a15505 --- /dev/null +++ b/99E4T4oBgHgl3EQf3g3K/content/tmp_files/2301.05307v1.pdf.txt @@ -0,0 +1,2725 @@ +Partial entropy decomposition reveals higher-order structures in human brain activity +Thomas F. Varley,1, 2, ∗ Maria Pope,3, 2 Maria Grazia Puxeddu,1 Joshua Faskowitz,1 and Olaf Sporns1 +1Department of Psychological and Brain Sciences, Indiana University, Bloomington, IN 47405 +2School of Informatics, Computing & Engineering, Indiana University, Bloomington, IN 47405 +3Program in Neuroscience, Indiana University, Bloomington, IN 47405 +(Dated: January 16, 2023) +The standard approach to modeling the human brain as a complex system is with a network, +where the basic unit of interaction is a pairwise link between two brain regions. While powerful, +this approach is limited by the inability to assess higher-order interactions involving three or more +elements directly. In this work, we present a method for capturing higher-order dependencies in +discrete data based on partial entropy decomposition (PED). Our approach decomposes the joint +entropy of the whole system into a set of strictly non-negative partial entropy atoms that describe the +redundant, unique, and synergistic interactions that compose the system’s structure. We begin by +showing how the PED can provide insights into the mathematical structure of both the FC network +itself, as well as established measures of higher-order dependency such as the O-information. When +applied to resting state fMRI data, we find robust evidence of higher-order synergies that are largely +invisible to standard functional connectivity analyses. +This synergistic structure is symmetrical +across hemispheres, largely conserved across individual subjects, and is distinct from structural +features based on redundancy that have previously dominated FC analyses. Our approach can also +be localized in time, allowing a frame-by-frame analysis of how the distributions of redundancies +and synergies change over the course of a recording. We find that different ensembles of regions can +transiently change from being redundancy-dominated to synergy-dominated, and that the temporal +pattern is structured in time. These results provide strong evidence that there exists a large space +of unexplored structures in human brain data that have been largely missed by a focus on bivariate +network connectivity models. This synergistic “shadow structures” is dynamic in time and, likely will +illuminate new and interesting links between brain and behavior. Beyond brain-specific application, +the PED provides a very general approach for understanding higher-order structures in a variety of +complex systems. +Keywords: Higher-Order Interactions, Entropy, Information Theory, Functional Connectivity, +fMRI, Neuroimaging +Since the notion of the “connectome” was first formal- +ized in neuroscience [1], network models of the nervous +system have become ubiquitous in the field [2, 3]. In a +network model, elements of a complex system (typically +neurons or brain regions) are modelled as a graph com- +posed of vertices (or nodes) connected by edges, which +denote some kind of connectivity or statistical depen- +dency between them. Arguably the most ubiquitous ap- +plication of network models to the brain is the “functional +connectivity” (FC) framework [3–5]. In whole-brain neu- +roimaging, FC networks generally define connections as +correlations between the associated regional time series +(e.g. fMRI BOLD signals, EEG waves, etc). The corre- +lation matrix is then cast as the adjacency matrix of a +weighted network, on which a wide number of network +measures can be computed [6]. +Despite the widespread adoption of functional con- +nectivity analyses, there remains a little-discussed, but +profound limitation inherent to the entire methodology: +the only statistical dependencies directly visible to pair- +wise correlation are bivariate, and in the most commonly +performed network analyses, every edge between pairs +Xi and Xj is treated as independent of any other edge. +There are no direct ways to infer statistical dependencies +∗ tvarley@indiana.edu +between three or more variables. “Higher order” interac- +tions are constructed by aggregating bivariate couplings +in analyses such as motifs [7] or community detection [8]. +One of the largest issues holding back the direct study of +higher-order interactions has been the lack of effective, +accessible mathematical tools with which such interac- +tions can be recognized [9]. Recently, however, work in +the field of multivariate information theory has enabled +the development of a plethora of different measures and +frameworks for capturing statistical dependencies beyond +the pairwise correlation [10]. +The few applications of these techniques to brain data +have suggested that higher-order dependencies can en- +code meaningful bio-markers (such as discriminating be- +tween health and pathological states induced by anes- +thesia or brain injury [11]) and reflect changes associated +with age [12]. Since the space of possible higher-order +structures is so much vaster than the space of pairwise +dependencies, the development of tools that make these +structures accessible opens the doors to a large number +of possible studies linking brain activity to cognition and +behavior. +Of the tools that have been applied, one of the most +well developed is the partial information decomposition +[13, 14] (PID), which reveals that multiple interact- +ing variables can participate in a variety of distinct +information-sharing relationships, including redundant, +arXiv:2301.05307v1 [q-bio.NC] 12 Jan 2023 + +2 +unique, and synergistic modes. Redundant and synergis- +tic information sharing represent two distinct, but related +“types” of higher order interaction:.Redundancy refers to +information that is “duplicated” over many elements, so +that the same information could be learned by observ- +ing X1 ∨ X2, ∨, . . . , ∨XN. In contrast, synergy refers to +information that is only accessible when considering the +joint-states of multiple elements and no simpler combi- +nations of sources. Synergistic information can only be +learned by observing X1 ∧ . . . ∧ XN. +Redundant, and synergistic information sharing modes +can be combined to create more complex relationships. +For example, given three variables X1, X2, and X3, +information can be redundantly common to all three, +which could be learned by observing X1 ∨ X2 ∨ X3. We +can also consider the information redundantly shared by +joint states: for example, the information that could be +learned by observing X1 ∨ (X2 ∧ X3) (i.e. observing X1 +or the joint state of X2 and X3). For a finite set of inter- +acting variables, it is possible to enumerate all possible +information-sharing modes, and given a formal definition +of “redundancy”, they can be calculated (for details see +below). +The identification of redundancy and synergy as pos- +sible families of statistical dependence raises questions +about how such relationships might be reflected (or +missed) by the standard, pairwise correlation-based ap- +proach for inferring networks. We propose two criteria +by which we might assess the performance of bivariate +functional connectivity. The first we call specificity: the +degree to which a pairwise correlation between some Xi +and Xj reports dependencies that are unique to Xi and +Xj alone, and not shared with any other edges. +In a +sense, it reflects how appropriate the ubiquitous assump- +tion that edges are independent is. The second criterion +we call completeness: whether all of the statistical de- +pendencies present in a data set are accounted for and +incorporated into the model, or if predictive structure is +“lost” when restrictive analyses are used. +We hypothesized that classical functional connectivity +would prove to be both non-specific (due to the presence +of multivariate redundancies that get repeatedly “seen” +by many pairwise correlations) and incomplete (due to +the presence of synergies). To test this hypothesis, we +used the a framework derived from the PID: the partial +entropy decomposition [15] (PED, explained in detail be- +low) to fully retrieve all components of statistical depen- +dencies in sets of three and four brain regions. As part of +this analysis, we propose a measure of redundant entropy +applicable to arbitrarily-sized collections, which allows us +to fully explore the space of higher order interactions. +We chose the PED over the PID because the PID re- +quires partitioning the system into predictors and “tar- +gets” (the elements whose behavior we are predicting). +This distinction is often artificial, and makes it difficult +to analyze the system itself as a structured whole. The +PED does not require making a source/target distinction, +and serves to generalize the PID to the analysis of whole +systems. +By computing the full PED for all triads of 200 +brain regions, and a subset of approximately two mil- +lion tetrads, we can provide a rich and detailed picture +of beyond-pairwise dependencies in the brain. Further- +more, by separately considering redundancy and synergy +instead of assessing just which one is dominant (as is +commonly done [12, 16]), we can reveal previously un- +seen structures in resting state brain activity. +I. +THEORY +A Note on Notation +In this paper, we will be making reference to multi- +ple different “kinds” of random variables. In general, we +will use uppercase italics to refer to single variables (e.g. +X). Sets of multiple variables will be denoted in boldface +(e.g. X = {X1, . . . , XN}, with subscript indexing). Spe- +cific instances of a variable will be denoted with lower +case font: X = x. Functions (such as the probability, +entropy, and mutual information), will be denoted using +caligraphic font. Finally, we will make a distinction be- +tween expected values of information-theoretic quantities +using upper case function notation (e.g. the Shannon en- +tropy of X is H(X), while the local entropy/surprisal is +h(x)). For a brief review of local information theory, see +the Supplementary Material Section S2. Finally, when +referring to the partial entropy function H∂ (described +below), we will use superscript index notation to indicate +the full set of variables that contextualizes the individual +atom. +For example, H123 +∂ +({1}{2}) refers to the infor- +mation redundantly shared by X1 and X2, when both +are considered as part of the triad X = {X1, X2, X3}, +while H12 +∂ ({1}{2}) refers to the information redundantly +shared by X1 and X2 qua themselves. +A. +Partial Entropy Decomposition +The partial entropy decomposition (PED) provides +a framework with which we can extract all of the +meaningful “structure” in a system of interacting ran- +dom variables [15]. +By “structure”, we are referring +to the (possibly higher-order) patterns of information- +sharing between elements. +Consider a system X = +{X1, X2, . . . , XN}, comprised of N interacting, discrete +random variables: the set of all informative relationships +between elements (and ensembles of elements) in X forms +its “structure.” We begin by defining the total entropy +of X using the Shannon entropy: +H(X) := − +� +x∈X +P(x) log2 P(x) +(1) +Where x indicates a particular configuration of X and + +3 +X is the support set of X. This joint entropy quantifies, +on average, how much it is possible to “know” about X +(i.e. how many bits of information would be required, on +average, to reduce our uncertainty to zero). The entropy +is a summary statistic describing an entire distribution +P(X): +H(X) = E[− log2 P(x)] +(2) +Where − log2 P(x) is the local entropy h(x). We can +intuitively understand the local entropy with the logic of +local probability mass exclusions [17, 18]. Suppose that +we observe X = x. Upon observing x, we can immedi- +ately rule out the possibility that X is in any state ¬x, +and by ruling out those possibilities, we exclude all the +probability mass associated with P(X = ¬x). If P(x) +is very low, then upon learning X = x, we exclude a +large amount of probability mass (1 − P(x)), and conse- +quently, h(x) is high. Conversely, if P(x) is large, then +only a small amount of probability mass is excluded, and +so h(x) is low. +1. +Quantifying Shared Entropy +The measure h(x) is a very crude one: it gives us a +single summary statistic that describes the behaviour of +the “whole” without making reference to the structure +of the relationships between x’s constituent elements. If +X has some non-trivial structure that integrates multi- +ple elements (or ensembles of elements), then we pro- +pose that those elements must “share” entropy. This no- +tion of shared entropy forms the cornerstone of the PED. +The way all of the parts of X share entropy forms the +“structure” of the system. +In the original proposal of +the PED by Ince [15], shared entropy (Hcs) was defined +using the local co-information, which treats the entropy +of variables as sets and defines the shared entropy using +inclusion-exclusion criteria. Unfortunately, as discussed +by Finn and Lizier, the set-theoretic interpretation of +mutlivariate mutual information is complex, as both the +expected and local co-information can be negative [19], +and the PED computed using Ince’s proposed method +can result in negative values that are difficult to inter- +pret. +Here, we propose an alternative way to operationalize +the notion of “redundant entropy” by saying that two +variables X1, X2 ∈ X share entropy if they induce the +same exclusions: i.e. if learning X1 or X2 rules out the +same configurations of the whole [17]. Our goal, then, +becomes to determine how the entropy of the whole is +parcellated out over (potentially multivariate) sharing +modes between parts. +In our toy system given by Table I, suppose we learn +that X1 = 0 OR X2 = 0. Only one global state is ex- +cluded: X = (1, 1) is incompatible with both possibil- +ities, regardless of which is true. Consequently we are +P +X1 X2 +P00 +0 +0 +P01 +0 +1 +P10 +1 +0 +P11 +1 +1 +Table I. Joint entropy of two discrete random variables +that together make up the macro-variable X. +only excluding P11 from the overall distribution. We can +quantify this “shared entropy” using the local entropy of +shared exclusions hsx: +hx +sx({1}{2}) = − log2 P(x1 ∪ x2) +(3) +Here, we are adapting the partial entropy notation first +introduced by Ince in [20]. +The function hx +sx({1}{2}) +quantifies the total probability mass of P(X) excluded +by learning either X1 = x1 or X2 = x2. Said differently, +it is the amount of information that could be learned from +either variable alone. Importantly, while it is a measure +of dependency, it is distinct from the classic mutual in- +formation. +We term this function hsx to indicate that it is the +shared entropy based on common exclusions (“entropy +of shared exclusions”) from some set of sources. We also +note that the form of hsx is equivalent to the informative +part of the local redundancy function derived by Makkeh +et al., [21], which they term isx. For a discussion of how +hsx is related to isx and the deeper connections between +partial entropy decomposition and partial information +decomposition, see Appendix 1. +So far, we have restricted our examples to the simple +case of two variables, x1 and x2, however, we are inter- +ested in the general case of information common to arbi- +trarily large, potentially overlapping subsets of a system +that has adopted a particular state x. This requires first +enumerating the set of subsets, s, which we will call the +set of sources. It is equivalent to the power set of x, ex- +cluding the empty set. For example, if x = {x1, x2, x3}, +then the source set s is equal to: +s = +� +� +� +� +� +{x1}, {x2}, {x3}, +{x1, x2}, {x1, x3}, {x2, x3}, +{x1, x2, x3} +� +� +� +� +� +(4) +We are interested in how collections of sources a ∈ s +might share entropy (i.e. to what extent the exclude the +same possible global configurations of x), which allows us +to write our redundant entropy function in full generality. +For a collection of sources {a1, . . . , ak}: +hsx(a1, . . . , ak) := log2 +1 +P(a1 ∪ . . . ∪ ak) +(5) + +4 +hsx can be interpreted in terms of logical conjunctions +and dysjunctions of variables [14]. Consider the example: +hsx({x1}{x2, x3}), which quantifies the amount of prob- +ability mass about the state of the “whole” that would +be excluded by observing just the part x1 or the joint +state of x2 and x3. This relationship between probabil- +ity mass exclusions on one hand, and formal logic on the +other, places hsx on a sound conceptual footing. While +initially defined locally, it is possible to compute an ex- +pected value Hsx for a joint distribution: +Hsx(A1, . . . , Ak) := E[hsx(a1, . . . , ak)] +(6) +2. +The Partial Entropy Lattice +Our function hsx has a number of appealing mathe- +matical properties, which collectively satisfy the set of +Axioms initially introduced by Williams & Beer for the +problem of information decomposition [13] as applied to +local information [18, 21]: +Symmetry: +hsx +is +invariant +under +permu- +tation +of +it’s +argument: +hsx(a1, . . . , ak) += +hsx(σ(a1), . . . , σ(ak)) +Monotonicity: hsx decreases as more sources are +added: hsx(a1, . . . , ak) ≤ hsx(a1, . . . , ak, ak+1) +Self-redundancy: In the special case of a single +source, hsx is equivalent to the classic local Shan- +non entropy: hsx(a) = h(a). +For proof of these, see [21] Appendix A. Based on these +properties, it is possible to specify the domain of hsx (all +non-degenerate combinations of sources) in terms of a +partially-ordered lattice structure A [13, 18]. Not every +combination of sources a1 . . . ak is a valid partial entropy +atom, only those where no source is a subset of any other: +A = {α ∈ P1(s) : ∀ai, aj ∈ α, ai ̸⊂ aj} +(7) +Where P1(s) indicates the power set of s, excluding +the empty set. For an in-depth derivation of the lattice, +see [13, 14, 18], for a visualization of the lattice, see Fig. +1. The value of any element h∂(α) on the lattice can be +computed via Mobius inversion: +hx +∂(α) = hsx(α) − +� +β⪯α +hx +∂(β) +(8) +The result is the entropy specific to a particular α +and no simpler combination of sources. +Furthermore, +the structure of the lattice and the properties of hsx en- +sure that hx +∂(α) will always be non-negative. +We can +re-compute the total joint entropy of x as: +h(x) = +|A| +� +i=1 +hx +∂(αi) +(9) +Like hsx, it is also possible to compute an expected +value of h∂ (which will also be strictly non-negative): +HX +∂ (α) = E[hx +∂(α)] +(10) +3. +Decomposing Marginal and Joint Entropies +Having defined hsx and the Mobius inversion on the +partial entropy lattice, we can now do a complete de- +composition of the joint entropy, and its marginal com- +ponents. +For example, consider the bivariate system +X = {X1, X2}. We can decompose the joint entropy: +H(X) = H12 +∂ ({1}{2}) + H12 +∂ ({1}) +(11) ++ H12 +∂ ({2}) + H12 +∂ ({1, 2}) +Furthermore, +we +can +decompose +the +associated +marginal entropies in a manner consistent with the par- +tial information decomposition [13]: +H(X1) = H12 +∂ ({1}{2}) + H12 +∂ ({1}) +(12) +H(X2) = H12 +∂ ({1}{2}) + H12 +∂ ({2}) +These decompositions can be done for larger ensem- +bles, or more statistical dependencies (see below) and +can reveal how higher-order interactions can complicate +(and in some cases, compromise) the standard bivariate +approaches to functional connectivity. +4. +Mathematical Analysis of the PED +The partial entropy decomposition reveals a rich and +complex structure of statistical dependencies even in +small systems. Before considering the empirical results, +it is worth discussing how the PED relates to classic mea- +sures from information theory and what it reveals about +the limitations of bivariate FC measures. +The first key finding is that the PED provides interest- +ing insights into the nature of bivariate mutual informa- +tion. Typically, mutual information is conflated with re- +dundancy at the outset (for example, in Venn diagrams), +however, when considering the PED of two variables X1 +and X2, it becomes clear that: +I(X1; X2) = H12 +∂ ({1}{2}) − H12 +∂ ({1, 2}) +(13) + +5 +Figure 1. The partial entropy lattice. The lattice of partial entropy atoms induced by the Hsx function. Each vertex +of the lattice corresponds to a single PE atom, and the Venn diagram describes the associated structure of probability mass +exclusions. The blue area indicates the probability mass from P(x) that is excluded by some combination of observations. For +example, in the legend, we can see the probability mass excluded by observing X1 ∨ X2. The blue area is all of the probability +mass one would exclude after learning the state of either component alone. The lowest atom is the entropy redundant to all +three elements (Hsx({1}{2}{3})), and the dependencies get increasingly synergistic higher on the lattice. +This relationship was originally noted by Ince [15] and +later re-derived by Finn and Lizier [19]. In a sense, the +higher-order information present in the joint-state of (X1 +and X2) “obscures” the lower-order structure. This is- +sue is also inherited by parametric correlation measures +based on the Pearson correlation coefficient, since the +mutual information is a deterministic function of Pear- +son’s ρ for Gaussian variables [22]. +When considering the decomposition of local mutual +information into informative and misinformative compo- +nents proposed by Finn and Lizier, it is clear that re- +dundancy corresponds to the informative component of +local mutual information, while synergy corresponds to +the misinformative component. +We can do a similar analysis extracting the bivariate +mutual information from the trivariate PED, which re- +veals that the bivariate correlation is not specific: +I(X1; X2) = H123 +∂ +({1}{2}{3}) + H123 +∂ +({1}{2}) +(14) +− H123 +∂ +({3}{1, 2}) − H123 +∂ +({1, 2}{1, 3}{2, 3}) +− H123 +∂ +({1, 2}{1, 3}) − H123 +∂ +({1, 2}{2, 3}) +− H123 +∂ +({1, 2}) +It is clear from Eq. 15 that the bivariate mutual infor- +mation incorporates information that is triple-redundant +across three variables (H123 +∂ +({1}{2}{3})), and if one were +to take the standard FC approach to a triad (pairwise +correlation between all three pairs of elements), that +the triple redundancy would be triple counted and er- +roneously ascribed to three separate interactions. Fur- + +(1] [2] +α= Xin X2 n X3 +X +d = X2n X3 +X3 +Xi +X6 +thermore, not only does bivariate mutual information +double-count redundancy, but it also penalizes higher- +order synergies. +Any higher-order atom that includes +the joint state of X1 ∧ X2 counts against I(X1; X2). +Having established that the presence of higher-order +redundancies explicitly precludes bivariate correlation +from being specific, we now ask: can we improve the +specificity using common statistical methods? One ap- +proach aimed at “controlling” for the context of addi- +tional variables in a bivariate correlation analysis is using +conditioning or partial correlation. Typically, these anal- +yses are assumed to improve the specificity of a pairwise +dependency by removing the influence of confounders, +however, by decomposing the conditional mutual infor- +mation between three variables, we can see that condi- +tioning does not ensure specificity: +I(X1; X2|X3) = H123 +∂ +({1}{2}) +(15) ++ H123 +∂ +({1}{2, 3}) + H123 +∂ +({2}{1, 3}) ++ H123 +∂ +({1, 2}{1, 3}{2, 3}) ++ H123 +∂ +({1, 3}{2, 3}) +− H123 +∂ +({1, 2}) − H123 +∂ +({1, 2, 3}) +The decomposition of I(X1; X2|X3) conflates the true +pairwise redundancy (H123 +∂ +{1}{2}) with the a higher- +order redundancy involving the joint state of X1∧X3 and +X2 ∧ X3: H123 +∂ +{1, 3}{2, 3}. Furthermore, the conditional +mutual information penalizes synergistic entropy shared +in the joint state of all three variables (H123 +∂ +{1, 2, 3}). +Consequently, we can conclude that the specificity of bi- +variate functional connectivity cannot be salvaged using +conditioning or partial correlation. Not only does con- +trolling fail to provide specificity, it also actively com- +promises completeness, since it brings in higher-order in- +teractions. +Given that conditional mutual information +and partial correlation are equivalent for Gaussian vari- +ables [23], this issue also affects standard, parametric ap- +proaches to conditional connectivity, just as with bivari- +ate mutual information/Pearson correlation. +It is important to understand that these analytic re- +sults are not a consequence of the particular form of hsx: +any shared entropy function that allows for the forma- +tion of a partial entropy lattice will produce these same +results (many were first derived by Ince, who used a dif- +ferent measure based on the local co-information [15]). +5. +Higher-Order Dependency Measures +In addition revealing the structure of commonly-used +correlations (bivariate and partial correlations), the PED +can also be used to develop intuitions about multivariate +generalizations of the mutual information. Many of these +generalizations exist, and here we will focus on four: the +total correlation [24], the dual total correlation [25], the +O-information [16, 26] (also called the “enigmatic” infor- +mation [27]) and the S-information [26] (also called the +“exogenous” information [27]). While useful, these mea- +sures are often difficult to intuitively understand, and can +display surprising behavior. Since they can all be writ- +ten in terms of sums and differences of joint and marginal +entropies, we can use the PED framework to more com- +pletely understand them. +The oldest measure is the total correlation, defined as: +T (X) := +|X| +� +i=1 +H(Xi) − H(X) +(16) +which is equivalent to the Kullback-Leibler divergence +between the true joint distribution P(X) and the product +of the marginals: +T (X) = DKL(P(X)|| +|X| +� +i=1 +P(Xi) +(17) +Based on equation 17, we can understand the total cor- +relation as the divergence from the maximum entropy dis- +tribution to the true distribution, implying that it might +be something like a measure of the “total” structure of +the system (as it’s name would suggest). +We can de- +compose the 3-variable case to get a full picture of the +structure of the TC: +T (X1, X2, X3) = (2 × {1}{2}{3}) +(18) ++ {1}{2} + {1}{3} + {2}{3} +− {1, 2}{1, 3}{2, 3} +− {1, 2}{1, 3} − {1, 2}{2, 3} − {1, 3}{2, 3} +− {1, 2} − {1, 3} − {2, 3} +− {1, 2, 3} +We can see that the total correlation is largely a mea- +sure of redundancy, sensitive to information shared be- +tween single elements, but penalizing higher-order infor- +mation present in joint states. This can be understood +by considering the lattice in Figure 1: each of the H(Xi) +terms will only incorporate atoms preceding (or equal +to) the unique entropy term H123 +∂ +(i) - anything that can +only be seen by considering the joint-state of X will be +negative. +The second generalization of mutual information is the +dual total correlation [25]. Defined in terms of entropies +by: +D(X) := H(X) − +|X| +� +i=1 +H(Xi|X−i) +(19) +where X−i refers to the set of every element of X ex- +cluding the ith. The dual total correlation can be under- + +7 +stood as the difference between the total entropy of X +and all of the entropy in each element of X that is “in- +trinsic” to it and not shared with any other part. When +we decompose the three-variable case, we find: +D(X1, X2, X3) = {1}{2}{3} +(20) ++ {1){2} + {1}{3} + {2}{3} ++ {1}{23} + {2}{1, 3} + {3}{1, 2} ++ {1, 2}{1, 3}{2, 3} +− {1, 2} − {1, 3} − {2, 3} − (2 × {1, 2, 3}) +This shows that dual total correlation is a much more +“complete” picture of the structure of a system than to- +tal correlation. It is sensitive to both shared redundan- +cies and synergies, penalizing only the un-shared, higher- +order synergy terms such as H123 +∂ +({1, 2}). +The sum of the total correlation and the dual total +correlation is the exogenous information [27], also called +by the S-information. +E(X) := T (X) + D(X) +(21) +Prior work has shown the exogenous entropy to be +very tightly correlated with the Tononi-Sporns-Edelman +complexity [16, 26, 28], a measure of global integra- +tion/segregation balance. +James also showed that the +S-information quantified the total information that ev- +ery element shares with every other element [27]. +We +can see that: +E(X1, X2, X3) = (3 × {1}{2}{3}) ++ 2 × ({1}{2} + {1}{3} + {2}{3})) ++ {1}{2, 3} + {2}{1, 3} + {3}{1, 2} +− {1, 2}{1, 3} − {1, 2}{2, 3} − {1, 3}{2, 3} +− 2 × ({1, 2} + {1, 3} + {2, 3}) +− (3 × {1, 2, 3}) +This reveals that S-information to be an unusual mea- +sure, in that it counts each redundancy term multiple +times (i.e. in the case of three variables, the triple redun- +dancy term appears three times, each double-redundancy +term appears twice, etc), and penalizes them likewise +when considering unshared synergies. +The final, and arguably most interesting measure is the +difference between the total correlation and the dual total +correlation is often referred to as the O-information [26], +and has been hypothesized to give a heuristic measure +of the extent to which a given system is dominated by +redundant or synergistic interactions: +O(X) := T (X) − D(X) +(22) +where O(X) > 0 implies a redundancy-dominated +structure and O(X) < 0 implies a synergy dominated +one. PED analysis reveals: +O(X1, X2, X3) = {1}{2}{3} +(23) +− {1}{2, 3} − {2}{1, 3} − {3}{1, 2} +− (2 × {1, 2}{1, 3}{2, 3}) +− {1, 2}{1, 3} − {2, 3}{1, 3} − {1, 2}{2, 3} ++ {1, 2, 3} +This shows that the O-information generally satis- +fies the intuitions proposed by Rosas et al., as it is +positively sensitive to the non-pairwise redundancy (in +this case just H123 +∂ +({1}{2}{3})) and negatively sensi- +tive to any higher-order shared information. Curiously, +O(X1, X2, X3) positively counts the highest, un-shared +synergy atom (H123 +∂ +({1, 2, 3}). +Conceivably, it may be +possible for a set of three variables with no redundancy +to return a positive O-information, although whether this +can actually occur is an area of future research. +For three-element systems, the O-information is also +equivalent to the co-information [26], which forms the +base of the original redundant entropy function Hcs pro- +posed by Ince [15]. From this we can see that, at least +for three variables, co-information is not a pure mea- +sure of redundancy, conflating the true redundancy and +the highest synergy term, as well as penalizing other +higher-order modes of information-sharing. A similar ar- +gument was made by Williams and Beer using the mu- +tual information-based interpretation of co-information +[13]. +While the O-information and co-information di- +verge for N > 3, we anticipate that the behavior of the +co-information will remain similarly complex at higher +N. +These results reveal how the PED framework can +provide clarity to the often-murky world of multivariate +information theory. +6. +Novel Higher-Order Measures +From these PED atoms, we can construct a novel mea- +sures of higher-order dependence that extends beyond +TC, DTC, O-Information and S-Information. +When considering higher-order redundancy, we are in- +terested in all of those atoms that duplicate information +over three or more individual elements. We define this +as the redundant structure. For a three element system: +SR(X1, X2, X3) = {1}{2}{3} +(24) +For a four-element system: +SR(X1, X2, X3, X4) = {1}{2}{3}{4} +(25) ++ {1}{2}{3} + {1}{2}{4} ++ {1}{3}{4} + {2}{3}{4} + +8 +And so on for larger systems. +We can also define an analogous measure of synergis- +tic structure: all those atoms representing information +shared over the joint state of two or more elements. For +example, for a three element system: +SS(X1, X2, X3) = {1}{2, 3} + {2}{1, 3} + {3}{1, 2} ++ {1, 2}{1, 3}{2, 3} ++ {1, 2}{1, 3} + {2, 3}{1, 3} ++ {1, 2}{2, 3} +(26) +For three element systems, the difference SR − SS +is analagous to a “corrected” O-information: the atom +{1, 2}{1, 3}{2, 3} is only counted once and the confound- +ing triple synergy {1, 2, 3} is not included. Finally, we +can define a measure of total (integrated) structure (i.e. +all shared information) as the sum of all atoms composed +of multiple sources: +S = +� +α∈A +α ⇐⇒ |α| > 1 +(27) +B. +Applications to the Brain +The mathematical structure of the PED is domain ag- +nostic: any complex system composed of discrete ran- +dom variables is amenable to this kind of information- +theoretic analysis. In this paper, we focus on data col- +lected from the human brain with functional magnetic +resonance imaging (fMRI). For detailed methods, see the +Materials & Methods section (V, but in brief, data from +ninety five human subjects resting quietly was recorded +as part of the Human Connectome Project [29]. All of +the scans were concatenated and each channel binarized +about the mean [30] to create multidimensional, binary +time series. We then computed the full PED for all tri- +ads, and approximatley two million tetrads, to compare +to the standard, bivariate functional connectivity net- +work (computed with mutual information). +By looking at the redundant and synergistic structures, +and relating them to the standard FC, we can explore +how higher-order dependencies are represented in bivari- +ate networks, as well as what brain regions participate in +more redundancy- or synergy-dominated ensembles. +II. +RESULTS +A. +PED Reveals the Limitations of Bivariate +Networks +We now discuss how the PED relates multivariate mea- +sures of bivariate network structure commonly used in +the functional connectivity literature. +These measures +describe statistical dependencies between ensembles of +regions, but mediated by the topology of bivariate con- +nections. We hypothesized that this emergence from bi- +variate dependencies would render them largely insen- +sitive to synergies, which in turn would mean that such +measures do not solve the issue of incompleteness in func- +tional connectivity. +Following [31], we compared the redundant and syn- +ergistic structure of triads and tetrads to a measure of +subgraph strength: the arithmetic mean of all edges in +the subgraph. We found that the arithmetic mean FC +density was positively correlated with redundancy for +triads (ρ = 0.999, p < 10−20) and tetrads (ρ = 0.995, +p < 10−20), indicating that information duplicated over +many brain regions contributes to multiple edges, lead- +ing to double-counting. In contrast, for triads, arithmetic +mean FC density was largely independent of synergistic +structure (ρ = −0.05, p < 10−20), but for tetrads they +were strongly anticorrelated (ρ = −0.988, p < 10−20). +In addition to subgraph structure, another common +method of assessing polyadic interactions in networks +is via community detection [8]. +Using the multi- +resolution consensus clustering algorithm [32], we clus- +tered the bivariate functional connectivity matrix into +non-overlapping communities. +We then looked at the +distributions of higher-order redundant and synergistic +structure for triads and tetrads that spanned different +numbers of consensus communities. We found that tri- +ads where all nodes were members of one community had +significantly less synergy than triads that spanned two +or three communities (Kolmogorov-Smirnov two sample +test, D = 0.44, p < 10−20). The pattern was more pro- +nounced when considering tetrads: tetrads that all be- +longed to one community had lower synergy than those +that spanned two communities (D = 0.45, p < 10−20), +who in turn had lower synergy than those that spanned +three communities (D = 0.37, p < 10−20). In Figure 2 +(top row), we show cumulative probability density plots +for the distribution of synergies for triads and tetrads +that spanned one, two, three, and four FC communities, +where it is clear that participation in increasingly diverse +communities is associated with greater synergistic struc- +ture. In contrast, redundant structure was higher in tri- +ads that were all members of a small number of commu- +nities. Triads that spanned three communities had lower +redundancy than triads that spanned two communities +(D = 0.48, p < 10−20), which in turn had lower redun- +dancy than those that were all members of one commu- +nity (D = 0.47, p < 10−20) (see Fig. 2, bottom row). +These results, coupled with the mathematical analysis of +the PED discussed in Section I provide strong theoret- +ical and empirical evidence that bivariate, correlation- +based FC measures are largely sensitive to redundant in- +formation duplicated over many individual brain regions, +but largely insensitive to (or even anti-correlated with) +higher-order synergies involving the joint state of mul- +tiple regions. +These results imply the possibility that +there is a vast space of neural dynamics and structures +that have not previously been captured in FC analyses. + +9 +Figure 2. +The limits of bivariate functional connectivity. +A. In triads, bivariate functional connectivity is largely +independent of synergistic structure, and B, is very positively correlated with redundant structure. C. In tetrads, bivariate +functional connectivity is strongly negatively correlated with synergistic structure and D, is strongly correlated with redundant +structure. +E-F. Triads that have all elements within one FC community have significantly less synergistic structure than +those that have elements with two communities, while for redundnat structure, there was a clear pattern that the more FC +communities a triad straddled, the lower it’s overall redundant structure. G-H. The same pattern was even more pronounced +in tetrads: as the number of FC communities a tetrad straddled increased, the expected synergistic structure climbed, while +expected redundant structure fell. +1. +PED with Hsx is consistent with O-information +To test whether the PED using the Hsx redundancy +function was consistent with other, information-theoretic +measures of redundancy and synergy, we compared the +average redundant and synergistic structures (as revealed +by PED), to the O-information. We hypothesized that +redundant structure would be positively correlated with +O-information (as O > 0 implies redundancy dominance) +and that synergistic structure would be negatively corre- +lated, for the same reason. +For both triads and tetrads, our hypothesis was bourne +out. The Pearson correlation between O-information and +redundant structure was significantly positive for both +triads (ρ = 0.72, p < 10−20) and tetrads (ρ = 0.82, +p < 10−20). Conversely, the Pearson correlation between +the O-information and the synergistic structure was sig- +nificantly negative (triads: ρ = −0.7, p < 10−20, tetrads: +ρ = −0.72, p < 10−20). +These results show that the +structures revealed by the PED are consistent with other, +non-decomposition-based inference methods and serves +to validate the overall framework. +Interestingly, +when +comparing +the +triadic +O- +information +to +the +corrected +triadic +O-information +(which does not double-count H123 +∂ +({1, 2}{1, 3}{2, 3}) +and does not add back in the atom H123 +∂ +({1, 2, 3})), we +can see that the addition of H123 +∂ +({1, 2, 3}) can lead to +erroneous conclusions. +Of all those triads that had a +negative corrected O-information (i.e. +had a greater +synergistic structure than redundant structure), 61.7% +had a positive O-information, which could only be +attributable to the presence of the triple-synergy being +(mis)interpreted as redundancy and overwhelming the +true difference. This suggests that, for small systems, the +O-information may not provide an unbiased estimator +of redundancy/synergy balance. +B. +Characterizing Higher-Order Brain Structures +Having established the presence of beyond-pairwise re- +dundancies and synergies in brain data, and shown that +standard, network-based approaches show an incomplete +picture of the overall architecture, we now describe the +distribution of redundancies and synergies across the hu- +man brain. +We began by applying a higher-order generalization of +the standard community detection approach using a hy- +pergraph modularity maximization algorithm [33]. this +algorithm partitions collections of (potentially overlap- +ping) sets of nodes called hyperedges into communities +that have a high degree of internal integration and a lower + +A +B +C +D +Triadic Synergy +Tetradic Synergy +Triadic Redundancy +Tetradic Redundancy +0.35 +0.35 +p=-0.054 +p=0.999 +p=-0.988 +p=0.995 +0.30 +0.30 +0.20 +0.20 +Arithmetic Mean FC +0.15 +0.15 +0.20 +0.20 + 0.15 + 0.15 +0.10 +0.10 +0.10 +0.05 +0.05 +0.05 +0.05 +0.00 +0.00 +0.00 +0.00 +0.37 +0.39 +0.41 +0.42 +0.2 +0.38 +0.3 +0.4 +0.5 +1.0 +1.1 +1.2 +1.3 +0.50 +0.55 +0.60 +0.65 +0.70 +Synergistic Structure +Redundant Structure +Synergistic Structure +Redundant Structure +E +F +G +H +1.0 +1.0 +1.0 +1.0 +1 Community +1 Community + 2 Communities + 2 Communities +3 Communities + 3 Communities +0.8 +0.8 +0.8 +0.8 +Average +4 Communities + Average +0.6 +0.6 +0.6 +1 Community + 2 Communities +1 Community +0.2 +0.2 +0.2 +0.2 +2 Communities + 3 Communities +3 Communities + 4 Communities +Average +Average +0.0 +0.0 +0.0 +0.0 +0.37 +0.38 +0.39 +0.40 +0.41 +0.42 +0.2 +0.3 +0.4 +0.5 +1.0 +1.1 +1.2 +1.3 +0.50 +0.55 +0.60 +0.65 +0.70 + Synergistic Structure +Redundant Structure +Synergistic Structure + Redundant Structure10 +Figure 3. Redundant and synergistic hypergraph community structure. A-B. Surface plots of the two communities +structures: on the left is the redundant structure and on the right is the synergistic structure. We can see that both patterns +are largely symmetrical for both information-sharing modes, although the synergistic structure has two large, lateralized +communities. C-D. The co-classification matrices for redundant structure (left) and the synergistic structure (right). The +higher the value of a pair, the more frequently the hypergraph modularity maximization [33] assigns those two regions to the +same hyper-community. The yellow squares indicate the seven canonical Yeo functional networks [34], and we can see that the +higher-order redundant structure matches the bivariate Yeo systems well (despite consisting of information shared redundantly +across three nodes). In contrast, the synergistic structure largely fails to match the canonical network structure at all. E. +For each of the 95 subjects and for each of the 1000 permutation nulls used to significance test the NMI between subject-level +community structure and the master level structure, we computed the log-ratio of the empirical NMI to the null NMI. For +redundancy, there was not a single null, over any subject, that was greater than the associated empirical NMI. For the case of +the synergy, only 0.6% of nulls were greater than their associated empirical NMI. +degree of between-community integration. We selected +all those triads that had a greater synergistic structure +than any of the one million maximum entropy null triads +(see Materials and Methods), which yielded a set of 3,746 +unique triads. From these, we constructed an unweighted +hypergraph with 200 nodes and 3,746 hyperedges (cast- +ing each triad as a hyperedge incident on three nodes). +We then performed 1,000 trials of the hypergraph cluster- +ing algorithm proposed by Kumar et al., [33], from which +we built a consensus matrix that tracked how frequently +two brain regions Xi and Xj were assigned to the same +hyper-community. We repeated the process for the 3,746 +maximally redundant triads to create two partitions: a +synergistic structure and a redundant structure. +In Figure 3 we show surface plots of the resulting com- +munities computed from the concatenated time series +comprising all ninety-five subjects and all 4 runs. The +redundant structure (left) is very similar to the canoni- +cal seven Yeo systems [34]: we can see a well-developed +DMN (orange), a distinct visual system (sky blue), a +somato-motor strip (violet), and a fronto-parietal net- +work (dark blue). In contrast, when considering the syn- + +A +B +Redundant Communities +Synergistic Communities +D +0 +1.0 +0 +1.0 +-. +25 +25 +0.8 +0.8 +50 +50 +Rate +Rate +99 251 94191 +75 +75 +0.6 + 0.6 +Co-occurance I +# +100 +100 +0.4 + 0.4 +125 +125 +# +150 +150 +0.10 38 +0.2 +0.2 +175 +175 +0.0 +0.0 +0 +50 +100 +150 +0 +50 +100 +150 +Regions +Regions +E + Redundancy +2.5 +Synergy +2.0 +Density +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.011 +ergistic structure (right), a strikingly different pattern is +apparent. Synergistic connectivity appears more lateral- +ized over left and right hemispheres (orange and violet +communities respectively), although there is a high de- +gree of symmetry along the cortical midline comprised of +apparently novel communities. These include a synergis- +tic coupling between visual and limbic regions (sky blue), +as well a occipital subset of the DMN (green) and a curi- +ous, symmetrical set of regions combining somato-motor +and DMN regions (red). +These results show two things: +the first is further +confirmation that the canonical structures studied in an +FC framework can be interpreted as reflecting primar- +ily patterns of redundant information. +The second is +that higher-order synergies are structured in non-random +ways, combining multiple brain regions into integrated +systems that are usually thought to be independent when +considering just correlation-based analyses. If the syner- +gistic structure were reflecting mere noise, then we would +not expect the high-degree of symmetry and structure we +observe. +To test whether the patterns we observed were con- +sistent across individuals, we re-ran the entire pipeline +(PED of all triads, hypergraph clustering of redundant +and synergistic triads, etc) for each of the 95 subjects +seperately. +Then, for each subject, we computed the +normalized mutual information (NMI) [6] between the +subject-level partition and the relevant master partition +(redundancy or synergy) created from the concatenated +time series of all four scans from each of the ninety-five +subjects. We significance tested each comparison with a +permutation null model. For each null, we permuted the +subject-level community assignment vector of nodes, re- +computing the NMI between the master partition and a +shuffled subject-level partition (1,000 permutations). In +the case of the redundant partition, we found that that +no subjects ever had a shuffled null that was greater than +the empirical NMI: all had significant NMI (0.52 ± 0.07). +In the case of the synergistic partition, 91 of the 95 +subjects showed significant NMI (0.1 ± 0.03, p < 0.05, +Benjamini-Hochberg FDR corrected). These results sug- +gest that both structures (redundant and synergistic) +are broadly conserved across individuals, however, it ap- +pears that the synergistic partitions are generally more +variable between subjects than the redundant partition +(which hews closer to the master partition constructed +by combining the data from all subjects). +When we +computed the normalized mutual information of all the +subject level redundancy partitions to the canonical Yeo +systems, we found a high degree of correlation (NMI = +0.6196±0.0117, p < 10−20). The same analysis with the +subject level synergy partitions found a much lower de- +gree of concordance (NMI = 0.2290±0.0117, p < 10−20). +1. +Redundancy-synergy gradient & time-resolved analysis +Thus far, we have analyzed higher-order redundancy +and synergy separately. To understand how they inter- +act, we began by replicating the analysis of Luppi et al., +[35]. We counted how many times each brain region ap- +peared in the set of 3,746 most synergistic and 3,746 most +redundant triads. We then ranked each node to create +two vectors which rank how frequently each region par- +ticipates in high-redundancy and high-synergy configu- +rations. By subtracting those two rank vectors, we get +a measure of relative redundancy/synergy dominance. A +value greater than zero indicates that a region’s relative +redundancy (compared to all other regions) is greater +than its relative synergy (compared to all other regions), +and vice versa. +By projecting the rank-differences onto the cortical +surface (Fig. 4A), we recover the same gradient-like pat- +tern first reported by Luppi et al., with relatively redun- +dant regions located in primary sensory and motor cor- +tex, and relatively synergistic regions located in multi- +modal and executive cortex. This replication is notewor- +thy, as Luppi et al., used an entirely different method of +computing synergy (based on the information flow from +past to future in pairs of brain regions), while we are +looking at generalizations of static FC for which dynamic +order does not matter. The fact that the same gradi- +ent appears when using both analytical methods strongly +suggests it is a robust feature of brain activity. +A limitation of the analysis by Luppi et al. is the re- +striction that only average values of synergy and redun- +dancy are accessible: the results describe expected values +over all TRs and obscure any local variability. The PED +analysis using hsx can be localized (see Sec. I) to individ- +ual frames. This allows us to see how the redundant and +synergistic structure fluctuate over the course of a resting +state scan, and how the distributions of relative synergies +and redundancies vary over the cortex. Figure 4B shows +how the redundant and synergistic structure fluctuate +over the course of 1100 TRs taken from a single subject +(for scans concatenated). +This allows us to probe the +information structure of previously identified patterns in +frame-wise dynamics. Analysis of instantaneous pairwise +co-fluctuations (also called “edge time series”) reveals a +highly structured pattern, with periods of relative disin- +tegration interspersed with high co-fluctuation “events” +[36, 37]. The distribution of these co-fluctuations reflect +various factors of cognition [38], generative structure [39], +functional network organization [30], and individual dif- +ferences [40]. By correlating the instantaneous average +whole-brain redundant and synergistic structures with in- +stantaneous whole-brain co-fluctuation amplitude (RSS), +we can get an understanding of the “informational struc- +ture” of high-RSS “events.” We found that redundancy +is positively correlated with co-fluctuation RSS (ρ = 0.6, +p < 10−50) and synergy is negatively correlated with co- +fluctuation amplitude (ρ = −0.43, p < 10−50). Given +that synergy is known to drive bivariate functional con- + +12 +Figure 4. Time-resolved analysis. A. Surface plots for the distributions of relative synergies and relative redundancies across +the human brain. These results match prior work by Luppi et al., [35], with primary sensory and motor cortex being relatively +redundant, while multi-modal association areas being relatively synergistic. B. Over the course of one subject’s scan (1100 +TRs), the total redundant and synergistic structure varies over time, although never so much that the curves cross (i.e. there is +never more redundant structure than synergistic structure present). C. Instantaneous redundant and synergistic structure are +anti-correlated (ρ = −0.83, p < 10−50). D. Redundancy is positively correlated with the amplitude of bivariate co-fluctuations +(ρ = 0.6, p < 10−50) and E. synergy is negatively correlated with co-fluctuation amplitude (ρ = −0.43, p < 10−50). F. For each +TR, we show the difference in the rank-redundancy and rank-synergy for each node (red indicates a higher rank-redundancy +than rank-synergy and vice versa for blue). When nodes are stratified by Yeo system [34] (grey, horizontal lines), it is clear +that different systems alternate between high-redundancy and high-synergy configurations in different ways. G. For every +pair of columns in Panel F. we compute the Pearson correlation between them to construct a time × time similarity matrix, +which we then clustered using the MRCC algorithm [32]. Note that rows and columns are not in time order, but rather, +re-ordered to reveal the state-structure of the time series. H. Five example states (centroids of each community show in Panel +G.) projected onto the cortical surface. It is clear that the instantaneous pattern of relative synergies and redundancies varies +from the average structure presented in Panel A. For example, in States 3 and 4, the visual system is highly redundant (as in +the average), however in state 5, the visual system is synergistic. +nectivity [36], this is again consistent with the hypothesis +that FC patterns largely reflect redundancy and are in- +sensitive to higher-order synergies. +With full PED analysis completed for every frame, it +is possible to compute the instantaneous distribution of +relative redundancies and synergies across the cortex for +every TR. The resulting multidimensional time-series can +be seen in Fig. 4F. When sorted by Yeo systems [34], +we can see that different systems show distinct relative +redundancy/synergy profiles. The nodes in the somato- +motor system had the highest median value (22.0 ± 73), +followed by the visual system (14.0±80), indicating that +they were, on-average relatively more redundant than +synergistic. In contrast, the ventral attentional system + +A +B +0.45 + 0.40 +0.35 +陆 +0.30 + 0.25 +S +0.20 +TRs +C +D +E +0.25 +0.24 +Synergy +0.42 +0.23 +0.23 +lergy +.i... +0.22 +0.40 + 0.22 +2.40 +.. +0.21 +0.21 +0.38 +0.38 +0.20 +0.20 +0.24 +200 +200 +400 +4100 +RSS +Redundancy +RSS +F +G +Vis. +Cent. +1 +150 +Som.Mot. +Cent. 2 +DAN +Ranks +Cent. 3 +VAN +Lim. +Cent. 4 +Cont. +-0.2 +100 +.50 +150 +DMN +Cent. 5 +200 +TRs +Rs +H +Centroid 1 +entroid +Centroid 3 +Centroid 4 +Centroid 513 +Figure 5. State-to-state transitions. For each of the nine distinct states, we can see how many times each state transitions +another (self-loops are not shown for visual clarity). We can see that the various states have meaningful differences between +each-other (e.g. +the visual system or the somato-motor systems both transition from redundancy- to synergy-dominated +configurations over time), however, within a state, the patterns are largely symmetrical across hemispheres. +had the lowest median value (−11.0 ± 66), indicating a +relatively synergistic dynamic. +Other systems seemed +largely balanced: with median values near zero but a +wide spread between them, such as the dorsal atten- +tion network (1.0 ± 70), fronto-parietal control system +(−5.0±56), and the DMN (−2.0±67). These are systems +that transiently shift from largely redundancy-dominated +to synergy-dominated regimes in equal measure. Finally, +the limbic system had small values and relatively lit- +tle spread (−5.0 ± 18), indicating a system that never +achieved either extreme. +We then correlated every TR against every other frame +to construct a weighted, signed recurrence network [41], +which we could then cluster using the MRCC algorithm +[32] (Fig. 4G). This allowed us to assign every TR to +one of nine discrete “states”, each of which can be rep- +resented by its centroid (for five examples see Fig 4H). +We can see that these states are generally symmetrical, +but show markedly different patterns relative redundancy +and synergy across the cortex, and some systems can +change valance entirely. For example, in states three and +four the visual system is highly redundant (consistent +with the average behavior), while in state five the same +regions are more synergy-dominated. In the same vein, +the somato-motor strip is highly redundant in state 4, +but slightly synergy-biased in state 3. This shows that +the dynamics of information processing are variable in +time, with different areas of cortex transiently becoming +more redundant or more synergistic in concert. +The sequence of states occupied at each TR is a dis- +crete time series which we can analyze as a finite-state +machine (for visualization, see Figure 5). Shannon tem- +poral mutual information found that the present state +was significantly predictive of the future state (1.59 bit, +p < 10−50), and that the transitions between states were +generally more deterministic [42, 43] (2.29 bit p < 10−50) +than would be expected by chance. While the sample +size is small (1099 transitions), these results suggest that + +3 +514 +the transition between states is structured in non-random +ways. +III. +DISCUSSION +In this paper, we have explored a novel framework +for extracting higher-order dependencies from data and +applied it to fMRI rcordings. +We found that the hu- +man brain is rich in beyond-pairwise, synergistic struc- +tures, as well as redundant information copied over many +brain regions. +Based on a partial entropy decomposi- +tion framework [15, 19] our method returns strictly non- +negative values, does not require grouping elements into +“sources” and “targets”, and is localizable, permitting a +time-resolved analysis of the system’s dynamics. +Prior work on the partial entropy decomposition has +analytically shown that the bivariate mutual information +between two elements incorporates non-local information +that is redundantly present over more than two elements +[15, 19]. This means that classic approaches to functional +connectivity are non-specific: the link between two ele- +ments does not reflect information uniquely shared by +those two but double (or triple-counts) higher-order re- +dundancies distributed over the system. We verified this +empirically by comparing the distribution of higher-order +(beyond pairwise) redundancies to a bivariate correlation +network and found that the redundancies closely matched +the classic network structure. +These non-local redundancies shed new light on a well- +documented feature of bivariate functional connectivity +networks: the transitivity of correlation [44]. +In func- +tional connectivity networks, if Xi and Xj are correlated, +as well as Xj and Xk, then there is a much higher-than +expected chance that Xi and Xk are correlated (even +though this is not theoretically necessary [45]). Since the +Pearson correlation related the mutual information under +Gaussian assumptions [22], we claim that the observed +transitivity of functional connectivity is a consequence of +previously-unrecognized, non-local redundancies copied +over ensembles of nodes. This hypothesis is consistent +with our findings that redundancies correlate with key +features of functional network topology, including sub- +graph density and community structure. +In addition to higher-order redundancies, we also found +strong evidence of higher order synergies: information +present in the joint states of multiple brain regions and +only accessible when considering “wholes” rather than +just “parts.” These synergies appear to be structured in +part by the physical brain (for example, being largely +symmetric across hemispheres), but also don’t readily +correspond to the standard functional connectivity net- +works previously explored in the literature. Since syn- +ergiestic structures appear to be largely anti-correlated +with the standard bivariate network structures, it is plau- +sible that these synergistic systems represent a novel or- +ganization of human brain activity. +These higher-order interactions represent a vast space +of largely unexplored, but potentially significant as- +pects of brain activity. +One possible avenue of study +is how higher-order synergies reflect individual differ- +ences [40, 46] and subject identifiability [47]. The find- +ing that the synergistic community structure was more +variable across subjects than the redundant structure +suggests that synergistic dependencies may reflect more +unique, individualized differences, while the redundant +structure (reflected in the functional connectivity) repre- +sents a more conserved architecture. This is consistent +with recent theoretical work linking synergy to individu- +ality [48], as well as empirical findings that the evolution +of humans is associated with an enrichment of synergis- +tic cortical structures [35]. +The ability to expand be- +yond pairwise network models of the brain into the much +richer space of beyond-pairwise structures offers a the op- +portunity to explore previously inaccessible relationships +between brain activity, cognition, and behavior. +Since normal cognitive functioning requires the coordi- +nation of many different brain regions [49–51], and patho- +logical states are associated with the dis-integrated dy- +namics [52–54], it is reasonable to assume that alterations +to higher-order, synergistic coordination may also reflect +clinically significant changes in cognition and health. Re- +cent work has already indicated that changes in bivariate +synergy track loss of consciousness under anesthesia and +following traumatic and anoxic brain injury [11] suggest- +ing that higher-order dependencies can encode clinically +significant biomarkers. +We hypothesize that beyond- +pairwise synergies in particular may be worth exploring +in the context of recognizing early signs of Alzheimer’s +and other neurodegenerative diseases, as synergy requires +the coordination of many regions simultaneously and may +begin to show signs of fragmentation earlier than stan- +dard, functional connectivity-based patterns (which are +dominated by non-local redundancies may obscure early +fragmentation of the system). +Finally, the localizable nature of the Hsx partial en- +tropy function allows us a high degree of temporal preci- +sion when analyzing brain dynamics. The standard ap- +proach to time-varying connectivity is using a sliding- +windows analysis, however, this approach blurs temporal +features and obscures higher-frequency events [55]. By +being able to localize the redundancies and synergies in +time, we can see that there is a complex interplay between +both “types” of integration. When considering expected +values, we find a distribution of redundancies and syn- +ergies that replicates the findings of Luppi et al., [35], +however, when we localize the analysis in time, we find a +high degree of variability between frames. It appears that +there are not consistently “redundant” or “synergistic” +brain regions (or ensembles), but rather, various brain +regions can transiently participate in highly synergistic +or highly redundant behaviors at different times. +The +structure of these dynamics appears to be non-random +(based on the structure of the state-transition matrix), +however, the significance of the various combinations of +redundancy and synergy remains a topic for much future + +15 +work. The fact that some systems (such as the visual sys- +tem) can be either redundancy- or synergy-dominated at +different times complicates the notion of a “synergistic +core”. Instead, there may be a “synergistic landscape” +of configurations that the system traverses, with differ- +ent configurations of brain regions transiently serving as +the core and providing a flexible architecture for neural +computation in response to different demands. +This analysis does have some limitations, however. +The most significant is that the size of the partial en- +tropy lattice grows explosively as the size of the system +increases: a system with only eight elements will have +a lattice with 5.6×1022 unique partial entropy atoms. +While our aggregated measures of redundant and syn- +ergistic structure can summarize the dependencies in a +principled way, simply computing that many atoms is +computationally prohibitive. +In this paper, we took a +large system of 200 nodes, and calculated every triad +and a large number of tetrads, however, this also quickly +runs into combinatorial difficulties, as the number of pos- +sible groups of size k one can make from N elements +grows with the binomial coefficient. Heuristic measures +such as the O-information can help, although as we have +seen, this measure can conflate redundancy and synergy +in sometimes surprising ways. One possible avenue of fu- +ture work could be to leverage optimization algorithms +to find small, tractable subsets of systems that show in- +teresting redundant or synergistic structure, as was done +in [16, 56, 57]. Alternately, coarse-graining approaches +that can reduce the dimensionality of the system while +preserving the informational or causal structure may al- +low the analysis of a compressed version of the system +small enough to be tractable [42, 58]. +In the context of this study, the use of fMRI BOLD +data presents some inherent limitations, such as a small +number of samples (TRs) from which to infer probabil- +ity distributions, and the necessity of binarizing a slow, +continuous signal. Generalizing the logic of shared prob- +ability mass exclusions remains an area of on-going work +[59], although for the time being, the hsx function re- +quires discrete random variables. +BOLD itself is also +fundamentally a proxy measure of brain activity based +on oxygenated blood flow and not a direct measure of +neural activity. Applying this work to electrophysciolog- +ical data (M/EEG, which can be discretized in princi- +pled ways to enable information-theoretic analysis [60]), +and naturally discrete spiking neural data [61], will help +deepen our understanding of how higher-order interac- +tions contribute to cognition and behavior. The applica- +bility of the PED to multiple scales of analysis highlights +one of the foundational strengths of the approach (and +information-theoretic frameworks more broadly): being +based on the fundamental logic of inferences under con- +ditions of uncertainty, the PED can be applied to a large +number of complex systems (beyond just the brain), or +to multiple scales within a single system to provide a +detailed, and holistic picture of the system’s structure. +IV. +CONCLUSIONS +In this work, we have shown how the joint entropy of +a complex system can be decomposed into atomic com- +ponents of redundancy and synergy, which reveal higher- +order, beyond-pairwise dependencies in the structure of +the system. +When applied to human brain data, this +partial entropy decomposition framework reveals previ- +ously unrecognized, higher-order structures in the human +brain. +We find that the well-known patterns of func- +tional connectivity networks largely reflect redundant in- +formation copied over many brain regions. In contrast, +the synergies for a kind of “shadow structure” that is +largely independent from, or anticorrelated with, the bi- +variate network and has consequently remained less well +explored. The patterns of redundancy and synergy over +the cortex are dynamic across time, with different en- +sembles of brain regions transiently forming redundancy- +or synergy-dominated structures. This space of beyond- +pairwise dynamics is likely rich in previously unidentified +links between brain activity and cognition. The PED can +also be applied to problems beyond neuroscience and may +provide a general tool with which higher-order structure +can be studied in any complex system. +V. +MATERIALS & METHODS +A. +Human Connectome Project fMRI Data +The data used in this study was taken from a set of 100 +unrelated subjects included in the Human Connectome +Project (HCP) [29]. Refs [29, 62] provide a detailed de- +scription of the acquisition and preprocessing of this data, +which have been used in many previous studies[30, 39]. +Briefly, all subjects gave informed consent to protocols +approved by the Washington University Institutional Re- +view Board. Data was collected with a Siemens 3T Con- +nectom Skyra using a head coil with 32 channels. Func- +tional data analysed here was acquired during resting +state with a gradient-echo echo-planar imaging (EPI) se- +quence. Collection occurred over four scans on two sep- +arate days (scan duration: 14:33 min; eyes open). The +main acquisition parameters included TR = 720 ms, TE += 33.1 ms, flip angle of 52°, 2 mm isotropic voxel resolu- +tion, and a multiband factor of 8. Resting state data was +mapped to a 200-node parcellation scheme [63] covering +the entire cerebral cortex. +Considerations for subject inclusion were established +before the study and are as follows. +The mean and +mean absolute deviation of the relative root mean square +(RMS) motion throughout any of the four resting scans +were calculated. Subjects that exceeded 1.5 times the in- +terquartile range in the adverse direction for two or more +measures they were excluded. This resulted in the exclu- +sion of four subjects, and an additional subject due to a +software error during diffusion MRI processing. The in- +cluded subjects had demographic characteristics of: 56% + +16 +female, mean age = 29.29 ± 3.66, age range = 22-36 +years. +1. +Preprocessing +The minimal preprocessing of HCP rs-fMRI data can +be found described in detail in ref. [62]. Five main steps +were followed: 1) susceptibility, distortion, and motion +correction; 2) registration to subject-specific T1-weighted +data; 3) bias and intensity normalization; 4) projection +onto the 32k fs LR mesh; and 5) alignment to common +space with a multimodal surface registration (81). This +pipeline produced an ICA+FIX time series in the CIFTI +grayordinate coordinate system. We included two addi- +tional preprocessing steps: 6) global signal regression and +7) detrending and band pass filtering (0.008 to 0.08 Hz) +[64]. We discarded the first and last 50 frames of each +time series after confound regression and filtering to pro- +duce final scans with length 13.2 min (1,100 frames). All +four scans from 95 subjects were then z-scored and con- +catenated to give a final time-series of 200 brain regions +and 418,000 time points. +2. +Discretizing BOLD Signals +Unfortunately, the Hsx measure is only well-defined for +discrete random variables. Consequently, we discretized +our data by binarizing the z-scored time series: setting +any value greater than zero to one and any value less than +zero to zero. Prior work has established that transform- +ing BOLD signals into binary point processes preserves +the majority of the total correlation structure [30, 65], +so we are confident that our analysis is robust, especially +considering the large number of samples. +We chose to binarize around the z-score (as opposed +to alternative point-processing techniques such as local +maxima), as the z-score ensures that each individual +channel is generally maximally entropic (i.e. +P(Xi = +1) ≈ P(Xi = 0) ≈ 1/2). This ensures that every indi- +vidual channel has approximately the same entropy, and +so deviations from maximum entropy at the level of the +entire triad or tetrad can only emerge from correlations +between two or more channels, rather than being influ- +enced by biases at the channel-level. The choice to bina- +rize about the mean also links this work to previous work +on decomposing functional connectivity into discrete par- +titions [30]. +B. +Statistical Analyses +1. +Triads & tetrads +In standard FC analysis, it is typical to compute the +pairwise correlation between all pairs of brain regions, re- +sulting +�N +2 +� +unique pairs. For this analysis, we computed +all triads of brain regions, resulting in +�200 +3 +� += 1, 313, 400 +unique triples. +For each triad, we computed the joint +entropy, and performed the full partial entropy decom- +position to compute each of the eighteen partial entropy +atoms. Finally, each of the atoms was normalized by the +total joint entropy, to give a measure of how much each +atom contributes to the whole entropy. This allows us +to directly compare triads that have different joint en- +tropies. +It was not feasible to brute-force all possible tetrads, +which is a set of approximately sixty-four million. In- +stead, we randomly sub-sampled sets of four randomly, +collecting 1954000 tetrads (≈ 3% of the total space) and +analyzing them. +2. +Bivariate functional connectivity networks +To directly compare the PED framework to the stan- +dard, correlation-based FC network framework, we con- +structed single, representative FC network by computing +the pairwise mutual information between every pair of +regions in the fMRI scan (as was done in [39]). +I(X; Y ) = H(X) + H(Y ) − H(X, Y ) +(28) +3. +Subgraph Analysis +Since we are interested in how the bivariate FC frame- +work reflects (or fails to reflect) higher-order redundan- +cies and synergies, we also compute a battery of struc- +ture metrics on matching subgraphs taken from the FC +network. Formally presented by Onnela et al., [31], we +consider arithmetic mean of the subgraph connectivity: +GA(X) = +� +i̸=j I(Xi; Xj) +|X|2 − |X| +(29) +For a given triad of tetrad X, we compared the mean +FC density to the various redundant and synergistic +information-sharing structures of X. +4. +Community Detection on Bivariate Matrices +Multi-resolution consensus clustering [32] was used to +detect network communities in the functional connectiv- +ity matrix across multiple scales. +The algorithm pro- +ceeds in three main stages. In the first stage, modular- +ity maximization using the Louvain method is performed +for 1,000 different values of the resolution parameter, γ. +This produced a range of γ values that resulted with par- +titions having between 2 and N communities. The sec- +ond stage consisted of a more fine-grained sweep (10,000 +steps) over the γ values defined in the first stage of the + +17 +process. We aggregate the partitions produced by this +sweep into a node-by-node co-classification matrix stor- +ing how frequently nodes are partitioned into the same +community. +A null model with expected values of co- +classification based on the size and number of commu- +nities was subtracted from the co-classification matrix +[32]. +Finally, in the third stage, the null-adjusted co- +classification matrix was clustered again using consensus +clustering with 100 repetitions and a consensus threshold +τ of 0 [66]. The resulting partition was used for analyses. +We assessed the similarity between single-subject par- +titions and consensus partitions using Normalized Mu- +tual Information (NMI). Each partition can be formal- +ized as a vector of integers of dimension N whose entries +denote the nodes’ allegiance to communities. NMI esti- +mates the similarity between two partitions by counting +co-occurrences in the two vectors. +We computed NMI between each one of the 95 single- +subject partitions and the consensus partition, in both +cases of redundancy and synergy hypergraphs. We as- +sessed the significance of NMI values by comparing them +with a null case obtained by randomly shuffling 1000 +times communities labels in the single-subject partitions. +The p-values of the statistical test, calculated as the frac- +tion of null-case NMI greater than the actual NMI, have +been corrected with a Benjamini-Hochberg test. +5. +Null Model +To ensure that the statistical dependencies we were +observing reflect non-trivial interactions, we significance- +tested triads and tetrads against a null distribution com- +posed of one million, maximum entropy null models. We +constructed sets of totally independent, maximum en- +tropy binary time series and computed the PED on each +set of three or four null channels. From this, we can con- +struct distributions of the expected null structure and +expected synergistic structure against which to compare +triads and tetrads. +6. +Hypergraph Community Detection +Each of the triads can be thought of as a hyper-edge on +a 3-uniform hypergraph of 200 nodes. For the synergistic +structure, we selected only those hyperedges who had a +greater synergistic structure than any of the one million +maximum-entropy nulls that formed our null distribu- +tion. This resulted in a hypergraph with 200 hundred +nodes and 3,746 regular hyper-edges. We used the same +criteria to build a redundant structure hypergraph using +the top 3,746 most redundant hyperedges. +Both hypergraphs were clustered using the HyperNetX +package +(available +on +Github: +https://github. +com/pnnl/HyperNetX) implementation of the hyper- +modularity optimization by Kumar and Vaidyanathan et +al., [33]. +Briefly, the algorithm by Kumar and Vaidyanathan et +al., takes a modularity maximization approach to parti- +tioning the vertices of a hypergraph into non-overlapping +communities. In dyadic networks, the modularity func- +tion compares the distribution of within- and between- +community edges to the expected distribution based on +a degree-preserving, configuration null model [67]. In the +case of hypergraphs, a hyper-configuration model can be +used instead. A generalized modularity metric can then +be used as an objective function in a Louvain-based, mod- +ularity maximization search. +7. +Temporal Structure +To explore the temporal structure of the state- +transition series, we used the active information storage +[68, 69] (a measure of how predictable is the future given +the past) and the determinism [42, 43], (a measure of +how constrained the future is given the past). +For a +one dimensional, discrete random variable X that evolves +through time, we can compute the information that the +past Xt−1 discloses about the future Xt with the mutual +information: +AIS(X) = I(Xt−1; Xt) +(30) +This measure quantifies the degree to which knowing +the past reduces our uncertainty about the future. This +term can be further decomposed into two components: +the determinism and the degeneracy [42]: +I(Xt−1; Xt) = Det(X) − Deg(X) +(31) +Where determinism is: +Det(X) = log2(N) − H(Xt|Xt−1) +(32) +And degeneracy is: +Deg(X) = log2(N) − H(Xt) +(33) +The determinism quantifies how reliably a given +past state xt−1 leads to a single future state xt. +If +P(xt|xt−1) ≈ 1, then we say that xt−1 deterministically +leads to xt. +We significance tested both the active information stor- +age and the determinism by comparing the empirical val- +ues to an ensemble of ten thousand randomly permuted +nulls generated by shuffling the time series. +Since the +degeneracy is unchanged by permutation of the temporal +structure (since the marginal entropy H(Xt) is the same), +any changes in active information storage produced by +shuffling must be driven by changes in the determinism. + +18 +C. +Software +All partial information/entropy decompositions were +done using the SxPID package released with [21] and +can be accessed on Github: +https://github.com/ +Abzinger/SxPID. 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Zomaya, A +Framework for the Local Information Dynamics of Dis- +tributed Computation in Complex Systems, in Guided +Self-Organization: +Inception, Emergence, Complexity +and Computation, edited by M. Prokopenko (Springer, +Berlin, Heidelberg, 2014) pp. 115–158. +[70] A. Kolchinsky, A Novel Approach to the Partial Informa- +tion Decomposition, Entropy 24, 403 (2022), number: 3 +Publisher: Multidisciplinary Digital Publishing Institute. +[71] D. J. C. MacKay, Information Theory, Inference and +Learning Algorithms (Cambridge University Press, 2003) +google-Books-ID: AKuMj4PN EMC. +SI 1. MATHEMATICAL PROPERTIES OF Hsx +Partial Entropy Decomposition & Partial +Information Decomposition +The redundant entropy function hsx is closely related +to the redundant information function isx proposed by +Makkeh et al., [21]. Our function hsx is defined: +hsx(a1, . . . , ak) = log +1 +P(a1 ∪ . . . ∪ ak) +(34) +This measure is equivalent to the informative compo- +nent of the measure isx proposed by Makkeh et al., [21] +in the context of single-target partial information decom- +position. The local redundant information function isx +is defined: +isx(a1, . . . , ak; y) := +(35) +log2 +P(y) − P(y ∩ (¯a1 ∩ . . . ∩ ¯ak)) +1 − P(¯a1 ∩ . . . ∩ ¯ak) +− log2 P(y) +(36) +Which can be further decomposed into informative and +misinformative components [17]: +i+ +sx(a1, . . . , ak; y) := log2 +1 +P(a1 ∪ . . . ∪ ak) +(37) +i− +sx(a1, . . . , ak; y) := log2 +P(y) +P(y ∩ (a1 ∪ . . . ∪ ak)) +(38) +isx(a1, . . . , ak; y) = i+ +sx(a1, . . . , ak; y) − i− +sx(a1, . . . , ak; y) +(39) +Where it is clear that hsx(·) = i+ +sx(·; y), with the sole +difference that i+ +sx(·; y) is implicitly defined with respect +to some target variable y (although y has no actual im- +pact on the value). Below, we show that, if the target y +is set to the joint state of the whole (x), then the par- +tial entropy decomposition of h(x) with hsx as the shard +entropy function becomes equivalent to the partial in- +formation decomposition i(x1, . . . , xN; x) with isx as the +redundant entropy function. The notion that the PED + +21 +is equivalent to doing the PID of the information all the +“parts” disclose about the “whole” was mentioned par- +enthetically in [21], although the finding that the infor- +mative component is all that is required is novel. +Given the equivalence between hsx(·) and i+ +sx(·; y), it +suffices to show that i− +sx(x1, . . . , xN; x) = 0 bit in all +cases. When y = x, we can re-write the function as: +i− +sx(x1, . . . , xN; x) = +(40) +log2 +P(x1 ∩ . . . ∩ xN) +P((x1 ∩ . . . ∩ xN) ∩ (x1 ∪ . . . , ∪xN)) +the union of x1 ∪ . . . ∪ xk is clearly a superset of x1 ∩ +. . . ∩ xk, so +i− +sx = log2 +P(x1 ∩ . . . ∩ xN) +P(x1 ∩ . . . ∩ xN) +(41) +Which is clearly log2(1) = 0 bit □ +We can understand the partial entropy decomposition +using hsx as being equivalent to the decomposition of +i(x1, . . . , xN; x). Intuitively, this is consistent with the +identity for discrete variables that I(X, X) = H(X). +This has curious implications: for example, while x1 +may misinform about x2, neither variable can misinform +about their contribution to the state of the whole x. +Example: Logical Exclusive-OR (XOR) Gate +XOR +AND +P +X1 ⊕ X2 = T +P +X1 ∧ X2 = T +1/4 +0 +0 +0 +1/4 +0 +0 +0 +1/4 +0 +1 +1 +1/4 +0 +1 +0 +1/4 +1 +0 +1 +1/4 +1 +0 +0 +1/4 +1 +1 +0 +1/4 +1 +1 +1 +Table S1. Logical XOR and AND gates. +To demonstrate how partial entropy decomposition can +be used to untangle higher-order interactions, consider +the logical exclusive-OR (XOR) gate (for the lookup ta- +ble, see Table S1). The XOR gate is an example of a +synergistic logic gate: the ability to predict the state of +the target T depends on having access to both X1 and +X2 jointly: the pariwise marginal mutual informations +are equal to 0: I(X1; T) = I(X2; T) = 0 bit, but the +joint mutual information is nonzero: I(X1, X2; T) = 1 +bit. +We +can +initially +see +that +the +triple-redundancy +H12T +∂ +({1}{2}{T}) = 0 bit. This is because any configu- +ration of logical disjunctions does not actually rule out +any states: for example, P(X1 = 0∪X2 = 0∪T = 0) = 1 +as there is no configuration (1, 1, 1) that can be excluded. +Other results can be unintuitive. +For example, most +of the partial entropy is shared between the three bi- +variate relationships H12T +∂ +({1}{2}), H12T +∂ +({1}{T}), and +H12T +∂ +({2}{T}). +How is this consistent with the fact +that the mutual information between any pair of vari- +ables is zero? +The bivariate redundancy can be non- +zero in this case because, on average, knowing the local +state of x1 ∨ x2 reduces our uncertainty about the joint +state of {x1, x2, t}. For example, suppose we learn that +x1 = 1 ∨ x2 = 1. This excludes the joint configuration +{x1 = 0, x2 = 0, t = 0}. This exclusion of the associated +probability mass is recognized by hsx(·) as informative, +in that it reduces our uncertainty about the joint-state +of the whole, despite the fact that, on average, X1 and +X2 disclose no information about T. There is no redun- +dant information common to X1, X2 and T, however, +and there a number of higher-order dependencies, such +as H12T +∂ +({1}{2, T}) and H12T +∂ +({1, 2}{1, T}{2, T}). +Atom +H12T +∂ +|| XOR +AND +MaxEnt +{1}{2}{T} +0.0 +0.208 +0.193 +{1}{2} +0.415 +0.208 +0.222 +{1}{T} +0.415 +0.25 +0.222 +{2}{T} +0.415 +0.25 +0.222 +{1}{2, T} +0.17 +0.04 +0.041 +{2}{1, T} +0.17 +0.04 +0.041 +{T}{1, 2} +0.17 +0.104 +0.041 +{1} +0.0 +0.292 +0.322 +{2} +0.0 +0.292 +0.322 +{T} +0.0 +0.0 +0.322 +{1, 2}{1, T}{2, T} +0.245 +0.0 +0.018 +{1, 2}{1, T} +0.0 +0.104 +0.093 +{1, 2}{2, T} +0.0 +0.104 +0.093 +{1, T}{2, T} +0.0 +0.0 +0.093 +{1, 2} +0.0 +0.104 +0.17 +{1, T} +0.0 +0.0 +0.17 +{2, T} +0.0 +0.0 +0.17 +{1, 2, T} +0.0 +0.0 +0.245 +Table S2. The Partial Entropy Decomposition for the +XOR, AND, and Maximum Entropy Gates. +S3: Independent Variables +One unusual property of hsx, as demonstrated by the +logical-XOR results is that independent variables can still +share entropy. This is a recognized feature of multiple +measures of redundant information/entropy and is gen- +erally considered to be an issue to be excised [70] (some +have gone so far as the suggest an axiom that such a +property must be disallowed from the outset [20]). While +we understand that shared entropy for random variables +may seem initially counter intuitive, it can be readily +understood when considering the problem of inference. +Let us return to our two element example (Table I), +and this time specify that X1⊥X2. We know then that + +22 +I(X1; X2) = 0 bit, however, hsx({X1}{X2}) ≈ 0.415 bit. +Why? The answer is that, while the two variables are in- +dependent, in all cases learning either X1 = x1∨X2 = x2 +is sufficient to exclude a single possible state: the case +where X1 = ¬x1 ∧ X2 = ¬x2. If we were to formalize +this in terms of a gambling problem, we would find that, +despite the independence of both variables, a player is, in +fact, more likely to win with a correct guess after learning +X1 ∨ X2. See Figure S1. +Figure S1. Utility of redundant information. Suppose an +agent plays a gambling game, where two independent, binary +variables are set at random (so all outcomes P(x1, x2) = 1/4 +for all configurations). If the agent guesses the correct vari- +able, they win $1 and if they guess wrong, they win nothing. +Clearly, the expected value of each trial is $0.25 (blue curve). +However, if another agent learns that X1 = x1 ∨ X2 = x2, +then they can do better at the game, with an expected value +of each trial of $0.33. +The difference between the two cu- +mulative distributions of 1000 trials is the extra “value” that +can be extracted from the redundant information. This shows +that, while counter-intuitive, the fact that H12 +∂ ({1}{2}) > 0 +even if X1⊥X2 is interpretable in practical contexts. +Furthermore, we can see that, while hsx will be greater +than zero for small, maximum entropy systems, as the +system gets larger, the redundancy will logarithmically +trend towards zero. +The proof for binary systems is +straightforward. For a discrete, maximum entropy sys- +tem with k elements, learning the state of X1 = x1 ∨ +. . . ∨ Xk = xk will always exclude a single state: the +state where X1 ̸= x1 ∧ . . . ∧ Xk ̸= xk. This single state +x∗ will have P(x∗) = 1/k (as all states have the same +probability by the maximum entropy constraint). The +union of all surviving configurations will be 1 − P(x∗). +Since limk→∞ 1/k = 0, then the union probability will +→ 1 and consequently hsx → 0 bit. This suggests that, +for very large, idealized systems (such as an ideal gas), +the redundancy does go to 0 bit for maximum entropy +systems. How other values (such as the redundant and +synergistic structure) behave remains an area of further +study, although we conjecture that, as k → ∞, redundan- +cies and synergies will vanish faster than unique terms. +S3. BASIC INFORMATION THEORY REVIEW +Here we will provide a basic overview of information +theory for unfamiliar readers. For a more comprehensive +treatment of the subject, see the textbooks by Cover & +Thomas [22] and/or MacKay [71]. +The basic object of study in information theory is the +entropy, which quantifies the total uncertainty that we, +as observers, have about the state of some variable X. +For the purposes of this paper, we will assume that X is +discrete, with a finite number of possible states that can +be pulled from the support set X. For every particular +state x ∈ X, there is an associated probability P(x). The +entropy of X is given by: +H(X) = − +� +x∈X +P(x) log P(x) +(42) +For multiple variables, we can define the joint entropy +as: +H(X1, X2) = − +� +x1∈X1 +x2∈X2 +P(x1, x2) log P(x1, x2) +(43) +We can also define the conditional entropy as the un- +certainty about X1 left over after accounting for the +knowledge that X2 = x2: +H(X1|X2) = − +� +x1∈X1 +x2∈X2 +P(x1, x2) log P(x1|x2) +(44) +From these basic components, we can define the mu- +tual information as the difference between our initial un- +certainty about the state of X1 the the remaining uncer- +tainty about X1 that is not resolved by learning the state +of X2: +I(X1; X2) = H(X1) − H(X1|X2) +(45) +The mutual information is symmetric in it’s argu- +ments: I(X1; X2) = I(X2; X1). If we have multiple Xs +disclosing information about a single target T, the joint +mutual information has the same form: +I(X1, X2; T) = H(T) − H(T|X1, X2) +(46) +The mutual information can also be written in terms +of probabilities: + +No Information +300 +Redundant Information +Utility of Redundancy +250 +200 +150 +100 +50 +0 +0 +200 +400 +600 +800 +1000 +Number of Trials23 +I(X1; X2) = +� +x1∈X1 +x2∈X2 +P(x1, x2) log P(x1|x2) +P(x1) +(47) +Local Information Theory +Both the entropy and the mutual information can be +understood as “expected values” over some (potentially +multivariate) distribution): +H(X) = E[− log P(x)] +(48) +The term − log P(x) is known as the local entropy or +the Shannon information content and it quantifies how +“surprised” we, as observers are to see that X = x. It is +typically denoted as h(x). +I(X1; X2) = E +� +log2 +P(x1|x2) +P(x1) +� +(49) +The term +P (x1|x2) +P (x1) +is known as the local mutual in- +formation and it quantifies the divergence between the +“prior” probability X1 = x1 and the posterior probabil- +ity X1 = x1 after accounting for the fact that X2 = x2. +It is typically denoted as i(x1; x2). Unlike the expected +mutual information, which is strictly non-negative, the +local mutual information can be either greater than, or +less than, zero. If P(x1|x2) < P(x1), then i(x1; x2) > 0, +and if P(x1|x2) < P(x1), then i(x1; x2) < 0. In the latter +case, we say that x1 misinforms on the state of x2. + diff --git a/99E4T4oBgHgl3EQf3g3K/content/tmp_files/load_file.txt b/99E4T4oBgHgl3EQf3g3K/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bdd472fe87ff350061bfcda59d36d9f8cf72aac5 --- /dev/null +++ b/99E4T4oBgHgl3EQf3g3K/content/tmp_files/load_file.txt @@ -0,0 +1,1621 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf,len=1620 +page_content='Partial entropy decomposition reveals higher-order structures in human brain activity Thomas F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Varley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ∗ Maria Pope,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 2 Maria Grazia Puxeddu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='1 Joshua Faskowitz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='1 and Olaf Sporns1 1Department of Psychological and Brain Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Indiana University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Bloomington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' IN 47405 2School of Informatics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Computing & Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Indiana University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Bloomington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' IN 47405 3Program in Neuroscience,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Indiana University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Bloomington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' IN 47405 (Dated: January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 2023) The standard approach to modeling the human brain as a complex system is with a network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' where the basic unit of interaction is a pairwise link between two brain regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' While powerful, this approach is limited by the inability to assess higher-order interactions involving three or more elements directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In this work, we present a method for capturing higher-order dependencies in discrete data based on partial entropy decomposition (PED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Our approach decomposes the joint entropy of the whole system into a set of strictly non-negative partial entropy atoms that describe the redundant, unique, and synergistic interactions that compose the system’s structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We begin by showing how the PED can provide insights into the mathematical structure of both the FC network itself, as well as established measures of higher-order dependency such as the O-information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' When applied to resting state fMRI data, we find robust evidence of higher-order synergies that are largely invisible to standard functional connectivity analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This synergistic structure is symmetrical across hemispheres, largely conserved across individual subjects, and is distinct from structural features based on redundancy that have previously dominated FC analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Our approach can also be localized in time, allowing a frame-by-frame analysis of how the distributions of redundancies and synergies change over the course of a recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We find that different ensembles of regions can transiently change from being redundancy-dominated to synergy-dominated, and that the temporal pattern is structured in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' These results provide strong evidence that there exists a large space of unexplored structures in human brain data that have been largely missed by a focus on bivariate network connectivity models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This synergistic “shadow structures” is dynamic in time and, likely will illuminate new and interesting links between brain and behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Beyond brain-specific application, the PED provides a very general approach for understanding higher-order structures in a variety of complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Keywords: Higher-Order Interactions, Entropy, Information Theory, Functional Connectivity, fMRI, Neuroimaging Since the notion of the “connectome” was first formal- ized in neuroscience [1], network models of the nervous system have become ubiquitous in the field [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In a network model, elements of a complex system (typically neurons or brain regions) are modelled as a graph com- posed of vertices (or nodes) connected by edges, which denote some kind of connectivity or statistical depen- dency between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Arguably the most ubiquitous ap- plication of network models to the brain is the “functional connectivity” (FC) framework [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In whole-brain neu- roimaging, FC networks generally define connections as correlations between the associated regional time series (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' fMRI BOLD signals, EEG waves, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The corre- lation matrix is then cast as the adjacency matrix of a weighted network, on which a wide number of network measures can be computed [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Despite the widespread adoption of functional con- nectivity analyses, there remains a little-discussed, but profound limitation inherent to the entire methodology: the only statistical dependencies directly visible to pair- wise correlation are bivariate, and in the most commonly performed network analyses, every edge between pairs Xi and Xj is treated as independent of any other edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' There are no direct ways to infer statistical dependencies ∗ tvarley@indiana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='edu between three or more variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' “Higher order” interac- tions are constructed by aggregating bivariate couplings in analyses such as motifs [7] or community detection [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' One of the largest issues holding back the direct study of higher-order interactions has been the lack of effective, accessible mathematical tools with which such interac- tions can be recognized [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Recently, however, work in the field of multivariate information theory has enabled the development of a plethora of different measures and frameworks for capturing statistical dependencies beyond the pairwise correlation [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The few applications of these techniques to brain data have suggested that higher-order dependencies can en- code meaningful bio-markers (such as discriminating be- tween health and pathological states induced by anes- thesia or brain injury [11]) and reflect changes associated with age [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Since the space of possible higher-order structures is so much vaster than the space of pairwise dependencies, the development of tools that make these structures accessible opens the doors to a large number of possible studies linking brain activity to cognition and behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Of the tools that have been applied, one of the most well developed is the partial information decomposition [13, 14] (PID), which reveals that multiple interact- ing variables can participate in a variety of distinct information-sharing relationships, including redundant, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='05307v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='NC] 12 Jan 2023 2 unique, and synergistic modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Redundant and synergis- tic information sharing represent two distinct, but related “types” of higher order interaction:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='Redundancy refers to information that is “duplicated” over many elements, so that the same information could be learned by observ- ing X1 ∨ X2, ∨, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , ∨XN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In contrast, synergy refers to information that is only accessible when considering the joint-states of multiple elements and no simpler combi- nations of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Synergistic information can only be learned by observing X1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ∧ XN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Redundant, and synergistic information sharing modes can be combined to create more complex relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For example, given three variables X1, X2, and X3, information can be redundantly common to all three, which could be learned by observing X1 ∨ X2 ∨ X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We can also consider the information redundantly shared by joint states: for example, the information that could be learned by observing X1 ∨ (X2 ∧ X3) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' observing X1 or the joint state of X2 and X3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For a finite set of inter- acting variables, it is possible to enumerate all possible information-sharing modes, and given a formal definition of “redundancy”, they can be calculated (for details see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The identification of redundancy and synergy as pos- sible families of statistical dependence raises questions about how such relationships might be reflected (or missed) by the standard, pairwise correlation-based ap- proach for inferring networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We propose two criteria by which we might assess the performance of bivariate functional connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The first we call specificity: the degree to which a pairwise correlation between some Xi and Xj reports dependencies that are unique to Xi and Xj alone, and not shared with any other edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In a sense, it reflects how appropriate the ubiquitous assump- tion that edges are independent is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The second criterion we call completeness: whether all of the statistical de- pendencies present in a data set are accounted for and incorporated into the model, or if predictive structure is “lost” when restrictive analyses are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We hypothesized that classical functional connectivity would prove to be both non-specific (due to the presence of multivariate redundancies that get repeatedly “seen” by many pairwise correlations) and incomplete (due to the presence of synergies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' To test this hypothesis, we used the a framework derived from the PID: the partial entropy decomposition [15] (PED, explained in detail be- low) to fully retrieve all components of statistical depen- dencies in sets of three and four brain regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' As part of this analysis, we propose a measure of redundant entropy applicable to arbitrarily-sized collections, which allows us to fully explore the space of higher order interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We chose the PED over the PID because the PID re- quires partitioning the system into predictors and “tar- gets” (the elements whose behavior we are predicting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This distinction is often artificial, and makes it difficult to analyze the system itself as a structured whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The PED does not require making a source/target distinction, and serves to generalize the PID to the analysis of whole systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' By computing the full PED for all triads of 200 brain regions, and a subset of approximately two mil- lion tetrads, we can provide a rich and detailed picture of beyond-pairwise dependencies in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Further- more, by separately considering redundancy and synergy instead of assessing just which one is dominant (as is commonly done [12, 16]), we can reveal previously un- seen structures in resting state brain activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' THEORY A Note on Notation In this paper, we will be making reference to multi- ple different “kinds” of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In general, we will use uppercase italics to refer to single variables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Sets of multiple variables will be denoted in boldface (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' X = {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , XN}, with subscript indexing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Spe- cific instances of a variable will be denoted with lower case font: X = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Functions (such as the probability, entropy, and mutual information), will be denoted using caligraphic font.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Finally, we will make a distinction be- tween expected values of information-theoretic quantities using upper case function notation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' the Shannon en- tropy of X is H(X), while the local entropy/surprisal is h(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For a brief review of local information theory, see the Supplementary Material Section S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Finally, when referring to the partial entropy function H∂ (described below), we will use superscript index notation to indicate the full set of variables that contextualizes the individual atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For example, H123 ∂ ({1}{2}) refers to the infor- mation redundantly shared by X1 and X2, when both are considered as part of the triad X = {X1, X2, X3}, while H12 ∂ ({1}{2}) refers to the information redundantly shared by X1 and X2 qua themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Partial Entropy Decomposition The partial entropy decomposition (PED) provides a framework with which we can extract all of the meaningful “structure” in a system of interacting ran- dom variables [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' By “structure”, we are referring to the (possibly higher-order) patterns of information- sharing between elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Consider a system X = {X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , XN}, comprised of N interacting, discrete random variables: the set of all informative relationships between elements (and ensembles of elements) in X forms its “structure.” We begin by defining the total entropy of X using the Shannon entropy: H(X) := − � x∈X P(x) log2 P(x) (1) Where x indicates a particular configuration of X and 3 X is the support set of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This joint entropy quantifies, on average, how much it is possible to “know” about X (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' how many bits of information would be required, on average, to reduce our uncertainty to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The entropy is a summary statistic describing an entire distribution P(X): H(X) = E[− log2 P(x)] (2) Where − log2 P(x) is the local entropy h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We can intuitively understand the local entropy with the logic of local probability mass exclusions [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Suppose that we observe X = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Upon observing x, we can immedi- ately rule out the possibility that X is in any state ¬x, and by ruling out those possibilities, we exclude all the probability mass associated with P(X = ¬x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' If P(x) is very low, then upon learning X = x, we exclude a large amount of probability mass (1 − P(x)), and conse- quently, h(x) is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Conversely, if P(x) is large, then only a small amount of probability mass is excluded, and so h(x) is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Quantifying Shared Entropy The measure h(x) is a very crude one: it gives us a single summary statistic that describes the behaviour of the “whole” without making reference to the structure of the relationships between x’s constituent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' If X has some non-trivial structure that integrates multi- ple elements (or ensembles of elements), then we pro- pose that those elements must “share” entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This no- tion of shared entropy forms the cornerstone of the PED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The way all of the parts of X share entropy forms the “structure” of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In the original proposal of the PED by Ince [15], shared entropy (Hcs) was defined using the local co-information, which treats the entropy of variables as sets and defines the shared entropy using inclusion-exclusion criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Unfortunately, as discussed by Finn and Lizier, the set-theoretic interpretation of mutlivariate mutual information is complex, as both the expected and local co-information can be negative [19], and the PED computed using Ince’s proposed method can result in negative values that are difficult to inter- pret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Here, we propose an alternative way to operationalize the notion of “redundant entropy” by saying that two variables X1, X2 ∈ X share entropy if they induce the same exclusions: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' if learning X1 or X2 rules out the same configurations of the whole [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Our goal, then, becomes to determine how the entropy of the whole is parcellated out over (potentially multivariate) sharing modes between parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In our toy system given by Table I, suppose we learn that X1 = 0 OR X2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Only one global state is ex- cluded: X = (1, 1) is incompatible with both possibil- ities, regardless of which is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Consequently we are P X1 X2 P00 0 0 P01 0 1 P10 1 0 P11 1 1 Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Joint entropy of two discrete random variables that together make up the macro-variable X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' only excluding P11 from the overall distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We can quantify this “shared entropy” using the local entropy of shared exclusions hsx: hx sx({1}{2}) = − log2 P(x1 ∪ x2) (3) Here, we are adapting the partial entropy notation first introduced by Ince in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The function hx sx({1}{2}) quantifies the total probability mass of P(X) excluded by learning either X1 = x1 or X2 = x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Said differently, it is the amount of information that could be learned from either variable alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Importantly, while it is a measure of dependency, it is distinct from the classic mutual in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We term this function hsx to indicate that it is the shared entropy based on common exclusions (“entropy of shared exclusions”) from some set of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We also note that the form of hsx is equivalent to the informative part of the local redundancy function derived by Makkeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=', [21], which they term isx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For a discussion of how hsx is related to isx and the deeper connections between partial entropy decomposition and partial information decomposition, see Appendix 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' So far, we have restricted our examples to the simple case of two variables, x1 and x2, however, we are inter- ested in the general case of information common to arbi- trarily large, potentially overlapping subsets of a system that has adopted a particular state x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This requires first enumerating the set of subsets, s, which we will call the set of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' It is equivalent to the power set of x, ex- cluding the empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For example, if x = {x1, x2, x3}, then the source set s is equal to: s = � � � � � {x1}, {x2}, {x3}, {x1, x2}, {x1, x3}, {x2, x3}, {x1, x2, x3} � � � � � (4) We are interested in how collections of sources a ∈ s might share entropy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' to what extent the exclude the same possible global configurations of x), which allows us to write our redundant entropy function in full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For a collection of sources {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , ak}: hsx(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , ak) := log2 1 P(a1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ∪ ak) (5) 4 hsx can be interpreted in terms of logical conjunctions and dysjunctions of variables [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Consider the example: hsx({x1}{x2, x3}), which quantifies the amount of prob- ability mass about the state of the “whole” that would be excluded by observing just the part x1 or the joint state of x2 and x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This relationship between probabil- ity mass exclusions on one hand, and formal logic on the other, places hsx on a sound conceptual footing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' While initially defined locally, it is possible to compute an ex- pected value Hsx for a joint distribution: Hsx(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , Ak) := E[hsx(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , ak)] (6) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The Partial Entropy Lattice Our function hsx has a number of appealing mathe- matical properties, which collectively satisfy the set of Axioms initially introduced by Williams & Beer for the problem of information decomposition [13] as applied to local information [18, 21]: Symmetry: hsx is invariant under permu- tation of it’s argument: hsx(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , ak) = hsx(σ(a1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , σ(ak)) Monotonicity: hsx decreases as more sources are added: hsx(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , ak) ≤ hsx(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , ak, ak+1) Self-redundancy: In the special case of a single source, hsx is equivalent to the classic local Shan- non entropy: hsx(a) = h(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For proof of these, see [21] Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Based on these properties, it is possible to specify the domain of hsx (all non-degenerate combinations of sources) in terms of a partially-ordered lattice structure A [13, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Not every combination of sources a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ak is a valid partial entropy atom, only those where no source is a subset of any other: A = {α ∈ P1(s) : ∀ai, aj ∈ α, ai ̸⊂ aj} (7) Where P1(s) indicates the power set of s, excluding the empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For an in-depth derivation of the lattice, see [13, 14, 18], for a visualization of the lattice, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The value of any element h∂(α) on the lattice can be computed via Mobius inversion: hx ∂(α) = hsx(α) − � β⪯α hx ∂(β) (8) The result is the entropy specific to a particular α and no simpler combination of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Furthermore, the structure of the lattice and the properties of hsx en- sure that hx ∂(α) will always be non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We can re-compute the total joint entropy of x as: h(x) = |A| � i=1 hx ∂(αi) (9) Like hsx, it is also possible to compute an expected value of h∂ (which will also be strictly non-negative): HX ∂ (α) = E[hx ∂(α)] (10) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Decomposing Marginal and Joint Entropies Having defined hsx and the Mobius inversion on the partial entropy lattice, we can now do a complete de- composition of the joint entropy, and its marginal com- ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For example, consider the bivariate system X = {X1, X2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We can decompose the joint entropy: H(X) = H12 ∂ ({1}{2}) + H12 ∂ ({1}) (11) + H12 ∂ ({2}) + H12 ∂ ({1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 2}) Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' we can decompose the associated marginal entropies in a manner consistent with the par- tial information decomposition [13]: H(X1) = H12 ∂ ({1}{2}) + H12 ∂ ({1}) (12) H(X2) = H12 ∂ ({1}{2}) + H12 ∂ ({2}) These decompositions can be done for larger ensem- bles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' or more statistical dependencies (see below) and can reveal how higher-order interactions can complicate (and in some cases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' compromise) the standard bivariate approaches to functional connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Mathematical Analysis of the PED The partial entropy decomposition reveals a rich and complex structure of statistical dependencies even in small systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Before considering the empirical results, it is worth discussing how the PED relates to classic mea- sures from information theory and what it reveals about the limitations of bivariate FC measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The first key finding is that the PED provides interest- ing insights into the nature of bivariate mutual informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Typically, mutual information is conflated with re- dundancy at the outset (for example, in Venn diagrams), however, when considering the PED of two variables X1 and X2, it becomes clear that: I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' X2) = H12 ∂ ({1}{2}) − H12 ∂ ({1, 2}) (13) 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The partial entropy lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The lattice of partial entropy atoms induced by the Hsx function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Each vertex of the lattice corresponds to a single PE atom, and the Venn diagram describes the associated structure of probability mass exclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The blue area indicates the probability mass from P(x) that is excluded by some combination of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For example, in the legend, we can see the probability mass excluded by observing X1 ∨ X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The blue area is all of the probability mass one would exclude after learning the state of either component alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The lowest atom is the entropy redundant to all three elements (Hsx({1}{2}{3})), and the dependencies get increasingly synergistic higher on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This relationship was originally noted by Ince [15] and later re-derived by Finn and Lizier [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In a sense, the higher-order information present in the joint-state of (X1 and X2) “obscures” the lower-order structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This is- sue is also inherited by parametric correlation measures based on the Pearson correlation coefficient, since the mutual information is a deterministic function of Pear- son’s ρ for Gaussian variables [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' When considering the decomposition of local mutual information into informative and misinformative compo- nents proposed by Finn and Lizier, it is clear that re- dundancy corresponds to the informative component of local mutual information, while synergy corresponds to the misinformative component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We can do a similar analysis extracting the bivariate mutual information from the trivariate PED, which re- veals that the bivariate correlation is not specific: I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' X2) = H123 ∂ ({1}{2}{3}) + H123 ∂ ({1}{2}) (14) − H123 ∂ ({3}{1, 2}) − H123 ∂ ({1, 2}{1, 3}{2, 3}) − H123 ∂ ({1, 2}{1, 3}) − H123 ∂ ({1, 2}{2, 3}) − H123 ∂ ({1, 2}) It is clear from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 15 that the bivariate mutual infor- mation incorporates information that is triple-redundant across three variables (H123 ∂ ({1}{2}{3})), and if one were to take the standard FC approach to a triad (pairwise correlation between all three pairs of elements), that the triple redundancy would be triple counted and er- roneously ascribed to three separate interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Fur- (1] [2] α= Xin X2 n X3 X d = X2n X3 X3 Xi X6 thermore, not only does bivariate mutual information double-count redundancy, but it also penalizes higher- order synergies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Any higher-order atom that includes the joint state of X1 ∧ X2 counts against I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Having established that the presence of higher-order redundancies explicitly precludes bivariate correlation from being specific, we now ask: can we improve the specificity using common statistical methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' One ap- proach aimed at “controlling” for the context of addi- tional variables in a bivariate correlation analysis is using conditioning or partial correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Typically, these anal- yses are assumed to improve the specificity of a pairwise dependency by removing the influence of confounders, however, by decomposing the conditional mutual infor- mation between three variables, we can see that condi- tioning does not ensure specificity: I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' X2|X3) = H123 ∂ ({1}{2}) (15) + H123 ∂ ({1}{2, 3}) + H123 ∂ ({2}{1, 3}) + H123 ∂ ({1, 2}{1, 3}{2, 3}) + H123 ∂ ({1, 3}{2, 3}) − H123 ∂ ({1, 2}) − H123 ∂ ({1, 2, 3}) The decomposition of I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' X2|X3) conflates the true pairwise redundancy (H123 ∂ {1}{2}) with the a higher- order redundancy involving the joint state of X1∧X3 and X2 ∧ X3: H123 ∂ {1, 3}{2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Furthermore, the conditional mutual information penalizes synergistic entropy shared in the joint state of all three variables (H123 ∂ {1, 2, 3}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Consequently, we can conclude that the specificity of bi- variate functional connectivity cannot be salvaged using conditioning or partial correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Not only does con- trolling fail to provide specificity, it also actively com- promises completeness, since it brings in higher-order in- teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Given that conditional mutual information and partial correlation are equivalent for Gaussian vari- ables [23], this issue also affects standard, parametric ap- proaches to conditional connectivity, just as with bivari- ate mutual information/Pearson correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' It is important to understand that these analytic re- sults are not a consequence of the particular form of hsx: any shared entropy function that allows for the forma- tion of a partial entropy lattice will produce these same results (many were first derived by Ince, who used a dif- ferent measure based on the local co-information [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Higher-Order Dependency Measures In addition revealing the structure of commonly-used correlations (bivariate and partial correlations), the PED can also be used to develop intuitions about multivariate generalizations of the mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Many of these generalizations exist, and here we will focus on four: the total correlation [24], the dual total correlation [25], the O-information [16, 26] (also called the “enigmatic” infor- mation [27]) and the S-information [26] (also called the “exogenous” information [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' While useful, these mea- sures are often difficult to intuitively understand, and can display surprising behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Since they can all be writ- ten in terms of sums and differences of joint and marginal entropies, we can use the PED framework to more com- pletely understand them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The oldest measure is the total correlation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' defined as: T (X) := |X| � i=1 H(Xi) − H(X) (16) which is equivalent to the Kullback-Leibler divergence between the true joint distribution P(X) and the product of the marginals: T (X) = DKL(P(X)|| |X| � i=1 P(Xi) (17) Based on equation 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' we can understand the total cor- relation as the divergence from the maximum entropy dis- tribution to the true distribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' implying that it might be something like a measure of the “total” structure of the system (as it’s name would suggest).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We can de- compose the 3-variable case to get a full picture of the structure of the TC: T (X1, X2, X3) = (2 × {1}{2}{3}) (18) + {1}{2} + {1}{3} + {2}{3} − {1, 2}{1, 3}{2, 3} − {1, 2}{1, 3} − {1, 2}{2, 3} − {1, 3}{2, 3} − {1, 2} − {1, 3} − {2, 3} − {1, 2, 3} We can see that the total correlation is largely a mea- sure of redundancy, sensitive to information shared be- tween single elements, but penalizing higher-order infor- mation present in joint states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This can be understood by considering the lattice in Figure 1: each of the H(Xi) terms will only incorporate atoms preceding (or equal to) the unique entropy term H123 ∂ (i) - anything that can only be seen by considering the joint-state of X will be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The second generalization of mutual information is the dual total correlation [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Defined in terms of entropies by: D(X) := H(X) − |X| � i=1 H(Xi|X−i) (19) where X−i refers to the set of every element of X ex- cluding the ith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The dual total correlation can be under- 7 stood as the difference between the total entropy of X and all of the entropy in each element of X that is “in- trinsic” to it and not shared with any other part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' When we decompose the three-variable case, we find: D(X1, X2, X3) = {1}{2}{3} (20) + {1){2} + {1}{3} + {2}{3} + {1}{23} + {2}{1, 3} + {3}{1, 2} + {1, 2}{1, 3}{2, 3} − {1, 2} − {1, 3} − {2, 3} − (2 × {1, 2, 3}) This shows that dual total correlation is a much more “complete” picture of the structure of a system than to- tal correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' It is sensitive to both shared redundan- cies and synergies, penalizing only the un-shared, higher- order synergy terms such as H123 ∂ ({1, 2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The sum of the total correlation and the dual total correlation is the exogenous information [27], also called by the S-information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' E(X) := T (X) + D(X) (21) Prior work has shown the exogenous entropy to be very tightly correlated with the Tononi-Sporns-Edelman complexity [16, 26, 28], a measure of global integra- tion/segregation balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' James also showed that the S-information quantified the total information that ev- ery element shares with every other element [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We can see that: E(X1, X2, X3) = (3 × {1}{2}{3}) + 2 × ({1}{2} + {1}{3} + {2}{3})) + {1}{2, 3} + {2}{1, 3} + {3}{1, 2} − {1, 2}{1, 3} − {1, 2}{2, 3} − {1, 3}{2, 3} − 2 × ({1, 2} + {1, 3} + {2, 3}) − (3 × {1, 2, 3}) This reveals that S-information to be an unusual mea- sure, in that it counts each redundancy term multiple times (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' in the case of three variables, the triple redun- dancy term appears three times, each double-redundancy term appears twice, etc), and penalizes them likewise when considering unshared synergies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The final, and arguably most interesting measure is the difference between the total correlation and the dual total correlation is often referred to as the O-information [26], and has been hypothesized to give a heuristic measure of the extent to which a given system is dominated by redundant or synergistic interactions: O(X) := T (X) − D(X) (22) where O(X) > 0 implies a redundancy-dominated structure and O(X) < 0 implies a synergy dominated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' PED analysis reveals: O(X1, X2, X3) = {1}{2}{3} (23) − {1}{2, 3} − {2}{1, 3} − {3}{1, 2} − (2 × {1, 2}{1, 3}{2, 3}) − {1, 2}{1, 3} − {2, 3}{1, 3} − {1, 2}{2, 3} + {1, 2, 3} This shows that the O-information generally satis- fies the intuitions proposed by Rosas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=', as it is positively sensitive to the non-pairwise redundancy (in this case just H123 ∂ ({1}{2}{3})) and negatively sensi- tive to any higher-order shared information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Curiously, O(X1, X2, X3) positively counts the highest, un-shared synergy atom (H123 ∂ ({1, 2, 3}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Conceivably, it may be possible for a set of three variables with no redundancy to return a positive O-information, although whether this can actually occur is an area of future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For three-element systems, the O-information is also equivalent to the co-information [26], which forms the base of the original redundant entropy function Hcs pro- posed by Ince [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' From this we can see that, at least for three variables, co-information is not a pure mea- sure of redundancy, conflating the true redundancy and the highest synergy term, as well as penalizing other higher-order modes of information-sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' A similar ar- gument was made by Williams and Beer using the mu- tual information-based interpretation of co-information [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' While the O-information and co-information di- verge for N > 3, we anticipate that the behavior of the co-information will remain similarly complex at higher N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' These results reveal how the PED framework can provide clarity to the often-murky world of multivariate information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Novel Higher-Order Measures From these PED atoms, we can construct a novel mea- sures of higher-order dependence that extends beyond TC, DTC, O-Information and S-Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' When considering higher-order redundancy, we are in- terested in all of those atoms that duplicate information over three or more individual elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We define this as the redundant structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For a three element system: SR(X1, X2, X3) = {1}{2}{3} (24) For a four-element system: SR(X1, X2, X3, X4) = {1}{2}{3}{4} (25) + {1}{2}{3} + {1}{2}{4} + {1}{3}{4} + {2}{3}{4} 8 And so on for larger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We can also define an analogous measure of synergis- tic structure: all those atoms representing information shared over the joint state of two or more elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For example, for a three element system: SS(X1, X2, X3) = {1}{2, 3} + {2}{1, 3} + {3}{1, 2} + {1, 2}{1, 3}{2, 3} + {1, 2}{1, 3} + {2, 3}{1, 3} + {1, 2}{2, 3} (26) For three element systems, the difference SR − SS is analagous to a “corrected” O-information: the atom {1, 2}{1, 3}{2, 3} is only counted once and the confound- ing triple synergy {1, 2, 3} is not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Finally, we can define a measure of total (integrated) structure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' all shared information) as the sum of all atoms composed of multiple sources: S = � α∈A α ⇐⇒ |α| > 1 (27) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Applications to the Brain The mathematical structure of the PED is domain ag- nostic: any complex system composed of discrete ran- dom variables is amenable to this kind of information- theoretic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In this paper, we focus on data col- lected from the human brain with functional magnetic resonance imaging (fMRI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For detailed methods, see the Materials & Methods section (V, but in brief, data from ninety five human subjects resting quietly was recorded as part of the Human Connectome Project [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' All of the scans were concatenated and each channel binarized about the mean [30] to create multidimensional, binary time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We then computed the full PED for all tri- ads, and approximatley two million tetrads, to compare to the standard, bivariate functional connectivity net- work (computed with mutual information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' By looking at the redundant and synergistic structures, and relating them to the standard FC, we can explore how higher-order dependencies are represented in bivari- ate networks, as well as what brain regions participate in more redundancy- or synergy-dominated ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' PED Reveals the Limitations of Bivariate Networks We now discuss how the PED relates multivariate mea- sures of bivariate network structure commonly used in the functional connectivity literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' These measures describe statistical dependencies between ensembles of regions, but mediated by the topology of bivariate con- nections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We hypothesized that this emergence from bi- variate dependencies would render them largely insen- sitive to synergies, which in turn would mean that such measures do not solve the issue of incompleteness in func- tional connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Following [31], we compared the redundant and syn- ergistic structure of triads and tetrads to a measure of subgraph strength: the arithmetic mean of all edges in the subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We found that the arithmetic mean FC density was positively correlated with redundancy for triads (ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='999, p < 10−20) and tetrads (ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='995, p < 10−20), indicating that information duplicated over many brain regions contributes to multiple edges, lead- ing to double-counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In contrast, for triads, arithmetic mean FC density was largely independent of synergistic structure (ρ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='05, p < 10−20), but for tetrads they were strongly anticorrelated (ρ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='988, p < 10−20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In addition to subgraph structure, another common method of assessing polyadic interactions in networks is via community detection [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Using the multi- resolution consensus clustering algorithm [32], we clus- tered the bivariate functional connectivity matrix into non-overlapping communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We then looked at the distributions of higher-order redundant and synergistic structure for triads and tetrads that spanned different numbers of consensus communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We found that tri- ads where all nodes were members of one community had significantly less synergy than triads that spanned two or three communities (Kolmogorov-Smirnov two sample test, D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='44, p < 10−20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The pattern was more pro- nounced when considering tetrads: tetrads that all be- longed to one community had lower synergy than those that spanned two communities (D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='45, p < 10−20), who in turn had lower synergy than those that spanned three communities (D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='37, p < 10−20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In Figure 2 (top row), we show cumulative probability density plots for the distribution of synergies for triads and tetrads that spanned one, two, three, and four FC communities, where it is clear that participation in increasingly diverse communities is associated with greater synergistic struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In contrast, redundant structure was higher in tri- ads that were all members of a small number of commu- nities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Triads that spanned three communities had lower redundancy than triads that spanned two communities (D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='48, p < 10−20), which in turn had lower redun- dancy than those that were all members of one commu- nity (D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='47, p < 10−20) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 2, bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' These results, coupled with the mathematical analysis of the PED discussed in Section I provide strong theoret- ical and empirical evidence that bivariate, correlation- based FC measures are largely sensitive to redundant in- formation duplicated over many individual brain regions, but largely insensitive to (or even anti-correlated with) higher-order synergies involving the joint state of mul- tiple regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' These results imply the possibility that there is a vast space of neural dynamics and structures that have not previously been captured in FC analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 9 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The limits of bivariate functional connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In triads, bivariate functional connectivity is largely independent of synergistic structure, and B, is very positively correlated with redundant structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In tetrads, bivariate functional connectivity is strongly negatively correlated with synergistic structure and D, is strongly correlated with redundant structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' E-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Triads that have all elements within one FC community have significantly less synergistic structure than those that have elements with two communities, while for redundnat structure, there was a clear pattern that the more FC communities a triad straddled, the lower it’s overall redundant structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' G-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The same pattern was even more pronounced in tetrads: as the number of FC communities a tetrad straddled increased, the expected synergistic structure climbed, while expected redundant structure fell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' PED with Hsx is consistent with O-information To test whether the PED using the Hsx redundancy function was consistent with other, information-theoretic measures of redundancy and synergy, we compared the average redundant and synergistic structures (as revealed by PED), to the O-information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We hypothesized that redundant structure would be positively correlated with O-information (as O > 0 implies redundancy dominance) and that synergistic structure would be negatively corre- lated, for the same reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For both triads and tetrads, our hypothesis was bourne out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The Pearson correlation between O-information and redundant structure was significantly positive for both triads (ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='72, p < 10−20) and tetrads (ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='82, p < 10−20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Conversely, the Pearson correlation between the O-information and the synergistic structure was sig- nificantly negative (triads: ρ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='7, p < 10−20, tetrads: ρ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='72, p < 10−20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' These results show that the structures revealed by the PED are consistent with other, non-decomposition-based inference methods and serves to validate the overall framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Interestingly, when comparing the triadic O- information to the corrected triadic O-information (which does not double-count H123 ∂ ({1, 2}{1, 3}{2, 3}) and does not add back in the atom H123 ∂ ({1, 2, 3})), we can see that the addition of H123 ∂ ({1, 2, 3}) can lead to erroneous conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Of all those triads that had a negative corrected O-information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' had a greater synergistic structure than redundant structure), 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='7% had a positive O-information, which could only be attributable to the presence of the triple-synergy being (mis)interpreted as redundancy and overwhelming the true difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This suggests that, for small systems, the O-information may not provide an unbiased estimator of redundancy/synergy balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Characterizing Higher-Order Brain Structures Having established the presence of beyond-pairwise re- dundancies and synergies in brain data, and shown that standard, network-based approaches show an incomplete picture of the overall architecture, we now describe the distribution of redundancies and synergies across the hu- man brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We began by applying a higher-order generalization of the standard community detection approach using a hy- pergraph modularity maximization algorithm [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' this algorithm partitions collections of (potentially overlap- ping) sets of nodes called hyperedges into communities that have a high degree of internal integration and a lower A B C D Triadic Synergy Tetradic Synergy Triadic Redundancy Tetradic Redundancy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='35 p=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='054 p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='999 p=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='988 p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='20 Arithmetic Mean FC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='70 Synergistic Structure Redundant Structure Synergistic Structure Redundant Structure E F G H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 1 Community 1 Community 2 Communities 2 Communities 3 Communities 3 Communities 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='8 Average 4 Communities Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='6 1 Community 2 Communities 1 Community 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='2 2 Communities 3 Communities 3 Communities 4 Communities Average Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='70 Synergistic Structure Redundant Structure Synergistic Structure Redundant Structure10 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Redundant and synergistic hypergraph community structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' A-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Surface plots of the two communities structures: on the left is the redundant structure and on the right is the synergistic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We can see that both patterns are largely symmetrical for both information-sharing modes, although the synergistic structure has two large, lateralized communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' C-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The co-classification matrices for redundant structure (left) and the synergistic structure (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The higher the value of a pair, the more frequently the hypergraph modularity maximization [33] assigns those two regions to the same hyper-community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The yellow squares indicate the seven canonical Yeo functional networks [34], and we can see that the higher-order redundant structure matches the bivariate Yeo systems well (despite consisting of information shared redundantly across three nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In contrast, the synergistic structure largely fails to match the canonical network structure at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For each of the 95 subjects and for each of the 1000 permutation nulls used to significance test the NMI between subject-level community structure and the master level structure, we computed the log-ratio of the empirical NMI to the null NMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For redundancy, there was not a single null, over any subject, that was greater than the associated empirical NMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For the case of the synergy, only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='6% of nulls were greater than their associated empirical NMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' degree of between-community integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We selected all those triads that had a greater synergistic structure than any of the one million maximum entropy null triads (see Materials and Methods), which yielded a set of 3,746 unique triads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' From these, we constructed an unweighted hypergraph with 200 nodes and 3,746 hyperedges (cast- ing each triad as a hyperedge incident on three nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We then performed 1,000 trials of the hypergraph cluster- ing algorithm proposed by Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=', [33], from which we built a consensus matrix that tracked how frequently two brain regions Xi and Xj were assigned to the same hyper-community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We repeated the process for the 3,746 maximally redundant triads to create two partitions: a synergistic structure and a redundant structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In Figure 3 we show surface plots of the resulting com- munities computed from the concatenated time series comprising all ninety-five subjects and all 4 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The redundant structure (left) is very similar to the canoni- cal seven Yeo systems [34]: we can see a well-developed DMN (orange), a distinct visual system (sky blue), a somato-motor strip (violet), and a fronto-parietal net- work (dark blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In contrast, when considering the syn- A B Redundant Communities Synergistic Communities D 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 25 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='8 50 50 Rate Rate 99 251 94191 75 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='6 Co-occurance I # 100 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='4 125 125 # 150 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='10 38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='2 175 175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0 50 100 150 0 50 100 150 Regions Regions E Redundancy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='5 Synergy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 Density 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='011 ergistic structure (right), a strikingly different pattern is apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Synergistic connectivity appears more lateral- ized over left and right hemispheres (orange and violet communities respectively), although there is a high de- gree of symmetry along the cortical midline comprised of apparently novel communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' These include a synergis- tic coupling between visual and limbic regions (sky blue), as well a occipital subset of the DMN (green) and a curi- ous, symmetrical set of regions combining somato-motor and DMN regions (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' These results show two things: the first is further confirmation that the canonical structures studied in an FC framework can be interpreted as reflecting primar- ily patterns of redundant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The second is that higher-order synergies are structured in non-random ways, combining multiple brain regions into integrated systems that are usually thought to be independent when considering just correlation-based analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' If the syner- gistic structure were reflecting mere noise, then we would not expect the high-degree of symmetry and structure we observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' To test whether the patterns we observed were con- sistent across individuals, we re-ran the entire pipeline (PED of all triads, hypergraph clustering of redundant and synergistic triads, etc) for each of the 95 subjects seperately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Then, for each subject, we computed the normalized mutual information (NMI) [6] between the subject-level partition and the relevant master partition (redundancy or synergy) created from the concatenated time series of all four scans from each of the ninety-five subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We significance tested each comparison with a permutation null model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For each null, we permuted the subject-level community assignment vector of nodes, re- computing the NMI between the master partition and a shuffled subject-level partition (1,000 permutations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In the case of the redundant partition, we found that that no subjects ever had a shuffled null that was greater than the empirical NMI: all had significant NMI (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='07).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In the case of the synergistic partition, 91 of the 95 subjects showed significant NMI (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='03, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='05, Benjamini-Hochberg FDR corrected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' These results sug- gest that both structures (redundant and synergistic) are broadly conserved across individuals, however, it ap- pears that the synergistic partitions are generally more variable between subjects than the redundant partition (which hews closer to the master partition constructed by combining the data from all subjects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' When we computed the normalized mutual information of all the subject level redundancy partitions to the canonical Yeo systems, we found a high degree of correlation (NMI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='6196±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0117, p < 10−20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The same analysis with the subject level synergy partitions found a much lower de- gree of concordance (NMI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='2290±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0117, p < 10−20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Redundancy-synergy gradient & time-resolved analysis Thus far, we have analyzed higher-order redundancy and synergy separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' To understand how they inter- act, we began by replicating the analysis of Luppi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=', [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We counted how many times each brain region ap- peared in the set of 3,746 most synergistic and 3,746 most redundant triads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We then ranked each node to create two vectors which rank how frequently each region par- ticipates in high-redundancy and high-synergy configu- rations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' By subtracting those two rank vectors, we get a measure of relative redundancy/synergy dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' A value greater than zero indicates that a region’s relative redundancy (compared to all other regions) is greater than its relative synergy (compared to all other regions), and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' By projecting the rank-differences onto the cortical surface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 4A), we recover the same gradient-like pat- tern first reported by Luppi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=', with relatively redun- dant regions located in primary sensory and motor cor- tex, and relatively synergistic regions located in multi- modal and executive cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This replication is notewor- thy, as Luppi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=', used an entirely different method of computing synergy (based on the information flow from past to future in pairs of brain regions), while we are looking at generalizations of static FC for which dynamic order does not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The fact that the same gradi- ent appears when using both analytical methods strongly suggests it is a robust feature of brain activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' A limitation of the analysis by Luppi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' is the re- striction that only average values of synergy and redun- dancy are accessible: the results describe expected values over all TRs and obscure any local variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The PED analysis using hsx can be localized (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' I) to individ- ual frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This allows us to see how the redundant and synergistic structure fluctuate over the course of a resting state scan, and how the distributions of relative synergies and redundancies vary over the cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Figure 4B shows how the redundant and synergistic structure fluctuate over the course of 1100 TRs taken from a single subject (for scans concatenated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This allows us to probe the information structure of previously identified patterns in frame-wise dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Analysis of instantaneous pairwise co-fluctuations (also called “edge time series”) reveals a highly structured pattern, with periods of relative disin- tegration interspersed with high co-fluctuation “events” [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The distribution of these co-fluctuations reflect various factors of cognition [38], generative structure [39], functional network organization [30], and individual dif- ferences [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' By correlating the instantaneous average whole-brain redundant and synergistic structures with in- stantaneous whole-brain co-fluctuation amplitude (RSS), we can get an understanding of the “informational struc- ture” of high-RSS “events.” We found that redundancy is positively correlated with co-fluctuation RSS (ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='6, p < 10−50) and synergy is negatively correlated with co- fluctuation amplitude (ρ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='43, p < 10−50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Given that synergy is known to drive bivariate functional con- 12 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Time-resolved analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Surface plots for the distributions of relative synergies and relative redundancies across the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' These results match prior work by Luppi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=', [35], with primary sensory and motor cortex being relatively redundant, while multi-modal association areas being relatively synergistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Over the course of one subject’s scan (1100 TRs), the total redundant and synergistic structure varies over time, although never so much that the curves cross (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' there is never more redundant structure than synergistic structure present).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Instantaneous redundant and synergistic structure are anti-correlated (ρ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='83, p < 10−50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Redundancy is positively correlated with the amplitude of bivariate co-fluctuations (ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='6, p < 10−50) and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' synergy is negatively correlated with co-fluctuation amplitude (ρ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='43, p < 10−50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For each TR, we show the difference in the rank-redundancy and rank-synergy for each node (red indicates a higher rank-redundancy than rank-synergy and vice versa for blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' When nodes are stratified by Yeo system [34] (grey, horizontal lines), it is clear that different systems alternate between high-redundancy and high-synergy configurations in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For every pair of columns in Panel F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' we compute the Pearson correlation between them to construct a time × time similarity matrix, which we then clustered using the MRCC algorithm [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Note that rows and columns are not in time order, but rather, re-ordered to reveal the state-structure of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Five example states (centroids of each community show in Panel G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=') projected onto the cortical surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' It is clear that the instantaneous pattern of relative synergies and redundancies varies from the average structure presented in Panel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For example, in States 3 and 4, the visual system is highly redundant (as in the average), however in state 5, the visual system is synergistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' nectivity [36], this is again consistent with the hypothesis that FC patterns largely reflect redundancy and are in- sensitive to higher-order synergies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' With full PED analysis completed for every frame, it is possible to compute the instantaneous distribution of relative redundancies and synergies across the cortex for every TR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The resulting multidimensional time-series can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 4F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' When sorted by Yeo systems [34], we can see that different systems show distinct relative redundancy/synergy profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The nodes in the somato- motor system had the highest median value (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 ± 73), followed by the visual system (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0±80), indicating that they were, on-average relatively more redundant than synergistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In contrast, the ventral attentional system A B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='35 陆 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='25 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='20 TRs C D E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='24 Synergy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='23 lergy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='. 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='24 200 200 400 4100 RSS Redundancy RSS F G Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 1 150 Som.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='Mot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 2 DAN Ranks Cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 3 VAN Lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 4 Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='2 100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='50 150 DMN Cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 5 200 TRs Rs H Centroid 1 entroid Centroid 3 Centroid 4 Centroid 513 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' State-to-state transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For each of the nine distinct states, we can see how many times each state transitions another (self-loops are not shown for visual clarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We can see that the various states have meaningful differences between each-other (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' the visual system or the somato-motor systems both transition from redundancy- to synergy-dominated configurations over time), however, within a state, the patterns are largely symmetrical across hemispheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' had the lowest median value (−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 ± 66), indicating a relatively synergistic dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Other systems seemed largely balanced: with median values near zero but a wide spread between them, such as the dorsal atten- tion network (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 ± 70), fronto-parietal control system (−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0±56), and the DMN (−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0±67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' These are systems that transiently shift from largely redundancy-dominated to synergy-dominated regimes in equal measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Finally, the limbic system had small values and relatively lit- tle spread (−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 ± 18), indicating a system that never achieved either extreme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We then correlated every TR against every other frame to construct a weighted, signed recurrence network [41], which we could then cluster using the MRCC algorithm [32] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 4G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This allowed us to assign every TR to one of nine discrete “states”, each of which can be rep- resented by its centroid (for five examples see Fig 4H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We can see that these states are generally symmetrical, but show markedly different patterns relative redundancy and synergy across the cortex, and some systems can change valance entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For example, in states three and four the visual system is highly redundant (consistent with the average behavior), while in state five the same regions are more synergy-dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In the same vein, the somato-motor strip is highly redundant in state 4, but slightly synergy-biased in state 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This shows that the dynamics of information processing are variable in time, with different areas of cortex transiently becoming more redundant or more synergistic in concert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The sequence of states occupied at each TR is a dis- crete time series which we can analyze as a finite-state machine (for visualization, see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Shannon tem- poral mutual information found that the present state was significantly predictive of the future state (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='59 bit, p < 10−50), and that the transitions between states were generally more deterministic [42, 43] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='29 bit p < 10−50) than would be expected by chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' While the sample size is small (1099 transitions), these results suggest that 3 514 the transition between states is structured in non-random ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' DISCUSSION In this paper, we have explored a novel framework for extracting higher-order dependencies from data and applied it to fMRI rcordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We found that the hu- man brain is rich in beyond-pairwise, synergistic struc- tures, as well as redundant information copied over many brain regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Based on a partial entropy decomposi- tion framework [15, 19] our method returns strictly non- negative values, does not require grouping elements into “sources” and “targets”, and is localizable, permitting a time-resolved analysis of the system’s dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Prior work on the partial entropy decomposition has analytically shown that the bivariate mutual information between two elements incorporates non-local information that is redundantly present over more than two elements [15, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This means that classic approaches to functional connectivity are non-specific: the link between two ele- ments does not reflect information uniquely shared by those two but double (or triple-counts) higher-order re- dundancies distributed over the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We verified this empirically by comparing the distribution of higher-order (beyond pairwise) redundancies to a bivariate correlation network and found that the redundancies closely matched the classic network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' These non-local redundancies shed new light on a well- documented feature of bivariate functional connectivity networks: the transitivity of correlation [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In func- tional connectivity networks, if Xi and Xj are correlated, as well as Xj and Xk, then there is a much higher-than expected chance that Xi and Xk are correlated (even though this is not theoretically necessary [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Since the Pearson correlation related the mutual information under Gaussian assumptions [22], we claim that the observed transitivity of functional connectivity is a consequence of previously-unrecognized, non-local redundancies copied over ensembles of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This hypothesis is consistent with our findings that redundancies correlate with key features of functional network topology, including sub- graph density and community structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In addition to higher-order redundancies, we also found strong evidence of higher order synergies: information present in the joint states of multiple brain regions and only accessible when considering “wholes” rather than just “parts.” These synergies appear to be structured in part by the physical brain (for example, being largely symmetric across hemispheres), but also don’t readily correspond to the standard functional connectivity net- works previously explored in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Since syn- ergiestic structures appear to be largely anti-correlated with the standard bivariate network structures, it is plau- sible that these synergistic systems represent a novel or- ganization of human brain activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' These higher-order interactions represent a vast space of largely unexplored, but potentially significant as- pects of brain activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' One possible avenue of study is how higher-order synergies reflect individual differ- ences [40, 46] and subject identifiability [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The find- ing that the synergistic community structure was more variable across subjects than the redundant structure suggests that synergistic dependencies may reflect more unique, individualized differences, while the redundant structure (reflected in the functional connectivity) repre- sents a more conserved architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This is consistent with recent theoretical work linking synergy to individu- ality [48], as well as empirical findings that the evolution of humans is associated with an enrichment of synergis- tic cortical structures [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The ability to expand be- yond pairwise network models of the brain into the much richer space of beyond-pairwise structures offers a the op- portunity to explore previously inaccessible relationships between brain activity, cognition, and behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Since normal cognitive functioning requires the coordi- nation of many different brain regions [49–51], and patho- logical states are associated with the dis-integrated dy- namics [52–54], it is reasonable to assume that alterations to higher-order, synergistic coordination may also reflect clinically significant changes in cognition and health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Re- cent work has already indicated that changes in bivariate synergy track loss of consciousness under anesthesia and following traumatic and anoxic brain injury [11] suggest- ing that higher-order dependencies can encode clinically significant biomarkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We hypothesize that beyond- pairwise synergies in particular may be worth exploring in the context of recognizing early signs of Alzheimer’s and other neurodegenerative diseases, as synergy requires the coordination of many regions simultaneously and may begin to show signs of fragmentation earlier than stan- dard, functional connectivity-based patterns (which are dominated by non-local redundancies may obscure early fragmentation of the system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Finally, the localizable nature of the Hsx partial en- tropy function allows us a high degree of temporal preci- sion when analyzing brain dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The standard ap- proach to time-varying connectivity is using a sliding- windows analysis, however, this approach blurs temporal features and obscures higher-frequency events [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' By being able to localize the redundancies and synergies in time, we can see that there is a complex interplay between both “types” of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' When considering expected values, we find a distribution of redundancies and syn- ergies that replicates the findings of Luppi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=', [35], however, when we localize the analysis in time, we find a high degree of variability between frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' It appears that there are not consistently “redundant” or “synergistic” brain regions (or ensembles), but rather, various brain regions can transiently participate in highly synergistic or highly redundant behaviors at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The structure of these dynamics appears to be non-random (based on the structure of the state-transition matrix), however, the significance of the various combinations of redundancy and synergy remains a topic for much future 15 work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The fact that some systems (such as the visual sys- tem) can be either redundancy- or synergy-dominated at different times complicates the notion of a “synergistic core”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Instead, there may be a “synergistic landscape” of configurations that the system traverses, with differ- ent configurations of brain regions transiently serving as the core and providing a flexible architecture for neural computation in response to different demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This analysis does have some limitations, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The most significant is that the size of the partial en- tropy lattice grows explosively as the size of the system increases: a system with only eight elements will have a lattice with 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='6×1022 unique partial entropy atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' While our aggregated measures of redundant and syn- ergistic structure can summarize the dependencies in a principled way, simply computing that many atoms is computationally prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In this paper, we took a large system of 200 nodes, and calculated every triad and a large number of tetrads, however, this also quickly runs into combinatorial difficulties, as the number of pos- sible groups of size k one can make from N elements grows with the binomial coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Heuristic measures such as the O-information can help, although as we have seen, this measure can conflate redundancy and synergy in sometimes surprising ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' One possible avenue of fu- ture work could be to leverage optimization algorithms to find small, tractable subsets of systems that show in- teresting redundant or synergistic structure, as was done in [16, 56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Alternately, coarse-graining approaches that can reduce the dimensionality of the system while preserving the informational or causal structure may al- low the analysis of a compressed version of the system small enough to be tractable [42, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In the context of this study, the use of fMRI BOLD data presents some inherent limitations, such as a small number of samples (TRs) from which to infer probabil- ity distributions, and the necessity of binarizing a slow, continuous signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Generalizing the logic of shared prob- ability mass exclusions remains an area of on-going work [59], although for the time being, the hsx function re- quires discrete random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' BOLD itself is also fundamentally a proxy measure of brain activity based on oxygenated blood flow and not a direct measure of neural activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Applying this work to electrophysciolog- ical data (M/EEG, which can be discretized in princi- pled ways to enable information-theoretic analysis [60]), and naturally discrete spiking neural data [61], will help deepen our understanding of how higher-order interac- tions contribute to cognition and behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The applica- bility of the PED to multiple scales of analysis highlights one of the foundational strengths of the approach (and information-theoretic frameworks more broadly): being based on the fundamental logic of inferences under con- ditions of uncertainty, the PED can be applied to a large number of complex systems (beyond just the brain), or to multiple scales within a single system to provide a detailed, and holistic picture of the system’s structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' CONCLUSIONS In this work, we have shown how the joint entropy of a complex system can be decomposed into atomic com- ponents of redundancy and synergy, which reveal higher- order, beyond-pairwise dependencies in the structure of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' When applied to human brain data, this partial entropy decomposition framework reveals previ- ously unrecognized, higher-order structures in the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We find that the well-known patterns of func- tional connectivity networks largely reflect redundant in- formation copied over many brain regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In contrast, the synergies for a kind of “shadow structure” that is largely independent from, or anticorrelated with, the bi- variate network and has consequently remained less well explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The patterns of redundancy and synergy over the cortex are dynamic across time, with different en- sembles of brain regions transiently forming redundancy- or synergy-dominated structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This space of beyond- pairwise dynamics is likely rich in previously unidentified links between brain activity and cognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The PED can also be applied to problems beyond neuroscience and may provide a general tool with which higher-order structure can be studied in any complex system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' MATERIALS & METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Human Connectome Project fMRI Data The data used in this study was taken from a set of 100 unrelated subjects included in the Human Connectome Project (HCP) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Refs [29, 62] provide a detailed de- scription of the acquisition and preprocessing of this data, which have been used in many previous studies[30, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Briefly, all subjects gave informed consent to protocols approved by the Washington University Institutional Re- view Board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Data was collected with a Siemens 3T Con- nectom Skyra using a head coil with 32 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Func- tional data analysed here was acquired during resting state with a gradient-echo echo-planar imaging (EPI) se- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Collection occurred over four scans on two sep- arate days (scan duration: 14:33 min;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' eyes open).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The main acquisition parameters included TR = 720 ms, TE = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='1 ms, flip angle of 52°, 2 mm isotropic voxel resolu- tion, and a multiband factor of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Resting state data was mapped to a 200-node parcellation scheme [63] covering the entire cerebral cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Considerations for subject inclusion were established before the study and are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The mean and mean absolute deviation of the relative root mean square (RMS) motion throughout any of the four resting scans were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Subjects that exceeded 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='5 times the in- terquartile range in the adverse direction for two or more measures they were excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This resulted in the exclu- sion of four subjects, and an additional subject due to a software error during diffusion MRI processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The in- cluded subjects had demographic characteristics of: 56% 16 female, mean age = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='29 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='66, age range = 22-36 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Preprocessing The minimal preprocessing of HCP rs-fMRI data can be found described in detail in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Five main steps were followed: 1) susceptibility, distortion, and motion correction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 2) registration to subject-specific T1-weighted data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 3) bias and intensity normalization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 4) projection onto the 32k fs LR mesh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' and 5) alignment to common space with a multimodal surface registration (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This pipeline produced an ICA+FIX time series in the CIFTI grayordinate coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We included two addi- tional preprocessing steps: 6) global signal regression and 7) detrending and band pass filtering (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='008 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='08 Hz) [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We discarded the first and last 50 frames of each time series after confound regression and filtering to pro- duce final scans with length 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='2 min (1,100 frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' All four scans from 95 subjects were then z-scored and con- catenated to give a final time-series of 200 brain regions and 418,000 time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Discretizing BOLD Signals Unfortunately, the Hsx measure is only well-defined for discrete random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Consequently, we discretized our data by binarizing the z-scored time series: setting any value greater than zero to one and any value less than zero to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Prior work has established that transform- ing BOLD signals into binary point processes preserves the majority of the total correlation structure [30, 65], so we are confident that our analysis is robust, especially considering the large number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We chose to binarize around the z-score (as opposed to alternative point-processing techniques such as local maxima), as the z-score ensures that each individual channel is generally maximally entropic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' P(Xi = 1) ≈ P(Xi = 0) ≈ 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This ensures that every indi- vidual channel has approximately the same entropy, and so deviations from maximum entropy at the level of the entire triad or tetrad can only emerge from correlations between two or more channels, rather than being influ- enced by biases at the channel-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The choice to bina- rize about the mean also links this work to previous work on decomposing functional connectivity into discrete par- titions [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Statistical Analyses 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Triads & tetrads In standard FC analysis, it is typical to compute the pairwise correlation between all pairs of brain regions, re- sulting �N 2 � unique pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For this analysis, we computed all triads of brain regions, resulting in �200 3 � = 1, 313, 400 unique triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For each triad, we computed the joint entropy, and performed the full partial entropy decom- position to compute each of the eighteen partial entropy atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Finally, each of the atoms was normalized by the total joint entropy, to give a measure of how much each atom contributes to the whole entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This allows us to directly compare triads that have different joint en- tropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' It was not feasible to brute-force all possible tetrads, which is a set of approximately sixty-four million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In- stead, we randomly sub-sampled sets of four randomly, collecting 1954000 tetrads (≈ 3% of the total space) and analyzing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Bivariate functional connectivity networks To directly compare the PED framework to the stan- dard, correlation-based FC network framework, we con- structed single, representative FC network by computing the pairwise mutual information between every pair of regions in the fMRI scan (as was done in [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Y ) = H(X) + H(Y ) − H(X, Y ) (28) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Subgraph Analysis Since we are interested in how the bivariate FC frame- work reflects (or fails to reflect) higher-order redundan- cies and synergies, we also compute a battery of struc- ture metrics on matching subgraphs taken from the FC network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Formally presented by Onnela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=', [31], we consider arithmetic mean of the subgraph connectivity: GA(X) = � i̸=j I(Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Xj) |X|2 − |X| (29) For a given triad of tetrad X, we compared the mean FC density to the various redundant and synergistic information-sharing structures of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Community Detection on Bivariate Matrices Multi-resolution consensus clustering [32] was used to detect network communities in the functional connectiv- ity matrix across multiple scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The algorithm pro- ceeds in three main stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In the first stage, modular- ity maximization using the Louvain method is performed for 1,000 different values of the resolution parameter, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This produced a range of γ values that resulted with par- titions having between 2 and N communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The sec- ond stage consisted of a more fine-grained sweep (10,000 steps) over the γ values defined in the first stage of the 17 process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We aggregate the partitions produced by this sweep into a node-by-node co-classification matrix stor- ing how frequently nodes are partitioned into the same community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' A null model with expected values of co- classification based on the size and number of commu- nities was subtracted from the co-classification matrix [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Finally, in the third stage, the null-adjusted co- classification matrix was clustered again using consensus clustering with 100 repetitions and a consensus threshold τ of 0 [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The resulting partition was used for analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We assessed the similarity between single-subject par- titions and consensus partitions using Normalized Mu- tual Information (NMI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Each partition can be formal- ized as a vector of integers of dimension N whose entries denote the nodes’ allegiance to communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' NMI esti- mates the similarity between two partitions by counting co-occurrences in the two vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We computed NMI between each one of the 95 single- subject partitions and the consensus partition, in both cases of redundancy and synergy hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We as- sessed the significance of NMI values by comparing them with a null case obtained by randomly shuffling 1000 times communities labels in the single-subject partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The p-values of the statistical test, calculated as the frac- tion of null-case NMI greater than the actual NMI, have been corrected with a Benjamini-Hochberg test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Null Model To ensure that the statistical dependencies we were observing reflect non-trivial interactions, we significance- tested triads and tetrads against a null distribution com- posed of one million, maximum entropy null models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We constructed sets of totally independent, maximum en- tropy binary time series and computed the PED on each set of three or four null channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' From this, we can con- struct distributions of the expected null structure and expected synergistic structure against which to compare triads and tetrads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Hypergraph Community Detection Each of the triads can be thought of as a hyper-edge on a 3-uniform hypergraph of 200 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For the synergistic structure, we selected only those hyperedges who had a greater synergistic structure than any of the one million maximum-entropy nulls that formed our null distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This resulted in a hypergraph with 200 hundred nodes and 3,746 regular hyper-edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We used the same criteria to build a redundant structure hypergraph using the top 3,746 most redundant hyperedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Both hypergraphs were clustered using the HyperNetX package (available on Github: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' com/pnnl/HyperNetX) implementation of the hyper- modularity optimization by Kumar and Vaidyanathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=', [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Briefly, the algorithm by Kumar and Vaidyanathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=', takes a modularity maximization approach to parti- tioning the vertices of a hypergraph into non-overlapping communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In dyadic networks, the modularity func- tion compares the distribution of within- and between- community edges to the expected distribution based on a degree-preserving, configuration null model [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In the case of hypergraphs, a hyper-configuration model can be used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' A generalized modularity metric can then be used as an objective function in a Louvain-based, mod- ularity maximization search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Temporal Structure To explore the temporal structure of the state- transition series, we used the active information storage [68, 69] (a measure of how predictable is the future given the past) and the determinism [42, 43], (a measure of how constrained the future is given the past).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For a one dimensional, discrete random variable X that evolves through time, we can compute the information that the past Xt−1 discloses about the future Xt with the mutual information: AIS(X) = I(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Xt) (30) This measure quantifies the degree to which knowing the past reduces our uncertainty about the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This term can be further decomposed into two components: the determinism and the degeneracy [42]: I(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Xt) = Det(X) − Deg(X) (31) Where determinism is: Det(X) = log2(N) − H(Xt|Xt−1) (32) And degeneracy is: Deg(X) = log2(N) − H(Xt) (33) The determinism quantifies how reliably a given past state xt−1 leads to a single future state xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' If P(xt|xt−1) ≈ 1, then we say that xt−1 deterministically leads to xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We significance tested both the active information stor- age and the determinism by comparing the empirical val- ues to an ensemble of ten thousand randomly permuted nulls generated by shuffling the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Since the degeneracy is unchanged by permutation of the temporal structure (since the marginal entropy H(Xt) is the same), any changes in active information storage produced by shuffling must be driven by changes in the determinism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 18 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Software All partial information/entropy decompositions were done using the SxPID package released with [21] and can be accessed on Github: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='com/ Abzinger/SxPID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' All scripts required to reproduce this analysis will be attached as supplementary material to the final published work.' metadata={'source': 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Computation, edited by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Prokopenko (Springer, Berlin, Heidelberg, 2014) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' 115–158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' [70] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Kolchinsky, A Novel Approach to the Partial Informa- tion Decomposition, Entropy 24, 403 (2022), number: 3 Publisher: Multidisciplinary Digital Publishing Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' [71] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' MacKay, Information Theory, Inference and Learning Algorithms (Cambridge University Press, 2003) google-Books-ID: AKuMj4PN EMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' SI 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' MATHEMATICAL PROPERTIES OF Hsx Partial Entropy Decomposition & Partial Information Decomposition The redundant entropy function hsx is closely related to the redundant information function isx proposed by Makkeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=', [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Our function hsx is defined: hsx(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , ak) = log 1 P(a1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ∪ ak) (34) This measure is equivalent to the informative compo- nent of the measure isx proposed by Makkeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=', [21] in the context of single-target partial information decom- position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The local redundant information function isx is defined: isx(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , ak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' y) := (35) log2 P(y) − P(y ∩ (¯a1 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ∩ ¯ak)) 1 − P(¯a1 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ∩ ¯ak) − log2 P(y) (36) Which can be further decomposed into informative and misinformative components [17]: i+ sx(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , ak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' y) := log2 1 P(a1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ∪ ak) (37) i− sx(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , ak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' y) := log2 P(y) P(y ∩ (a1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ∪ ak)) (38) isx(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , ak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' y) = i+ sx(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , ak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' y) − i− sx(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , ak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' y) (39) Where it is clear that hsx(·) = i+ sx(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' y), with the sole difference that i+ sx(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' y) is implicitly defined with respect to some target variable y (although y has no actual im- pact on the value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Below, we show that, if the target y is set to the joint state of the whole (x), then the par- tial entropy decomposition of h(x) with hsx as the shard entropy function becomes equivalent to the partial in- formation decomposition i(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , xN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' x) with isx as the redundant entropy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The notion that the PED 21 is equivalent to doing the PID of the information all the “parts” disclose about the “whole” was mentioned par- enthetically in [21], although the finding that the infor- mative component is all that is required is novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Given the equivalence between hsx(·) and i+ sx(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' y), it suffices to show that i− sx(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , xN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' x) = 0 bit in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' When y = x, we can re-write the function as: i− sx(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , xN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' x) = (40) log2 P(x1 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ∩ xN) P((x1 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ∩ xN) ∩ (x1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , ∪xN)) the union of x1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ∪ xk is clearly a superset of x1 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ∩ xk, so i− sx = log2 P(x1 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ∩ xN) P(x1 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ∩ xN) (41) Which is clearly log2(1) = 0 bit □ We can understand the partial entropy decomposition using hsx as being equivalent to the decomposition of i(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' , xN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Intuitively, this is consistent with the identity for discrete variables that I(X, X) = H(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This has curious implications: for example, while x1 may misinform about x2, neither variable can misinform about their contribution to the state of the whole x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Example: Logical Exclusive-OR (XOR) Gate XOR AND P X1 ⊕ X2 = T P X1 ∧ X2 = T 1/4 0 0 0 1/4 0 0 0 1/4 0 1 1 1/4 0 1 0 1/4 1 0 1 1/4 1 0 0 1/4 1 1 0 1/4 1 1 1 Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Logical XOR and AND gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' To demonstrate how partial entropy decomposition can be used to untangle higher-order interactions, consider the logical exclusive-OR (XOR) gate (for the lookup ta- ble, see Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The XOR gate is an example of a synergistic logic gate: the ability to predict the state of the target T depends on having access to both X1 and X2 jointly: the pariwise marginal mutual informations are equal to 0: I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' T) = I(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' T) = 0 bit, but the joint mutual information is nonzero: I(X1, X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' T) = 1 bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We can initially see that the triple-redundancy H12T ∂ ({1}{2}{T}) = 0 bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This is because any configu- ration of logical disjunctions does not actually rule out any states: for example, P(X1 = 0∪X2 = 0∪T = 0) = 1 as there is no configuration (1, 1, 1) that can be excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Other results can be unintuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For example, most of the partial entropy is shared between the three bi- variate relationships H12T ∂ ({1}{2}), H12T ∂ ({1}{T}), and H12T ∂ ({2}{T}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' How is this consistent with the fact that the mutual information between any pair of vari- ables is zero?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The bivariate redundancy can be non- zero in this case because, on average, knowing the local state of x1 ∨ x2 reduces our uncertainty about the joint state of {x1, x2, t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For example, suppose we learn that x1 = 1 ∨ x2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This excludes the joint configuration {x1 = 0, x2 = 0, t = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This exclusion of the associated probability mass is recognized by hsx(·) as informative, in that it reduces our uncertainty about the joint-state of the whole, despite the fact that, on average, X1 and X2 disclose no information about T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' There is no redun- dant information common to X1, X2 and T, however, and there a number of higher-order dependencies, such as H12T ∂ ({1}{2, T}) and H12T ∂ ({1, 2}{1, T}{2, T}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Atom H12T ∂ || XOR AND MaxEnt {1}{2}{T} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='193 {1}{2} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='415 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='222 {1}{T} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='415 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='222 {2}{T} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='415 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='222 {1}{2, T} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='041 {2}{1, T} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='041 {T}{1, 2} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='041 {1} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='292 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='322 {2} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='292 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='322 {T} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='322 {1, 2}{1, T}{2, T} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='018 {1, 2}{1, T} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='093 {1, 2}{2, T} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='093 {1, T}{2, T} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='093 {1, 2} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='17 {1, T} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='17 {2, T} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='17 {1, 2, T} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='245 Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The Partial Entropy Decomposition for the XOR, AND, and Maximum Entropy Gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' S3: Independent Variables One unusual property of hsx, as demonstrated by the logical-XOR results is that independent variables can still share entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This is a recognized feature of multiple measures of redundant information/entropy and is gen- erally considered to be an issue to be excised [70] (some have gone so far as the suggest an axiom that such a property must be disallowed from the outset [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' While we understand that shared entropy for random variables may seem initially counter intuitive, it can be readily understood when considering the problem of inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Let us return to our two element example (Table I), and this time specify that X1⊥X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' We know then that 22 I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' X2) = 0 bit, however, hsx({X1}{X2}) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='415 bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The answer is that, while the two variables are in- dependent, in all cases learning either X1 = x1∨X2 = x2 is sufficient to exclude a single possible state: the case where X1 = ¬x1 ∧ X2 = ¬x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' If we were to formalize this in terms of a gambling problem, we would find that, despite the independence of both variables, a player is, in fact, more likely to win with a correct guess after learning X1 ∨ X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' See Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Utility of redundant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Suppose an agent plays a gambling game, where two independent, binary variables are set at random (so all outcomes P(x1, x2) = 1/4 for all configurations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' If the agent guesses the correct vari- able, they win $1 and if they guess wrong, they win nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Clearly, the expected value of each trial is $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='25 (blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' However, if another agent learns that X1 = x1 ∨ X2 = x2, then they can do better at the game, with an expected value of each trial of $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The difference between the two cu- mulative distributions of 1000 trials is the extra “value” that can be extracted from the redundant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This shows that, while counter-intuitive, the fact that H12 ∂ ({1}{2}) > 0 even if X1⊥X2 is interpretable in practical contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Furthermore, we can see that, while hsx will be greater than zero for small, maximum entropy systems, as the system gets larger, the redundancy will logarithmically trend towards zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The proof for binary systems is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For a discrete, maximum entropy sys- tem with k elements, learning the state of X1 = x1 ∨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ∨ Xk = xk will always exclude a single state: the state where X1 ̸= x1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' ∧ Xk ̸= xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This single state x∗ will have P(x∗) = 1/k (as all states have the same probability by the maximum entropy constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The union of all surviving configurations will be 1 − P(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Since limk→∞ 1/k = 0, then the union probability will → 1 and consequently hsx → 0 bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' This suggests that, for very large, idealized systems (such as an ideal gas), the redundancy does go to 0 bit for maximum entropy systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' How other values (such as the redundant and synergistic structure) behave remains an area of further study, although we conjecture that, as k → ∞, redundan- cies and synergies will vanish faster than unique terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' BASIC INFORMATION THEORY REVIEW Here we will provide a basic overview of information theory for unfamiliar readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For a more comprehensive treatment of the subject, see the textbooks by Cover & Thomas [22] and/or MacKay [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The basic object of study in information theory is the entropy, which quantifies the total uncertainty that we, as observers, have about the state of some variable X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For the purposes of this paper, we will assume that X is discrete, with a finite number of possible states that can be pulled from the support set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' For every particular state x ∈ X, there is an associated probability P(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' The entropy of X is given by: H(X) = − � x∈X P(x) log P(x) (42) For multiple variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' we can define the joint entropy as: H(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' X2) = − � x1∈X1 x2∈X2 P(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' x2) log P(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' x2) (43) We can also define the conditional entropy as the un- certainty about X1 left over after accounting for the knowledge that X2 = x2: H(X1|X2) = − � x1∈X1 x2∈X2 P(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' x2) log P(x1|x2) (44) From these basic components,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' we can define the mu- tual information as the difference between our initial un- certainty about the state of X1 the the remaining uncer- tainty about X1 that is not resolved by learning the state of X2: I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' X2) = H(X1) − H(X1|X2) (45) The mutual information is symmetric in it’s argu- ments: I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' X2) = I(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' X1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' If we have multiple Xs disclosing information about a single target T, the joint mutual information has the same form: I(X1, X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' T) = H(T) − H(T|X1, X2) (46) The mutual information can also be written in terms of probabilities: No Information 300 Redundant Information Utility of Redundancy 250 200 150 100 50 0 0 200 400 600 800 1000 Number of Trials23 I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' X2) = � x1∈X1 x2∈X2 P(x1, x2) log P(x1|x2) P(x1) (47) Local Information Theory Both the entropy and the mutual information can be understood as “expected values” over some (potentially multivariate) distribution): H(X) = E[− log P(x)] (48) The term − log P(x) is known as the local entropy or the Shannon information content and it quantifies how “surprised” we, as observers are to see that X = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' It is typically denoted as h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' X2) = E � log2 P(x1|x2) P(x1) � (49) The term P (x1|x2) P (x1) is known as the local mutual in- formation and it quantifies the divergence between the “prior” probability X1 = x1 and the posterior probabil- ity X1 = x1 after accounting for the fact that X2 = x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' It is typically denoted as i(x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' Unlike the expected mutual information, which is strictly non-negative, the local mutual information can be either greater than, or less than, zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' If P(x1|x2) < P(x1), then i(x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' x2) > 0, and if P(x1|x2) < P(x1), then i(x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' x2) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQf3g3K/content/2301.05307v1.pdf'} +page_content=' In the latter case, we say that x1 misinforms on the state of x2.' 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VASSILIEV AND M. SUMETSKY* +Aston Institute of Photonic Technologies, Aston University, Birmingham B4 7ET, UK +*Email: m.sumetsky@aston.ac.uk + + +We experimentally demonstrate that side-coupling of coplanar bent optical fibers can induce a high Q-factor whispering +gallery mode (WGM) optical microresonator. To explain the effect, we consider WGMs with wavelengths close to the cutoff +wavelengths (CWs) of these fibers which slowly propagate along the fiber axes. In the vicinity of the touching region, WGMs +of adjacent fibers are coupled to each other, and CWs experience sub-nanoscale axial variation proportional to the coupling +strength. We show that in certain cases the CW variation leads to full localization of the WGMs and the creation of an optical +microresonator. By varying the characteristic curvature fiber radius from the centimeter order to millimeter order, we +demonstrate fully mechanically reconfigurable high Q-factor optical microresonators with dimensions varying from the +millimeter order to 100-micron order and free spectral range varying from a picometer to hundreds of picometers. The new +microresonators may find applications in cavity QED, microresonator optomechanics, frequency comb generation with +tunable repetition rate, tunable lasing, and tunable processing and delay of optical pulses. + + +1. Introduction +Microphotonic devices and circuits commonly consist of one or +multiple connected basic elements, such as waveguides, couplers, +and ring resonators [1, 2]. In addition to the requirements of high +fabrication precision and low losses [2, 3], the tunability of these +circuits and devices is of critical importance for a variety of +applications [4, 5]. While more complex tunable microphotonics +circuits are targeted at tunability enabling quite arbitrary +predetermined signal processing (see e.g., [1]), simple microdevices, +such as standing along tunable three-dimensional microresonators, +allow for unique functionalities not possible to achieve by other +means. For a variety of applications, the tunability of spherical, +toroidal, and bottle microresonators has been demonstrated using +mechanical stretching, heating, and nonlinear light effects including +those in monolithic and specially coated microresonators [6-10]. In +most of these approaches, it is only possible to tune series of +wavelength eigenvalues simultaneously without noticeable change +in their separation. + However, for several applications, which include cavity QED [8, +11, 12], optomechanics [13, 14], frequency microcomb generation +[15, 16], optical signal processing and delay [4, 5, 17], and lasing +[18-21], it is critical to have microresonators with tunable +eigenwavelength separation. For example, the latter allows the +creation of optical frequency microcomb generators and microlasers +with continuously tunable repetition rate and wavelength and to tune +the microresonator eigenfrequency separation in resonance with the +frequency of its mechanical oscillations. Considerable variation of +the eigenwavelength separation commonly requires the variation of +microresonator dimension and/or its refractive index parameters by +the quantity comparable with their original values. One approach to +solve this problem consists in using Fabry-Perot microresonators +with tunable mirror separation which contain the optical materials +under interest [12, 21, 22]. Additional flexibility of tuning can be +achieved by employing Fabry-Perot microresonators with a liquid +material inside [21] or translating a wedge-shaped solid optical +material to vary its dimensions inside the Fabry-Perot +microresonator [23]. + +Fig. 1. (a) Coplanar bent optical fibers touching each other. The fiber +profile is manipulated by bending and translation of the fiber tails +indicated by curved and straight arrows. (b) Illustration of coupling +between the input-output microfiber and WGMs in Fiber 1 and Fiber 2 +near cutoff wavelengths. + Alternatively, of special interest is attaining the eigenwavelength +separation tunability in three-dimensional monolithic high Q-factor +microresonators, e.g., those with spherical, toroidal, and bottle + +(a) +Fiber 2 +Fiber 1 +(b) +Coupled +WGMs +(12) +Fiber 2 +(z) +2n2 +nin? +Direct +Fiber 1 +l1n1 +contact +x +Microfiber +Kn1 +Zshapes. This will allow us to add tunability to the emerging +applications of these microresonators in QED, optomechanics, +lasing, and frequency comb generation noted above. However, the +deformation of most of these monolithic microresonators to achieve +significant change of their eigenwavelength separation is unfeasible. + A unique exception, though, is exhibited by SNAP (Surface +Nanoscale Axial Photonics) microresonators [24]. These +microresonators are introduced at the surface of an optical fiber by +its nanoscale deformation, which causes the nanoscale variation of +the cutoff wavelengths (CWs) controlling the slow propagation of +whispering gallery modes (WGMs) along the fiber axis (see [24, 25] +and references therein). In Ref. [26], a SNAP microresonator +induced and fully reconfigurable by local heating of an optical fiber +was demonstrated. In Ref. [27], it was shown that it is possible to +create a SNAP microresonator and control its dimensions by local +bending of an optical fiber. Both approaches allow for tuning of +eigenwavelength separation of microresonators by the quantity +comparable to or larger than its original value. However, in both +approaches, the induced microresonator shapes had limited +flexibility and their characteristic axial dimensions could not be +reduced below several millimeters. In the first case, this restriction +was caused by the imposed length of the characteristic heat +distribution along the fiber. In the second case, the reduction of +microresonator size was limited by the smallest curvature radius +corresponding to the fiber breakage threshold. + In this paper we report on our discovery of a new type of WGM +optical microresonators which belongs to the group of SNAP +microresonators. We show that side coupled coplanar bent fibers +(Fig. 1) can induce a high Q-factor SNAP microresonator localized +in the region of fiber coupling. The configuration of fibers shown in +Fig. 1 allows us to flexibly tune the shape of the induced SNAP +microresonators and their axial dimensions from several tens of +microns to several millimeters and, respectively, tune their +eigenwavelength separation from hundreds of picometers to a +picometer. +2. Cutoff wavelengths of uncoupled and side- +coupled straight fibers +First, it is instructive to consider the behavior of CWs for +uncoupled and side-coupled straight optical fibers. For this +purpose, we cleave a 125-micron diameter uncoated +commercial optical fiber into two pieces (Fiber 1 and Fiber 2), +which are then coaxially aligned and put into contact along +3.5 mm of their length as shown in Fig. 2(a). Light is launched +into Fiber 1 by a transversely oriented taper with the +micrometer +diameter +waist +(input-output +microfiber) +connected to the Optical Spectrum Analyzer (OSA). After +coupling into Fiber 1, light forms WGMs propagating along +the fiber surface. In the region of direct contact of fibers (Fig. +2(a)), WGMs in Fiber 1 and Fiber 2 are coupled to each other. + To characterize the effect of interfiber coupling, we measured the +spectrograms of the configured fiber system. For this purpose, the +input-output microfiber was translated along Fiber 1 (Figs. 1(b) and +2(a)) touching it periodically with the spatial resolution of 2 µm. At +the cut end of Fiber 1, the microfiber was moved towards Fiber 2 and +continued scanning Fiber 2. The spectrograms of transmission +power 𝑃�𝜆, 𝑧� were measured as a function of wavelength  and +microfiber position z along the axis of Fiber 1. + +Fig. 2. (a) Illustration of side-coupled straight optical fiber configuration. +(b) Spectrogram of this configuration. (c) Magnified section outlined in +the spectrogram (b). + The measured spectrogram of our fiber system is shown in Fig. +2(b). The left- and right-hand sides of this spectrogram show the +spectrograms of uncoupled Fiber 1 and Fiber 2, respectively. Lines +in spectrogram shown in Fig. 2(b) indicate the CWs of uncoupled +and coupled fibers. These CWs correspond to WGMs with different +azimuthal and radial quantum numbers. The magnified copy of the +section outlined in Fig. 2(b) is shown in Fig. 2(c). It is seen that the +CWs appear as straight lines slightly tilted with respect to the +horizontal direction. From the measured magnitude of tilt, ε� � +0.015 nm/mm, we determine the linear variation of the fiber radius +∆𝑟� � 𝑟�ε�/𝜆� � 0.6 nm/mm [28]. In the latter rescaling relation, +we used 𝑟� � 62.5 µm and 𝜆� � 1.55 nm. By linear extrapolation +of CWs of Fiber 1 and Fiber 2 (dashed white lines), we confirm that, +as expected, their positions (horizontal black dashed line) coincide at +the cut ends of these fibers. + At the 3.5 mm long region of fiber touching, WGMs in Fiber 1 +couple to WGMs in Fiber 2 and the corresponding CWs split. The +structure and positions of CWs in the touching region depend on the +magnitude of coupling and will be further discussed in Section 4. +Here we note that the value of CW splitting found, e.g., from Fig. +2(c) is ~ 0.1 nm, which coincides with characteristic values of CW +variation in SNAP microresonators [24, 25]. In particular, the +positive CW shift in the coupling region leads to the WGM +localization and creation of a microresonator which can be tuned by +changing the length of the side-coupled fiber segment. In our current +experiment, the Q-factor of the induced SNAP resonator was poor +due to the scattering of light at the imperfectly cleaved fiber ends, + +(a) +Nottoscale +3.5 mm +Fiber2 +Fiber1 +125μm +Microfiber +1549.00 +(b) +1548.50 +-5 +TransmissionPower (dB) +(wu) +1548.00 +length +1547.50 +10 +Wavel +1547.00 +15 +1546.50 +1546.00 +-20 +1545.50 +0 +1 +2 +3 +4 +5 +6 +Distance along fiber (mm) +0 +1548.50 +(c) +1548.40 +-5 +length +1548.20 +-10 +1548.10 +Javel +15 +1548.00 +1547.90 +-20 +1547.80 +0 +1 +2 +3 +4 +5 +6 +Distancealongfiber(mm)which, typically, ensure around 70% WGM reflectivity [29]. +Nevertheless, we suggest that the demonstrated resonator can be +directly used to create miniature broadly tunable optical delay lines +generalizing our previous results based on the SNAP +microresonators with fixed dimensions [30, 31]. Indeed, in these +devices the WGM pulses complete only a single round trip along the +fiber axis and therefore their attenuation at the fiber facets may +reduce the output light power by around 50% only. We also suggest +that, after feasible improvement, the Q-factor of these +microresonators can be significantly improved as further discussed +in Section 6. +3. Basic experiment +In our proof-of-concept experiments, we used 125-micron diameter +uncoated commercial silica optical fibers touching each other as +shown in Fig. 1(a). The ends of Fiber 1 and Fiber 2 were bent and +translated to arrive at the required profile of these fibers near their +coupling region illustrated in Fig. 1(b). The fibers used were either +originally straight or preliminary softened in a flame and bent +permanently. As described in the previous section, WGMs were +launched into Fiber 1 by a transversely oriented microfiber +connected to the OSA. If the separation between Fiber 1 and Fiber 2 +is small enough, WGMs penetrate from Fiber 1 into Fiber 2. + In the simplest configuration considered in this Section, Fiber 1 +was straight, and coplanar Fiber 2 was bent. The fibers were put in +contact and then slightly pushed towards each other to increase the +coupling region. The photograph of the fiber configuration used in +this experiment is shown in Fig. 3(a). From this picture, we estimated +the curvature radius of the bent fiber as 𝑅~30 mm (see further +discussion of the fiber profile in Section 4). Fig. 3(b) shows the +spectrogram of the configured structure measured along the 3.5 nm +bandwidth within the 700 µm axial length of Fiber 1. At the edges of +the scanned region, the interfiber coupling is negligible. In these +regions, CWs do not noticeably change with distance 𝑧 and, thus, +correspond to Fiber 1 only. The arrangement of CWs in these regions +is similar to that in Fig. 2(b). + +Fig. 3. (a) Photograph of the side-coupled fibers used in the experiment. The upper fiber is bent with the curvature radius 𝑅~30 mm and the lower +fiber has the curvature radius greater than 1 m. (b) The spectrogram measured in the vicinity of the coupling region of these fibers. (b1) and (b2) +Spectrograms showing the magnified sections outlined in the spectrogram (b). (c1) and (c2) Spectrograms of the microresonators numerically +calculated in the two-mode approximation detailed in the text, which replicate the experimental spectrograms in Figs. (b1) and (b2), respectively. + +(a) +R~30 mm +Experiment +3-10 +nsertion +1mm +-15 +(b1) +1549.65 +1549.46 +1549.48 +1547.45 +(b) +(b2) +Wavelength (um) +1549.60 +1550.00 +-2 +1547.40 +1549.55 +(b2) +-4 +1549.50 +1549.50 +-6 +Transmission Power (dB) +1549.45 +Wavelength (nm) +1549.00 +-8 +1549.40 +1547.20 +1549.35 +1548.50 +-10 +1547.15 +1549.30 +-12 +1548.00 +200 +400 +600 +200 +400 +600 +-14 +Distancealongfiber(um) +Distance along fiber (μm) +1547.50 +(b1) +-16 +-18 +1547.00 +Theory +-20 +0 +200 +400 +600 +Distance along fiber (μm) +(c1) +(c2)0.4 +0.4 +-5 +Wavelengthvariation(nm) +Wavelength variation (nm) +0.3 +0.3 +0.2 + 0.2 +-15 +0.1 +0.1 +0.0 +0.0 +-20 +0 +200 +400 +600 +0 +200 +400 +600 +Distancealongfiber(um) +Distance along fiber (μm) The effect of coupling shows up in the central region of the +spectrogram in Fig. 3(b). In this region, different CWs exhibit +different positive and negative variations along the axial length 𝑧. +The exemplary regions of this spectrogram named (b1) and (b2) are +magnified in Figs. 3(b1) and 3(b2), respectively. It is seen that, as +expected, in contrast to negative variations, positive CW variations +lead to the WGM confinement and the creation of microresonators. +Our estimates illustrated in the inset of Fig. 3(b2) show that the Q- +factor of the created microresonator (which measurement was +limited by the 1.3 pm resolution of the OSA used) exceeds 10�. The +observed CW variations in Figs. 3(b1) and (b2) can be explained by +the theory described below. +4. Basic theory +We assume that the fiber bending is small enough so that the +propagation of light along the axial direction of side-coupled +fibers (Fig. 1(b)) can be considered as propagation along a +single waveguide with asymmetric cross-section including +both fibers. The wavelengths of slow WGMs are close to the +CWs 𝜆��𝑧� of this compound waveguide. To determine the +complex-valued CWs 𝜆��𝑧�, we introduce the original CWs +𝜆��� � � +�𝛾��� and 𝜆��� � � +�𝛾��� of unbent Fiber 1 and Fiber 2 +with the imaginary parts determined primarily by material +losses and scattering of light at the fiber surface. We assume +that there are 𝑁� and 𝑁� cutoff wavelengths in Fibers 1 and +Fiber 2, respectively, which contribute to the resonant +transmission, so that 𝑛� � 1,2, … , 𝑁�, 𝑗 � 1,2. We refer to the +integers 𝑛, 𝑛� and 𝑛� as to the transverse quantum numbers. +Variation of 𝜆��𝑧� is caused by bending of fibers [27] and, in +our case, primarily by their coupling. In the absence of the +input-output fiber, the CWs of our system, 𝜆 � 𝜆��𝑧�, 𝑛 � +1,2, … , 𝑁� � 𝑁�, are determined as the roots of the +determinant: + + + +det +( ) +0 +z +  + +I +Ξ + (1) + +Here 𝐈 is the unitary �𝑁� � 𝑁�� � �𝑁� � 𝑁�� matrix and +matrix + +1 +1 +12 +† +12 +2 +2 +( ) +( ) +( ) +( ) +( ) +z +z +z +z +z + + + +  + + + + +Λ +Δ +Δ +Ξ +Δ +Λ +Δ +. (2) + +The submatrices in Eq. (2) determine the original CWs of +Fiber 1 and Fiber 2, 𝚲� � �𝜆��� � � +�𝛾����, couplings inside +each of the fiber caused by bending, 𝚫��𝑧� � �δ���� +��� +�𝑧��, and +interfiber +couplings +𝚫���𝑧� � �δ���� +���� �𝑧�� +, +𝑚�, 𝑛� � +1,2, … 𝑁�. + As in SNAP [24], dramatically small nanometer and sub- +nanometer scale variations of CWs 𝜆��𝑧� along the compound +fiber waveguide can localize WGMs and induce an optical +microresonator having eigenwavelengths 𝜆�� with axial +quantum numbers 𝑞. Due to the smooth and small CW variation +and proximity of the localized WGM wavelengths 𝜆�� to +𝜆��𝑧�, the corresponding eigenmode can be presented as +𝐸���𝑥, 𝑦, 𝑧� � Ψ���𝑧�Ω��𝑥, 𝑦, 𝑧� where the transverse WGM +distribution Ω��𝑥, 𝑦, 𝑧� is calculated at the CW 𝜆��𝑧� and +depends on 𝑧 parametrically slow [32], and function Ψ���𝑧� +determines the axial dependence of the microresonator +eigenmode amplitude and satisfies the one-dimensional wave +equation [24] + +2 +3/2 +2 +2 +3/ 2 +2 +( , ) +0, +( , ) +( ) +. +n +r +n +n +n +n +n +d +n +z +z +z +dz + + + + + + + + +  +  + + + (3) + +where 𝑛� is the refractive index of the fibers. + The coupling parameters 𝜅���𝑧�between WGM 𝐸���𝑥, 𝑦, 𝑧� +and the input-output wave in the microfiber is determined by +their overlap integral. Commonly, the microfiber diameter is +much smaller than the characteristic axial variation length of +𝐸���𝑥, 𝑦, 𝑧�. For this reason, similar to the analogous +approximation in the SNAP platform [24, 33], the coupling +parameters 𝜅���𝑧� are proportional to the values of +𝐸���𝑥, 𝑦, 𝑧� at the axial coordinate 𝑧 of the input-output +microfiber. Then, calculations based on the Mahaux- +Weidenmüller theory [34-36] presented in Supplementary +Material allowed us to express the transmission power 𝑃�𝜆, 𝑧� +through the input-output microfiber coupled to the considered +fiber configuration (Fig. 1(b)) as + +1 +2 +1 +2 +2 +* +1 +1 +1 +( ) +( , , ) +( , ) +1 +( ) +( , , ) +N +N +n +n +n +N +N +n +n +n +D z G z z +P z +D z G z z + + + + + + + + + + + + +. (4) + +Here 𝐺��𝑧�, 𝑧�, 𝜆) is the Green’s function of Eq. (3). Eq. (4) +generalizes the expression for the transmission power +previously derived in Ref. [24]. As shown below, functions +𝐷��𝑧� can be expressed through and have characteristic values +similar +to +the +coupling +D-parameters +which +were +experimentally measured previously and typically have the real +and imaginary parts ~ 0.01 µm-1 [24, 33]. Close to the resonance +condition, 𝜆 � 𝜆��, for sufficiently small losses and coupling, and +separated CWs 𝜆��𝑧�, only one Green’s function with number 𝑛 +contributes to the sums in Eq. (4). Then, Eq. (4) coincides with that +previously derived in Ref. [24]. However, generally, the +contribution of more than one term to the sums in Eq. (4) may be +significant. +Before the detailed description of the spectrograms in Figs. +2(b) and 3(b), we note that the transmission power plots in +these figures characterize the CWs of the coupled fiber system +determined by Eq. (1) viewed by the input-output microfiber +and, subsequently, OSA. Therefore, the CWs of Fiber 2, which are +the solutions of Eq. (1) but uncoupled from Fiber 1 cannot be +seen by the OSA. On the other hand, the number of CWs which +can show up in the coupling region can be as many as 𝑁� � 𝑁�, +i.e., significantly greater than the number 𝑁� of visible +uncoupled CWs of Fiber 1 (see Fig. 2(b) as an example). + To clarify the effect of coupling between WGMs in adjacent +fibers, we consider the two-mode approximation, 𝑁� � 𝑁� � 1, +assuming that the wavelength 𝜆 of the input light is close to an + +unperturbed single WGM CW 𝜆�� � � +�𝛾 of Fiber 1 and a single CW +𝜆�� � � +�𝛾 of Fiber 2 having the same imaginary part. Consequently, +in Fig. 1(b) we now set 𝑛� � 𝑛� � 1. We neglect the effect of the +CW variation due to the fiber bending [27], which is usually smaller +than the effect of fiber coupling, setting 𝛿�� +��� � 0. Then, the CWs +𝜆��𝑧� and 𝜆��𝑧� of the compound fiber are found from Eq. (1) as + + + + + + + +2 +2 +(12) +1,2 +11 +21 +11 +21 +11 +1 +1 +( ) +( ) +2 +4 +z +i +z + + + + + + + + + + + + + + (5) + + The dependence on the transverse coordinates 𝑥 and 𝑦 (Fig. 1(b)) +of the compound WGM corresponding to CWs 𝜆��𝑧� can be +calculated as follows. We introduce the unperturbed WGMs in Fiber +1 and 2 (considered unbent and uncoupled) calculated at their CWs +𝜆�� and 𝜆�� as Ω� +����𝑥, 𝑦� and Ω� +����𝑥, 𝑦�. Then, in the two-mode +approximation, the compound modes generated by weak coupling +of modes Ω� +����𝑥, 𝑦� and Ω� +����𝑥, 𝑦� are determined as [37] + +(1) +(2) +1 +1 +1 +2 +2 +(1) +(2) +2 +1 +1 +2 +2 +(12) +11 +11 +21 +1 +( ) +( , , ) +( , ) +( , ), +1 +( ) +1 +( ) +( ) +1 +( , , ) +( , ) +( , ), +1 +( ) +1 +( ) +( ) + ( ) +. +z +x y z +x y +x y +z +z +z +x y z +x y +x y +z +z +z +z + + + + + + + + + + + + + + + + + + +  + + + + + + + + + + + + + + + (6) + +Consequently, the coupling parameters to the microfiber entering +Eq. (4) at coordinate 𝑧 are + +1 +2 +2 +2 +( ) +( ) +, +( ) +, +1 +( ) +1 +( ) +D +D +z +D z +D z +z +z + + + + +  + + + + + + (7) + +where 𝐷 is the z-independent coupling parameter between the input- +output microfiber and Fiber 1 [24, 33]. + To map the bent fiber axial profile ℎ�𝑧� to the CW envelope +profiles of the induced microresonators, we have to determine the +relation between ℎ�𝑧� and coupling coefficient 𝛿�� +�����𝑧�. Similar +to calculations in Refs. [38, 39], for the smooth and small ℎ�𝑧� +considered here, we find + + + +1/2 +(12) +2 +11 +0 +2 +( ) +exp +1 +( ) +r +z +n +h z + + + + + + + + + + + + + +, (8) + +where 𝛿� is 𝑧-independent. Assuming the simplest profile of the +bent fiber having the curvature radius 𝑅 as + +ℎ�𝑧� � 𝑧�/2𝑅 (9) + +for silica fibers with 𝑛� �1.44, we estimate the FWHM of 𝛿�� +�����𝑧� +as 𝑧����~0.5�𝜆𝑅��/�. At 𝜆~1.55 µm and 𝑅~30 mm of our +experiment, we have 𝑧����~100 µm. From Eqs. (5) and (8), we +find that the FWHM of the CW, depending on the value of 𝜆�� � +𝜆��, is between 𝑧���� and 2𝑧���� which is only in qualitative +agreement with the microresonator FWHM 𝑧����~ 250 µm +found from experimental data in Figs. 3(b1) and (b2). + The results of our numerical modeling in the two-mode +approximation considered based on Eqs. (3)-(9) are shown in Figs. +3(c1) and 3(c2). To fit the experimental data, we set the average CW +0.5(𝜆�� � 𝜆��) = 1.55 µm, the CW difference 𝜆�� � 𝜆�� � 0.05 +nm in Fig. 3(c1) and 𝜆�� � 𝜆�� � �0.05 nm in Fig. 3(c2), +coupling parameter 𝐷 � �0.01 � 0.01𝑖 µm-1 [24, 33], Q-factor +𝑄 � 10�, the microresonator FWHM 𝑧����~ 250 µm and its +spectral height ~ 0.15 nm, similar to these values found from Figs. +2(b1) and (b2). + The experimental spectrograms in Fig. 3(b1) and (b2) and +theoretical spectrograms in Figs. 3(c1) and (c2) look nicely similar. +However, important differences between them should be noted. +From Eqs. (8) and (9), the FWHM value 𝑧����~ 250 µm +corresponds to the Fiber 2 curvature radius 𝑅~66 mm, which is +twice as large as that measured from the fiber image shown in Fig. +3(a). We suggest that the difference is caused by the deviation of the +shape of Fiber 2 from parabolic in the coupling region as well as by +the fiber misalignment. The additional deformation of fibers may be +induced by their electrostatic attraction and pressuring, which are not +visible in Fig. 3(a). Our suggestion is confirmed by the experimental +profiles of the induced microresonator envelopes and CW shapes in +Figs. 3(b1) and (b2) which, as compared to those in the theoretical +spectrograms in Figs. 3(c1) and (c2), have larger side slopes and are +flatter in the middle. Next, we notice that, in the theoretical +spectrograms, the CW wavelength profiles are more mirror- +symmetric to the microresonator envelopes with respect to the +horizontal line (following Eq. (5)), while, in the experimental +spectrograms, the lower CW profiles are shallower than the +microresonator envelopes. We suggest that this deviation can be +eliminated by taking into account the coupling with other WGMs +ignored in the two-mode approximation considered. +5. Tunability +Bending and translating the tails of Fiber 1 and Fiber 2 side- +coupled to each other as illustrated in Fig. 1 allowed us to tune +the dimensions of the fiber coupling region and thereby tune the +dimensions of created microresonators. As in the previous +sections, in our experiments we used 125 µm optical fibers. We +investigated the cases of the smallest microresonators +containing a few wavelength eigenvalues and having the +characteristic axial dimensions of hundred microns (Figs. 5(a1)- +(a4)), as well as larger microresonators with dimensions of +several hundred microns (Figs. 5(b1)-(b4) and (c1)-(c3)) and +the largest microresonator having the axial length of 5 +millimeters (Fig. 4(d)). +Considering the smallest microresonators, we monitored the +process of their creation. Side-coupling of a straight Fiber 1 and +Fiber 2 bent with a sufficiently small curvature radius of ~ 1 mm +introduced small perturbation in CWs shown in the +spectrogram in Fig. 4(a1). Increasing the fiber radius further, we +arrived at the microresonator with a single eigenwavelength +(Fig. 4(a2)). The inset inside the Fig. 4(a2) spectrogram, which +magnifies the region near this eigenwavelength, shows that the +axial dimension of the corresponding eigenmode is ~ 200 µm. +Remarkably, except for the axial dimension of localized WGMs +with +uniform +magnitude +in +specially +designed +bat + +microresonators [39, 40], this dimension (which expansion is +critical, e.g., for QED applications [41]) is the record large +characteristic +WGM +dimension +demonstrated +in +microresonators to date. The measured Q-factor of this +microresonator (limited by the 1.3 pm resolution of the OSA +used) was slightly greater than 10�. + Larger bending radii of Fiber 2 having the order of 10 mm led to +the creation of microresonators with millimeter-order axial +dimensions having the spectrograms shown in Figs. 5(b1)-(b4) and +(c1)-(c3). The close to parabolic shape of these microresonators +suggests that they can be used, e.g., as tunable optical frequency +comb generators [42]. We note that the behavior of the CWs and +microresonators envelopes in most of these spectrograms cannot be +accurately described by the two-mode approximations of Section 4. +Of particular interest is the spectrogram shown in Fig. 4(c2). At first +sight, the envelop of the microresonator in this spectrogram is the +continuation of the CW of Fiber 1 (compare with Figs. 3(b1) and +(c1)). Unexpectedly, the axial WGM localization in this +microresonator (caused by the WGM reflection from the CW- +generated turning points [24]) sharply dissolves inside the +microresonator area. + +Fig. 4. Tunability of microresonators. (a1)-(a4) Spectrograms of induced microresonators for small curvature radius of Fiber 2 ~ 1 mm. (b1)-(b4) and +(c1)-(c3) spectrograms of induced microresonators for a lager radius of Fiber 2 ~ 10 mm. (d) Spectrogram of a 5 mm long microresonator induced by +touching straight Fiber 1 and Fiber 2 which was preliminary permanently bent at the ends as shown in the inset. + +(a1) +(a2) +(a3) +(a4) +R=1.2mm +R=1.5mm +R=1.6mm +1.7mm +1546.80 +1546.80 +1546.80 +1546.80 +ap +(w +(wu +1546.6 +(wu +(wu +1546.70 +1546.70 +-5 +-45 +-5 +1546.60 +1546.60 +-6 +1546.60 +1546.60 +0200400600 +0200400600 +0200400600 +0200400600 +Distancealongfiber(μm) +Distancealongfiber(um) +Distancealongfiber(μm) +Distancealongfiber(um) +(b1) +(b2) +(b3) +(b4) +R=6.1mm +R=7.7mm +R=8.1mm +R=16.3mm +1551.90 +1551.90 +1551.90 +1551.90 +10 +10 +1551.80 +-20 +1551.80 +32 +1551.70 +1551.70 +1551.70 +1551.70 +lavele +4 +-5 +1551.60 +1551.60 +≤1551.60 +1551.60 +-5 +1551.50 +1551.50 +1551.50 +200400600800 +200400600800 +200400600800 +1551.50 +0 +200400600800 +Distancealongfiber(um) +Distancealongfibor(μm) +Distancealongfiber(um) +Distancealong fiber(um) +(c1) +(c2) +(c3) +R=18mm +R~30mm +R~30mm +1551.80 +1551.80 +Transmission Power (dB) +1551.80 +2 +(dB) +-2 +1551.70 +4 +-4 +nbu +.6 +6 +1551.60 +1551.60 +AeM +-8 +8 +1551.50 +1551.50 +1551.50 +-10 +-10 +-10 +0 +400 +800 +1200 +1600 +400 +800 +1200 +1600 +0 +400 +800 +1200 +1600 +Distancealong fiber(um) +Distancealongfiber(um) +Distancealongfiber (um) +(d) +1548.60 +length +4 +6 +-10 +1548.30 +0 +1000 +2000 +3000 +4000 +5000 +6000 +Distancealong fiber(um)To create longer microresonators, we, first, permanently bent +the tails of Fiber 2 as illustrated in the inset of Fig. 4(d). This +allowed us to arrive at an arbitrarily large curvature radius of +this fiber including its straight shape between the bent tails. As +an example, Fig. 4(d) shows the spectrogram of a 5 mm long +microresonator. Though the eigenwavelength width of this +microresonator is greater than its free spectral range, we +suggest that, in contrast to the lossy microresonators induced by +side-coupled cleaved straight fibers demonstrated in Section 2, +its Q-factor is similar to that of the smaller microresonators +considered in this section and Section 3. +6. Discussion +The effect of induction of high Q-factor WGM tunable optical +microresonators in side-coupled optical fibers discovered in this +paper enables a range of exciting generalizations and applications. +Further extension of tuning flexibility can be achieved by enabling +different boundary conditions at the fiber tails (Fig. 1(a)), different +interfiber touching stresses, and different preliminary permanent +fiber bending. + Configurations of fibers, which are potentially attractive for future +research and applications, are illustrated in Fig. 5. Fig. 5(a) shows a +way to create long microresonators alternative to the method +utilizing fibers with permanently bent tails illustrated in Fig. 4(d). In +the configuration of Fig. 5(a), the length of the induced +microresonator increases as the curvature radii of touching fibers +approach each other. Provided that the variation of the fiber radii can +be performed so that the parabolicity of the induced microresonators +was maintained, the configuration of Fig. 5(a) can serve for the +generation of the optical frequency combs with a tunable repetition +rate. + +Fig. 5. (a) Bent fibers with increased coupling region. (b) Bent fibers +with increased coupling region and abrupt side of the induced +microresonator. (c) A bottle microresonator side-coupled to a fiber. (d) +Side coupled straight fibers with tapered facets forming a rectangular +microresonator. (e) Two straight fibers with tapered facets coupled to +the third straight fiber forming a rectangular microresonator. (f) +Twisted side-coupled fibers. (g) A microcapillary fiber filled with liquid +and side coupled to a bent fiber. (h) Three straight coupled fibers. + In Fig. 5(b), the lower fiber is terminated with a short taper, which +can be introduced using, e.g., a CO2 laser. Simple estimates show +that a taper with a characteristic length of 100 µm at the end of a 125 +µm diameter optical fiber creates an abrupt CW barrier with a slope +of ~ 100 nm/µm at 1.5 µm wavelength. The steepness of the slope +of this barrier (critical for impedance matching of light from the +input-output microfiber [43]) is 100 times greater than that +demonstrated in Ref. [44] with the femtosecond laser inscription. +The configuration shown in Fig. 5(b) can be used for the creation of +miniature dispersionless tunable optical delay lines provided that the +shape of the induced microresonator is kept semi-parabolic in the +process of tuning [43]. + Experimental investigation and development of the theory of +WGMs in a microresonator side-coupled to an optical fiber is of +particular interest. Fig. 5(c) illustrates the side coupling of a fiber and +a bottle microresonator. While the fiber is open-ended, coupling of +the bottle microresonator to the straight fiber can cause the +localization of light in the fiber, similar to the coupling between bent +optical fibers considered above. The configuration shown in Fig. 5(c) +suggests a way of tuning the microresonator eigenwavelengths. + The fiber configuration shown in Fig. 5(d) is similar to two +straight side-coupled fibers considered in Section 2. To improve the +Q-factor of the microresonator induced along the coupling region, +the cleaved ends of fibers shown in Fig. 2(a) are modified by the +tapered ends. The configuration of fibers shown in Fig. 5(e) +illustrates an alternative way to create tunable microresonators when +the position of both their sides can be tuned. The rectangular +microresonators induced in both configurations can be used for the +creation of tunable delay lines which, as shown in Ref. [31], can be +dispersionless with a good accuracy. + The coupling of twisted optical fibers illustrated in Fig. 5(f) is +interesting to investigate both theoretically and experimentally. In +the cylindrical coordinates �𝑧, 𝜌, 𝜑� of one of the fibers, the curve +along which the fibers touch each other corresponds to the azimuthal +angle 𝜑 � 𝜑� � 𝛼𝑧 , where 𝛼 is the twisting coefficient. The +corresponding value of the WGM field is proportional to +exp �𝑖𝛽𝑧 � 𝑖𝑚�𝜑� � 𝛼𝑧�� where 𝛽 is the propagation constant. +From this expression, a WGM at CW corresponding to 𝛽 � 0 is +seen by another fiber as a mode with nonzero propagation constant. +Thus, in contrast to the untwisted fibers, coupling between the side- +coupled twisted fibers is essentially three dimensional. + Fig. 5(g) shows a microcapillary fiber filled with liquid and side- +coupled to a bent fiber. For the microcapillary with sufficiently thin +walls, a microresonator induced inside it by the side-coupled fiber +performs nonlocal sensing of liquid [45]. In Refs. [46] and [47], such +microresonators were introduced with the CO2 laser and slow +cooking methods. Fig. 5(g) suggests the simplest approach for the +realization of nonlocal microfluidic sensing. + Fig. 5(h) illustrates three straight side-coupled fibers. In contrast +to two coupled fibers, WGMs launched into this configuration will +propagate into both azimuthal direction and, in particular, into the +positive and negative directions of the input-output microfiber with +approximately the same amplitudes. The channel formed between +these fibers can be used for gas and microfluidic sensing. Unlike the +microcapillary illustrated in Fig. 5(g), no ultrathin wall enabling the +WGM sensing of the internal channel is required in this case. +While the model of two coupled CWs developed here +qualitatively explains some characteristic features of the + +(a) +(f) +(b) +(g) +(c) +(d) +(h) +(e)experimentally measured spectrograms, the complete explanation +and quantitative fitting of the experimental data should include the +effect of several CWs and be based on the further development of +the coupled wave theory. The future theory should also allow us +to express the fiber profiles and deformation in the region of +coupling through the values of forces and moments applied to +the fiber tails (Fig.1(a)) including the effect of electrostatic fiber +attraction. +We suggest that the fixed submicron-wide gaps between +coupled fibers and input-output microfiber, rather than their +direct contact considered here, will allow us to demonstrate the +proposed microresonators with the Q-factor exceeding 108 [8]. +While such large Q-factors are not required for the realization of +tunable delay lines [43], signal processors [25], and microlasers +[19-21], they may be important for the realization of frequency +comb generators with tunable repetition rate [15, 16, 42], as +well as for the cavity QED [8, 11,12] and optomechanical +applications [13, 14]. + + + +Supplementary material +Expression for the transmission power +We introduce the discrete eigenwavelengths of the microresonator +in the compound fiber system, 𝜆� � � +�𝛾� , 𝑚 � 1,2, … , 𝑀 and +coupling +coefficients 𝜅��𝑧� between +the +corresponding +eigenmodes and the input-output microfiber positioned at axial +coordinate 𝑧. We calculate the transmission power 𝑃�𝜆, 𝑧� of our +system by applying the Mahaux-Weidenmüller formula [34-36]: + + + +2 +1 +† +† +2 +( , ) +1 +( , ) , +( , ) +( ) +( ) +( ) +( ) +( ) +i +P +z +iT +z +T +z +z +z +z +z + + + + + + + + + +Κ +Δ +Κ +Κ +Κ +, (S1) + +where + +1 +2 +† +2 +1 +1 +2 +1 +1 +2 +2 +( ) +( ) +( ) +( ) +( ) +( ) , +( ) +, +... +( ) +0 +... +0 +0 +... +0 +( ) +. +... +... +... +... +0 +0 +... +i +M +i +i +i +M +M +z +z +z +z +z +z +z + + + + + + + + + + + + + + + + + + + + + + +  + + + + + + + + + + + + + + + + + +  + + + + + + + + + +Θ +Δ +Κ +Κ +Κ +Δ + (S2) + +It is assumed in Eq. (S1) that the coupling to the input-output +waveguide +does +not +introduce +the +shifts +of +the +eigenwavelengths [35] which will be added later. We simplify +the expression for the transmission power by expanding the +inverse matrix in Eq. (S1) as follows: + + + +   + +  +  + + +  +1 +† +2 +1 +† +1 +2 +0 +1 +† +2 +2 +1 +† +1 +† +2 +1 +1 +† +1 +† +1 +2 +2 +1 +† +2 +2 +2 +( ) +( ) +( ) +( ) +( ) +( ) +( ) +1 +( ) +( ) +( ) +( ) +( ) +( ) ( ) +( ) +( ) +... +( ) +( ) +( ) ( ) +( ) +( ) +... +( ) +( ) +1 +( ) +( ) +( ) +i +n +n +i +n +i +i +n +n +i +n +m +i +i +i +m +m +m +z +z +z +z +z +z +z +z +z +z +z +z +z +z +z +z +z + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +Δ +Κ +Κ +Δ +Κ +Κ +Δ +Δ +Κ +Κ +Δ +Κ +Κ +Δ +Κ +Κ +Δ +Κ +Κ +Δ +Κ +Κ +Δ +Δ +Κ +Κ +1 +0 +1 +1 +† +1 +2 +2 +2 +1 +2 +( ) +( ) +( ) +( ) +1 +( ) +( ) +1 +n +M +n +i +M +m +i +i +m +m +m +z +z +z + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +Δ +Δ +Κ +Κ +Δ + + +Substituting this expression into Eq. (S1), we find: + +2 +2 +2 +1 +2 +2 +2 +1 +2 +( ) +1 +( , ) +( ) +1 +M +m +i +i +m +m +m +M +m +i +i +m +m +m +z +P +z +z + + + + + + + + + + + + + + + + + + + + + (S4) + +We separate the series of eigenwavelengths 𝜆� � � +�𝛾� and +coupling coefficients 𝜅��𝑧� by their correspondence to CWs 𝜆��𝑧� +entering Eq. (3) of the main text. For this purpose, we rewrite these +parameters as 𝜆�� � � +�𝛾� and 𝜅���𝑧�, where 𝑞 is the axial quantum +number of the eigenmode 𝐸���𝑥, 𝑦, 𝑧� � Ψ���𝑧�Ω��𝑥, 𝑦, 𝑧�. +Here Ψ���𝑧� satisfies Eq. (3) and Ω��𝑥, 𝑦, 𝑧� is a parametrically +slow function of the axial coordinate 𝑧. Substituting 𝛾� → 𝛾� we +assume that the material losses do not depend on the axial quantum +number 𝑞. Then, similar to the arguments of Ref. [24] (see Eq. (13) +in this reference), the coupling coefficients can be factorized as +�𝜅���𝑧�� +� � 2𝑖𝐷��𝑧��Ψ���𝑧�� +� . Using the expression for the +Green’s function of Eq. (3), + +2 +2 +( ) +( , , ) +qn +n +i +q +qn +n +z +G z z  + + + + + + + + + , (S5) + +we rewrite Eq. (S4) as + +2 +* +1 +1 +1 +( ) +( , , ) +( , ) +1 +( ) +( , , ) +N +n +n +n +N +n +n +n +D z G z z +P +z +D z G z z + + + + + + + + + + +. (S6) + +To identify the physical meaning of parameters 𝐷��𝑧�, we recall the +expression for the transmission power of a SNAP microresonator +under the assumption of a single CW contribution (𝑁 � 1) and + +lossless coupling to the input-output microfiber [24]: + + +2 +* +1 +1 +1 +1 +1 +1 +( , , ) +( , ) +1 +( , , ) +D G z z +P +z +D G z z + + + + + + + (S7) + +Here complex parameter 𝐷�, which was experimentally +measured and analyzed previously [24, 33], determines the +coupling to the input-output microfiber as well as the WGM +phase shift due to this coupling. Importantly, while the +imaginary part of 𝐷��𝑧� contributes to the widths of the +resonances, its real part (not taken into account in the original Eq. +(S1)) determines the WGM phase shifts caused by the coupling to +the input-output microfiber. + +Funding. The Engineering and Physical Sciences Research Council +(EPSRC), grants EP/P006183/1 and EP/W002868/1. Horizon 2020 +MSCA-ITN-EID grant 814147. +Disclosures. The authors declare no conflicts of interest. +Data availability. Data underlying the results presented in +this paper are not publicly available at this time but may be +obtained from the authors upon reasonable request. +References +1. W. Bogaerts, D. Pérez, J. Capmany, D. A. B. Miller, J. Poon, D. Englund, F. +Morichetti, and A. Melloni, “Programmable photonic circuits,” Nature +586, 207 (2020). +2. S. Y. Siew, B. Li, F. Gao, H. Y. Zheng, W. Zhang, P. Guo, S. W. Xie, A. Song, +B. Dong, L. W. Luo, C. Li, X. Luo, and G.-Q. 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Sumetsky, “Photonic +Microresonators Created by Slow Optical Cooking,” ACS Photonics +8, 436 (2021). + + + + + + diff --git a/EdAzT4oBgHgl3EQfUPw6/content/tmp_files/load_file.txt b/EdAzT4oBgHgl3EQfUPw6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..463810836f099131c712f6ed6db65159e6d3cf3f --- /dev/null +++ b/EdAzT4oBgHgl3EQfUPw6/content/tmp_files/load_file.txt @@ -0,0 +1,1148 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf,len=1147 +page_content='Reconfigurable microresonators induced in side-coupled optical fibers V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' VASSILIEV AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' SUMETSKY* Aston Institute of Photonic Technologies, Aston University, Birmingham B4 7ET, UK Email: m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='sumetsky@aston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='uk We experimentally demonstrate that side-coupling of coplanar bent optical fibers can induce a high Q-factor whispering gallery mode (WGM) optical microresonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' To explain the effect, we consider WGMs with wavelengths close to the cutoff wavelengths (CWs) of these fibers which slowly propagate along the fiber axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In the vicinity of the touching region, WGMs of adjacent fibers are coupled to each other, and CWs experience sub-nanoscale axial variation proportional to the coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' We show that in certain cases the CW variation leads to full localization of the WGMs and the creation of an optical microresonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' By varying the characteristic curvature fiber radius from the centimeter order to millimeter order, we demonstrate fully mechanically reconfigurable high Q-factor optical microresonators with dimensions varying from the millimeter order to 100-micron order and free spectral range varying from a picometer to hundreds of picometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The new microresonators may find applications in cavity QED, microresonator optomechanics, frequency comb generation with tunable repetition rate, tunable lasing, and tunable processing and delay of optical pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Introduction Microphotonic devices and circuits commonly consist of one or multiple connected basic elements, such as waveguides, couplers, and ring resonators [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In addition to the requirements of high fabrication precision and low losses [2, 3], the tunability of these circuits and devices is of critical importance for a variety of applications [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' While more complex tunable microphotonics circuits are targeted at tunability enabling quite arbitrary predetermined signal processing (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=', [1]), simple microdevices, such as standing along tunable three-dimensional microresonators, allow for unique functionalities not possible to achieve by other means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' For a variety of applications, the tunability of spherical, toroidal, and bottle microresonators has been demonstrated using mechanical stretching, heating, and nonlinear light effects including those in monolithic and specially coated microresonators [6-10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In most of these approaches, it is only possible to tune series of wavelength eigenvalues simultaneously without noticeable change in their separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' However, for several applications, which include cavity QED [8, 11, 12], optomechanics [13, 14], frequency microcomb generation [15, 16], optical signal processing and delay [4, 5, 17], and lasing [18-21], it is critical to have microresonators with tunable eigenwavelength separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' For example, the latter allows the creation of optical frequency microcomb generators and microlasers with continuously tunable repetition rate and wavelength and to tune the microresonator eigenfrequency separation in resonance with the frequency of its mechanical oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Considerable variation of the eigenwavelength separation commonly requires the variation of microresonator dimension and/or its refractive index parameters by the quantity comparable with their original values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' One approach to solve this problem consists in using Fabry-Perot microresonators with tunable mirror separation which contain the optical materials under interest [12, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Additional flexibility of tuning can be achieved by employing Fabry-Perot microresonators with a liquid material inside [21] or translating a wedge-shaped solid optical material to vary its dimensions inside the Fabry-Perot microresonator [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (a) Coplanar bent optical fibers touching each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The fiber profile is manipulated by bending and translation of the fiber tails indicated by curved and straight arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (b) Illustration of coupling between the input-output microfiber and WGMs in Fiber 1 and Fiber 2 near cutoff wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Alternatively, of special interest is attaining the eigenwavelength separation tunability in three-dimensional monolithic high Q-factor microresonators, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=', those with spherical, toroidal, and bottle (a) Fiber 2 Fiber 1 (b) Coupled WGMs (12) Fiber 2 (z) 2n2 nin?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Direct Fiber 1 l1n1 contact x Microfiber Kn1 Zshapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' This will allow us to add tunability to the emerging applications of these microresonators in QED, optomechanics, lasing, and frequency comb generation noted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' However, the deformation of most of these monolithic microresonators to achieve significant change of their eigenwavelength separation is unfeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' A unique exception, though, is exhibited by SNAP (Surface Nanoscale Axial Photonics) microresonators [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' These microresonators are introduced at the surface of an optical fiber by its nanoscale deformation, which causes the nanoscale variation of the cutoff wavelengths (CWs) controlling the slow propagation of whispering gallery modes (WGMs) along the fiber axis (see [24, 25] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' [26], a SNAP microresonator induced and fully reconfigurable by local heating of an optical fiber was demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' [27], it was shown that it is possible to create a SNAP microresonator and control its dimensions by local bending of an optical fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Both approaches allow for tuning of eigenwavelength separation of microresonators by the quantity comparable to or larger than its original value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' However, in both approaches, the induced microresonator shapes had limited flexibility and their characteristic axial dimensions could not be reduced below several millimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In the first case, this restriction was caused by the imposed length of the characteristic heat distribution along the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In the second case, the reduction of microresonator size was limited by the smallest curvature radius corresponding to the fiber breakage threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In this paper we report on our discovery of a new type of WGM optical microresonators which belongs to the group of SNAP microresonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' We show that side coupled coplanar bent fibers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 1) can induce a high Q-factor SNAP microresonator localized in the region of fiber coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The configuration of fibers shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 1 allows us to flexibly tune the shape of the induced SNAP microresonators and their axial dimensions from several tens of microns to several millimeters and, respectively, tune their eigenwavelength separation from hundreds of picometers to a picometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Cutoff wavelengths of uncoupled and side- coupled straight fibers First, it is instructive to consider the behavior of CWs for uncoupled and side-coupled straight optical fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' For this purpose, we cleave a 125-micron diameter uncoated commercial optical fiber into two pieces (Fiber 1 and Fiber 2), which are then coaxially aligned and put into contact along 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='5 mm of their length as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Light is launched into Fiber 1 by a transversely oriented taper with the micrometer diameter waist (input-output microfiber) connected to the Optical Spectrum Analyzer (OSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' After coupling into Fiber 1, light forms WGMs propagating along the fiber surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In the region of direct contact of fibers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 2(a)), WGMs in Fiber 1 and Fiber 2 are coupled to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' To characterize the effect of interfiber coupling, we measured the spectrograms of the configured fiber system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' For this purpose, the input-output microfiber was translated along Fiber 1 (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 1(b) and 2(a)) touching it periodically with the spatial resolution of 2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' At the cut end of Fiber 1, the microfiber was moved towards Fiber 2 and continued scanning Fiber 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The spectrograms of transmission power 𝑃�𝜆, 𝑧� were measured as a function of wavelength \uf06c and microfiber position z along the axis of Fiber 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (a) Illustration of side-coupled straight optical fiber configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (b) Spectrogram of this configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (c) Magnified section outlined in the spectrogram (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The measured spectrogram of our fiber system is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The left- and right-hand sides of this spectrogram show the spectrograms of uncoupled Fiber 1 and Fiber 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Lines in spectrogram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 2(b) indicate the CWs of uncoupled and coupled fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' These CWs correspond to WGMs with different azimuthal and radial quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The magnified copy of the section outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 2(b) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' It is seen that the CWs appear as straight lines slightly tilted with respect to the horizontal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' From the measured magnitude of tilt, ε� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='015 nm/mm, we determine the linear variation of the fiber radius ∆𝑟� � 𝑟�ε�/𝜆� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='6 nm/mm [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In the latter rescaling relation, we used 𝑟� � 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='5 µm and 𝜆� � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='55 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' By linear extrapolation of CWs of Fiber 1 and Fiber 2 (dashed white lines), we confirm that, as expected, their positions (horizontal black dashed line) coincide at the cut ends of these fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' At the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='5 mm long region of fiber touching, WGMs in Fiber 1 couple to WGMs in Fiber 2 and the corresponding CWs split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The structure and positions of CWs in the touching region depend on the magnitude of coupling and will be further discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Here we note that the value of CW splitting found, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=', from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 2(c) is ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='1 nm, which coincides with characteristic values of CW variation in SNAP microresonators [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In particular, the positive CW shift in the coupling region leads to the WGM localization and creation of a microresonator which can be tuned by changing the length of the side-coupled fiber segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In our current experiment, the Q-factor of the induced SNAP resonator was poor due to the scattering of light at the imperfectly cleaved fiber ends, (a) Nottoscale 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='5 mm Fiber2 Fiber1 125μm Microfiber 1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='00 (b) 1548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='50 5 TransmissionPower (dB) (wu) 1548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='00 length 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='50 10 Wavel 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='00 15 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='50 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='00 20 1545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='50 0 1 2 3 4 5 6 Distance along fiber (mm) 0 1548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='50 (c) 1548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='40 5 length 1548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='20 10 1548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='10 Javel 15 1548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='00 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='90 20 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='80 0 1 2 3 4 5 6 Distancealongfiber(mm)which, typically, ensure around 70% WGM reflectivity [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Nevertheless, we suggest that the demonstrated resonator can be directly used to create miniature broadly tunable optical delay lines generalizing our previous results based on the SNAP microresonators with fixed dimensions [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Indeed, in these devices the WGM pulses complete only a single round trip along the fiber axis and therefore their attenuation at the fiber facets may reduce the output light power by around 50% only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' We also suggest that, after feasible improvement, the Q-factor of these microresonators can be significantly improved as further discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Basic experiment In our proof-of-concept experiments, we used 125-micron diameter uncoated commercial silica optical fibers touching each other as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The ends of Fiber 1 and Fiber 2 were bent and translated to arrive at the required profile of these fibers near their coupling region illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The fibers used were either originally straight or preliminary softened in a flame and bent permanently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' As described in the previous section, WGMs were launched into Fiber 1 by a transversely oriented microfiber connected to the OSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' If the separation between Fiber 1 and Fiber 2 is small enough, WGMs penetrate from Fiber 1 into Fiber 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In the simplest configuration considered in this Section, Fiber 1 was straight, and coplanar Fiber 2 was bent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The fibers were put in contact and then slightly pushed towards each other to increase the coupling region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The photograph of the fiber configuration used in this experiment is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' From this picture, we estimated the curvature radius of the bent fiber as 𝑅~30 mm (see further discussion of the fiber profile in Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(b) shows the spectrogram of the configured structure measured along the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='5 nm bandwidth within the 700 µm axial length of Fiber 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' At the edges of the scanned region, the interfiber coupling is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In these regions, CWs do not noticeably change with distance 𝑧 and, thus, correspond to Fiber 1 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The arrangement of CWs in these regions is similar to that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (a) Photograph of the side-coupled fibers used in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The upper fiber is bent with the curvature radius 𝑅~30 mm and the lower fiber has the curvature radius greater than 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (b) The spectrogram measured in the vicinity of the coupling region of these fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (b1) and (b2) Spectrograms showing the magnified sections outlined in the spectrogram (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (c1) and (c2) Spectrograms of the microresonators numerically calculated in the two-mode approximation detailed in the text, which replicate the experimental spectrograms in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (b1) and (b2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (a) R~30 mm Experiment 3-10 nsertion 1mm 15 (b1) 1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='65 1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='46 1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='48 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='45 (b) (b2) Wavelength (um) 1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='60 1550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='00 2 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='40 1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='55 (b2) 4 1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='50 1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='50 6 Transmission Power (dB) 1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='45 Wavelength (nm) 1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='00 8 1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='40 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='20 1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='35 1548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='50 10 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='15 1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='30 12 1548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='00 200 400 600 200 400 600 14 Distancealongfiber(um) Distance along fiber (μm) 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='50 (b1) 16 18 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='00 Theory 20 0 200 400 600 Distance along fiber (μm) (c1) (c2)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='4 5 Wavelengthvariation(nm) Wavelength variation (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='2 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='0 20 0 200 400 600 0 200 400 600 Distancealongfiber(um) Distance along fiber (μm) The effect of coupling shows up in the central region of the spectrogram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In this region, different CWs exhibit different positive and negative variations along the axial length 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The exemplary regions of this spectrogram named (b1) and (b2) are magnified in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(b1) and 3(b2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' It is seen that, as expected, in contrast to negative variations, positive CW variations lead to the WGM confinement and the creation of microresonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Our estimates illustrated in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(b2) show that the Q- factor of the created microresonator (which measurement was limited by the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='3 pm resolution of the OSA used) exceeds 10�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The observed CW variations in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(b1) and (b2) can be explained by the theory described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Basic theory We assume that the fiber bending is small enough so that the propagation of light along the axial direction of side-coupled fibers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 1(b)) can be considered as propagation along a single waveguide with asymmetric cross-section including both fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The wavelengths of slow WGMs are close to the CWs 𝜆��𝑧� of this compound waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' To determine the complex-valued CWs 𝜆��𝑧�, we introduce the original CWs 𝜆��� � � �𝛾��� and 𝜆��� � � �𝛾��� of unbent Fiber 1 and Fiber 2 with the imaginary parts determined primarily by material losses and scattering of light at the fiber surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' We assume that there are 𝑁� and 𝑁� cutoff wavelengths in Fibers 1 and Fiber 2, respectively, which contribute to the resonant transmission, so that 𝑛� � 1,2, … , 𝑁�, 𝑗 � 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' We refer to the integers 𝑛, 𝑛� and 𝑛� as to the transverse quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Variation of 𝜆��𝑧� is caused by bending of fibers [27] and, in our case, primarily by their coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In the absence of the input-output fiber, the CWs of our system, 𝜆 � 𝜆��𝑧�, 𝑛 � 1,2, … , 𝑁� � 𝑁�, are determined as the roots of the determinant: \uf028 \uf029 det ( ) 0 z \uf06c \uf02d \uf03d I Ξ (1) Here 𝐈 is the unitary �𝑁� � 𝑁�� � �𝑁� � 𝑁�� matrix and matrix 1 1 12 † 12 2 2 ( ) ( ) ( ) ( ) ( ) z z z z z \uf02b \uf0e6 \uf0f6 \uf03d \uf0e7 \uf0f7 \uf02b \uf0e8 \uf0f8 Λ Δ Δ Ξ Δ Λ Δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (2) The submatrices in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (2) determine the original CWs of Fiber 1 and Fiber 2, 𝚲� � �𝜆��� � � �𝛾����, couplings inside each of the fiber caused by bending, 𝚫��𝑧� � �δ���� ��� �𝑧��, and interfiber couplings 𝚫���𝑧� � �δ���� ���� �𝑧�� , 𝑚�, 𝑛� � 1,2, … 𝑁�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' As in SNAP [24], dramatically small nanometer and sub- nanometer scale variations of CWs 𝜆��𝑧� along the compound fiber waveguide can localize WGMs and induce an optical microresonator having eigenwavelengths 𝜆�� with axial quantum numbers 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Due to the smooth and small CW variation and proximity of the localized WGM wavelengths 𝜆�� to 𝜆��𝑧�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' the corresponding eigenmode can be presented as 𝐸���𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 𝑧� � Ψ���𝑧�Ω��𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 𝑧� where the transverse WGM distribution Ω��𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 𝑧� is calculated at the CW 𝜆��𝑧� and depends on 𝑧 parametrically slow [32],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' and function Ψ���𝑧� determines the axial dependence of the microresonator eigenmode amplitude and satisfies the one-dimensional wave equation [24] 2 3/2 2 2 3/ 2 2 ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' ) 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' ) ( ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' n r n n n n n d n z z z dz \uf070 \uf062 \uf06c \uf062 \uf06c \uf06c \uf06c \uf06c \uf059 \uf02b \uf059 \uf03d \uf03d \uf02d (3) where 𝑛� is the refractive index of the fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The coupling parameters 𝜅���𝑧�between WGM 𝐸���𝑥, 𝑦, 𝑧� and the input-output wave in the microfiber is determined by their overlap integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Commonly, the microfiber diameter is much smaller than the characteristic axial variation length of 𝐸���𝑥, 𝑦, 𝑧�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' For this reason, similar to the analogous approximation in the SNAP platform [24, 33], the coupling parameters 𝜅���𝑧� are proportional to the values of 𝐸���𝑥, 𝑦, 𝑧� at the axial coordinate 𝑧 of the input-output microfiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Then, calculations based on the Mahaux- Weidenmüller theory [34-36] presented in Supplementary Material allowed us to express the transmission power 𝑃�𝜆, 𝑧� through the input-output microfiber coupled to the considered fiber configuration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 1(b)) as 1 2 1 2 2 1 1 1 ( ) ( , , ) ( , ) 1 ( ) ( , , ) N N n n n N N n n n D z G z z P z D z G z z \uf06c \uf06c \uf06c \uf02b \uf03d \uf02b \uf03d \uf02b \uf03d \uf02b \uf0e5 \uf0e5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (4) Here 𝐺��𝑧�, 𝑧�, 𝜆) is the Green’s function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (4) generalizes the expression for the transmission power previously derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' As shown below, functions 𝐷��𝑧� can be expressed through and have characteristic values similar to the coupling D-parameters which were experimentally measured previously and typically have the real and imaginary parts ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='01 µm-1 [24, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Close to the resonance condition, 𝜆 � 𝜆��, for sufficiently small losses and coupling, and separated CWs 𝜆��𝑧�, only one Green’s function with number 𝑛 contributes to the sums in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (4) coincides with that previously derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' However, generally, the contribution of more than one term to the sums in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (4) may be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Before the detailed description of the spectrograms in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 2(b) and 3(b), we note that the transmission power plots in these figures characterize the CWs of the coupled fiber system determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (1) viewed by the input-output microfiber and, subsequently, OSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Therefore, the CWs of Fiber 2, which are the solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (1) but uncoupled from Fiber 1 cannot be seen by the OSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' On the other hand, the number of CWs which can show up in the coupling region can be as many as 𝑁� � 𝑁�, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=', significantly greater than the number 𝑁� of visible uncoupled CWs of Fiber 1 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 2(b) as an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' To clarify the effect of coupling between WGMs in adjacent fibers, we consider the two-mode approximation, 𝑁� � 𝑁� � 1, assuming that the wavelength 𝜆 of the input light is close to an unperturbed single WGM CW 𝜆�� � � �𝛾 of Fiber 1 and a single CW 𝜆�� � � �𝛾 of Fiber 2 having the same imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Consequently, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 1(b) we now set 𝑛� � 𝑛� � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' We neglect the effect of the CW variation due to the fiber bending [27], which is usually smaller than the effect of fiber coupling, setting 𝛿�� ��� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Then, the CWs 𝜆��𝑧� and 𝜆��𝑧� of the compound fiber are found from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (1) as \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 2 (12) 1,2 11 21 11 21 11 1 1 ( ) ( ) 2 4 z i z \uf06c \uf06c \uf06c \uf067 \uf06c \uf06c \uf064 \uf03d \uf02b \uf02b \uf0b1 \uf02d \uf02b (5) The dependence on the transverse coordinates 𝑥 and 𝑦 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 1(b)) of the compound WGM corresponding to CWs 𝜆��𝑧� can be calculated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' We introduce the unperturbed WGMs in Fiber 1 and 2 (considered unbent and uncoupled) calculated at their CWs 𝜆�� and 𝜆�� as Ω� ����𝑥, 𝑦� and Ω� ����𝑥, 𝑦�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Then, in the two-mode approximation, the compound modes generated by weak coupling of modes Ω� ����𝑥, 𝑦� and Ω� ����𝑥, 𝑦� are determined as [37] (1) (2) 1 1 1 2 2 (1) (2) 2 1 1 2 2 (12) 11 11 21 1 ( ) ( , , ) ( , ) ( , ), 1 ( ) 1 ( ) ( ) 1 ( , , ) ( , ) ( , ), 1 ( ) 1 ( ) ( ) ( ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' z x y z x y x y z z z x y z x y x y z z z z \uf064 \uf064 \uf064 \uf064 \uf064 \uf064 \uf064 \uf064 \uf06c \uf06c \uf057 \uf03d \uf057 \uf02b \uf057 \uf02b \uf02b \uf057 \uf03d \uf02d \uf057 \uf02b \uf057 \uf02b \uf02b \uf03d \uf02d \uf025 \uf025 \uf025 \uf025 \uf025 \uf025 \uf025 (6) Consequently, the coupling parameters to the microfiber entering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (4) at coordinate 𝑧 are 1 2 2 2 ( ) ( ) , ( ) , 1 ( ) 1 ( ) D D z D z D z z z \uf064 \uf064 \uf064 \uf03d \uf03d \uf02d \uf02b \uf02b \uf025 \uf025 \uf025 (7) where 𝐷 is the z-independent coupling parameter between the input- output microfiber and Fiber 1 [24, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' To map the bent fiber axial profile ℎ�𝑧� to the CW envelope profiles of the induced microresonators, we have to determine the relation between ℎ�𝑧� and coupling coefficient 𝛿�� �����𝑧�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Similar to calculations in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' [38, 39], for the smooth and small ℎ�𝑧� considered here, we find \uf028 \uf029 1/2 (12) 2 11 0 2 ( ) exp 1 ( ) r z n h z \uf070 \uf064 \uf064 \uf06c \uf0e6 \uf0f6 \uf03d \uf02d \uf02d \uf0e7 \uf0f7 \uf0e8 \uf0f8 , (8) where 𝛿� is 𝑧-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Assuming the simplest profile of the bent fiber having the curvature radius 𝑅 as ℎ�𝑧� � 𝑧�/2𝑅 (9) for silica fibers with 𝑛� �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='44, we estimate the FWHM of 𝛿�� �����𝑧� as 𝑧����~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='5�𝜆𝑅��/�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' At 𝜆~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='55 µm and 𝑅~30 mm of our experiment, we have 𝑧����~100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (5) and (8), we find that the FWHM of the CW, depending on the value of 𝜆�� � 𝜆��, is between 𝑧���� and 2𝑧���� which is only in qualitative agreement with the microresonator FWHM 𝑧����~ 250 µm found from experimental data in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(b1) and (b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The results of our numerical modeling in the two-mode approximation considered based on Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (3)-(9) are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(c1) and 3(c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' To fit the experimental data, we set the average CW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='5(𝜆�� � 𝜆��) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='55 µm, the CW difference 𝜆�� � 𝜆�� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='05 nm in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(c1) and 𝜆�� � 𝜆�� � �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='05 nm in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(c2), coupling parameter 𝐷 � �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='01 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='01𝑖 µm-1 [24, 33], Q-factor 𝑄 � 10�, the microresonator FWHM 𝑧����~ 250 µm and its spectral height ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='15 nm, similar to these values found from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 2(b1) and (b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The experimental spectrograms in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(b1) and (b2) and theoretical spectrograms in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(c1) and (c2) look nicely similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' However, important differences between them should be noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (8) and (9), the FWHM value 𝑧����~ 250 µm corresponds to the Fiber 2 curvature radius 𝑅~66 mm, which is twice as large as that measured from the fiber image shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' We suggest that the difference is caused by the deviation of the shape of Fiber 2 from parabolic in the coupling region as well as by the fiber misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The additional deformation of fibers may be induced by their electrostatic attraction and pressuring, which are not visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Our suggestion is confirmed by the experimental profiles of the induced microresonator envelopes and CW shapes in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(b1) and (b2) which, as compared to those in the theoretical spectrograms in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(c1) and (c2), have larger side slopes and are flatter in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Next, we notice that, in the theoretical spectrograms, the CW wavelength profiles are more mirror- symmetric to the microresonator envelopes with respect to the horizontal line (following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (5)), while, in the experimental spectrograms, the lower CW profiles are shallower than the microresonator envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' We suggest that this deviation can be eliminated by taking into account the coupling with other WGMs ignored in the two-mode approximation considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Tunability Bending and translating the tails of Fiber 1 and Fiber 2 side- coupled to each other as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 1 allowed us to tune the dimensions of the fiber coupling region and thereby tune the dimensions of created microresonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' As in the previous sections, in our experiments we used 125 µm optical fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' We investigated the cases of the smallest microresonators containing a few wavelength eigenvalues and having the characteristic axial dimensions of hundred microns (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(a1)- (a4)), as well as larger microresonators with dimensions of several hundred microns (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(b1)-(b4) and (c1)-(c3)) and the largest microresonator having the axial length of 5 millimeters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 4(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Considering the smallest microresonators, we monitored the process of their creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Side-coupling of a straight Fiber 1 and Fiber 2 bent with a sufficiently small curvature radius of ~ 1 mm introduced small perturbation in CWs shown in the spectrogram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 4(a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Increasing the fiber radius further, we arrived at the microresonator with a single eigenwavelength (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 4(a2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The inset inside the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 4(a2) spectrogram, which magnifies the region near this eigenwavelength, shows that the axial dimension of the corresponding eigenmode is ~ 200 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Remarkably, except for the axial dimension of localized WGMs with uniform magnitude in specially designed bat microresonators [39, 40], this dimension (which expansion is critical, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=', for QED applications [41]) is the record large characteristic WGM dimension demonstrated in microresonators to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The measured Q-factor of this microresonator (limited by the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='3 pm resolution of the OSA used) was slightly greater than 10�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Larger bending radii of Fiber 2 having the order of 10 mm led to the creation of microresonators with millimeter-order axial dimensions having the spectrograms shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(b1)-(b4) and (c1)-(c3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The close to parabolic shape of these microresonators suggests that they can be used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=', as tunable optical frequency comb generators [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' We note that the behavior of the CWs and microresonators envelopes in most of these spectrograms cannot be accurately described by the two-mode approximations of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Of particular interest is the spectrogram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 4(c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' At first sight, the envelop of the microresonator in this spectrogram is the continuation of the CW of Fiber 1 (compare with Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 3(b1) and (c1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Unexpectedly, the axial WGM localization in this microresonator (caused by the WGM reflection from the CW- generated turning points [24]) sharply dissolves inside the microresonator area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Tunability of microresonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (a1)-(a4) Spectrograms of induced microresonators for small curvature radius of Fiber 2 ~ 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (b1)-(b4) and (c1)-(c3) spectrograms of induced microresonators for a lager radius of Fiber 2 ~ 10 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (d) Spectrogram of a 5 mm long microresonator induced by touching straight Fiber 1 and Fiber 2 which was preliminary permanently bent at the ends as shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (a1) (a2) (a3) (a4) R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='2mm R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='5mm R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='6mm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='7mm 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='80 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='80 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='80 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='80 ap (w (wu 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='6 (wu (wu 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='70 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='70 5 45 5 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='60 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='60 6 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='60 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='60 0200400600 0200400600 0200400600 0200400600 Distancealongfiber(μm) Distancealongfiber(um) Distancealongfiber(μm) Distancealongfiber(um) (b1) (b2) (b3) (b4) R=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='1mm R=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='7mm R=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='1mm R=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='3mm 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='90 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='90 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='90 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='90 10 10 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='80 20 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='80 32 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='70 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='70 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='70 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='70 lavele 4 5 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='60 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='60 ≤1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='60 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='60 5 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='50 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='50 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='50 200400600800 200400600800 200400600800 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='50 0 200400600800 Distancealongfiber(um) Distancealongfibor(μm) Distancealongfiber(um) Distancealong fiber(um) (c1) (c2) (c3) R=18mm R~30mm R~30mm 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='80 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='80 Transmission Power (dB) 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='80 2 (dB) 2 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='70 4 4 nbu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='6 6 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='60 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='60 AeM 8 8 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='50 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='50 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='50 10 10 10 0 400 800 1200 1600 400 800 1200 1600 0 400 800 1200 1600 Distancealong fiber(um) Distancealongfiber(um) Distancealongfiber (um) (d) 1548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='60 length 4 6 10 1548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='30 0 1000 2000 3000 4000 5000 6000 Distancealong fiber(um)To create longer microresonators, we, first, permanently bent the tails of Fiber 2 as illustrated in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' This allowed us to arrive at an arbitrarily large curvature radius of this fiber including its straight shape between the bent tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' As an example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 4(d) shows the spectrogram of a 5 mm long microresonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Though the eigenwavelength width of this microresonator is greater than its free spectral range, we suggest that, in contrast to the lossy microresonators induced by side-coupled cleaved straight fibers demonstrated in Section 2, its Q-factor is similar to that of the smaller microresonators considered in this section and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Discussion The effect of induction of high Q-factor WGM tunable optical microresonators in side-coupled optical fibers discovered in this paper enables a range of exciting generalizations and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Further extension of tuning flexibility can be achieved by enabling different boundary conditions at the fiber tails (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 1(a)), different interfiber touching stresses, and different preliminary permanent fiber bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Configurations of fibers, which are potentially attractive for future research and applications, are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(a) shows a way to create long microresonators alternative to the method utilizing fibers with permanently bent tails illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In the configuration of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(a), the length of the induced microresonator increases as the curvature radii of touching fibers approach each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Provided that the variation of the fiber radii can be performed so that the parabolicity of the induced microresonators was maintained, the configuration of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(a) can serve for the generation of the optical frequency combs with a tunable repetition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (a) Bent fibers with increased coupling region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (b) Bent fibers with increased coupling region and abrupt side of the induced microresonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (c) A bottle microresonator side-coupled to a fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (d) Side coupled straight fibers with tapered facets forming a rectangular microresonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (e) Two straight fibers with tapered facets coupled to the third straight fiber forming a rectangular microresonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (f) Twisted side-coupled fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (g) A microcapillary fiber filled with liquid and side coupled to a bent fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (h) Three straight coupled fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(b), the lower fiber is terminated with a short taper, which can be introduced using, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=', a CO2 laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Simple estimates show that a taper with a characteristic length of 100 µm at the end of a 125 µm diameter optical fiber creates an abrupt CW barrier with a slope of ~ 100 nm/µm at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='5 µm wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The steepness of the slope of this barrier (critical for impedance matching of light from the input-output microfiber [43]) is 100 times greater than that demonstrated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' [44] with the femtosecond laser inscription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The configuration shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(b) can be used for the creation of miniature dispersionless tunable optical delay lines provided that the shape of the induced microresonator is kept semi-parabolic in the process of tuning [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Experimental investigation and development of the theory of WGMs in a microresonator side-coupled to an optical fiber is of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(c) illustrates the side coupling of a fiber and a bottle microresonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' While the fiber is open-ended, coupling of the bottle microresonator to the straight fiber can cause the localization of light in the fiber, similar to the coupling between bent optical fibers considered above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The configuration shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(c) suggests a way of tuning the microresonator eigenwavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The fiber configuration shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(d) is similar to two straight side-coupled fibers considered in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' To improve the Q-factor of the microresonator induced along the coupling region, the cleaved ends of fibers shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 2(a) are modified by the tapered ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The configuration of fibers shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(e) illustrates an alternative way to create tunable microresonators when the position of both their sides can be tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The rectangular microresonators induced in both configurations can be used for the creation of tunable delay lines which, as shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' [31], can be dispersionless with a good accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The coupling of twisted optical fibers illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(f) is interesting to investigate both theoretically and experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In the cylindrical coordinates �𝑧, 𝜌, 𝜑� of one of the fibers, the curve along which the fibers touch each other corresponds to the azimuthal angle 𝜑 � 𝜑� � 𝛼𝑧 , where 𝛼 is the twisting coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The corresponding value of the WGM field is proportional to exp �𝑖𝛽𝑧 � 𝑖𝑚�𝜑� � 𝛼𝑧�� where 𝛽 is the propagation constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' From this expression, a WGM at CW corresponding to 𝛽 � 0 is seen by another fiber as a mode with nonzero propagation constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Thus, in contrast to the untwisted fibers, coupling between the side- coupled twisted fibers is essentially three dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(g) shows a microcapillary fiber filled with liquid and side- coupled to a bent fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' For the microcapillary with sufficiently thin walls, a microresonator induced inside it by the side-coupled fiber performs nonlocal sensing of liquid [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' [46] and [47], such microresonators were introduced with the CO2 laser and slow cooking methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(g) suggests the simplest approach for the realization of nonlocal microfluidic sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(h) illustrates three straight side-coupled fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' In contrast to two coupled fibers, WGMs launched into this configuration will propagate into both azimuthal direction and, in particular, into the positive and negative directions of the input-output microfiber with approximately the same amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The channel formed between these fibers can be used for gas and microfluidic sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Unlike the microcapillary illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 5(g), no ultrathin wall enabling the WGM sensing of the internal channel is required in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' While the model of two coupled CWs developed here qualitatively explains some characteristic features of the (a) (f) (b) (g) (c) (d) (h) (e)experimentally measured spectrograms, the complete explanation and quantitative fitting of the experimental data should include the effect of several CWs and be based on the further development of the coupled wave theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The future theory should also allow us to express the fiber profiles and deformation in the region of coupling through the values of forces and moments applied to the fiber tails (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='1(a)) including the effect of electrostatic fiber attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' We suggest that the fixed submicron-wide gaps between coupled fibers and input-output microfiber, rather than their direct contact considered here, will allow us to demonstrate the proposed microresonators with the Q-factor exceeding 108 [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' While such large Q-factors are not required for the realization of tunable delay lines [43], signal processors [25], and microlasers [19-21], they may be important for the realization of frequency comb generators with tunable repetition rate [15, 16, 42], as well as for the cavity QED [8, 11,12] and optomechanical applications [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Supplementary material Expression for the transmission power We introduce the discrete eigenwavelengths of the microresonator in the compound fiber system, 𝜆� � � �𝛾� , 𝑚 � 1,2, … , 𝑀 and coupling coefficients 𝜅��𝑧� between the corresponding eigenmodes and the input-output microfiber positioned at axial coordinate 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' We calculate the transmission power 𝑃�𝜆, 𝑧� of our system by applying the Mahaux-Weidenmüller formula [34-36]: \uf028 \uf029 2 1 † † 2 ( , ) 1 ( , ) , ( , ) ( ) ( ) ( ) ( ) ( ) i P z iT z T z z z z z \uf06c \uf06c \uf06c \uf06c \uf02d \uf03d \uf02b \uf03d \uf02d Κ Δ Κ Κ Κ , (S1) where 1 2 † 2 1 1 2 1 1 2 2 ( ) ( ) ( ) ( ) ( ) ( ) , ( ) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' ( ) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 0 ( ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' i M i i i M M z z z z z z z \uf06b \uf06b \uf06c \uf06b \uf06c \uf06c \uf067 \uf06c \uf06c \uf067 \uf06c \uf06c \uf06c \uf067 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf03d \uf02b \uf03d \uf0e7 \uf0f7 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='\uf02d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='\uf0e8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='\uf0f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='\uf0e5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='\uf0e5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='\uf0e5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='Δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='Δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='Κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='Κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='Δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content='Substituting this expression into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (S1), we find: 2 2 2 1 2 2 2 1 2 ( ) 1 ( , ) ( ) 1 M m i i m m m M m i i m m m z P z z \uf06b \uf06c \uf06c \uf067 \uf06c \uf06b \uf06c \uf06c \uf067 \uf03d \uf03d \uf02b \uf02d \uf02d \uf03d \uf02d \uf02d \uf02d \uf0e5 \uf0e5 (S4) We separate the series of eigenwavelengths 𝜆� � � �𝛾� and coupling coefficients 𝜅��𝑧� by their correspondence to CWs 𝜆��𝑧� entering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (3) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' For this purpose, we rewrite these parameters as 𝜆�� � � �𝛾� and 𝜅���𝑧�, where 𝑞 is the axial quantum number of the eigenmode 𝐸���𝑥, 𝑦, 𝑧� � Ψ���𝑧�Ω��𝑥, 𝑦, 𝑧�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Here Ψ���𝑧� satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (3) and Ω��𝑥, 𝑦, 𝑧� is a parametrically slow function of the axial coordinate 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Substituting 𝛾� → 𝛾� we assume that the material losses do not depend on the axial quantum number 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Then, similar to the arguments of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' [24] (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (13) in this reference), the coupling coefficients can be factorized as �𝜅���𝑧�� � � 2𝑖𝐷��𝑧��Ψ���𝑧�� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Using the expression for the Green’s function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (3), 2 2 ( ) ( , , ) qn n i q qn n z G z z \uf06c \uf06c \uf06c \uf067 \uf059 \uf03d \uf02d \uf02d \uf0e5 , (S5) we rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (S4) as 2 1 1 1 ( ) ( , , ) ( , ) 1 ( ) ( , , ) N n n n N n n n D z G z z P z D z G z z \uf06c \uf06c \uf06c \uf03d \uf03d \uf02b \uf03d \uf02b \uf0e5 \uf0e5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (S6) To identify the physical meaning of parameters 𝐷��𝑧�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' we recall the expression for the transmission power of a SNAP microresonator under the assumption of a single CW contribution (𝑁 � 1) and lossless coupling to the input-output microfiber [24]: 2 1 1 1 1 1 1 ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' ) ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' ) 1 ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' ) D G z z P z D G z z \uf06c \uf06c \uf06c \uf02b \uf03d \uf02b (S7) Here complex parameter 𝐷�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' which was experimentally measured and analyzed previously [24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' 33],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' determines the coupling to the input-output microfiber as well as the WGM phase shift due to this coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Importantly, while the imaginary part of 𝐷��𝑧� contributes to the widths of the resonances, its real part (not taken into account in the original Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' (S1)) determines the WGM phase shifts caused by the coupling to the input-output microfiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The Engineering and Physical Sciences Research Council (EPSRC), grants EP/P006183/1 and EP/W002868/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Horizon 2020 MSCA-ITN-EID grant 814147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Disclosures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Bogaerts, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfUPw6/content/2301.01262v1.pdf'} +page_content=' Pérez, J.' 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How loud does the sun shine? We developed a multisensory and multimodal tool, the Loaded Dice, for use in +co-design workshops to research the design space of IoT usage scenarios. The Loaded Dice incorporate the principle of a technical +synesthesia, being able to map any of the included sensors to any of the included actuators. With just a turn of one of the cubical +devices it is possible to create a new combination. We discuss the core principles of the Loaded Dice, what sensors and actuators are +included, how they relate to human senses, and how we realized a meaningful mapping between sensors and actuators. We further +discuss where we see additional potential in the Loaded Dice to support synesthetic exploration – as Synesthetic Dice – so that you can +eventually find out who cries brighter. +CCS CONCEPTS • Human-centered computing~Human computer interaction (HCI) +Additional Keywords and Phrases: multisensory, multimodal, synesthesia, design, ideation, tools, methods, IoT, +Internet of Things, haptic technology, cubic shape, tangible interactive devices, input and output devices, tangibles +ACM Reference Format: +Albrecht Kurze. 2022. Synesthetic Dice: Sensors, Actuators, And Mappings. In Workshop Sensory Sketching (CHI’22). +April 22, 2022. 4 pages. +1 +INTRODUCTION +Some years ago we designed and developed the Loaded Dice [8,9], a multisensory and multimodal hybrid toolkit to +ideate Internet of Things (IoT) devices and scenarios, e.g. for the ‘smart’ home, and with different groups of co- +designers [3,7,8]. The Loaded Dice filled a gap between analog, non-functional tools, often card-based, e.g. KnowCards +[1], and functional but tinkering based tools, e.g. littleBits [2], for multisensory und multimodal exploration, ideation +and prototyping. +We introduce the Loaded Dice, the core concepts that they are built on, the used sensors and actuators, and how +they map to different human senses. We will then continue to discuss how we realized mappings between sensed raw +value, normalized intermediate values, and actuated values. While the mappings that we currently use are sufficiently +good enough for current purposes, we see big potential in some extended uses as ‘Synesthetic Dice’. +This brings us to our core question: How can the Loaded Dice be used for exploration and research of synesthetic +mappings between sensors and actuators, e.g. for innovative interactions and non-verbal communication? + +SENSORDIE +TemperatureSensor +Light Sensor +Microphone +MovementSensor +Potentiometer +Distance Sensor +ACTUATORDIE +Vibration +Heating Surface +LED-Bargraph +Loudspeaker +Power-LEDs +FanFigure 1: The Loaded Dice; left: example of devices in use, turning heat into light (sensor die with temperature sensor active and +actuator die with power LED active) [9]; right faces and functions – sensors and actuators [8] +2 +THE LOADED DICE - SENSES, SENSORS, ACTUATORS +The Loaded Dice1 are a set of two cubical devices wirelessly connected (fig. 2a). Each cube has six sides, offering in one +cube six sensors and in the other cube six actuators, one on each side, suitable for multisensory and multimodal +environmental and user interactions. The sensor cube normalizes a raw sensor value meaningfully, transmits it, and +then the other cube actuates it mapped on an output. The cubical shape communicates the intuitive reading that the +top side is active, like a die, offering an easy and spontaneous way to re-combine sensors and actuators. Every sensor- +in and actuator-out combination is possible resulting in 36 combinations in total. [5] +The “traditional five” human senses are sight, hearing, taste, smell and touch. Secondary senses are temperature, +pain, proprioception and balance. Due to the constraints of the technical platform we could not address all human +senses with sensors and actuators. Overall the Loaded Dice holds sensors and actuators equivalent to some human +senses directly (see fig. 1 and table 1 for details). It is also possible to think about effects to address other senses using +the given sensors and actuators, e.g. to inflict pain via the Peltier element through excessive heat or cold (not intended +nor recommended). It is also possible (but currently not implemented) to use the internal inertial measurement unit +(IMU), consisting of an accelerometer and gyrometer, not only for interaction controls but also as a sense, as an +equivalent to proprioception and balance (movement and position). +New multisensory interaction modalities are possible but not yet implemented, e.g. olfactory / smell. They have the +potential to broaden interaction qualities even further and especially in an emotional way [6]. +Table 1. Human senses vs. sensors and actuators in the Loaded Dice +Human Sense +Sensor +Actuator +sight +(visual stimuli) +luxmeter (visible light luminosity/ brightness) +passive infrared detector (PIR movement) +ultrasonic transceiver (distance) +power LED (brightness) +LED ring-graph (count, overall brightness, color) +hearing +(auditive stimuli) +microphone (amplitude) +sound (modulated note for instrument) +(vibration motor, rattling noise) +(fan, air flow noise) +touch +(tactile stimuli) +potentiometer (manual angular dial of 270°) +vibration motor (vibration) +fan (mechanical stimulation on hairs) +temperature +(thermal stimuli) +infrared thermometer (thermopile / thermal +radiation) +Peltier element (cooling and heating plate) +fan (cooling by chill effect on skin) +1 video demonstrating the Loaded Dice: https://www.youtube.com/watch?v=-E5aUiktCic +2 + +SENSORDIE +TemperatureSensor +Light Sensor +Microphone +MovementSensor +Potentiometer +Distance Sensor +ACTUATORDIE +Vibration +Heating Surface +LED-Bargraph +Loudspeaker +Power-LEDs +Fan3 +SYNESTHESIA - MAPPING SENSES AND MODALITIES +Synesthesia describes the phenomenon of an event being experienced by another, separate sensory modality [4]. While +medical not exact, in principle, this means a sound might not only be heard but also be seen as a color (as an example). +Most existing tools, i.e. for IoT ideation, do not employ synesthesia effects as a design opportunity in order to break +with existing sensing stereotypes for framing design spaces. Such a stereotype could be e.g. that making noise should +always be connected with hearing noise. While most related digital (IoT) ideation tools do allow for flexible +combinations of different sensors and actuators in principle, this is not ad hoc possible. Instead they require necessary +steps in combining parts or mapping sensor values to actuator values. Thus, they demand an initial idea of how the +combination should play out. Our tool allows users to explore such synesthetic effects ad hoc. +We implemented a meaningful mapping between every sensor and actuator that is used in the Loaded Dice. This +includes reasonably chosen sampling rates, ranges and steppings for raw input values, their normalization on internal +values and the conversion back to meaningful output values. All this is done internally in hard- and software, without +the need of user intervention. Selecting a new sensor-actuator combination just requires bringing another side to the +top. Based on the presented design rationale, a co-designer can transport heat over a distance by choosing the infrared +thermometer and Peltier element sides of both cubes. Rotating the actuator cube to the power-LEDs would transform +the temperature into light, thus mimicking synesthesia-like perception. +The possibilities of the Loaded Dice can be used in a framed scenario-driven co-design approach, in open +exploration or even just for ‘sensory sketching’, even for ‘weird’ synesthetic combinations, e.g: + +to try out how bright sunlight sounds or feels as vibration + +what temperature a loud cry has + +how much air-flow half a meter distance is + +whether you can feel the flickering of light … +We use meaningful but simple functions for preprocessing of raw sensor values and normalization to an +intermediate data value as well back to actuations (table 2). Overall, the mappings are done in a predefined ‘static’ +way. However, static does not mean one fits all. It is necessary to consider non-linearities and dynamics, e.g. for light +and sound, as these senses are not perceived in a linear or static manner by humans. However, we applied ‘just good +enough’ assumptions for meaningfulness without the claim of physical or psychometric correctness, sometimes even a +bit off to make effects clearer. Currently also the sensor as well as the selected actuator are considered for the mapping +in addition to the normalized value. We do this mainly for technical reasons as the different modalities operate at +different speeds. Currently, only the LED ring graphic signals which sensor has sampled the data by changing color. +Table 2: Current mapping from sensed values to intermediate values and then to actuated values +Sensor +Sensor Mapping +Value +Actuator Mapping +Actuator +potentiometer +0..270° AD sampling 0..1023  linear  0..24 +0..24 + Neopixels count 0..24, color coded by sensor, +brightness per pixel static +ring-graph +thermometer +digital read-out 0..50 °C  linear  0..24 + sqr  0..576  0..255 RGB brightness +power LED +microphone +50ms window AD sampling 0..1023  max-min +difference  0..1023  linear  0..24 +0  0; 1..24  MIDI noteOn(value+50) +sound +distance +0  0; 1..72 cm  linear  1..24 +0..12  -255..0 (cooling) 12..24  0..255 (heating) PWM +or 0..24  0..255 (from neutral to heating only) PWM +Peltier thermo +PIR movement +binary 0  0; 1  24 +0  0; 1..24  64..255 PWM +vibration +light +digital read-out 0..65535 lx  sqrt  0..48  0..24 +0  0; 1..24  160..255 PWM +fan +Every combination is possible, alignment in lines just as examples. AD: analogdigital conversion, PWM: pulse width modulation +3 + +SENSORDIE +TemperatureSensor +Light Sensor +Microphone +MovementSensor +Potentiometer +Distance Sensor +ACTUATORDIE +Vibration +Heating Surface +LED-Bargraph +Loudspeaker +Power-LEDs +FanWhile we are quite satisfied what the Loaded Dice can already do there are some new possibilities at hand: + +more use of colors: for power LED element and LED ring-graph (NeoPixels are colorful…) + +other use of sound: other (music/midi) instruments, modulation of velocity and pitch, other sounds +(artificial or sampled in nature) + +use of spatial component: position of the LEDs of the ring-graph, color fades, patterns + +use of temporal components: from time static value to dynamic patterns for sound, vibration, light, air +flow etc. +A flexible “sketching” of a new mapping function would allow to bring in completely new synesthesia effects, also not +necessarily only limited to one input sensor and one output actuator at one time. +4 +CONCLUSION +While the Loaded Dice can already be used meaningfully for activities associated with synesthesia, e.g. for ideation, we +see a lot of potential in more flexible mappings and even other creative uses of what the sensors and actuators might +do. We are open for inspirations and ideas. +ACKNOWLEDGMENTS +This research is funded by the German Ministry of Education and Research (BMBF), grant FKZ 16SV7116. +References +[1] +Tina Aspiala and Alexandra Deschamps-Sonsino. 2016. Know Cards: Learn. Play. Collect. Know Cards. Retrieved December 6, 2016 from +http://know-cards.myshopify.com/ +[2] +Ayah Bdeir. 2009. Electronics As Material: LittleBits. In Proceedings of the 3rd International Conference on Tangible and Embedded Interaction (TEI +’09), 397–400. https://doi.org/10.1145/1517664.1517743 +[3] +Arne Berger, William Odom, Michael Storz, Andreas Bischof, Albrecht Kurze, and Eva Hornecker. 2019. The Inflatable Cat: Idiosyncratic Ideation +Of Smart Objects For The Home. In CHI Conference on Human Factors in Computing Systems Proceedings. https://doi.org/10.1145/3290605.3300631 +[4] +Peter G. Grossenbacher and Christopher T. Lovelace. 2001. Mechanisms of synesthesia: cognitive and physiological constraints. Trends in cognitive +sciences 5, 1: 36–41. Retrieved December 15, 2016 from http://www.sciencedirect.com/science/article/pii/S1364661300015710 +[5] +Albrecht Kurze. 2021. Interaction Qualities For Interactions With, Between, And Through IoT Devices. In 11th International Conference on the +Internet of Things (IoT ‘21), November 08-12, 2021, St.Gallen, Switzerland. https://doi.org/10.1145/3494322.3494348 +[6] +Albrecht Kurze. 2021. Scented Dice: New interaction qualities for ideating connected devices. In Workshop Smell, Taste, and Temperature Interfaces +at Conference on Human Factors in Computing Systems (CHI ’21). Retrieved from https://arxiv.org/abs/2201.10484 +[7] +Albrecht Kurze, Kevin Lefeuvre, Michael Storz, Andreas Bischof, Sören Totzauer, and Arne Berger. 2016. Explorative Co-Design-Werkzeuge zum +Entwerfen von Smart Connected Things am Beispiel eines Workshops mit Blinden und Sehbehinderten. In Technische Unterstützungssysteme, die +die Menschen wirklich wollen, 395–400. Retrieved January 19, 2017 from http://tinyurl.com/janya26 +[8] +Kevin Lefeuvre, Sören Totzauer, Andreas Bischof, Albrecht Kurze, Michael Storz, Lisa Ullmann, and Arne Berger. 2016. Loaded Dice: Exploring the +Design Space of Connected Devices with Blind and Visually Impaired People. In Proceedings of the 9th Nordic Conference on Human-Computer +Interaction (NordiCHI ’16), 31:1-31:10. https://doi.org/10.1145/2971485.2971524 +[9] +Kevin Lefeuvre, Sören Totzauer, Andreas Bischof, Michael Storz, Albrecht Kurze, and Arne Berger. 2017. Loaded Dice: How to cheat your way to +creativity. In Proceedings of the 3rd Biennial Research Through Design Conference. https://doi.org/10.6084/m9.figshare.4746976.v1 +4 + +SENSORDIE +TemperatureSensor +Light Sensor +Microphone +MovementSensor +Potentiometer +Distance Sensor +ACTUATORDIE +Vibration +Heating Surface +LED-Bargraph +Loudspeaker +Power-LEDs +Fan \ No newline at end of file diff --git a/GdFJT4oBgHgl3EQfECxK/content/tmp_files/load_file.txt b/GdFJT4oBgHgl3EQfECxK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..044f7f5abe4e64789259bac6efc58e2c60492c99 --- /dev/null +++ b/GdFJT4oBgHgl3EQfECxK/content/tmp_files/load_file.txt @@ -0,0 +1,205 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf,len=204 +page_content='Synesthetic Dice: Sensors, Actuators, And Mappings Albrecht Kurze Chemnitz University of Technology, Albrecht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='Kurze@informatik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='tu-chemnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='de How bright can you cry?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' How loud does the sun shine?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' We developed a multisensory and multimodal tool, the Loaded Dice, for use in co-design workshops to research the design space of IoT usage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' The Loaded Dice incorporate the principle of a technical synesthesia, being able to map any of the included sensors to any of the included actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' With just a turn of one of the cubical devices it is possible to create a new combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' We discuss the core principles of the Loaded Dice, what sensors and actuators are included, how they relate to human senses, and how we realized a meaningful mapping between sensors and actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' We further discuss where we see additional potential in the Loaded Dice to support synesthetic exploration – as Synesthetic Dice – so that you can eventually find out who cries brighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' CCS CONCEPTS • Human-centered computing~Human computer interaction (HCI) Additional Keywords and Phrases: multisensory, multimodal, synesthesia, design, ideation, tools, methods, IoT, Internet of Things, haptic technology, cubic shape, tangible interactive devices, input and output devices, tangibles ACM Reference Format: Albrecht Kurze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Synesthetic Dice: Sensors, Actuators, And Mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' In Workshop Sensory Sketching (CHI’22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' April 22, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' 4 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' 1 INTRODUCTION Some years ago we designed and developed the Loaded Dice [8,9], a multisensory and multimodal hybrid toolkit to ideate Internet of Things (IoT) devices and scenarios, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' for the ‘smart’ home, and with different groups of co- designers [3,7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' The Loaded Dice filled a gap between analog, non-functional tools, often card-based, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' KnowCards [1], and functional but tinkering based tools, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' littleBits [2], for multisensory und multimodal exploration, ideation and prototyping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' We introduce the Loaded Dice, the core concepts that they are built on, the used sensors and actuators, and how they map to different human senses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' We will then continue to discuss how we realized mappings between sensed raw value, normalized intermediate values, and actuated values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' While the mappings that we currently use are sufficiently good enough for current purposes, we see big potential in some extended uses as ‘Synesthetic Dice’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' This brings us to our core question: How can the Loaded Dice be used for exploration and research of synesthetic mappings between sensors and actuators, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' for innovative interactions and non-verbal communication?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' SENSORDIE TemperatureSensor Light Sensor Microphone MovementSensor Potentiometer Distance Sensor ACTUATORDIE Vibration Heating Surface LED-Bargraph Loudspeaker Power-LEDs FanFigure 1: The Loaded Dice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' left: example of devices in use, turning heat into light (sensor die with temperature sensor active and actuator die with power LED active) [9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' right faces and functions – sensors and actuators [8] 2 THE LOADED DICE - SENSES, SENSORS, ACTUATORS The Loaded Dice1 are a set of two cubical devices wirelessly connected (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Each cube has six sides, offering in one cube six sensors and in the other cube six actuators, one on each side, suitable for multisensory and multimodal environmental and user interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' The sensor cube normalizes a raw sensor value meaningfully, transmits it, and then the other cube actuates it mapped on an output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' The cubical shape communicates the intuitive reading that the top side is active, like a die, offering an easy and spontaneous way to re-combine sensors and actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Every sensor- in and actuator-out combination is possible resulting in 36 combinations in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' [5] The “traditional five” human senses are sight, hearing, taste, smell and touch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Secondary senses are temperature, pain, proprioception and balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Due to the constraints of the technical platform we could not address all human senses with sensors and actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Overall the Loaded Dice holds sensors and actuators equivalent to some human senses directly (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' 1 and table 1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' It is also possible to think about effects to address other senses using the given sensors and actuators, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' to inflict pain via the Peltier element through excessive heat or cold (not intended nor recommended).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' It is also possible (but currently not implemented) to use the internal inertial measurement unit (IMU), consisting of an accelerometer and gyrometer, not only for interaction controls but also as a sense, as an equivalent to proprioception and balance (movement and position).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' New multisensory interaction modalities are possible but not yet implemented, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' olfactory / smell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' They have the potential to broaden interaction qualities even further and especially in an emotional way [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Human senses vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' sensors and actuators in the Loaded Dice Human Sense Sensor Actuator sight (visual stimuli) luxmeter (visible light luminosity/ brightness) passive infrared detector (PIR movement) ultrasonic transceiver (distance) power LED (brightness) LED ring-graph (count,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' overall brightness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' color) hearing (auditive stimuli) microphone (amplitude) sound (modulated note for instrument) (vibration motor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' rattling noise) (fan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' air flow noise) touch (tactile stimuli) potentiometer (manual angular dial of 270°) vibration motor (vibration) fan (mechanical stimulation on hairs) temperature (thermal stimuli) infrared thermometer (thermopile / thermal radiation) Peltier element (cooling and heating plate) fan (cooling by chill effect on skin) 1 video demonstrating the Loaded Dice: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='v=-E5aUiktCic 2 SENSORDIE TemperatureSensor Light Sensor Microphone MovementSensor Potentiometer Distance Sensor ACTUATORDIE Vibration Heating Surface LED-Bargraph Loudspeaker Power-LEDs Fan3 SYNESTHESIA - MAPPING SENSES AND MODALITIES Synesthesia describes the phenomenon of an event being experienced by another, separate sensory modality [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' While medical not exact, in principle, this means a sound might not only be heard but also be seen as a color (as an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Most existing tools, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' for IoT ideation, do not employ synesthesia effects as a design opportunity in order to break with existing sensing stereotypes for framing design spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Such a stereotype could be e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' that making noise should always be connected with hearing noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' While most related digital (IoT) ideation tools do allow for flexible combinations of different sensors and actuators in principle, this is not ad hoc possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Instead they require necessary steps in combining parts or mapping sensor values to actuator values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Thus, they demand an initial idea of how the combination should play out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Our tool allows users to explore such synesthetic effects ad hoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' We implemented a meaningful mapping between every sensor and actuator that is used in the Loaded Dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' This includes reasonably chosen sampling rates, ranges and steppings for raw input values, their normalization on internal values and the conversion back to meaningful output values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' All this is done internally in hard- and software, without the need of user intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Selecting a new sensor-actuator combination just requires bringing another side to the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Based on the presented design rationale, a co-designer can transport heat over a distance by choosing the infrared thermometer and Peltier element sides of both cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Rotating the actuator cube to the power-LEDs would transform the temperature into light, thus mimicking synesthesia-like perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' The possibilities of the Loaded Dice can be used in a framed scenario-driven co-design approach, in open exploration or even just for ‘sensory sketching’, even for ‘weird’ synesthetic combinations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='g: to try out how bright sunlight sounds or feels as vibration what temperature a loud cry has how much air-flow half a meter distance is whether you can feel the flickering of light … We use meaningful but simple functions for preprocessing of raw sensor values and normalization to an intermediate data value as well back to actuations (table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Overall, the mappings are done in a predefined ‘static’ way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' However, static does not mean one fits all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' It is necessary to consider non-linearities and dynamics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' for light and sound, as these senses are not perceived in a linear or static manner by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' However, we applied ‘just good enough’ assumptions for meaningfulness without the claim of physical or psychometric correctness, sometimes even a bit off to make effects clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Currently also the sensor as well as the selected actuator are considered for the mapping in addition to the normalized value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' We do this mainly for technical reasons as the different modalities operate at different speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Currently, only the LED ring graphic signals which sensor has sampled the data by changing color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Table 2: Current mapping from sensed values to intermediate values and then to actuated values Sensor Sensor Mapping Value Actuator Mapping Actuator potentiometer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.270° AD sampling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.1023 \uf0e0 linear \uf0e0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.24 \uf0e0 Neopixels count 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.24, color coded by sensor, brightness per pixel static ring-graph thermometer digital read-out 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.50 °C \uf0e0 linear \uf0e0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.24 \uf0e0 sqr \uf0e0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.576 \uf0e0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.255 RGB brightness power LED microphone 50ms window AD sampling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.1023 \uf0e0 max-min difference \uf0e0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.1023 \uf0e0 linear \uf0e0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.24 0 \uf0e0 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.24 \uf0e0 MIDI noteOn(value+50) sound distance 0 \uf0e0 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.72 cm \uf0e0 linear \uf0e0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.12 \uf0e0 -255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.0 (cooling) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.24 \uf0e0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.255 (heating) PWM or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.24 \uf0e0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.255 (from neutral to heating only) PWM Peltier thermo PIR movement binary 0 \uf0e0 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' 1 \uf0e0 24 0 \uf0e0 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.24 \uf0e0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.255 PWM vibration light digital read-out 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.65535 lx \uf0e0 sqrt \uf0e0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.48 \uf0e0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.24 0 \uf0e0 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.24 \uf0e0 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='.255 PWM fan Every combination is possible, alignment in lines just as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' AD: analog\uf0e0digital conversion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' PWM: pulse width modulation 3 SENSORDIE TemperatureSensor Light Sensor Microphone MovementSensor Potentiometer Distance Sensor ACTUATORDIE Vibration Heating Surface LED-Bargraph Loudspeaker Power-LEDs FanWhile we are quite satisfied what the Loaded Dice can already do there are some new possibilities at hand: more use of colors: for power LED element and LED ring-graph (NeoPixels are colorful…) other use of sound: other (music/midi) instruments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' modulation of velocity and pitch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' other sounds (artificial or sampled in nature) use of spatial component: position of the LEDs of the ring-graph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' color fades,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' patterns use of temporal components: from time static value to dynamic patterns for sound,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' vibration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' air flow etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' A flexible “sketching” of a new mapping function would allow to bring in completely new synesthesia effects, also not necessarily only limited to one input sensor and one output actuator at one time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' 4 CONCLUSION While the Loaded Dice can already be used meaningfully for activities associated with synesthesia, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' for ideation, we see a lot of potential in more flexible mappings and even other creative uses of what the sensors and actuators might do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' We are open for inspirations and ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' ACKNOWLEDGMENTS This research is funded by the German Ministry of Education and Research (BMBF), grant FKZ 16SV7116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' References [1] Tina Aspiala and Alexandra Deschamps-Sonsino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Know Cards: Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Know Cards.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='1145/3494322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='3494348 [6] Albrecht Kurze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Scented Dice: New interaction qualities for ideating connected devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' In Workshop Smell, Taste, and Temperature Interfaces at Conference on Human Factors in Computing Systems (CHI ’21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Retrieved from https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='org/abs/2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='10484 [7] Albrecht Kurze, Kevin Lefeuvre, Michael Storz, Andreas Bischof, Sören Totzauer, and Arne Berger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Explorative Co-Design-Werkzeuge zum Entwerfen von Smart Connected Things am Beispiel eines Workshops mit Blinden und Sehbehinderten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' In Technische Unterstützungssysteme, die die Menschen wirklich wollen, 395–400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Retrieved January 19, 2017 from http://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='com/janya26 [8] Kevin Lefeuvre, Sören Totzauer, Andreas Bischof, Albrecht Kurze, Michael Storz, Lisa Ullmann, and Arne Berger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Loaded Dice: Exploring the Design Space of Connected Devices with Blind and Visually Impaired People.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' In Proceedings of the 9th Nordic Conference on Human-Computer Interaction (NordiCHI ’16), 31:1-31:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='1145/2971485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='2971524 [9] Kevin Lefeuvre, Sören Totzauer, Andreas Bischof, Michael Storz, Albrecht Kurze, and Arne Berger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' Loaded Dice: How to cheat your way to creativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' In Proceedings of the 3rd Biennial Research Through Design Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='6084/m9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='4746976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} +page_content='v1 4 SENSORDIE TemperatureSensor Light Sensor Microphone MovementSensor Potentiometer Distance Sensor ACTUATORDIE Vibration Heating Surface LED-Bargraph Loudspeaker Power-LEDs Fan' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'} diff --git a/GtAyT4oBgHgl3EQfrfmY/content/tmp_files/2301.00562v1.pdf.txt b/GtAyT4oBgHgl3EQfrfmY/content/tmp_files/2301.00562v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..27d45cbbaf713940e7d28d7ce0f9f1faabac0ead --- /dev/null +++ b/GtAyT4oBgHgl3EQfrfmY/content/tmp_files/2301.00562v1.pdf.txt @@ -0,0 +1,1507 @@ +Age-Optimal Multi-Channel-Scheduling under +Energy and Tolerance Constraints +Xujin Zhou, Irem Koprulu, Atilla Eryilmaz +Electrical and Computer Engineering +The Ohio State University +Columbus, US +{zhou.2400@osu.edu, irem.koprulu@gmail.com, eryilmaz.2@osu.edu} +Abstract—We study the optimal scheduling problem where n +source nodes attempt to transmit updates over L shared wireless +on/off fading channels to optimize their age performance under +energy and age-violation tolerance constraints. Specifically, we +provide a generic formulation of age-optimization in the form of +a constrained Markov Decision Processes (CMDP), and obtain +the optimal scheduler as the solution of an associated Linear +Programming problem. We investigate the characteristics of the +optimal single-user multi-channel scheduler for the important +special cases of average-age and violation-rate minimization. +This leads to several key insights on the nature of the optimal +allocation of the limited energy, where a usual threshold-based +policy does not apply and will be useful in guiding scheduler +designers. We then investigate the stability region of the optimal +scheduler for the multi-user case. We also develop an online +scheduler using Lyapunov-drift-minimization methods that do +not require the knowledge of channel statistics. Our numerical +studies compare the stability region of our online scheduler to the +optimal scheduler to reveal that it performs closely with unknown +channel statistics. +I. INTRODUCTION +In recent years, the Internet of Things (IoT) has become +one of the most important frameworks of the next-generation +wireless networks, whereby a large number of mobile devices +need to be supported over an ultra-wide frequency spectrum +(see, for example, [1]). In particular, for many real-time IoT +applications, it is necessary for the devices to send fresh +updates over the shared spectrum. To measure the freshness +of data, the concept of Age of Information (AoI) has been +introduced over the last decade (see, for example, [2]–[4]), +which is defined concisely as the elapsed time since the +generation time of the last received status update. Since +the introduction of the AoI metric, numerous related studies +emerged in various networking scenarios, including wireless +random access networks (e.g., [5], [6]), content distribution +networks (e.g., [7], [8]), scheduling (e.g., [9]–[13]), queuing +networks (e.g., [14], [15]), and vehicular networks (e.g., [16]). +Recently, other AoI related metrics have been developed in +order to address more generalized or different forms of ageing, +such as: non-linear AoI (e.g., [4], [17]), peak AoI (e.g., [18]), +time-since-last-service (e.g., [19]), age upon decisions (e.g., +[20]), to name a few. Among them, the metric, called the age- +violation-rate (see [15], [21], [22]) is of particular interest for +real-time IoT services that have hard age-deadline constraints +and a limited tolerance to violating this deadline (see [23], +[24] for further motivation of this metric). +In view of its significance for next generation IoT networks, +in this paper, we study the general optimal multi-channel +scheduling problem to optimize varying forms of age perfor- +mances under energy and age-violation tolerance constraints. +Our contributions can be listed as: +• We provide a generic formulation of age-optimization +as a Constrained Markov Decision Problem (CMDP) +(see [25]–[27]) and obtain the age-optimal multi-channel +scheduler as the solution of an associated Linear Pro- +gramming problem, first for the single-source (in Sec- +tion III) and then for general the multi-source (in Sec- +tion IV) scenarios. +• For the single-source multi-channel scenario, we in- +vestigate the characteristics of the optimal schedulers +under energy constraints for two age metrics that are +important for IoT applications: (i) average-age mini- +mization; and (ii) age-violation-rate minimization, a non- +convex/concave metric (in Section III-C). Our investiga- +tions reveal various insights on different energy allocation +structures, as well as the common monotonicity proper- +ties of the optimal schedulers for minimizing these two +metrics, which is useful for guiding scheduler designers. +• For the multi-source age optimal scheduling problem, +we also study the feasibility region of the average- +age-optimal scheduler under age-violation-rate tolerance +constraints to contrast its results with those of related +earlier works that are developed for the single-channel +multi-user scenario (see Section IV-C and Section VI). +• Moreover, we develop (in Section V) an online scheduler +using Lyapunov-drift-minimization methods (e.g., [28]) +that does not require the knowledge of channel statistics, +and compare its performance to the optimal and earlier +designs to reveal how much the knowledge of channel +statistics affects the feasibility region (see Section VI). +Our work relates to, but also differs from several other +related works in this domain. Many early works (e.g., [9], +[12], [29]) aim to minimize AoI under power constraints +but with the assumption of reliable channels as opposed to +the fading channels that we consider. More recent works +arXiv:2301.00562v1 [cs.IT] 2 Jan 2023 + +(e.g., [10], [30]) aim to minimize AoI-related costs based +on max-age matching, while other works (e.g., [29], [31]) +proposed AoI minimization schedulers based on Whittle Index +approach. However, to the best of our knowledge, prior works +predominantly assume that one source can choose at most +one channel, which is an important factor in proving the +Whittle Indexability of the corresponding problems they solve. +In contrast, one of the key features our setting is the possibility +of each user to transmit over multiple channels as enabled by +new wireless technologies. Furthermore, most of the above +mentioned works have average or peak AoI as the objective +function, while we consider more general age-based objective +functions, which for example allows the objective function to +be a non-convex metric such as the age-violation-rate. In this +multi-channel setting with general objectives, we observe (cf. +Section III-C) that the optimal solution can in fact possess non- +monotone characteristics, which make the Whittle Indexability +approach infeasible in general. The work in [21] has con- +sidered the multi-source single-channel scheduling problem +under tolerance constraints, which is a special case of our +setting. We would like to note that this interesting work +[21] has been a primary motivation for our current work in +exploring a different approach based on the CMDP framework +that guarantees optimality and applies to more general multi- +channel scenarios with additional energy constraints. There +are also works (e.g., [32], [33]) that focus on learning-based +approaches which can be considered as complementary to the +focus of this work. +II. SYSTEM MODEL +We consider the operation of a discrete-time wireless access +system, whereby n source nodes share L on/off fading wireless +channels to update their ageing status at a receiver (such as a +base station) under energy and violation tolerance constraints +(see Figure 1). +Figure 1. n sources share L on-off fading channels to update their status to +a receiver under energy and tolerance constraints in order to keep their age +levels low. +Our goal is to develop generic solution strategies to find +optimal schedulers that can optimize diverse age-based metrics +while meeting certain requirements on energy consumption +and tolerance levels. We describe the key terminology and the +essential system dynamics in the rest of this section. Then, +in the following sections we formulate and solve classes of +age-optimization problems for single and multi-source cases, +subsequently. +Scheduling policy and age-violation-tolerance: We assume +that each source node i ∈ {1, · · · , n} refreshes its status +and creates a new packet at the beginning of every time +slot t ∈ {1, 2, 3, · · · }. Source nodes attempt to transmit their +freshest packet to the receiver, for example a base station +(BS), whenever they get a chance to transmit. Every time the +BS successfully receives a new status from source node i, +it saves the current status and discards all previous packets +received from that node. As such, the BS keeps only one +packet from each source node, namely the freshest one. We +use Xi[t] to denote the generation time of the packet stored +at the BS from source i at time t. We define the age Ai[t] +of source node i at time t as the time that has elapsed since +the generation of its last received packet1: Ai[t] ≜ t − Xi[t]. +We use2 A[t] ≜ (A1[t], · · · , An[t]) to denote the ages of all +sources at time slot t. +At the beginning of each time slot, the centralized scheduler +decides which channels each of the source nodes will use to +transmit to the base station based on the ages A[t] of all source +nodes. Let ui(A[t]) be the number of channels source node +i uses to transmit at time t. Each transmission attempt can +resolve in success or failure which we will describe below +as part of the channel success model. If the base station +successfully receives the packet from source i at time t, then +its age at time t + 1 will reset to 1, otherwise its age will +increase by one, i.e., +Ai[t + 1] = +� +1, if transmission of source i succeeds +Ai[t] + 1, +otherwise. +We allow each source i to have a desired age thresh- +old/deadline τi. The information of source i is up-to-date if +its age is less than or equal to this threshold τi. Otherwise, we +speak of an age violation in that slot. In particular, we define +the age-violation-rate of source i as the long-term average +fraction of time slots when the source’s age Ai[t] exceeds +its threshold τi, i.e., lim +T →∞ +1 +T +T +� +t=1 +1 {Ai[t] > τi}. We use ϵi ∈ +[0, 1] to indicate the tolerance of source i that measures the +maximum allowed age-violation-rate for its updates. (ϵi = 1 +indicates that there is no violation rate constraint, and ϵi = 0 +indicates that we do not allow any deadline violation.) When +the age violation rate is no greater than the tolerance rate, the +age violation tolerance constraint is satisfied. +Channel success model and energy constraints: The n +source nodes share L wireless on/off fading channels, each +of which can accommodate at most one packet transmission. +However, even when there is a single transmission over +a channel, a successful transmission is not guaranteed. In +1This metric is also referred to as Age-of-Information (AoI) and Time-Since- +Last-Service (TSLS) in different contexts. In the rest of the paper, we will refer +to it as AoI or simple as age, interchangeably. +2We will consistently use bold symbols to represent vectors. + +particular, source node i has a channel success probability of +µi when transmitting over each of its assigned channels3. +We call the update of source i in a slot to be a success +if any one of its transmissions over its assigned channels is +successful. Since the channel is a collision channel, for an +optimal scheduler we always have +n +� +i=1 +ui(A[t]) ≤ L. Once +the value of ui(A[t]) is decided for all i, the scheduler will +assign different channels to different sources, so that no two +sources transmit over the same channel. Also, note that under +the described channel success model, the probability for the +BS to successfully receive an update from source node i when +the node uses l channels is 1 − (1 − µi)l. +We assume that each transmission over a channel comes +with an energy cost of 1 unit4. We require that the aggregate +time-average energy cost for source i is not greater than a +given constraint bi channels per slot, i.e., we require +lim +T →∞ +1 +T +T +� +t=1 +ui (A[t]) ≤ bi, +bi ∈ R+. +It is obvious that transmitting over more channels will +increase the success probability of a source, but increase +energy consumption. We are interested in finding the number +of channels that when allocated to sources optimize the desired +age performance given the current age state, as well as energy +and and tolerance constraints discussed above. In the next +section, we attack this problem within the constrained Markov +Decision Process (MDP) framework first for a single user, and +then extend our approach to cover the multi-user setting. +III. AGE-OPTIMAL MULTI-CHANNEL SCHEDULING FOR A +SINGLE USER +In this section, we first consider the single-user age-optimal +multi-channel scheduling problem. This not only allows us to +simplify the notation by omitting the subscripts, but also is of +particular interest for the next generation ultra-wideband wire- +less communication technologies that are expected to support +low-delay access over multiple fading channels. We formulate +a general age-optimal optimization problem which can be +used in different scenarios in Section III-A and following the +analysis of the performance in Section III-B. To that end, +in Section III-C, we study the characterization and insights +of the optimal schedulers for two important special cases of +minimizing the average-age and the age-violation-rate, which +will be useful in guiding scheduler designers. +A. Problem formulation +The problem of minimizing time-averaged age-based ob- +jectives under average energy and tolerance constraints can +3All our development can be generalized to the case when the success +probability between source i and channel j is allowed to be different as µij. +However, this is omitted here as it increases the complexity of the exposition +without adding to the substance. +4This can also be generalized to non-uniform energy costs over different +channels, but omitted to avoid cumbersome notation. +be generally formulated as the following constrained Markov +decision problem [25]: +min +u(A) +lim +T →∞ +1 +T +T +� +t=1 +E [ω0(A[t])] +(1) +s.t : +lim +T →∞ +1 +T +T +� +t=1 +E [u (A[t])] ≤ b, +(2) +lim +T →∞ +1 +T +T +� +t=1 +E [ωk (A[t])] ≤ ck, k = 1, · · · , K, +u(A[t]) ∈ {0, 1, · · · , L}. +The optimization is performed over Markovian policies +described by a function u(·) that maps age levels to number +of channels. It is known that such Markovian policies are +sufficient for optimal operation [25]. +The first constraint on the time-averaged u(·) captures the +average energy constraint discussed in the system model. The +functions ωk(·) serve as general functions that map the current +state A[t] to a value that measures the cost of that age with +respect to various measures5 By setting different mappings for +the weight function ω0(A[t]), the objective can be changed +into different commonly used age-related objectives: letting +ω0(a) = −1{a = 1} transform the objective to maximizing +the average throughput; letting ω0(a) = a makes the objective +minimize the average AoI; letting ω0(a) = 1{a ≥ d} make the +objective minimize the average age-violation rate. Note that +this allows the objective function to be a non-convex/concave +function. +B. Performance analysis +Next, we will analyze the generic constrained optimization +problem under energy constraint by showing that the problem +is equivalent to a Linear Programming (LP) problem and thus +describe the optimal policy. +Theorem 1: The solution of the generic age-optimization +problem (1) can be obtained by solving the following linear +programming problem: +min +yla +D +� +a=1 +L +� +l=0 +yl +aω0(a) +s.t: +D +� +a=1 +L +� +l=0 +yl +a · l ≤ b, +D +� +a=1 +L +� +l=0 +yl +aωk(a) ≤ ck, k = 1, · · · , K, +0 ≤ yl +a ≤ 1 +∀1 ≤ a ≤ D, 0 ≤ l ≤ L, +D +� +a=1 +L +� +l=0 +yl +a = 1, +Qy = 0, +where y is a column vector of size DL with y += +(y1 +1, · · · , yL +1 , · · · , y1 +D, · · · , yL +D)T as its components; D is an +5We note that the problem can also solved with the same approach +(but heavier notation) by more generally defining ωk(A[t], u(A[t])) to be +functions of both the age and the action. + +upper bound on the age state in the system which can be +set sufficiently large so that the probability of reaching D +is vanishing.6 Qy = 0 is the matrix representation of the +following (global balance) equations: +L +� +l=0 +yl +a+1 − +L +� +l=0 +yl +a(1 − µ)l = 0 +∀a = 1, · · · , D − 2, +L +� +l=0 +� +1 − (1 − µ)l� +yl +D − +L +� +l=0 +yl +D−1(1 − µ)l = 0, +− +L +� +l=0 +yl +1(1 − µ)l + +D +� +a=2 +L +� +l=0 +yl +a +� +1 − (1 − µ)l� += 0. +If this LP is feasible, and y is an optimal solution, then the +optimal policy u∗(a) is a probabilistic policy, whereby the +probability f l +a of choosing l channels when the age is at state +a equals: +f l +a = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +yl +a +L +� +l=0 +yl +a +, +if +L +� +l=0 +yl +a ̸= 0 +1 +L, +if +� +l +yl +a = 0 +(3) +for l = 0, 1, · · · , L and a = 1, 2, · · · , D. +Proof: +As shown in [25], it is enough for us to optimize +over the Markovian policies for Problem 1. Since the process +is not affected by a shift in time, we can define the probabilistic +scheduling policy where f l +a denotes the probability of choosing +l channels when the AoI of single source is at state a. The +normalization constraint of the probabilistic scheduling policy +requires +L +� +l=0 +f l +a = 1 and f l +a ⩾ 0 for all a. +Notice that the system state can be fully characterized by a +one-dimensional Markov chain with age A[t] as state. Given +the current state information A[t], the system state at the next +time slot A[t+1] depends only on the current state A[t] (with +no dependence on earlier states) and the current action u[t]. +In addition, the objective and constraints only depend on the +current state and action. So an equivalent MDP problem can +be formulated. Let λa2 +a1 denote the transition probability from +state a1 to a2, and define ¯µ ≜ 1 − µ as the probability of +channel failure. Then based on the channel success model, +λa2 +a1 = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +L +� +l=1 +f l +a1 ¯µl, +1 ≤ a1 ≤ D − 1, a2 = a1 + 1 +L +� +l=1 +f l +a(1 − ¯µl), +a1 = 1, · · · , D, a2 = 1 +L +� +l=1 +f l +D(1 − ¯µl), +a1 = D, a2 = D +0, +otherwise. +(4) +Since there are finitely many states, there exists a stationary +distribution π(a) for every a. Let C be the set of all recurrent +6In practice, moderate level of D is enough so that the dimension of LP +won’t be large. Also, when there is only age violation related objective and +constraints, it’s enough to set D = d + 1. See III-C and IV-C for references. +states, then C is irreducible and closed, thus C is positive +recurrent. When a ∈ C the stationary distribution π(a) is equal +to the long term average lim +T →∞ +1 +T +T +� +t=1 +1{A[t] = a} independent +of the starting point. When state a /∈ C, then both the stationary +distribution and the long term average are equal to zero. So the +optimization problem is equivalent to the following constraint +MDP problem: +min +f la +D +� +a=1 +π(a)ω0(a) +s.t: +D +� +a=1 +L +� +l=0 +π(a)f l +al ≤ b +D +� +a=1 +π(a)ωk(a) ≤ ck, k = 1, · · · , K +(5) +L +� +l=0 +f l +a = 1, f l +a ⩾ 0 +∀a ≤ D, l ≤ L +(6) +H · Π = Π, +1 · Π = 1 +(7) +where Π = [π(1), · · · , π(D)]T is the stationary distribution of +the Markov Chain and H is the D × D transition matrix with +hij = λi +j. Let us define yl +a = π(a)f l +a, then π(a) = +L +� +l=0 +yl +a for +a ≤ D. Then the constraint 5 becomes: +D +� +a=1 +L +� +l=0 +yl +aωk(a) ≤ ck, k = 1, · · · , K. +The +normalization +constraint +in +Equation +7 +requires +D +� +a=1 +L +� +l=0 +yl +a = 1. Substituting yl +a into the CMDP problem and +after simplifying, we establish the equivalency of the Linear +Programming problem. After obtaining the solution y, we +let f l +a = yl +a/π(a) for π(a) ̸= 0.States a with π(a) = 0, are +transient states, and the actions at these states do not affect +the average results. For those states we adopt a simple policy +as in Equation 3, then the constraint 7 is also satisfied. +C. Characterization and Insights on Age-Optimal Schedulers +Our general framework encompasses a wide range of objec- +tives and constraints for different choices of ωk(·) functions +using different age and age-violation metrics. In this section, +we focus on two important problems that can be expressed +within our framework: average age minimization and age- +violation-rate minimization. This effort will enable us to +characterize their optimal schedulers and gain insights into +their nature. +Optimal scheduler minimizing average age: When we set +ω0(a) = a in (1), the objective of the optimization problem +becomes to minimize the average age +lim +T →∞ +1 +T +T +� +t=1 +E{A[t]} = +D +� +a=1 +a π(a). + +0 +5 +10 +15 +AoI +0 +2 +4 +6 +8 +Average number of activated channels +=0.12 +=0.1 +=0.08 +=0.04 +Figure 2. Optimal number of channels to choose to minimize average AoI +when b = 2. +For this problem formulation, we retain the energy constraint +lim +T →∞ +1 +T +T +� +t=1 +E [u (A[t])] ≤ b; but do not need additional age +constraints. Hence, ωk(a) = 0 and ck = 0, for all k and a. +Figure 2 depicts the average number of activated channels +of the average-age optimal scheduler as a function of the age +states under different channel success probabilities µ for the +energy constraint b = 2. We will further discuss these results +at the end of this section in comparison with the next scheduler +of interest. +Optimal scheduler minimizing age-violation-rate: Setting +ω0(a) = 1{a > τ} n (1), the objective becomes minimizing +the average age-violation-rate +lim +T →∞ +1 +T +T +� +t=1 +E{1{A[t] > τ}} = +D +� +a=τ+1 +π(a). +As before, we keep the energy constraint, but do not need +additional age constraints. Hence, ωk(a) = 0 and ck = 0, for +all k and a. +With this, the problem becomes minimizing the age- +violation-rate under an energy constraint. Unlike in the previ- +ous problem, our goal is not to minimize the average age but +to avoid age-violation events. In this scenario, we can view +all the states with a > τ as state τ + 1, so it’s enough to set +D = τ + 1. +Figure 3 depicts the average number of activated channels +of the violation-rate optimal scheduler as a function of the age +states under different channel success probabilities µ for age +threshold τ = 8 and the same energy constraint b = 2. Next, +we compare the optimal policies of these two schedulers and +discuss the insights that can be gained from their study. +Insights on the two optimal schedulers: We start by noting +the similarities of the optimal policy under both scenarios: +(i) Each optimal policy is a probabilistic combination of at +most two deterministic policies, which matches the result +that the number of randomization is at most the number +of constraints, as shown in [25]. +(ii) For each scenario, as the channel success probability +increases, the corresponding optimal policy starts trans- +2 +4 +6 +8 +AoI +0 +5 +10 +15 +20 +25 +Average number of activated channels +=0.12 +=0.1 +=0.08 +=0.04 +Figure 3. Optimal number of channels to choose to minimize AoI violation +rate when b = 2 and τ = 8. +mitting at lower age levels, and also tends to choose +more channels at the same age level. This is a somewhat +counter-intuitive characteristic that indicates that the opti- +mal policy should be more active and active earlier when +the channels are more reliable. +(iii) The optimal policy in each scenario is idle when AoI is +relatively small. This is meaningful once we observe that, +when the age is relatively small, a successful transmission +will not benefit the objective as much as when the age +is large. Hence, the optimal scheduler saves energy for +larger age states. +However, we also notice differences between the two sets +of schedulers: +(i) The optimal policy in the average age minimization prob- +lem has an activation function u∗(·) that is monotone non- +decreasing with increasing age state. On the other hand, +the monotonicity does not hold in the age violation rate +minimization problem. This difference comes from the +non-convex nature of the the age violation rate function +in the latter case. In [25] and many related works (e.g., +[9], [34]), the authors exploit the monotone structure and +threshold nature of the optimal scheduling policy for solv- +ing the CMDP, revealing insights as well as simplifying +the algorithm by using the convexity or concavity of the +objective functions. However, in our general treatment, +the objective functions, such as age violation rate, are not +necessarily convex or concave, which prevents us from +using the same approach. Hence, to obtain the optimal +policy, we use the generally applicable LP method despite +the higher computational complexity that it may require +in order to develop insights about the optimal solution. +(ii) In the average age minimization problem, the number +of activated channels of the optimal policy experiences +a sub-linear/concave like increase with respect to ages +after the age level that the number of activated channels +starts to be above zero. In contrast, the age violation rate +minimizing schedulers experience a super-linear/convex +like increasing with respect to age until the deadline level +τ. This difference can be interpreted as follows: in the age + +1 +2 +3 +4 +5 +6 +AoI +0 +5 +10 +15 +20 +25 +Average number of activated channels +=0.1 +=0.001 +=0.0001 +Figure 4. +Optimal number of channels to choose to minimize average age +under violation rate constraint when τ = 5, b = 3, µ = 0.2 +violation rate minimization problem, the penalty happens +only when the age is beyond the age deadline, and hence +the optimal scheduler will be more aggressive as the +threshold level is approached from below. In contrast, +for the average age minimization problem, the number +of activated channels increases more gradually to balance +the tradeoff between consuming energy unnecessarily at +very low age levels and waiting too long to consume the +available energy, which yields an indefinitely increasing +cost. +These insights on the structure of the allocation functions of +the optimal schedulers can guide designers in restricting their +search to classes of functions with sufficiently flexible but also +tractable forms whenever the solution through the LP strategy +is not possible due to lack of prior statistical information as +well as computational resources. +To demonstrate how the age violation rate constraint effects +the shape of the scheduler more clearly, in Figure 4 we set +the objective function to be ω0(a) = a, the energy constraint +to be b = 3, and the channel success probability to be µ = +0.2. In addition, we set ω1(a) = 1{a > τ}, where the age +deadline τ = 5. We set c1 = ϵ and show how the number of +activated channels changes over age states under different ϵ +levels. By adding and tightening the tolerance constraint, we +can see the transition from concave (or sublinear) to convex +(or superlinear) form. As such, the optimal scheduler becomes +more aggressive when the age increases. This reveals a trade- +off between the average age and the age-violation-rate, namely +that reducing the age violation rate calls for an increasingly +more aggressive allocation function. +IV. AGE-OPTIMAL MULTI-CHANNEL SCHEDULING FOR +MULTIPLE USERS +In this section, we extend our framework to the general +multi-user multi-channel age-optimal scheduling problem. As +before, this formulation allows us to cover a range of scenarios +depending on the choice for objective function and constraints. +To that end, we investigate the feasibility and stability region +of the optimal policy along with alternatives from related +literature associated with multi-user settings. +A. Problem Formulation +The formulation of the optimization problem for the multi- +user case is similar to single user case (1): +min +u(A) +lim +T →∞ +1 +T +T +� +t=1 +E [ω0(A[t])] +(8) +s.t : +lim +T →∞ +1 +T +T +� +t=1 +E [ui (A[t])] ≤ bi, i = 1, · · · , n, +lim +T →∞ +1 +T +T +� +t=1 +E [ωk (A[t])] ≤ ck, k = 1, · · · , K, +ui(A[t]) ∈ {0, 1, · · · , L}, i = 1, · · · , n, +n +� +i=1 +ui(A[t]) ≤ L +where = (u1(A), · · · , un(A)) denotes the scheduling policy +at state A with ui(A) as the number of channels allocated to +source i. The weight functions ωk(·), k = 0, 1, · · · , K, map +the age states to cost values that capture age-related objectives +and constraints. Source nodes can have heterogeneous energy +constraints bi, which means node i can transmit over at most +bi channels per slot on average. +B. Performance analysis +Next, we establish the equivalence of the multi-user problem +formulation to a linear programming (LP) problem, as we +did for the single user case in Section III-B. To enable a +more compact notation, we will use a ≜ (a1, a2, · · · , an) +and l ≜ (l1, l2, · · · , ln) to denote values of A[t] and u(A), +respectively. We further define sets A ≜ {1, · · · , D}n, L ≜ +{1, · · · , L}n, and L1 ≜ {l : lΣ ≤ L} where lΣ ≜ +n +� +i=1 +li. +Theorem 2: The solution of the multi-user age-optimization +problem (8) can be obtained by solving the following linear +programming problem: +min +yl +a +� +a∈A +� +l∈L1 +yl +aω0(a) +s.t: +� +a∈A +� +l∈L1 +yl +ali ≤ bi, i = 1, 2, · · · , n +0 ≤ yl +a ≤ 1 +∀l ∈ L, a ∈ A +yl +a = 0 +∀l ∈ L/L1 +� +a∈A +� +l∈L1 +yl +a = 1 +� +a∈A +� +l∈L1 +yl +aωk(a) ≤ ck, k = 1, · · · , K +(9) +Qy = 0 +where y is a column vector with yl +a as components and +Q represents the transition matrix associated with the age + +dynamics, exactly in the same form as in the single-user case +(cf. Theorem 1). +If this LP is feasible and y is an optimal solution, then +the optimal policy u∗ +i (a) is a probabilistic policy, whereby +the probability f l +a of choosing l channels for source nodes +i = 1, · · · , n when the AoI is at state a equals: +f l +a = +� +� +� +� +� +� +� +� +� +yl +a +� +l∈L +yl +a +, +if +� +l∈L +yl +a ̸= 0 +1 +|L|, +if +� +l∈L +yl +a = 0 +for l ∈ L and a ∈ A. +Proof: +We will use f l +a to denote the probability of choosing +l = (l1, · · · , ln) channels for source nodes (1, · · · , n) when +the AoI is at state a. Thus � +l∈L +f l +a = 1, and f l +a ≥ 0 for all a. +Similarly as in Theorem 1, the constraint MDP problem with +n−dimensional Markov Chains for multi-user scheduling can +be generally formulated as: +min +� +a +π(a)ω0(a) +s.t: +� +a +� +l +π(a)f l +ali ≤ bi, i = 1, 2, · · · , n +(10) +f l +a = 0 +∀l ∈ L/L1 +� +a +π(a)ωk(a) ≤ ck +k = 1, · · · , K +H · Π = Π, +1 · Π = 1, +(11) +where the indices range over a ∈ A and l ∈ L; π(a) is the +stationary distribution of state a; and ωk(a), k = 0, 1, · · · , K, +are age related objective and cost functions. The constraints 10 +bound the average energy of nodes i by bi for i = 1, · · · , n. +In the constraint 11, Π is a Dn × 1 stationary distribution +vector with π(a), a +∈ +A as entries.7 H represents the +Dn×Dn transaction matrix with hi,j equals the probability of +transaction from the jth state in Π to the ith state in Π, which +can be detailed by using the age evolution and channel success +probability equations similarly as in Equation 4. Similarly, we +will define +yl +a ≜ yl1,l2,··· ,ln +a1,a2,··· ,an = π(a)f l +a. +By changing the value of the weight functions, we can get +different AoI related metrics, but all are linear with respect to +yl +a. Then, +π(a) = +� +l +yl +a, +and the normalization constraint requires: +� +a +� +l +yl +a = 1. +Substituting yl +a into the CMDP problem, we obtain the equiv- +alence of the LP problem. +7The existence of the stationary distribution follows by the same proof as +in Theorem 1. +C. Characterization and insights on multi-user scheduling +problem with violation tolerance Constraints +Since there is no closed form solution to the general age- +optimal problem, we will study the multi-user single-channel +scheduling feasibility problem with age-violation tolerance +constraint as a common setting to investigate its performance +and characteristics. +In particular, we will compare the stability region of the +optimal scheduler with a previously developed algorithm that +was developed for the special case of multi-user single-channel +setting [21]. To that end, we set L = 1 and bi > 1. Thus, all +the energy constraints will be inactive, and we can focus on the +tolerance constraint, as in [21]. Since we are only interested +in feasibility, we set ω0(a) = 1 for all a. To express the +age-violation rate constraints we define the weight functions +ωk(a) = +� +0, +if ak ≤ τk +1, +if ak ≥ τk + 1, +and set ck = ϵk for k = 1, 2, · · · , K = n, to represent the +heterogeneous age-violation tolerance level for the kth source. +Then the constraint +� +a +π(a)ωk(a) ≤ ck becomes +πk(τk + 1) ≤ ϵk +∀k = 1, · · · , K = n, +where πk(τk + 1) denotes the total probability (under the +stationary distribution) that source k violates its age threshold +τk. Since +πk(τk+1) = +� +j1,...,jk−1,jk+1,...,jn +π(j1, ...jk−1, τk+1, jk+1..., jn), +the constraint (9) in the linear programming problem becomes +� +j1,...,jk−1,jk+1,...,jn +� +l +yl +j1,...,jk−1,τk+1,jk+1,...,jn ≤ ϵk. +For the sake of easy visualization, we study the case with +n = 2 users. In this case, the LP problem is formulated as: +min +1 +s.t: +0 ≤ yl1,l2 +a1,a2 ≤ 1 +∀l1, l2 = 0, 1 +yl1,l2 +a1,a2 = 0 +∀l1 + l2 > 1 +� +j +� +l1,l2 +yl1,l2 +τ1+1,j ≤ ϵ1 +� +j +� +l1,l2 +yl1,l2 +j,τ2+1 ≤ ϵ2 +The numerical results can be seen in Figures5 and 6 for +different parameters where the upper right area of the solid +blue line is the stability region of the optimal scheduler. +These typical examples reveal the non-negligible gap between +the performance of the optimal scheduler and the previously +proposed design, even for a small two user setting. +This motivates the search for new algorithms that can +perform closer to the optimal scheduler, even when the channel +statistics are unknown a priori. This is performed in the next +section along with further discussion about these numerical +results after we discuss our online scheduling algorithm. + +Before we proceed, we note even the above numerical +results are for two-user single-channel scheduling problem +under tolerance constraints for visualization purposes, our +methods apply to the more general multi-user multi-channel +scheduling problem under violation tolerance and energy +constraints. Although the computational complexity may be +relatively high for the LP solution compared to other solutions +that exploit the special structure of particular problems, as we +mentioned above, due to the non-convexity and non-concavity +of the tolerance constraints, the monotone and threshold +structure of the optimal policy does not hold. The Whittle +Index approach (used, for example, in [29], [31]) which have +relatively low complexity also does not apply to our multi- +channel scheduling problems since each user in our setting +is allowed to transmit over multiple channels simultaneously, +whereby the Whittle’s Indexability condition does not hold. +Using the generally applicable LP-based approach reveals key +insights that can guide the designers in developing efficient +schedulers for future multi-channel wireless technologies. +V. ONLINE SCHEDULING UNDER UNKNOWN CHANNEL +STATISTICS +Until this point, we have assumed that the channel success +probabilities are known when solving the optimization prob- +lems. In this section, we use a Lyapunov-drift-plus-penalty +approach(see [28]) to solve the multi-user online age related +optimization problem in the scenario when only the current +channel states are known, but the channel statistics are un- +known. +We will transfer all the energy and age-related constraints +into the virtual queues and view the objective as a penalty term +with parameter M. For the energy constraint of the source i, +let us define the corresponding virtual queue as Q1,i[t], whose +initial value is Q1,i[0] = 0 and update equation is: +Q1,i[t + 1] = (Q1,i[t] + ui (A[t]) − bi)+ . +Similarly, we define the virtual queue Q2,k[t] for the kth age- +related constraint, whose initial value is Q2,k[0] = 0 and +update equation is: +Q2,k[t + 1] = (Q2,k[t] + ωk (A[t]) − ck)+ . +Generically, if the virtual queue Q1,i[t] is stable, then its input +rate lim +T →∞ +1 +T +T +� +t=1 +E [ui (A[t])] will be less than its output rate +bi [28], so that the corresponding constraint can be satisfied. +Define the state of both virtual queues and age at time t +as Q[t] = (Q1,1[t], · · · , Q1,n[t], Q2,1[t], · · · , Q2,K[t], A[t]). +Based on the virtual queues, we will define the quadratic +Lyapunov function as: +V [t] = 1 +2( +n +� +i=1 +Q2 +1,i[t] + +K +� +k=1 +Q2 +2,k[t]), +and develop an online algorithm to greedily minimize +the upper bound of the Lyapunov-drift-plus-penalty func- +tion ∆V (q) + ME[ω0(a)] given the current state q += +(q1,1, · · · , q1,n, q2,1, · · · , q2,K, a), where: +∆V (q) = E[V [t] − V [t − 1]|Q[t] = q]. +We consider the multi-user single-channel scheduling prob- +lem under tolerance constraints as a specific example to +present the design. Since there are no energy constraints, +we do not need the set of virtual queues {Q1,i[t]}i. In +order to express the kth violation rate constraint for source +k = 1, · · · , n, we let ωk (A[t]) = 1 (Ak[t + 1] > τk) and +ck = ϵk. Then the virtual queue Q2,k[t], whose initial value +is Q2,k[t] = 0, updates as follows: +Q2,k[t + 1] = (Q2,k[t] + 1 (Ak[t + 1] > τk) − ϵk)+ , +where Ak[t + 1] = 1 + Ak[t](1 − Sk[t]Uk[t]); Sk[t] represents +the channel success; Uk[t] represents whether the source is +scheduled to transmit or not. If virtual queue Q2,k[t] is stable, +its input rate, the threshold violation rate πk(τk + 1) = +limT →∞ 1 +T +�T +t=1 1 (Ak[t + 1] > τk) , will be less than its +output rate ϵk. +The conditional Lyapunov drift can be bounded as follows: +∆V (q) +≤ +n +� +k=1 +q2,kE [Rk − ϵk|q2,k] + +n +� +k=1 +E +� +(Rk − ϵk)2 +2 +|q2,k +� +, +where Rk +∆= 1{1 + Ak (1 − SkCk) > τk}. At every time slot +t, we can develop an online algorithm as summarized below +to greedily minimize the upper bound of the Lyapunov drift +given the queue lengths Q[t − 1] and A[t − 1] since there is +no objective or penalty term in this case. +Algorithm 1 A Heuristic Scheduling Policy +1: Input current system state: Ai[t],Qi[t]. +2: Define available transmission decision set: only one Ui[t] +can be 1. +3: Choose U[t] to minimize the upper bound of Lyapunov +drift function in the above inequality. +4: Update queue lengths for next time slot. +Again, for the sake of easy visualization, we will only +present the simulation results for the two-user online schedul- +ing problem under age tolerance constraints, but the online +algorithm can be simply applied to any number of sources. +The simulation results are illustrated in Fig 5 and Fig 6 for +different parameters where the upper right area of the dash-dot +purple line is the stability region of the online scheduler when +the channel condition µi. The comparison will be in the next +section. +VI. COMPARISON OF STABILITY REGIONS UNDER AGE +VIOLATION CONSTRAINTS +In this section, we compare the performance of three dif- +ferent algorithms for the two-user single channel scheduling +feasibility problem under age violation tolerance constraints. +These are: the optimal scheduler from Section IV; the prior + +design from [21] developed for a single-channel multi-user +setting; and our online scheduler from Section V that does +not require channel statistics. +We first focus on the case when the two source nodes are +symmetric. In Figure 5, there are two source nodes with the +same age thresholds of τ1 = τ2 = 2 and the same channel +success probabilities of µ1 = µ2 = 0.85. The upper right +area of the blue line is the stability region for the optimal +scheduling algorithm in Section IV-C. The yellow and orange +lines correspond to the algorithm in [21] and capture the two +cases when the rate vector does or does not possess a special +property (called step-down rate vector). The purple line marks +the stability region for the online algorithm when the channel +conditions µ1, µ2 are unknown. Several observations are in +order from these simulation results: +(i) The stability regions are all symmetric, as can be expected +due to the homogeneous deadline thresholds and channel +conditions. +(ii) The optimum policy (blue line) outperforms other poli- +cies, with markedly better performance in cases where +the tolerance levels are greatly different from each other. +(iii) The online algorithm (purple line) performs very closely +to the optimal policy, experiencing a small performance +loss only at some extreme range of tolerance levels. +(iv) When compared with the algorithms from [21](yellow +and red lines), the online algorithm performs particularly +better when one of the tolerance rates is smaller than the +corresponding channel loss probability, as observed by +the vertical gap between purple and yellow lines. +(v) The online and optimal policies are continuous with +respect to the tolerance level, which eliminates the need +to check if the tolerance rate vector satisfies certain +properties, such as the step-down rate condition in [21]. +To compare the advantages and disadvantages of the al- +gorithms under non-homogeneous scenarios, in Figure 6, we +consider two source nodes with asymmetric age thresholds of +τ1 = 2, τ2 = 4 and a common channel success probability of +µ1 = µ2 = 0.85. Since the violation rate depends on both +the age thresholds and the channel success probabilities, this +is a non-homogeneous scenario even though µ1 = µ2. In this +figure, in contrast to the previous figure, we can further see +that the optimal policy outperforms others when one of the +0 +0.2 +0.4 +0.6 +0.8 +1 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +2 +Optimal scheduling algrithm +Algorithm proposed in [21] +under general rate vector +Algorithm proposed in [21] +under step-down rate vector +Online scheduling algorithm +Figure 5. Stability region (upper-righter) comparison for symmetric case. +0 +0.2 +0.4 +0.6 +0.8 +1 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +2 +Optimal scheduling algrithm +Algorithm proposed in [21] +under general rate vector +Algorithm proposed in [21] +under step-down rate vector +Online scheduling algorithm +Figure 6. Stability region (upper-righter) comparison for asymmetric case. +tolerance constraints is very strict, namely when ϵ1 approaches +1. In this regime, the feasible tolerance level ϵ2 of user 2 +other algorithms is bounded away from zero while the optimal +algorithm decreases towards zero. +These simulation results are typical of other circumstances, +with the common observation that our online scheduler per- +forms close to the optimal scheduler and typically non- +negligibly better than the most closely related state-of-art +algorithm from [21], despite the fact that it operates without +the knowledge of channel statistics that is assumed in the other +designs. +VII. CONCLUSIONS +In this paper, we considered a general class of age-optimal +scheduling problems for multi-source multi-channel commu- +nication. We formulated the generic age-optimization problem +with flexible weight functions ωk under energy and tolerance +constraints in the form of a CMDP. We solved this generic +problem, which a usual threshold-based structure policy does +not apply, by relating it to the solution an associated linear +programming problem using the powerful theory of CMDPs. +Then, we focused on the special case of single-source multi- +channel scenario to investigate the characteristics of optimal +scheduler for the important special cases of average-age and +violation-rate minimization. +Our investigations revealed several interesting insights, in- +cluding the observation that age-violation-rate minimizing +scheduler employs a super-linearly like growing energy al- +location strategy with increasing age, as opposed to the +sub-linearly like growing allocation for the average-age- +minimizing scheduler. 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Chiang, “Minimizing age-of- +information in heterogeneous multi-channel systems: A new partial- +index approach,” in Proceedings of the Twenty-second International +Symposium on Theory, Algorithmic Foundations, and Protocol Design +for Mobile Networks and Mobile Computing, 2021, pp. 11–20. +[32] A. Elgabli, H. Khan, M. Krouka, and M. Bennis, “Reinforcement +learning based scheduling algorithm for optimizing age of information +in ultra reliable low latency networks,” in ISCC. +IEEE, 2019, pp. 1–6. +[33] M. Li, C. Chen, C. Hua, and X. Guan, “Learning-based autonomous +scheduling for aoi-aware industrial wireless networks,” IEEE Internet of +Things Journal, vol. 7, no. 9, pp. 9175–9188, 2020. +[34] H. Tang, J. Wang, L. Song, and J. Song, “Minimizing age of infor- +mation with power constraints: Multi-user opportunistic scheduling in +multi-state time-varying channels,” IEEE Journal on Selected Areas in +Communications, vol. 38, no. 5, pp. 854–868, 2020. + diff --git a/GtAyT4oBgHgl3EQfrfmY/content/tmp_files/load_file.txt b/GtAyT4oBgHgl3EQfrfmY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a599988ffa98666b7beade9a1fbc262805353c7 --- /dev/null +++ b/GtAyT4oBgHgl3EQfrfmY/content/tmp_files/load_file.txt @@ -0,0 +1,603 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf,len=602 +page_content='Age-Optimal Multi-Channel-Scheduling under Energy and Tolerance Constraints Xujin Zhou, Irem Koprulu, Atilla Eryilmaz Electrical and Computer Engineering The Ohio State University Columbus, US {zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='2400@osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='edu, irem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='koprulu@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='com, eryilmaz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='2@osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='edu} Abstract—We study the optimal scheduling problem where n source nodes attempt to transmit updates over L shared wireless on/off fading channels to optimize their age performance under energy and age-violation tolerance constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Specifically, we provide a generic formulation of age-optimization in the form of a constrained Markov Decision Processes (CMDP), and obtain the optimal scheduler as the solution of an associated Linear Programming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We investigate the characteristics of the optimal single-user multi-channel scheduler for the important special cases of average-age and violation-rate minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' This leads to several key insights on the nature of the optimal allocation of the limited energy, where a usual threshold-based policy does not apply and will be useful in guiding scheduler designers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We then investigate the stability region of the optimal scheduler for the multi-user case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We also develop an online scheduler using Lyapunov-drift-minimization methods that do not require the knowledge of channel statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Our numerical studies compare the stability region of our online scheduler to the optimal scheduler to reveal that it performs closely with unknown channel statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' INTRODUCTION In recent years, the Internet of Things (IoT) has become one of the most important frameworks of the next-generation wireless networks, whereby a large number of mobile devices need to be supported over an ultra-wide frequency spectrum (see, for example, [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In particular, for many real-time IoT applications, it is necessary for the devices to send fresh updates over the shared spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' To measure the freshness of data, the concept of Age of Information (AoI) has been introduced over the last decade (see, for example, [2]–[4]), which is defined concisely as the elapsed time since the generation time of the last received status update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Since the introduction of the AoI metric, numerous related studies emerged in various networking scenarios, including wireless random access networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', [5], [6]), content distribution networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', [7], [8]), scheduling (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', [9]–[13]), queuing networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', [14], [15]), and vehicular networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Recently, other AoI related metrics have been developed in order to address more generalized or different forms of ageing, such as: non-linear AoI (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', [4], [17]), peak AoI (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', [18]), time-since-last-service (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', [19]), age upon decisions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', [20]), to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Among them, the metric, called the age- violation-rate (see [15], [21], [22]) is of particular interest for real-time IoT services that have hard age-deadline constraints and a limited tolerance to violating this deadline (see [23], [24] for further motivation of this metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In view of its significance for next generation IoT networks, in this paper, we study the general optimal multi-channel scheduling problem to optimize varying forms of age perfor- mances under energy and age-violation tolerance constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Our contributions can be listed as: We provide a generic formulation of age-optimization as a Constrained Markov Decision Problem (CMDP) (see [25]–[27]) and obtain the age-optimal multi-channel scheduler as the solution of an associated Linear Pro- gramming problem, first for the single-source (in Sec- tion III) and then for general the multi-source (in Sec- tion IV) scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' For the single-source multi-channel scenario, we in- vestigate the characteristics of the optimal schedulers under energy constraints for two age metrics that are important for IoT applications: (i) average-age mini- mization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' and (ii) age-violation-rate minimization, a non- convex/concave metric (in Section III-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Our investiga- tions reveal various insights on different energy allocation structures, as well as the common monotonicity proper- ties of the optimal schedulers for minimizing these two metrics, which is useful for guiding scheduler designers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' For the multi-source age optimal scheduling problem, we also study the feasibility region of the average- age-optimal scheduler under age-violation-rate tolerance constraints to contrast its results with those of related earlier works that are developed for the single-channel multi-user scenario (see Section IV-C and Section VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Moreover, we develop (in Section V) an online scheduler using Lyapunov-drift-minimization methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', [28]) that does not require the knowledge of channel statistics, and compare its performance to the optimal and earlier designs to reveal how much the knowledge of channel statistics affects the feasibility region (see Section VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Our work relates to, but also differs from several other related works in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Many early works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', [9], [12], [29]) aim to minimize AoI under power constraints but with the assumption of reliable channels as opposed to the fading channels that we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' More recent works arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='00562v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='IT] 2 Jan 2023 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', [10], [30]) aim to minimize AoI-related costs based on max-age matching, while other works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', [29], [31]) proposed AoI minimization schedulers based on Whittle Index approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' However, to the best of our knowledge, prior works predominantly assume that one source can choose at most one channel, which is an important factor in proving the Whittle Indexability of the corresponding problems they solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In contrast, one of the key features our setting is the possibility of each user to transmit over multiple channels as enabled by new wireless technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Furthermore, most of the above mentioned works have average or peak AoI as the objective function, while we consider more general age-based objective functions, which for example allows the objective function to be a non-convex metric such as the age-violation-rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In this multi-channel setting with general objectives, we observe (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Section III-C) that the optimal solution can in fact possess non- monotone characteristics, which make the Whittle Indexability approach infeasible in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' The work in [21] has con- sidered the multi-source single-channel scheduling problem under tolerance constraints, which is a special case of our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We would like to note that this interesting work [21] has been a primary motivation for our current work in exploring a different approach based on the CMDP framework that guarantees optimality and applies to more general multi- channel scenarios with additional energy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' There are also works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', [32], [33]) that focus on learning-based approaches which can be considered as complementary to the focus of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' SYSTEM MODEL We consider the operation of a discrete-time wireless access system, whereby n source nodes share L on/off fading wireless channels to update their ageing status at a receiver (such as a base station) under energy and violation tolerance constraints (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' n sources share L on-off fading channels to update their status to a receiver under energy and tolerance constraints in order to keep their age levels low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Our goal is to develop generic solution strategies to find optimal schedulers that can optimize diverse age-based metrics while meeting certain requirements on energy consumption and tolerance levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We describe the key terminology and the essential system dynamics in the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Then, in the following sections we formulate and solve classes of age-optimization problems for single and multi-source cases, subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Scheduling policy and age-violation-tolerance: We assume that each source node i ∈ {1, · · · , n} refreshes its status and creates a new packet at the beginning of every time slot t ∈ {1, 2, 3, · · · }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Source nodes attempt to transmit their freshest packet to the receiver, for example a base station (BS), whenever they get a chance to transmit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Every time the BS successfully receives a new status from source node i, it saves the current status and discards all previous packets received from that node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' As such, the BS keeps only one packet from each source node, namely the freshest one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We use Xi[t] to denote the generation time of the packet stored at the BS from source i at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We define the age Ai[t] of source node i at time t as the time that has elapsed since the generation of its last received packet1: Ai[t] ≜ t − Xi[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We use2 A[t] ≜ (A1[t], · · · , An[t]) to denote the ages of all sources at time slot t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' At the beginning of each time slot, the centralized scheduler decides which channels each of the source nodes will use to transmit to the base station based on the ages A[t] of all source nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Let ui(A[t]) be the number of channels source node i uses to transmit at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Each transmission attempt can resolve in success or failure which we will describe below as part of the channel success model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' If the base station successfully receives the packet from source i at time t, then its age at time t + 1 will reset to 1, otherwise its age will increase by one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', Ai[t + 1] = � 1, if transmission of source i succeeds Ai[t] + 1, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We allow each source i to have a desired age thresh- old/deadline τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' The information of source i is up-to-date if its age is less than or equal to this threshold τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Otherwise, we speak of an age violation in that slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In particular, we define the age-violation-rate of source i as the long-term average fraction of time slots when the source’s age Ai[t] exceeds its threshold τi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', lim T →∞ 1 T T � t=1 1 {Ai[t] > τi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We use ϵi ∈ [0, 1] to indicate the tolerance of source i that measures the maximum allowed age-violation-rate for its updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' (ϵi = 1 indicates that there is no violation rate constraint, and ϵi = 0 indicates that we do not allow any deadline violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=') When the age violation rate is no greater than the tolerance rate, the age violation tolerance constraint is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Channel success model and energy constraints: The n source nodes share L wireless on/off fading channels, each of which can accommodate at most one packet transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' However, even when there is a single transmission over a channel, a successful transmission is not guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In 1This metric is also referred to as Age-of-Information (AoI) and Time-Since- Last-Service (TSLS) in different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In the rest of the paper, we will refer to it as AoI or simple as age, interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' 2We will consistently use bold symbols to represent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' particular, source node i has a channel success probability of µi when transmitting over each of its assigned channels3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We call the update of source i in a slot to be a success if any one of its transmissions over its assigned channels is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Since the channel is a collision channel, for an optimal scheduler we always have n � i=1 ui(A[t]) ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Once the value of ui(A[t]) is decided for all i, the scheduler will assign different channels to different sources, so that no two sources transmit over the same channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Also, note that under the described channel success model, the probability for the BS to successfully receive an update from source node i when the node uses l channels is 1 − (1 − µi)l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We assume that each transmission over a channel comes with an energy cost of 1 unit4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We require that the aggregate time-average energy cost for source i is not greater than a given constraint bi channels per slot, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', we require lim T →∞ 1 T T � t=1 ui (A[t]) ≤ bi, bi ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' It is obvious that transmitting over more channels will increase the success probability of a source, but increase energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We are interested in finding the number of channels that when allocated to sources optimize the desired age performance given the current age state, as well as energy and and tolerance constraints discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In the next section, we attack this problem within the constrained Markov Decision Process (MDP) framework first for a single user, and then extend our approach to cover the multi-user setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' AGE-OPTIMAL MULTI-CHANNEL SCHEDULING FOR A SINGLE USER In this section, we first consider the single-user age-optimal multi-channel scheduling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' This not only allows us to simplify the notation by omitting the subscripts, but also is of particular interest for the next generation ultra-wideband wire- less communication technologies that are expected to support low-delay access over multiple fading channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We formulate a general age-optimal optimization problem which can be used in different scenarios in Section III-A and following the analysis of the performance in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' To that end, in Section III-C, we study the characterization and insights of the optimal schedulers for two important special cases of minimizing the average-age and the age-violation-rate, which will be useful in guiding scheduler designers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Problem formulation The problem of minimizing time-averaged age-based ob- jectives under average energy and tolerance constraints can 3All our development can be generalized to the case when the success probability between source i and channel j is allowed to be different as µij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' However, this is omitted here as it increases the complexity of the exposition without adding to the substance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' 4This can also be generalized to non-uniform energy costs over different channels, but omitted to avoid cumbersome notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' be generally formulated as the following constrained Markov decision problem [25]: min u(A) lim T →∞ 1 T T � t=1 E [ω0(A[t])] (1) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='t : lim T →∞ 1 T T � t=1 E [u (A[t])] ≤ b, (2) lim T →∞ 1 T T � t=1 E [ωk (A[t])] ≤ ck, k = 1, · · · , K, u(A[t]) ∈ {0, 1, · · · , L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' The optimization is performed over Markovian policies described by a function u(·) that maps age levels to number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' It is known that such Markovian policies are sufficient for optimal operation [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' The first constraint on the time-averaged u(·) captures the average energy constraint discussed in the system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' The functions ωk(·) serve as general functions that map the current state A[t] to a value that measures the cost of that age with respect to various measures5 By setting different mappings for the weight function ω0(A[t]), the objective can be changed into different commonly used age-related objectives: letting ω0(a) = −1{a = 1} transform the objective to maximizing the average throughput;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' letting ω0(a) = a makes the objective minimize the average AoI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' letting ω0(a) = 1{a ≥ d} make the objective minimize the average age-violation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Note that this allows the objective function to be a non-convex/concave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Performance analysis Next, we will analyze the generic constrained optimization problem under energy constraint by showing that the problem is equivalent to a Linear Programming (LP) problem and thus describe the optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Theorem 1: The solution of the generic age-optimization problem (1) can be obtained by solving the following linear programming problem: min yla D � a=1 L � l=0 yl aω0(a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='t: D � a=1 L � l=0 yl a · l ≤ b, D � a=1 L � l=0 yl aωk(a) ≤ ck, k = 1, · · · , K, 0 ≤ yl a ≤ 1 ∀1 ≤ a ≤ D, 0 ≤ l ≤ L, D � a=1 L � l=0 yl a = 1, Qy = 0, where y is a column vector of size DL with y = (y1 1, · · · , yL 1 , · · · , y1 D, · · · , yL D)T as its components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' D is an 5We note that the problem can also solved with the same approach (but heavier notation) by more generally defining ωk(A[t], u(A[t])) to be functions of both the age and the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' upper bound on the age state in the system which can be set sufficiently large so that the probability of reaching D is vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='6 Qy = 0 is the matrix representation of the following (global balance) equations: L � l=0 yl a+1 − L � l=0 yl a(1 − µ)l = 0 ∀a = 1, · · · , D − 2, L � l=0 � 1 − (1 − µ)l� yl D − L � l=0 yl D−1(1 − µ)l = 0, − L � l=0 yl 1(1 − µ)l + D � a=2 L � l=0 yl a � 1 − (1 − µ)l� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' If this LP is feasible, and y is an optimal solution, then the optimal policy u∗(a) is a probabilistic policy, whereby the probability f l a of choosing l channels when the age is at state a equals: f l a = � � � � � � � � � � � � � � � yl a L � l=0 yl a , if L � l=0 yl a ̸= 0 1 L, if � l yl a = 0 (3) for l = 0, 1, · · · , L and a = 1, 2, · · · , D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Proof: As shown in [25], it is enough for us to optimize over the Markovian policies for Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Since the process is not affected by a shift in time, we can define the probabilistic scheduling policy where f l a denotes the probability of choosing l channels when the AoI of single source is at state a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' The normalization constraint of the probabilistic scheduling policy requires L � l=0 f l a = 1 and f l a ⩾ 0 for all a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Notice that the system state can be fully characterized by a one-dimensional Markov chain with age A[t] as state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Given the current state information A[t], the system state at the next time slot A[t+1] depends only on the current state A[t] (with no dependence on earlier states) and the current action u[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In addition, the objective and constraints only depend on the current state and action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' So an equivalent MDP problem can be formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Let λa2 a1 denote the transition probability from state a1 to a2, and define ¯µ ≜ 1 − µ as the probability of channel failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Then based on the channel success model, λa2 a1 = � � � � � � � � � � � � � � � � � � � � � � � � � � � L � l=1 f l a1 ¯µl, 1 ≤ a1 ≤ D − 1, a2 = a1 + 1 L � l=1 f l a(1 − ¯µl), a1 = 1, · · · , D, a2 = 1 L � l=1 f l D(1 − ¯µl), a1 = D, a2 = D 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' (4) Since there are finitely many states, there exists a stationary distribution π(a) for every a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Let C be the set of all recurrent 6In practice, moderate level of D is enough so that the dimension of LP won’t be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Also, when there is only age violation related objective and constraints, it’s enough to set D = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' See III-C and IV-C for references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' states, then C is irreducible and closed, thus C is positive recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' When a ∈ C the stationary distribution π(a) is equal to the long term average lim T →∞ 1 T T � t=1 1{A[t] = a} independent of the starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' When state a /∈ C, then both the stationary distribution and the long term average are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' So the optimization problem is equivalent to the following constraint MDP problem: min f la D � a=1 π(a)ω0(a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='t: D � a=1 L � l=0 π(a)f l al ≤ b D � a=1 π(a)ωk(a) ≤ ck, k = 1, · · · , K (5) L � l=0 f l a = 1, f l a ⩾ 0 ∀a ≤ D, l ≤ L (6) H · Π = Π, 1 · Π = 1 (7) where Π = [π(1), · · · , π(D)]T is the stationary distribution of the Markov Chain and H is the D × D transition matrix with hij = λi j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Let us define yl a = π(a)f l a, then π(a) = L � l=0 yl a for a ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Then the constraint 5 becomes: D � a=1 L � l=0 yl aωk(a) ≤ ck, k = 1, · · · , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' The normalization constraint in Equation 7 requires D � a=1 L � l=0 yl a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Substituting yl a into the CMDP problem and after simplifying, we establish the equivalency of the Linear Programming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' After obtaining the solution y, we let f l a = yl a/π(a) for π(a) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='States a with π(a) = 0, are transient states, and the actions at these states do not affect the average results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' For those states we adopt a simple policy as in Equation 3, then the constraint 7 is also satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Characterization and Insights on Age-Optimal Schedulers Our general framework encompasses a wide range of objec- tives and constraints for different choices of ωk(·) functions using different age and age-violation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In this section, we focus on two important problems that can be expressed within our framework: average age minimization and age- violation-rate minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' This effort will enable us to characterize their optimal schedulers and gain insights into their nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Optimal scheduler minimizing average age: When we set ω0(a) = a in (1), the objective of the optimization problem becomes to minimize the average age lim T →∞ 1 T T � t=1 E{A[t]} = D � a=1 a π(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' 0 5 10 15 AoI 0 2 4 6 8 Average number of activated channels =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='12 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='1 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='08 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='04 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Optimal number of channels to choose to minimize average AoI when b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' For this problem formulation, we retain the energy constraint lim T →∞ 1 T T � t=1 E [u (A[t])] ≤ b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' but do not need additional age constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Hence, ωk(a) = 0 and ck = 0, for all k and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Figure 2 depicts the average number of activated channels of the average-age optimal scheduler as a function of the age states under different channel success probabilities µ for the energy constraint b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We will further discuss these results at the end of this section in comparison with the next scheduler of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Optimal scheduler minimizing age-violation-rate: Setting ω0(a) = 1{a > τ} n (1), the objective becomes minimizing the average age-violation-rate lim T →∞ 1 T T � t=1 E{1{A[t] > τ}} = D � a=τ+1 π(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' As before, we keep the energy constraint, but do not need additional age constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Hence, ωk(a) = 0 and ck = 0, for all k and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' With this, the problem becomes minimizing the age- violation-rate under an energy constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Unlike in the previ- ous problem, our goal is not to minimize the average age but to avoid age-violation events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In this scenario, we can view all the states with a > τ as state τ + 1, so it’s enough to set D = τ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Figure 3 depicts the average number of activated channels of the violation-rate optimal scheduler as a function of the age states under different channel success probabilities µ for age threshold τ = 8 and the same energy constraint b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Next, we compare the optimal policies of these two schedulers and discuss the insights that can be gained from their study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Insights on the two optimal schedulers: We start by noting the similarities of the optimal policy under both scenarios: (i) Each optimal policy is a probabilistic combination of at most two deterministic policies, which matches the result that the number of randomization is at most the number of constraints, as shown in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' (ii) For each scenario, as the channel success probability increases, the corresponding optimal policy starts trans- 2 4 6 8 AoI 0 5 10 15 20 25 Average number of activated channels =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='12 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='1 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='08 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='04 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Optimal number of channels to choose to minimize AoI violation rate when b = 2 and τ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' mitting at lower age levels, and also tends to choose more channels at the same age level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' This is a somewhat counter-intuitive characteristic that indicates that the opti- mal policy should be more active and active earlier when the channels are more reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' (iii) The optimal policy in each scenario is idle when AoI is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' This is meaningful once we observe that, when the age is relatively small, a successful transmission will not benefit the objective as much as when the age is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Hence, the optimal scheduler saves energy for larger age states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' However, we also notice differences between the two sets of schedulers: (i) The optimal policy in the average age minimization prob- lem has an activation function u∗(·) that is monotone non- decreasing with increasing age state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' On the other hand, the monotonicity does not hold in the age violation rate minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' This difference comes from the non-convex nature of the the age violation rate function in the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In [25] and many related works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', [9], [34]), the authors exploit the monotone structure and threshold nature of the optimal scheduling policy for solv- ing the CMDP, revealing insights as well as simplifying the algorithm by using the convexity or concavity of the objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' However, in our general treatment, the objective functions, such as age violation rate, are not necessarily convex or concave, which prevents us from using the same approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Hence, to obtain the optimal policy, we use the generally applicable LP method despite the higher computational complexity that it may require in order to develop insights about the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' (ii) In the average age minimization problem, the number of activated channels of the optimal policy experiences a sub-linear/concave like increase with respect to ages after the age level that the number of activated channels starts to be above zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In contrast, the age violation rate minimizing schedulers experience a super-linear/convex like increasing with respect to age until the deadline level τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' This difference can be interpreted as follows: in the age 1 2 3 4 5 6 AoI 0 5 10 15 20 25 Average number of activated channels =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='1 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='001 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='0001 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Optimal number of channels to choose to minimize average age under violation rate constraint when τ = 5, b = 3, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='2 violation rate minimization problem, the penalty happens only when the age is beyond the age deadline, and hence the optimal scheduler will be more aggressive as the threshold level is approached from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In contrast, for the average age minimization problem, the number of activated channels increases more gradually to balance the tradeoff between consuming energy unnecessarily at very low age levels and waiting too long to consume the available energy, which yields an indefinitely increasing cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' These insights on the structure of the allocation functions of the optimal schedulers can guide designers in restricting their search to classes of functions with sufficiently flexible but also tractable forms whenever the solution through the LP strategy is not possible due to lack of prior statistical information as well as computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' To demonstrate how the age violation rate constraint effects the shape of the scheduler more clearly, in Figure 4 we set the objective function to be ω0(a) = a, the energy constraint to be b = 3, and the channel success probability to be µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In addition, we set ω1(a) = 1{a > τ}, where the age deadline τ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We set c1 = ϵ and show how the number of activated channels changes over age states under different ϵ levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' By adding and tightening the tolerance constraint, we can see the transition from concave (or sublinear) to convex (or superlinear) form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' As such, the optimal scheduler becomes more aggressive when the age increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' This reveals a trade- off between the average age and the age-violation-rate, namely that reducing the age violation rate calls for an increasingly more aggressive allocation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' AGE-OPTIMAL MULTI-CHANNEL SCHEDULING FOR MULTIPLE USERS In this section, we extend our framework to the general multi-user multi-channel age-optimal scheduling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' As before, this formulation allows us to cover a range of scenarios depending on the choice for objective function and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' To that end, we investigate the feasibility and stability region of the optimal policy along with alternatives from related literature associated with multi-user settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Problem Formulation The formulation of the optimization problem for the multi- user case is similar to single user case (1): min u(A) lim T →∞ 1 T T � t=1 E [ω0(A[t])] (8) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='t : lim T →∞ 1 T T � t=1 E [ui (A[t])] ≤ bi, i = 1, · · · , n, lim T →∞ 1 T T � t=1 E [ωk (A[t])] ≤ ck, k = 1, · · · , K, ui(A[t]) ∈ {0, 1, · · · , L}, i = 1, · · · , n, n � i=1 ui(A[t]) ≤ L where = (u1(A), · · · , un(A)) denotes the scheduling policy at state A with ui(A) as the number of channels allocated to source i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' The weight functions ωk(·), k = 0, 1, · · · , K, map the age states to cost values that capture age-related objectives and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Source nodes can have heterogeneous energy constraints bi, which means node i can transmit over at most bi channels per slot on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Performance analysis Next, we establish the equivalence of the multi-user problem formulation to a linear programming (LP) problem, as we did for the single user case in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' To enable a more compact notation, we will use a ≜ (a1, a2, · · · , an) and l ≜ (l1, l2, · · · , ln) to denote values of A[t] and u(A), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We further define sets A ≜ {1, · · · , D}n, L ≜ {1, · · · , L}n, and L1 ≜ {l : lΣ ≤ L} where lΣ ≜ n � i=1 li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Theorem 2: The solution of the multi-user age-optimization problem (8) can be obtained by solving the following linear programming problem: min yl a � a∈A � l∈L1 yl aω0(a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='t: � a∈A � l∈L1 yl ali ≤ bi, i = 1, 2, · · · , n 0 ≤ yl a ≤ 1 ∀l ∈ L, a ∈ A yl a = 0 ∀l ∈ L/L1 � a∈A � l∈L1 yl a = 1 � a∈A � l∈L1 yl aωk(a) ≤ ck, k = 1, · · · , K (9) Qy = 0 where y is a column vector with yl a as components and Q represents the transition matrix associated with the age dynamics, exactly in the same form as in the single-user case (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' If this LP is feasible and y is an optimal solution, then the optimal policy u∗ i (a) is a probabilistic policy, whereby the probability f l a of choosing l channels for source nodes i = 1, · · · , n when the AoI is at state a equals: f l a = � � � � � � � � � yl a � l∈L yl a , if � l∈L yl a ̸= 0 1 |L|, if � l∈L yl a = 0 for l ∈ L and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Proof: We will use f l a to denote the probability of choosing l = (l1, · · · , ln) channels for source nodes (1, · · · , n) when the AoI is at state a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Thus � l∈L f l a = 1, and f l a ≥ 0 for all a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Similarly as in Theorem 1, the constraint MDP problem with n−dimensional Markov Chains for multi-user scheduling can be generally formulated as: min � a π(a)ω0(a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='t: � a � l π(a)f l ali ≤ bi, i = 1, 2, · · · , n (10) f l a = 0 ∀l ∈ L/L1 � a π(a)ωk(a) ≤ ck k = 1, · · · , K H · Π = Π, 1 · Π = 1, (11) where the indices range over a ∈ A and l ∈ L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' π(a) is the stationary distribution of state a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' and ωk(a), k = 0, 1, · · · , K, are age related objective and cost functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' The constraints 10 bound the average energy of nodes i by bi for i = 1, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In the constraint 11, Π is a Dn × 1 stationary distribution vector with π(a), a ∈ A as entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='7 H represents the Dn×Dn transaction matrix with hi,j equals the probability of transaction from the jth state in Π to the ith state in Π, which can be detailed by using the age evolution and channel success probability equations similarly as in Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Similarly, we will define yl a ≜ yl1,l2,··· ,ln a1,a2,··· ,an = π(a)f l a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' By changing the value of the weight functions, we can get different AoI related metrics, but all are linear with respect to yl a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Then, π(a) = � l yl a, and the normalization constraint requires: � a � l yl a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Substituting yl a into the CMDP problem, we obtain the equiv- alence of the LP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' 7The existence of the stationary distribution follows by the same proof as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Characterization and insights on multi-user scheduling problem with violation tolerance Constraints Since there is no closed form solution to the general age- optimal problem, we will study the multi-user single-channel scheduling feasibility problem with age-violation tolerance constraint as a common setting to investigate its performance and characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In particular, we will compare the stability region of the optimal scheduler with a previously developed algorithm that was developed for the special case of multi-user single-channel setting [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' To that end, we set L = 1 and bi > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Thus, all the energy constraints will be inactive, and we can focus on the tolerance constraint, as in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Since we are only interested in feasibility, we set ω0(a) = 1 for all a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' To express the age-violation rate constraints we define the weight functions ωk(a) = � 0, if ak ≤ τk 1, if ak ≥ τk + 1, and set ck = ϵk for k = 1, 2, · · · , K = n, to represent the heterogeneous age-violation tolerance level for the kth source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Then the constraint � a π(a)ωk(a) ≤ ck becomes πk(τk + 1) ≤ ϵk ∀k = 1, · · · , K = n, where πk(τk + 1) denotes the total probability (under the stationary distribution) that source k violates its age threshold τk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Since πk(τk+1) = � j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=',jk−1,jk+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=',jn π(j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='jk−1, τk+1, jk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=', jn), the constraint (9) in the linear programming problem becomes � j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=',jk−1,jk+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=',jn � l yl j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=',jk−1,τk+1,jk+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=',jn ≤ ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' For the sake of easy visualization, we study the case with n = 2 users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In this case, the LP problem is formulated as: min 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='t: 0 ≤ yl1,l2 a1,a2 ≤ 1 ∀l1, l2 = 0, 1 yl1,l2 a1,a2 = 0 ∀l1 + l2 > 1 � j � l1,l2 yl1,l2 τ1+1,j ≤ ϵ1 � j � l1,l2 yl1,l2 j,τ2+1 ≤ ϵ2 The numerical results can be seen in Figures5 and 6 for different parameters where the upper right area of the solid blue line is the stability region of the optimal scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' These typical examples reveal the non-negligible gap between the performance of the optimal scheduler and the previously proposed design, even for a small two user setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' This motivates the search for new algorithms that can perform closer to the optimal scheduler, even when the channel statistics are unknown a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' This is performed in the next section along with further discussion about these numerical results after we discuss our online scheduling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Before we proceed, we note even the above numerical results are for two-user single-channel scheduling problem under tolerance constraints for visualization purposes, our methods apply to the more general multi-user multi-channel scheduling problem under violation tolerance and energy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Although the computational complexity may be relatively high for the LP solution compared to other solutions that exploit the special structure of particular problems, as we mentioned above, due to the non-convexity and non-concavity of the tolerance constraints, the monotone and threshold structure of the optimal policy does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' The Whittle Index approach (used, for example, in [29], [31]) which have relatively low complexity also does not apply to our multi- channel scheduling problems since each user in our setting is allowed to transmit over multiple channels simultaneously, whereby the Whittle’s Indexability condition does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Using the generally applicable LP-based approach reveals key insights that can guide the designers in developing efficient schedulers for future multi-channel wireless technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' ONLINE SCHEDULING UNDER UNKNOWN CHANNEL STATISTICS Until this point, we have assumed that the channel success probabilities are known when solving the optimization prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In this section, we use a Lyapunov-drift-plus-penalty approach(see [28]) to solve the multi-user online age related optimization problem in the scenario when only the current channel states are known, but the channel statistics are un- known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We will transfer all the energy and age-related constraints into the virtual queues and view the objective as a penalty term with parameter M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' For the energy constraint of the source i, let us define the corresponding virtual queue as Q1,i[t], whose initial value is Q1,i[0] = 0 and update equation is: Q1,i[t + 1] = (Q1,i[t] + ui (A[t]) − bi)+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Similarly, we define the virtual queue Q2,k[t] for the kth age- related constraint, whose initial value is Q2,k[0] = 0 and update equation is: Q2,k[t + 1] = (Q2,k[t] + ωk (A[t]) − ck)+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Generically, if the virtual queue Q1,i[t] is stable, then its input rate lim T →∞ 1 T T � t=1 E [ui (A[t])] will be less than its output rate bi [28], so that the corresponding constraint can be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Define the state of both virtual queues and age at time t as Q[t] = (Q1,1[t], · · · , Q1,n[t], Q2,1[t], · · · , Q2,K[t], A[t]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Based on the virtual queues, we will define the quadratic Lyapunov function as: V [t] = 1 2( n � i=1 Q2 1,i[t] + K � k=1 Q2 2,k[t]), and develop an online algorithm to greedily minimize the upper bound of the Lyapunov-drift-plus-penalty func- tion ∆V (q) + ME[ω0(a)] given the current state q = (q1,1, · · · , q1,n, q2,1, · · · , q2,K, a), where: ∆V (q) = E[V [t] − V [t − 1]|Q[t] = q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We consider the multi-user single-channel scheduling prob- lem under tolerance constraints as a specific example to present the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Since there are no energy constraints, we do not need the set of virtual queues {Q1,i[t]}i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In order to express the kth violation rate constraint for source k = 1, · · · , n, we let ωk (A[t]) = 1 (Ak[t + 1] > τk) and ck = ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Then the virtual queue Q2,k[t], whose initial value is Q2,k[t] = 0, updates as follows: Q2,k[t + 1] = (Q2,k[t] + 1 (Ak[t + 1] > τk) − ϵk)+ , where Ak[t + 1] = 1 + Ak[t](1 − Sk[t]Uk[t]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Sk[t] represents the channel success;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Uk[t] represents whether the source is scheduled to transmit or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' If virtual queue Q2,k[t] is stable, its input rate, the threshold violation rate πk(τk + 1) = limT →∞ 1 T �T t=1 1 (Ak[t + 1] > τk) , will be less than its output rate ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' The conditional Lyapunov drift can be bounded as follows: ∆V (q) ≤ n � k=1 q2,kE [Rk − ϵk|q2,k] + n � k=1 E � (Rk − ϵk)2 2 |q2,k � , where Rk ∆= 1{1 + Ak (1 − SkCk) > τk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' At every time slot t, we can develop an online algorithm as summarized below to greedily minimize the upper bound of the Lyapunov drift given the queue lengths Q[t − 1] and A[t − 1] since there is no objective or penalty term in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Algorithm 1 A Heuristic Scheduling Policy 1: Input current system state: Ai[t],Qi[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' 2: Define available transmission decision set: only one Ui[t] can be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' 3: Choose U[t] to minimize the upper bound of Lyapunov drift function in the above inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' 4: Update queue lengths for next time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Again, for the sake of easy visualization, we will only present the simulation results for the two-user online schedul- ing problem under age tolerance constraints, but the online algorithm can be simply applied to any number of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' The simulation results are illustrated in Fig 5 and Fig 6 for different parameters where the upper right area of the dash-dot purple line is the stability region of the online scheduler when the channel condition µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' The comparison will be in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' COMPARISON OF STABILITY REGIONS UNDER AGE VIOLATION CONSTRAINTS In this section, we compare the performance of three dif- ferent algorithms for the two-user single channel scheduling feasibility problem under age violation tolerance constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' These are: the optimal scheduler from Section IV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' the prior design from [21] developed for a single-channel multi-user setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' and our online scheduler from Section V that does not require channel statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We first focus on the case when the two source nodes are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In Figure 5, there are two source nodes with the same age thresholds of τ1 = τ2 = 2 and the same channel success probabilities of µ1 = µ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' The upper right area of the blue line is the stability region for the optimal scheduling algorithm in Section IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' The yellow and orange lines correspond to the algorithm in [21] and capture the two cases when the rate vector does or does not possess a special property (called step-down rate vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' The purple line marks the stability region for the online algorithm when the channel conditions µ1, µ2 are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Several observations are in order from these simulation results: (i) The stability regions are all symmetric, as can be expected due to the homogeneous deadline thresholds and channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' (ii) The optimum policy (blue line) outperforms other poli- cies, with markedly better performance in cases where the tolerance levels are greatly different from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' (iii) The online algorithm (purple line) performs very closely to the optimal policy, experiencing a small performance loss only at some extreme range of tolerance levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' (iv) When compared with the algorithms from [21](yellow and red lines), the online algorithm performs particularly better when one of the tolerance rates is smaller than the corresponding channel loss probability, as observed by the vertical gap between purple and yellow lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' (v) The online and optimal policies are continuous with respect to the tolerance level, which eliminates the need to check if the tolerance rate vector satisfies certain properties, such as the step-down rate condition in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' To compare the advantages and disadvantages of the al- gorithms under non-homogeneous scenarios, in Figure 6, we consider two source nodes with asymmetric age thresholds of τ1 = 2, τ2 = 4 and a common channel success probability of µ1 = µ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Since the violation rate depends on both the age thresholds and the channel success probabilities, this is a non-homogeneous scenario even though µ1 = µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In this figure, in contrast to the previous figure, we can further see that the optimal policy outperforms others when one of the 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='8 1 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='8 1 2 Optimal scheduling algrithm Algorithm proposed in [21] under general rate vector Algorithm proposed in [21] under step-down rate vector Online scheduling algorithm Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Stability region (upper-righter) comparison for symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='8 1 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content='8 1 2 Optimal scheduling algrithm Algorithm proposed in [21] under general rate vector Algorithm proposed in [21] under step-down rate vector Online scheduling algorithm Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Stability region (upper-righter) comparison for asymmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' tolerance constraints is very strict, namely when ϵ1 approaches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' In this regime, the feasible tolerance level ϵ2 of user 2 other algorithms is bounded away from zero while the optimal algorithm decreases towards zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' These simulation results are typical of other circumstances, with the common observation that our online scheduler per- forms close to the optimal scheduler and typically non- negligibly better than the most closely related state-of-art algorithm from [21], despite the fact that it operates without the knowledge of channel statistics that is assumed in the other designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' CONCLUSIONS In this paper, we considered a general class of age-optimal scheduling problems for multi-source multi-channel commu- nication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We formulated the generic age-optimization problem with flexible weight functions ωk under energy and tolerance constraints in the form of a CMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We solved this generic problem, which a usual threshold-based structure policy does not apply, by relating it to the solution an associated linear programming problem using the powerful theory of CMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Then, we focused on the special case of single-source multi- channel scenario to investigate the characteristics of optimal scheduler for the important special cases of average-age and violation-rate minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Our investigations revealed several interesting insights, in- cluding the observation that age-violation-rate minimizing scheduler employs a super-linearly like growing energy al- location strategy with increasing age, as opposed to the sub-linearly like growing allocation for the average-age- minimizing scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' These insights may provide useful guidelines for IoT network designers in developing effective update strategies based on different sensitivities of applications to age performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' We also studied the special case of multi-source single- channel scheduling problem with age violation rate constraints to investigate the feasibility region of the optimal scheduler together with that of most closely related prior works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQfrfmY/content/2301.00562v1.pdf'} +page_content=' Finally, we have developed an online scheduler that does not require the knowledge of channel statistics, and compared its perfor- mance to the optimal scheduler through simulations to observe that it performs closely to the optimal scheduler despite its lack of information on channel statistics.' metadata={'source': 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b/H9FJT4oBgHgl3EQfuy1Y/content/tmp_files/2301.11623v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..18c1f59f03d476dc39d15292addfa74dea26240d --- /dev/null +++ b/H9FJT4oBgHgl3EQfuy1Y/content/tmp_files/2301.11623v1.pdf.txt @@ -0,0 +1,1297 @@ +Higher-Order Patterns Reveal Causal Timescales +of Complex Systems +Luka V. Petrovi´c1, Anatol Wegner2, and Ingo Scholtes1,2 +1Data Analytics Group, University of Zurich, Z¨urich, Switzerland +2Center for Artificial Intelligence and Data Science (CAIDAS), +Julius-Maximilians-Universit¨at W¨urzburg, W¨urzburg, Germany +January 30, 2023 +Abstract +The analysis of temporal networks heavily depends on the analysis of time-respecting paths. +However, before being able to model and analyze the time-respecting paths, we have to infer +the timescales at which the temporal edges influence each other. In this work we introduce +temporal path entropy, an information theoretic measure of temporal networks, with the aim to +detect the timescales at which the causal influences occur in temporal networks. The measure +can be used on temporal networks as a whole, or separately for each node. We find that the +temporal path entropy has a non-trivial dependency on the causal timescales of synthetic and +empirical temporal networks. +Furthermore, we notice in both synthetic and empirical data +that the temporal path entropy tends to decrease at timescales that correspond to the causal +interactions. Our results imply that timescales relevant for the dynamics of complex systems +can be detected in the temporal networks themselves, by measuring temporal path entropy. +This is crucial for the analysis of temporal networks where inherent timescales are unavailable +and hard to measure. +1 +Introduction +The research of dynamic complex systems has in recent years advanced beyond static graph repre- +sentations [Lambiotte et al., 2019, Battiston et al., 2020]. The focus has shifted to various general- +izations of diadic interactions in graphs: multiple types of interactions in multilayer network [Kivel¨a +et al., 2014], multibody interactions in the form of simplicial complexes and hypergraphs [Petri and +Barrat, 2018] and models that incorporate concepts of memory [Scholtes et al., 2014, Lambiotte +et al., 2015, Williams et al., 2022]. Such generalized relationships allow us to better model complex +systems, because they can represent richer data. +Temporal networks are one kind of such rich data; they record not only who interacted with +whom, but also when each interaction happened. They bring us closer to understanding the dynamics +of complex systems, but require us to perform analysis beyond the static networks approach [Holme +and Saram¨aki, 2012, Holme, 2015]. The time information can yield valuable insights on its own [Goh +and Barab´asi, 2008], and, although initially the temporal and topological aspects of temporal net- +works were mostly studied independently, even richer insights are hidden in the coupling of the +temporal and topological patterns [Ceria et al., 2022]. Such coupling can affect the statistics of +1 +arXiv:2301.11623v1 [physics.soc-ph] 27 Jan 2023 + +time-respecting paths [Holme and Saram¨aki, 2012] in temporal networks, which impacts e.g., anal- +ysis of accessibility [Lentz et al., 2013], reachability [Badie-Modiri et al., 2020], spreading [Masuda +et al., 2013, Scholtes et al., 2014, Lambiotte et al., 2015, Badie-Modiri et al., 2022], clustering [Ros- +vall et al., 2014], centralities [Scholtes et al., 2016], and visualization [Perri and Scholtes, 2020]. +Although there are many possible ways in which temporal and topological patterns can couple in +complex systems, one of the most basic cases is when an incoming temporal edge to a node causes +a change of frequencies of the edges emanating from the node within a given time-window. For +instance, in a communication network we expect an incoming message to induce a outgoing message +on the same topic, e.g. in the form of a reply, within a certain time window reflecting the minimal +reaction time and memory of the recipient. However, information on the timescales relevant for the +temporal network dynamics is rarely available in real world settings. +In this work, we define an information theoretic measure to detect timescales at which interactions +in a complex system cause each other. We demonstrate its effectiveness in both synthetic and real +world data. +2 +Background +Let Γ = (V, E) be a temporal network consisting of a set of nodes V and a set of time-stamped edges +E ⊆ V × V × R. We denote the set of unique edges with E ⊂ V × V . A temporal edge (v, w, t) ∈ E +represents a direct link from node v to node w at time t. For simplicity, we assume that the temporal +edges are instantaneous, however the method and algorithms can be modified in a straightforward +fashion to the case where edges have finite duration. Formally, we call a sequence of time-stamped +edges (v1, w1, t1), . . . , (vk, wk, tk) a time-respecting path iff for all i ∈ {2, . . . , k} they satisfy the +following conditions [Pan and Saram¨aki, 2011, Holme and Saram¨aki, 2012, Casteigts et al., 2021]: +wi−1 = vi +(1) +τmin < ti − ti−1 < τmax. +(2) +The parameters τmin and τmax naturally introduce a timescale that affects all analyses of temporal +networks that are based on time-respecting paths. +The timescale has to be defined differently for processes on the temporal network or the processes +of the temporal network [Holme and Saram¨aki, 2012]. In the former case, the timescale is defined by +the process running on the temporal network, e.g. in the case of an epidemic that is spreading over +a temporal network of contacts, the timescale is a property of a disease, related to the time interval +in which a person is contagious and not related to the timescales at which contacts occur 1. In the +latter case, the timescale is part of the process of edge activation, and thus shapes the temporal +network itself. For example, information that is spreading between individuals is also affecting the +individuals’ choice with whom to share the information: a person would be more likely to share the +family-related information with a family member and work-related information with a colleague. In +this letter, we investigate the latter case, more specifically, we investigate whether interactions in a +complex system induce one another at a given timescale τ = [τmin, τmax]. +In the literature, there exist a variety of definitions of timescales in temporal networks, as well as +a variety of methods aimed at detecting them. The various definitions of timescales are based on the +different structural features of temporal networks. One popular definition of timescales in temporal +networks is the approach based on splitting the network into time-slices and aggregating the edges +1We note that the processes on and of the temporal network may interact [Gross and Sayama, 2009], and thus blur +the distinction. +2 + +inside the time-interval [Caceres and Berger-Wolf, 2013, Darst et al., 2016]. In the same framework, +Ghasemian et al. [2016] and Taylor et al. [2016] investigate the limitations of detectability of cluster +structures dependent on the timescales of aggregation. Since this framework is based on aggregating +the temporal network into a sequence of static time-aggregated networks, it loses information of +the time-respecting paths and is therefore not in line with our aims. Other lines of research often +related to timescale detection are change point detection [Peixoto and Gauvin, 2018], and analysis +of large-scale structures. Gauvin et al. [2014] detects clusters and their temporal activations in a +temporal network using tensor decomposition. Similarly, Peixoto [2015] proposed a method to detect +the change points of cluster structure in a temporal network. Peixoto and Rosvall [2017] proposed +a method to simultaneously detect the clusters and timescales in temporal network, however, they +model the temporal network as a single sequence of tokens (similar to [Peixoto and Gauvin, 2018]) +that represent temporal edges, and their timescale inference refers to the number of tokens in the +memory of a Markov chain that models such a sequence. In our view, these works focus on mesoscale +structures, and take a coarse grained view of temporal networks, while in this work, we propose a +complementary approach by focusing on local correlations between temporal edges incident on a +node and subsequent temporal edges emanating from it. Among the works that took a fine-grained +view, Williams et al. [2022] investigated pairwise correlations between the temporal edges. Different +from the approach that we took, they aggregate the network in time-slices as a preprocessing step, +and the timescale is defined as a maximum number of time slices back in time at which correlations +are detectable. Scholtes et al. [2016] found that correlations between edges on time-respecting paths +affect centralities; they modeled the time-respecting paths with higher-order models and found that +this approach improves the centrality rankings. They identified the issue of timescale detection in +the context of time-respecting paths, which our work addresses. Our work also complements the +work of Pfitzner et al. [2013] which introduces betweenness preference that can be used to study +over- and under-represented time-respected paths in temporal networks, but does not address the +problem of detecting the timescales at which these paths occur. To the best of our knowledge, our +work is the first to address the issue of timescale detection for time-respecting paths in temporal +networks. +3 +Temporal Path Entropy +We address the issue of timescale detection by analysing the statistics of time-respecting paths +Pk +τ (Γ) of length k at timescales τ = [τmin, τmax] in a temporal network Γ. We define “temporal path +entropy” H for paths (v0, v1, . . . , vk) of length k as the entropy of the last node vk conditional on +the sub-path (v0, v1, . . . , vk−1): +H = H(vk|v0, . . . , vk−1) +(3) += H(v0, . . . , vk) − H(v0, . . . , vk−1), +(4) +where H(P) = − � +i pi ln(pi) is the Shannon entropy. +The identity in Eq. 4 can be obtained +by applying the chain rule (see Appendix for derivation). By definition, temporal path entropy +H measures uncertainty in the last step of time-respecting paths given the k − 1 previous steps. +A lower value of the entropy indicates a high correlation between the memory of time-respecting +paths and subsequent steps. Hence the τ for which the entropy reaches its minimum gives us the +timescale for which time-respecting paths become most predictable, i.e. +where the correlations +between subsequent temporal edges are the most pronounced. The entropy can also be defined for +a single node v, by simply fixing vk−1 = v, allowing for a more fine grained analysis that could be +important if nodes differ significantly with respect to the timescales they operate on. The intuition +3 + +behind the temporal path entropy is to measure how much the target vk of an edge emanating +form the node vk−1 depends on the incoming paths that influenced it in the past. Testing those +dependencies at different timescales would thus point to the timescales at which the dependencies +are most pronounced. When we compute temporal path entropy for the whole temporal network, +we use all time-respecting paths of length k in the temporal network, while when we compute it for +a node v, we select only the paths where vk−1 = v. +1.75 +2.00 +2.25 +H[ nat ] +Synthetic-2 +0 +1 +2 +WB-DE +2 +3 +HC-email +50 +100 +150 +200 +250 +0.0 +0.5 +1.0 +counts [103] +100 +102 +104 +106 +time [s] +0 +20 +103 +105 +0.00 +0.05 +0.10 +m +m +h +d +w +s +m +h +d +w +original +shuffled +inter-event times +Figure 1: +Top: temporal path entropy as a function of causal temporal scales in datasets Synthetic- +2, WB-DE, and HC-email (transparent red) and in the temporal networks with shuffled timestamps +(transparent blue). +The height of a bar represents temporal path entropy (error bars represent +the estimation error) and the x-limits of a bar represent the interval τ = [τmin, τmax] on which the +temporal path entropy was measured. We indicate on x-axis the timescales of one minute (m), +hour (h), day (d), week (w), and year (y). We observe that the temporal path entropy differs more +between the original and the shuffled network at causal timescales. Bottom: histogram of causal +inter-event times. +In practice, the temporal path entropy can be estimated from the counts of time-respecting +paths Pk +τ (Γ) by assuming multinomial distributions with respective probabilities p(v0, . . . , vk−1) and +p(v0, . . . , vk). The counts of time-respecting paths can be computed e.g. using the methods from +[Kivel¨a et al., 2018, Petrovi´c and Scholtes, 2021]. The estimation of the entropy can be challenging +especially for small ranges of timescales, since the temporal network can get temporally disconnected, +resulting in very few paths of order k being observed. As a result we require an efficient method for +estimating the entropy that performs well even in such under-sampled regimes. The simplest estima- +tor of a multinomial distribution, called the plug-in estimator, is based on the maximum likelihood +estimation, which, however, is known to severely underestimate the entropy in the undersampled +regime and has various corrections [e.g. Miller, 1955, Grassberger, 2003]). An alternative to the +plug-in estimator is to follow a Bayesian approach which results in entropy estimators that strongly +depend on the choice of prior. To counteract this dependency, the NSB estimator [Nemenman et al., +2001] directly infers the entropy from the counts by averaging over different priors for the transition +probabilities, rather than inferring the transition probabilities themselves. Being a Bayesian method, +the NSB estimator can also be used to quantify the uncertainty of the estimates. Assuming that +the estimates of H(v0, . . . , vk) and H(v0, . . . , vk−1) have independent errors σk and σk−1, we can +approximate the total error of the estimate as σ = (σ2 +k + σ2 +k−1)1/2. As the NSB estimator requires +the size of the alphabet to be known, it is most suitable for cases where the number of nodes is fixed +and improves further if the set of edges that can occur are known a priory as this further restricts +the number of potential paths. In cases when the number of nodes in the system is unknown, the +4 + +0 +2 +EU-email-A +100 +101 +102 +103 +104 +105 +106 +0.0 +2.5 +DNC-16 +s +m +h +d +w +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +time [s] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +H[ nat ] +original +shuffled +Figure 2: Temporal path entropy as a function of the timescale τ in EU-email-A and DNC-16 and in +timestamp shuffled networks. Timescale τ is represented with the x-limits of the bar, and temporal +path entropy is represented as the height of the bar. Error bars indicate the error of the temporal +path entropy estimates. +Pitman-Yor Mixture entropy estimator [Archer et al., 2014] could be used instead. +Finally, we address testing whether an interval τ is a causal timescale of a temporal network Γ. +To do so, we need to assume the null hypothesis that there are no temporal correlations between +temporal edges, but the main issue is to obtain a sample of temporal networks under this assump- +tion. To resolve this issue, we repeatedly randomize the observed temporal network Γ by randomly +permuting timestamps between its temporal edges [Holme and Saram¨aki, 2012]. These samples of +temporal networks would preserve both the edge frequencies and timestamp distribution while de- +stroying the correlations between temporal edges. We can use the samples to determine whether +temporal path entropy of the observed network has an unexpected value under the null assumption. +4 +Experiments +In the following part, we first show the behavior of temporal path entropy in synthetically generated +temporal networks with known causal timescales (the description of the generation process can +be found in the Appendix); we then present how it behaves in two real-world networks with the +information about the ground truth timescales, and two real world networks without the information +about the ground truth timescales. +In the top left panel of Fig. 1, we present the temporal path entropy H (y-axis) for various +timescales (x-axis): the left and right x limit of a bar represents τmin and τmax, and the height of +the bar represents H. The results are shown both for the synthetic network and its shuffled network. +In the bottom left panel, we show the histogram of inter-event times on causal paths. We observe +that the temporal path entropy behaves as expected and decreases in accordance with the planted +timescale at which the interactions cause one another. Moreover, this pattern disappears when the +timestamps of edges are shuffled, demonstrating that temporal path entropy captures the interplay +of temporal and topological patterns. +We consider here two empirical temporal networks where we have information about the ground +truth causal structure and two empirical temporal networks where we have no information about +5 + +the ground truth causal structure. As a first data set, we consider the bipartite temporal network of +German Wikibooks co-editing patterns (WB-DE) [Wikimedia Foundation, Peixoto, 2020]. This data +contains information about edits on the Wikibooks website: for each edit, we know the editor, the +article that was edited, and the time at which the edit occurred. We preprocess this data to obtain +a temporal network of editors: if editor v edited an article prior to editor w who edited the same +article at time t, we assume that a link (v, w, t) occurred in the temporal network of editors. We +define causal inter-event times based on the articles: we extract the time intervals between successive +edits of each article. In WB-DE data, we analyze the timescales of the whole temporal network. +As a second data set, we consider public data set of Hillary Clinton’s emails (HC-email) [Kaggle, +2022], which contains the sender, the receiver, the timestamp, and the subject of each email. In this +data set we analyze the timescales of node representing Hillary Clinton. While sender, receiver and +the timestamp constitute a temporal network, email subjects allow us to obtain causal inter-event +times: for each incoming email, we extract the time duration until an email with the same subject +was sent. We use the inter-event times between emails with the same subject and the inter-event +times of articles for evaluation; the temporal networks contain only the temporal edges and not any +additional information about the ground truth timescales. We also use two email data sets without a +ground truth timescales: EU-email-A [Paranjape et al., 2017], which contains email correspondence +between researchers of an EU institution from four deparments, and DNC-16 [Rossi and Ahmed, +2015], which contains emails of the US Democratic National Committee. Results on other datasets +as well as details of all datasets are in the Appendix. Reproducibility package is available at [Petrovi´c +et al., 2023]. +Results of the WB-DE and HC-email data are in Fig. 1 (middle and right, respectively). When +we compare the histogram of causal inter-event times with the temporal path entropy at different +timescales of the temporal network, we see that increased number of causal interactions increases the +difference in temporal path entropy between the original and the shuffled network. The temporal +path entropy converges for large timescales because the interval sizes increase, the density of causal +interactions decreases, and the noise increases. In Fig. 2, although we do not have the ground truth, +we see that the largest difference between the original and the shuffled datasets are at timescales +between a minute a day, which is what we would expect from email correspondence. +We identify four limitations of our approach. First, our base assumption is that the interactions, +represented by edges, cause one another, and our measure can not separate that case that from +the case when edges are generated by some common factor. Second, being based on directed paths +the current method is restricted in the types of causal interactions it considers namely interactions +where a incoming link into a vertex effects the subsequent links emanating from the vertex. The +method could potentially be generalized to other types of interactions by considering other patterns +to alleviate this shortcoming. Third, our method cannot detect timescales at which the incoming +edges to a node change the overall activity of the node without changing the relative frequencies of +the outgoing edges. Detecting timescales of such causal influences is thus an open problem. Fourth, +real data can contain time-varying timescales, e.g. during day or night, which would probably require +an application of time warping techniques. +5 +Conclusion +To summarize, the analysis of temporal networks heavily depends on the analysis of time-respecting +paths [Holme and Saram¨aki, 2012, Holme, 2015, Pan and Saram¨aki, 2011, Masuda et al., 2013, +Scholtes et al., 2016, Kivel¨a et al., 2018]. However, in order to model and analyze the time-respecting +paths, we first need to identify the correct timescale. +In this work we address this problem by +6 + +introducing an information theoretic measure, the temporal path entropy, that is able to can identify +timescales at which the influences are highly correlated. Using real world data we demonstrated that +the measure can be applied to temporal networks as a whole as well as to a single node. We showed +that the temporal path entropy can capture the causal timescales in both synthetic and empirical +temporal networks. We further support our findings by observing that the differences in the temporal +path entropy between the original and shuffled networks coincide with increases in the number of +causal paths. The temporal path entropy allows system-relevant timescales to be inferred from the +temporal networks themselves which is crucial for the analysis of temporal networks where inherent +timescales are unavailable and hard to measure. +Acknowledgments +The authors would like to thank Christopher Bl¨ocker, Chester Tan, and Franziska Heeg for valu- +able comments on the manuscript. LP and IS acknowledge support by the Swiss National Science +Foundation, grant 176938. +References +E. Archer, I. M. Park, and J. W. Pillow. Bayesian entropy estimation for countable discrete distri- +butions. The Journal of Machine Learning Research, 15(1):2833–2868, 2014. +A. Badie-Modiri, M. Karsai, and M. Kivel¨a. Efficient limited-time reachability estimation in temporal +networks. 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To add each path (v0, v1, v2) +to the temporal network, we sample a random starting time t uniformly from [0, Ttotal − ¯τmax] and +create a temporal edge (v0, v1, t); we then sample temporal distance δ between edges on the path +(inter-event time) uniformly from ¯τ and create the temporal edge (v1, v2, t + δ). We choose ¯τ with +¯τmin = 100 and ¯τmax = 200. To add some noise to the system, we uniformly sample 20000 edges +from the static graph, and sample their timestamps uniformly from [0, Ttotal]. The temporal network +Synthetic-4 contains two time-scales relevant for the dynamics. To do so, we generated two different +temporal networks based on two random graphs of 50 nodes (with the same node names) and 500 +edges and based on the different timescales τ 1 = [50, 100] and τ 2 = [150, 200]. We used the same +procedure as above with parameters nu.p. = 500; np = 5000; Ttotal = 105; nr.e. = 10000. We merged +the two temporal networks into one; the details of the resulting network are in Table 1. The dataset +Synthetic-5 contains paths of length three. Again, there are 50 nodes and 500 edges in the static +Erd˝os R´enyi graph. +We sample nu.p. = 20 unique paths, we sample np = 20000 of them, and +spread them across Ttotal = 105 using the same procedure and timescale τ = [100, 200]. We add +nr.e = 10000 random edges to the network as noise. +We also use empirical dataset where can get access to the ground truth causal path structure. We +consider the bipartite temporal network of Wikibooks co-edits in Arabic (WB-AR), French (WB- +FR) and German (WB-DE) [Wikimedia Foundation, Peixoto, 2020]. This data contains information +about edits on the Wikibooks website: for each edit, we know the editor, the article that was edited, +and the time at which the edit occurred. We preprocess this data to obtain a temporal network +of editors: if editor v edited an article prior to editor w who edited the same article at time t, we +assume that a link (v, w, t) occurred in the temporal network of editors. We define causal inter-event +times based on the articles: we extract the time intervals between successive edits of each article. +In these data, we analyze the timescales of the whole temporal network. Another dataset where we +10 + +dataset +|V | +|E| +|E| +Ttotal [s] +Ants-1-1 +89 +947 +1911 +1.44e+03 +Ants-1-2 +72 +862 +1820 +1.75e+03 +Ants-2-1 +71 +636 +975 +1.44e+03 +Ants-2-2 +69 +769 +1917 +1.8e+03 +Ants-3-1 +11 +37 +78 +1.13e+03 +Ants-3-2 +6 +21 +104 +1.42e+03 +DNC-16 +1891 +5598 +39264 +8.49e+07 +EU-email-1 +309 +3031 +61046 +6.94e+07 +EU-email-2 +162 +1772 +46772 +6.94e+07 +EU-email-3 +89 +1506 +12216 +6.93e+07 +EU-email-4 +142 +1375 +48141 +6.94e+07 +EU-email-A +986 +24929 +332334 +6.95e+07 +Gallery +10972 +89034 +831824 +6.95e+06 +HC-email +326 +385 +8313 +1.19e+08 +Hospital +75 +2278 +64848 +3.48e+05 +Hypertext +113 +4392 +41636 +2.12e+05 +OSS +5789 +6888 +12583 +3.54e+08 +Primary +242 +16634 +251546 +1.17e+05 +School-13 +327 +11636 +377016 +3.64e+05 +Synthetic-1 +50 +500 +30000 +1e+05 +Synthetic-2 +50 +500 +40000 +1e+05 +Synthetic-3 +50 +500 +60000 +1e+05 +Synthetic-4 +50 +898 +40000 +1e+05 +Synthetic-5 +50 +500 +50000 +1e+05 +WB-AR +1124 +3334 +27166 +3.89e+08 +WB-DE +10999 +54700 +464089 +4.87e+08 +WB-FR +9735 +53606 +362094 +4.88e+08 +Work-13 +92 +1510 +19654 +9.88e+05 +Table 1: The sizes of the sets of nodes V , unique edges E, and temporal edges E of temporal +networks that we analyzed in the experiments. Datasets synth-2, HC email and WB DE are in the +main paper. The other datasets are shown in the Appendix. +can get access to the ground truth causal structure is the public data set of Hillary Clinton’s emails +(HC-email) [Kaggle, 2022], which contains the sender, the receiver, the timestamp, and the subject +of each email. In this data set we analyze the timescales of node representing Hillary Clinton. While +sender, receiver and the timestamp form a temporal network, email subjects allow us to obtain +causal inter-event times: for each incoming email, we extract the time duration until an email with +the same subject was sent. We use the inter-event times between emails with the same subject and +the inter-event times of articles for evaluation; the temporal networks contain only the temporal +edges and not any additional information about the ground truth timescales. The details of each +data-set are in Table 1. +Finally, we also use empirical temporal networks where we do not know the ground truth causal +path structure. Datasets Ants-1-1, Ants-1-2, Ants-2-1, Ants-2-2, Ants-3-1, and Ants-3-3 [Blonder +and Dornhaus, 2011] contain antenna contacts in ant colonies. Dataset DNC-16 [Rossi and Ahmed, +2015] contains emails of the US Democratic National Committee leaked in 2016. +Datasets EU- +11 + +email-1, EU-email-2, EU-email-3, EU-email-4, and EU-email-A [Paranjape et al., 2017] contain +email correspondence between researchers of an EU institution from first, second, third, fourth and +all departments, respectively. Datasets Gallery [Isella et al., 2011], Hospital [Vanhems et al., 2013], +Hypertext [Isella et al., 2011], Primary [Gemmetto et al., 2014, Stehl´e et al., 2011], Work-13 [G´enois +et al., 2015] and School-13 [Mastrandrea et al., 2015] contain human face-to-face interactions in +different settings measured by the SocioPatterns collaborations. Dataset OSS [Zanetti et al., 2013] +contains ASSIGN relationships between members of the Open Source Software community Apache. +7 +Results: Synthetic Data +We present results for datasets Synthetic-1, Synthetic-3, Synthetic-4, Synthetic-5. +1.5 +2.0 +H[ nat ] +Synthetic-1 +50 +100 +150 +200 +250 +time [s] +0.0 +0.2 +0.4 +counts [103] +original +shuffled +inter-event times +Figure 3: Top: temporal path entropy as a function of the timescale τ in temporal network Synthetic- +1 and in Synthetic-1 with shuffled timestamps. Timescale τ is represented with the x-limits of the +bar, and temporal path entropy is represented as the height of the bar. Error bars indicate the error +of the temporal path entropy estimates. Bottom: histogram of inter-event times of synthetic causal +interactions. +8 +Results: Empirical Data with Ground Truth +In this section we show results on other Wikibooks datasets that we used to test the method. In +Fig. 7, we test temporal path entropy on the WB-AR dataset, and in Fig. 8, we test it on the WB-FR +dataset. Similar to the WB-DE in the main paper, the bottom panel shows the yellow histogram of +inter-event times of edits per article for all articles. +9 +Empirical data without the ground truth +In this section, we show multiple datasets in which we do not have access to the ground truth +temporal scales. Although the lack of ground truth in these datasets makes objective evaluation +12 + +1.5 +2.0 +H[ nat ] +Synthetic-3 +50 +100 +150 +200 +250 +time [s] +0 +1 +2 +counts [103] +original +shuffled +inter-event times +Figure 4: Equivalent of Fig. 3, for Synthetic-3. +2 +3 +H[ nat ] +Synthetic-4 +50 +100 +150 +200 +250 +time [s] +0.0 +0.5 +1.0 +counts [103] +original +shuffled +inter-event times +Figure 5: Equivalent of Fig. 3, for Synthetic-4. +of the method difficult, the results across datasets are consistent and in accordance with what one +would expect: e.g. in the email datasets, temporal path entropy is different between the original +and the shuffled network for timescales between one minute and a few days, which corresponds to +what we would expect to be the interval in which emails are responded to. +13 + +1.0 +1.2 +H[ nat ] +k = 2 +Synthetic-5 +0.75 +1.00 +1.25 +H[ nat ] +k = 3 +50 +75 +100 +125 +150 +175 +200 +225 +250 +time [s] +0.0 +2.5 +counts [103] +original +shuffled +inter-event times +Figure 6: Temporal path entropy as a function of the timescale τ in temporal network Synthetic-5 +and in Synthetic-5 with shuffled timestamps for orders k = 2 (top) and k = 3 (middle). Timescale τ +is represented with the x-limits of the bar, and temporal path entropy is represented as the height of +the bar. Error bars indicate the error of the temporal path entropy estimates. Bottom: histogram +of inter-event times of synthetic causal interactions. +0.0 +0.5 +1.0 +H[ nat ] +WB-AR +100 +102 +104 +106 +time [s] +0 +2 +counts [103] +s +m +h +d +w +original +shuffled +inter-event times +Figure 7: +Top: temporal path entropy as a function of the timescale τ in WB-AR temporal network +and of WB-AR temporal network with shuffled timestamps. Timescale τ is represented with the +x-limits of the bar, and temporal path entropy is represented as the height of the bar. Error bars +indicate the error of the temporal path entropy estimates. Bottom: histogram of inter-event times +for all articles of edits of the same article. +14 + +0.0 +0.5 +1.0 +H[ nat ] +WB-FR +100 +102 +104 +106 +time [s] +0 +20 +counts [103] +s +m +h +d +w +original +shuffled +inter-event times +Figure 8: +Equivalent of Fig. 7 for WB-FR. +15 + +100 +101 +102 +103 +time [s] +0 +1 +2 +3 +4 +5 +H[ nat ] +Ants-1-1 +original +shuffled +s +m +100 +101 +102 +103 +time [s] +0 +1 +2 +3 +4 +H[ nat ] +Ants-1-2 +original +shuffled +s +m +100 +101 +102 +103 +time [s] +0 +1 +2 +3 +4 +H[ nat ] +Ants-2-1 +original +shuffled +s +m +100 +101 +102 +103 +time [s] +0 +1 +2 +3 +H[ nat ] +Ants-2-2 +original +shuffled +s +m +100 +101 +102 +103 +time [s] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +H[ nat ] +Ants-3-1 +original +shuffled +s +m +100 +101 +102 +103 +time [s] +0.0 +0.5 +1.0 +1.5 +2.0 +H[ nat ] +Ants-3-2 +original +shuffled +s +m +Figure 9: Temporal path entropy as a function of the timescale τ in temporal networks of antenna +contacts in ant collonies. For each temporal network, we show the temporal path entropy of the +original and of a shuffled network. Timescale τ is represented with the x-limits of the bar, and +temporal path entropy is represented as the height of the bar. Error bars indicate the error of the +temporal path entropy estimates. +16 + +100 +101 +102 +103 +104 +105 +106 +time [s] +0 +1 +2 +3 +4 +H[ nat ] +DNC-16 +original +shuffled +s +m +h +d +w +100 +101 +102 +103 +104 +105 +106 +time [s] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +H[ nat ] +EU-email-A +original +shuffled +s +m +h +d +w +100 +101 +102 +103 +104 +105 +106 +time [s] +0 +1 +2 +3 +H[ nat ] +EU-email-1 +original +shuffled +s +m +h +d +w +100 +101 +102 +103 +104 +105 +106 +time [s] +0 +1 +2 +3 +4 +5 +H[ nat ] +EU-email-2 +original +shuffled +s +m +h +d +w +100 +101 +102 +103 +104 +105 +106 +time [s] +0 +1 +2 +3 +4 +5 +H[ nat ] +EU-email-3 +original +shuffled +s +m +h +d +w +100 +101 +102 +103 +104 +105 +106 +time [s] +0 +1 +2 +3 +4 +5 +6 +H[ nat ] +EU-email-4 +original +shuffled +s +m +h +d +w +Figure 10: Temporal path entropy as a function of the timescale τ in temporal networks of email +correspondence. For each temporal network, we show the temporal path entropy of the original and +of a shuffled network. Timescale τ is represented with the x-limits of the bar, and temporal path +entropy is represented as the height of the bar. Error bars indicate the error of the temporal path +entropy estimates. +17 + +102 +103 +104 +time [s] +0 +2 +4 +6 +H[ nat ] +Gallery +original +shuffled +m +h +102 +103 +104 +105 +time [s] +0 +1 +2 +3 +4 +5 +6 +H[ nat ] +School-13 +original +shuffled +m +h +d +102 +103 +104 +105 +time [s] +0 +1 +2 +3 +4 +5 +6 +H[ nat ] +Hospital +original +shuffled +m +h +d +102 +103 +104 +105 +time [s] +0 +1 +2 +3 +4 +5 +6 +H[ nat ] +Hypertext +original +shuffled +m +h +d +102 +103 +104 +105 +time [s] +0 +2 +4 +6 +H[ nat ] +Primary +original +shuffled +m +h +d +102 +103 +104 +105 +time [s] +0 +1 +2 +3 +4 +5 +H[ nat ] +Work-13 +original +shuffled +m +h +d +w +Figure 11: Temporal path entropy as a function of the timescale τ in temporal networks of human +face-to-face interactions measured by the SocioPatterns collaboration. For each temporal network, +we show the temporal path entropy of the original and of a shuffled network. +Timescale τ is +represented with the x-limits of the bar, and temporal path entropy is represented as the height +of the bar. Error bars indicate the error of the temporal path entropy estimates. +18 + +100 +101 +102 +103 +104 +105 +106 +time [s] +0 +1 +2 +3 +4 +H[ nat ] +OSS +original +shuffled +s +m +h +d +w +Figure 12: Temporal path entropy as a function of the timescale τ in temporal networks ASSIGN +relationships between members of the Open Source Software community Apache. +We show the +temporal path entropy of the original and of a shuffled network. Timescale τ is represented with the +x-limits of the bar, and temporal path entropy is represented as the height of the bar. Error bars +indicate the error of the temporal path entropy estimates. +19 + +10 +Conditional entropy: The chain rule +For discrete random variables X and Y , the definition of the entropy (in nats) is +H(X) = − +� +x +p(X = x) ln p(X = x) +and the definition of conditional entropy (in nats) H(Y |X) is: +H(Y |X) = − +� +x,y +p(X = x, Y = y) ln p(X = x, Y = y) +p(X = x) +In the following, we use the above definitions to derive the chain rule of conditional entropy: +H(Y |X) = − +� +x,y +p(X = x, Y = y) (ln p(X = x, Y = y) − ln p(X = x)) = += − +� +x,y +p(X = x, Y = y) ln p(X = x, Y = y) − +� +− +� +x,y +p(X = x, Y = y) ln(p(X = x))) +� += += H(X, Y ) − +� +− +� +x,y +p(Y = y|X = x)p(X = x) ln(p(X = x))) +� += += H(X, Y ) − +� +�− +� +x +p(X = x) ln(p(X = x))) + +:1 +�� +y +p(Y = y|X = x) +� +� +� = += H(X, Y ) − H(X). +(5) +20 + diff --git a/H9FJT4oBgHgl3EQfuy1Y/content/tmp_files/load_file.txt b/H9FJT4oBgHgl3EQfuy1Y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d083238fcc964badb949a9eddca653e3734fb997 --- /dev/null +++ b/H9FJT4oBgHgl3EQfuy1Y/content/tmp_files/load_file.txt @@ -0,0 +1,1031 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf,len=1030 +page_content='Higher-Order Patterns Reveal Causal Timescales of Complex Systems Luka V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Petrovi´c1, Anatol Wegner2, and Ingo Scholtes1,2 1Data Analytics Group, University of Zurich, Z¨urich, Switzerland 2Center for Artificial Intelligence and Data Science (CAIDAS), Julius-Maximilians-Universit¨at W¨urzburg, W¨urzburg, Germany January 30, 2023 Abstract The analysis of temporal networks heavily depends on the analysis of time-respecting paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' However, before being able to model and analyze the time-respecting paths, we have to infer the timescales at which the temporal edges influence each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In this work we introduce temporal path entropy, an information theoretic measure of temporal networks, with the aim to detect the timescales at which the causal influences occur in temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The measure can be used on temporal networks as a whole, or separately for each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We find that the temporal path entropy has a non-trivial dependency on the causal timescales of synthetic and empirical temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Furthermore, we notice in both synthetic and empirical data that the temporal path entropy tends to decrease at timescales that correspond to the causal interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Our results imply that timescales relevant for the dynamics of complex systems can be detected in the temporal networks themselves, by measuring temporal path entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' This is crucial for the analysis of temporal networks where inherent timescales are unavailable and hard to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 1 Introduction The research of dynamic complex systems has in recent years advanced beyond static graph repre- sentations [Lambiotte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2019, Battiston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The focus has shifted to various general- izations of diadic interactions in graphs: multiple types of interactions in multilayer network [Kivel¨a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2014], multibody interactions in the form of simplicial complexes and hypergraphs [Petri and Barrat, 2018] and models that incorporate concepts of memory [Scholtes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2014, Lambiotte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2015, Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Such generalized relationships allow us to better model complex systems, because they can represent richer data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Temporal networks are one kind of such rich data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' they record not only who interacted with whom, but also when each interaction happened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' They bring us closer to understanding the dynamics of complex systems, but require us to perform analysis beyond the static networks approach [Holme and Saram¨aki, 2012, Holme, 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The time information can yield valuable insights on its own [Goh and Barab´asi, 2008], and, although initially the temporal and topological aspects of temporal net- works were mostly studied independently, even richer insights are hidden in the coupling of the temporal and topological patterns [Ceria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Such coupling can affect the statistics of 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='11623v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='soc-ph] 27 Jan 2023 time-respecting paths [Holme and Saram¨aki, 2012] in temporal networks, which impacts e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', anal- ysis of accessibility [Lentz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2013], reachability [Badie-Modiri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2020], spreading [Masuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2013, Scholtes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2014, Lambiotte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2015, Badie-Modiri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2022], clustering [Ros- vall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2014], centralities [Scholtes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2016], and visualization [Perri and Scholtes, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Although there are many possible ways in which temporal and topological patterns can couple in complex systems, one of the most basic cases is when an incoming temporal edge to a node causes a change of frequencies of the edges emanating from the node within a given time-window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' For instance, in a communication network we expect an incoming message to induce a outgoing message on the same topic, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' in the form of a reply, within a certain time window reflecting the minimal reaction time and memory of the recipient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' However, information on the timescales relevant for the temporal network dynamics is rarely available in real world settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In this work, we define an information theoretic measure to detect timescales at which interactions in a complex system cause each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We demonstrate its effectiveness in both synthetic and real world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 2 Background Let Γ = (V, E) be a temporal network consisting of a set of nodes V and a set of time-stamped edges E ⊆ V × V × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We denote the set of unique edges with E ⊂ V × V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' A temporal edge (v, w, t) ∈ E represents a direct link from node v to node w at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' For simplicity, we assume that the temporal edges are instantaneous, however the method and algorithms can be modified in a straightforward fashion to the case where edges have finite duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Formally, we call a sequence of time-stamped edges (v1, w1, t1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' , (vk, wk, tk) a time-respecting path iff for all i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' , k} they satisfy the following conditions [Pan and Saram¨aki, 2011, Holme and Saram¨aki, 2012, Casteigts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2021]: wi−1 = vi (1) τmin < ti − ti−1 < τmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' (2) The parameters τmin and τmax naturally introduce a timescale that affects all analyses of temporal networks that are based on time-respecting paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The timescale has to be defined differently for processes on the temporal network or the processes of the temporal network [Holme and Saram¨aki, 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In the former case, the timescale is defined by the process running on the temporal network, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' in the case of an epidemic that is spreading over a temporal network of contacts, the timescale is a property of a disease, related to the time interval in which a person is contagious and not related to the timescales at which contacts occur 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In the latter case, the timescale is part of the process of edge activation, and thus shapes the temporal network itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' For example, information that is spreading between individuals is also affecting the individuals’ choice with whom to share the information: a person would be more likely to share the family-related information with a family member and work-related information with a colleague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In this letter, we investigate the latter case, more specifically, we investigate whether interactions in a complex system induce one another at a given timescale τ = [τmin, τmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In the literature, there exist a variety of definitions of timescales in temporal networks, as well as a variety of methods aimed at detecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The various definitions of timescales are based on the different structural features of temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' One popular definition of timescales in temporal networks is the approach based on splitting the network into time-slices and aggregating the edges 1We note that the processes on and of the temporal network may interact [Gross and Sayama, 2009], and thus blur the distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 2 inside the time-interval [Caceres and Berger-Wolf, 2013, Darst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In the same framework, Ghasemian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' [2016] and Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' [2016] investigate the limitations of detectability of cluster structures dependent on the timescales of aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Since this framework is based on aggregating the temporal network into a sequence of static time-aggregated networks, it loses information of the time-respecting paths and is therefore not in line with our aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Other lines of research often related to timescale detection are change point detection [Peixoto and Gauvin, 2018], and analysis of large-scale structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Gauvin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' [2014] detects clusters and their temporal activations in a temporal network using tensor decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Similarly, Peixoto [2015] proposed a method to detect the change points of cluster structure in a temporal network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Peixoto and Rosvall [2017] proposed a method to simultaneously detect the clusters and timescales in temporal network, however, they model the temporal network as a single sequence of tokens (similar to [Peixoto and Gauvin, 2018]) that represent temporal edges, and their timescale inference refers to the number of tokens in the memory of a Markov chain that models such a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In our view, these works focus on mesoscale structures, and take a coarse grained view of temporal networks, while in this work, we propose a complementary approach by focusing on local correlations between temporal edges incident on a node and subsequent temporal edges emanating from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Among the works that took a fine-grained view, Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' [2022] investigated pairwise correlations between the temporal edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Different from the approach that we took, they aggregate the network in time-slices as a preprocessing step, and the timescale is defined as a maximum number of time slices back in time at which correlations are detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Scholtes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' [2016] found that correlations between edges on time-respecting paths affect centralities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' they modeled the time-respecting paths with higher-order models and found that this approach improves the centrality rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' They identified the issue of timescale detection in the context of time-respecting paths, which our work addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Our work also complements the work of Pfitzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' [2013] which introduces betweenness preference that can be used to study over- and under-represented time-respected paths in temporal networks, but does not address the problem of detecting the timescales at which these paths occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' To the best of our knowledge, our work is the first to address the issue of timescale detection for time-respecting paths in temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 3 Temporal Path Entropy We address the issue of timescale detection by analysing the statistics of time-respecting paths Pk τ (Γ) of length k at timescales τ = [τmin, τmax] in a temporal network Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We define “temporal path entropy” H for paths (v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' , vk) of length k as the entropy of the last node vk conditional on the sub-path (v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' , vk−1): H = H(vk|v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' , vk−1) (3) = H(v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' , vk) − H(v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' , vk−1), (4) where H(P) = − � i pi ln(pi) is the Shannon entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The identity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 4 can be obtained by applying the chain rule (see Appendix for derivation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' By definition, temporal path entropy H measures uncertainty in the last step of time-respecting paths given the k − 1 previous steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' A lower value of the entropy indicates a high correlation between the memory of time-respecting paths and subsequent steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Hence the τ for which the entropy reaches its minimum gives us the timescale for which time-respecting paths become most predictable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' where the correlations between subsequent temporal edges are the most pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The entropy can also be defined for a single node v, by simply fixing vk−1 = v, allowing for a more fine grained analysis that could be important if nodes differ significantly with respect to the timescales they operate on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The intuition 3 behind the temporal path entropy is to measure how much the target vk of an edge emanating form the node vk−1 depends on the incoming paths that influenced it in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Testing those dependencies at different timescales would thus point to the timescales at which the dependencies are most pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' When we compute temporal path entropy for the whole temporal network, we use all time-respecting paths of length k in the temporal network, while when we compute it for a node v, we select only the paths where vk−1 = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='25 H[ nat ] Synthetic-2 0 1 2 WB-DE 2 3 HC-email 50 100 150 200 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 counts [103] 100 102 104 106 time [s] 0 20 103 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='10 m m h d w s m h d w original shuffled inter-event times Figure 1: Top: temporal path entropy as a function of causal temporal scales in datasets Synthetic- 2, WB-DE, and HC-email (transparent red) and in the temporal networks with shuffled timestamps (transparent blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The height of a bar represents temporal path entropy (error bars represent the estimation error) and the x-limits of a bar represent the interval τ = [τmin, τmax] on which the temporal path entropy was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We indicate on x-axis the timescales of one minute (m), hour (h), day (d), week (w), and year (y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We observe that the temporal path entropy differs more between the original and the shuffled network at causal timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Bottom: histogram of causal inter-event times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In practice, the temporal path entropy can be estimated from the counts of time-respecting paths Pk τ (Γ) by assuming multinomial distributions with respective probabilities p(v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' , vk−1) and p(v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' , vk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The counts of time-respecting paths can be computed e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' using the methods from [Kivel¨a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2018, Petrovi´c and Scholtes, 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The estimation of the entropy can be challenging especially for small ranges of timescales, since the temporal network can get temporally disconnected, resulting in very few paths of order k being observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' As a result we require an efficient method for estimating the entropy that performs well even in such under-sampled regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The simplest estima- tor of a multinomial distribution, called the plug-in estimator, is based on the maximum likelihood estimation, which, however, is known to severely underestimate the entropy in the undersampled regime and has various corrections [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Miller, 1955, Grassberger, 2003]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' An alternative to the plug-in estimator is to follow a Bayesian approach which results in entropy estimators that strongly depend on the choice of prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' To counteract this dependency, the NSB estimator [Nemenman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2001] directly infers the entropy from the counts by averaging over different priors for the transition probabilities, rather than inferring the transition probabilities themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Being a Bayesian method, the NSB estimator can also be used to quantify the uncertainty of the estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Assuming that the estimates of H(v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' , vk) and H(v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' , vk−1) have independent errors σk and σk−1, we can approximate the total error of the estimate as σ = (σ2 k + σ2 k−1)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' As the NSB estimator requires the size of the alphabet to be known, it is most suitable for cases where the number of nodes is fixed and improves further if the set of edges that can occur are known a priory as this further restricts the number of potential paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In cases when the number of nodes in the system is unknown, the 4 0 2 EU-email-A 100 101 102 103 104 105 106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='5 DNC-16 s m h d w 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 time [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 H[ nat ] original shuffled Figure 2: Temporal path entropy as a function of the timescale τ in EU-email-A and DNC-16 and in timestamp shuffled networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Timescale τ is represented with the x-limits of the bar, and temporal path entropy is represented as the height of the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Error bars indicate the error of the temporal path entropy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Pitman-Yor Mixture entropy estimator [Archer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2014] could be used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Finally, we address testing whether an interval τ is a causal timescale of a temporal network Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' To do so, we need to assume the null hypothesis that there are no temporal correlations between temporal edges, but the main issue is to obtain a sample of temporal networks under this assump- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' To resolve this issue, we repeatedly randomize the observed temporal network Γ by randomly permuting timestamps between its temporal edges [Holme and Saram¨aki, 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' These samples of temporal networks would preserve both the edge frequencies and timestamp distribution while de- stroying the correlations between temporal edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We can use the samples to determine whether temporal path entropy of the observed network has an unexpected value under the null assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 4 Experiments In the following part, we first show the behavior of temporal path entropy in synthetically generated temporal networks with known causal timescales (the description of the generation process can be found in the Appendix);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' we then present how it behaves in two real-world networks with the information about the ground truth timescales, and two real world networks without the information about the ground truth timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In the top left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 1, we present the temporal path entropy H (y-axis) for various timescales (x-axis): the left and right x limit of a bar represents τmin and τmax, and the height of the bar represents H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The results are shown both for the synthetic network and its shuffled network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In the bottom left panel, we show the histogram of inter-event times on causal paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We observe that the temporal path entropy behaves as expected and decreases in accordance with the planted timescale at which the interactions cause one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Moreover, this pattern disappears when the timestamps of edges are shuffled, demonstrating that temporal path entropy captures the interplay of temporal and topological patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We consider here two empirical temporal networks where we have information about the ground truth causal structure and two empirical temporal networks where we have no information about 5 the ground truth causal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' As a first data set, we consider the bipartite temporal network of German Wikibooks co-editing patterns (WB-DE) [Wikimedia Foundation, Peixoto, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' This data contains information about edits on the Wikibooks website: for each edit, we know the editor, the article that was edited, and the time at which the edit occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We preprocess this data to obtain a temporal network of editors: if editor v edited an article prior to editor w who edited the same article at time t, we assume that a link (v, w, t) occurred in the temporal network of editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We define causal inter-event times based on the articles: we extract the time intervals between successive edits of each article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In WB-DE data, we analyze the timescales of the whole temporal network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' As a second data set, we consider public data set of Hillary Clinton’s emails (HC-email) [Kaggle, 2022], which contains the sender, the receiver, the timestamp, and the subject of each email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In this data set we analyze the timescales of node representing Hillary Clinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' While sender, receiver and the timestamp constitute a temporal network, email subjects allow us to obtain causal inter-event times: for each incoming email, we extract the time duration until an email with the same subject was sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We use the inter-event times between emails with the same subject and the inter-event times of articles for evaluation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' the temporal networks contain only the temporal edges and not any additional information about the ground truth timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We also use two email data sets without a ground truth timescales: EU-email-A [Paranjape et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2017], which contains email correspondence between researchers of an EU institution from four deparments, and DNC-16 [Rossi and Ahmed, 2015], which contains emails of the US Democratic National Committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Results on other datasets as well as details of all datasets are in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Reproducibility package is available at [Petrovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2023].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Results of the WB-DE and HC-email data are in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 1 (middle and right, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' When we compare the histogram of causal inter-event times with the temporal path entropy at different timescales of the temporal network, we see that increased number of causal interactions increases the difference in temporal path entropy between the original and the shuffled network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The temporal path entropy converges for large timescales because the interval sizes increase, the density of causal interactions decreases, and the noise increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 2, although we do not have the ground truth, we see that the largest difference between the original and the shuffled datasets are at timescales between a minute a day, which is what we would expect from email correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We identify four limitations of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' First, our base assumption is that the interactions, represented by edges, cause one another, and our measure can not separate that case that from the case when edges are generated by some common factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Second, being based on directed paths the current method is restricted in the types of causal interactions it considers namely interactions where a incoming link into a vertex effects the subsequent links emanating from the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The method could potentially be generalized to other types of interactions by considering other patterns to alleviate this shortcoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Third, our method cannot detect timescales at which the incoming edges to a node change the overall activity of the node without changing the relative frequencies of the outgoing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Detecting timescales of such causal influences is thus an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Fourth, real data can contain time-varying timescales, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' during day or night, which would probably require an application of time warping techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 5 Conclusion To summarize, the analysis of temporal networks heavily depends on the analysis of time-respecting paths [Holme and Saram¨aki, 2012, Holme, 2015, Pan and Saram¨aki, 2011, Masuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2013, Scholtes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2016, Kivel¨a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' However, in order to model and analyze the time-respecting paths, we first need to identify the correct timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In this work we address this problem by 6 introducing an information theoretic measure, the temporal path entropy, that is able to can identify timescales at which the influences are highly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Using real world data we demonstrated that the measure can be applied to temporal networks as a whole as well as to a single node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We showed that the temporal path entropy can capture the causal timescales in both synthetic and empirical temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We further support our findings by observing that the differences in the temporal path entropy between the original and shuffled networks coincide with increases in the number of causal paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The temporal path entropy allows system-relevant timescales to be inferred from the temporal networks themselves which is crucial for the analysis of temporal networks where inherent timescales are unavailable and hard to measure.' metadata={'source': 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+page_content=' Pinton, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Khanafer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' R´egis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Kim, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Comte, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Voirin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Estimating potential infection transmission routes in hospital wards using wearable proximity sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' PloS one, 8(9):e73970, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Wikimedia Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Wikimedia downloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' URL http://dumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='wikimedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Williams, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Lacasa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Mill´an, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Latora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The shape of memory in temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Nature communications, 13(1):1–8, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Zanetti, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Scholtes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Tessone, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Schweitzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Categorizing bugs with social networks: a case study on four open source software communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In 2013 35th International Conference on Software Engineering (ICSE), pages 1032–1041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' IEEE, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 6 Datasets In this work we considered synthetic and empirical temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' To generate synthetic temporal networks Synthetic-1, Synthetic-2 and Synthetic-3 with a ground truth timescale ¯τ = [¯τmin, ¯τmax], we start from a static Erd˝os-R´enyi random graph with 50 nodes and 500 directed edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We sample a random subset Pcausal of nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' = 500 unique paths of length k = 2 in the static network which correspond to causal influences in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We sample with repetition np = 5000 paths from Pcausal to generate dataset Synthetic-1, np = 10000 paths to generate dataset Synthetic-2 and np = 20000 paths to generate dataset Synthetic-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' To add each path (v0, v1, v2) to the temporal network, we sample a random starting time t uniformly from [0, Ttotal − ¯τmax] and create a temporal edge (v0, v1, t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' we then sample temporal distance δ between edges on the path (inter-event time) uniformly from ¯τ and create the temporal edge (v1, v2, t + δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We choose ¯τ with ¯τmin = 100 and ¯τmax = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' To add some noise to the system, we uniformly sample 20000 edges from the static graph, and sample their timestamps uniformly from [0, Ttotal].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The temporal network Synthetic-4 contains two time-scales relevant for the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' To do so, we generated two different temporal networks based on two random graphs of 50 nodes (with the same node names) and 500 edges and based on the different timescales τ 1 = [50, 100] and τ 2 = [150, 200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We used the same procedure as above with parameters nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' = 500;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' np = 5000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Ttotal = 105;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' = 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We merged the two temporal networks into one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' the details of the resulting network are in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The dataset Synthetic-5 contains paths of length three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Again, there are 50 nodes and 500 edges in the static Erd˝os R´enyi graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We sample nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' = 20 unique paths, we sample np = 20000 of them, and spread them across Ttotal = 105 using the same procedure and timescale τ = [100, 200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We add nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='e = 10000 random edges to the network as noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We also use empirical dataset where can get access to the ground truth causal path structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We consider the bipartite temporal network of Wikibooks co-edits in Arabic (WB-AR), French (WB- FR) and German (WB-DE) [Wikimedia Foundation, Peixoto, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' This data contains information about edits on the Wikibooks website: for each edit, we know the editor, the article that was edited, and the time at which the edit occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We preprocess this data to obtain a temporal network of editors: if editor v edited an article prior to editor w who edited the same article at time t, we assume that a link (v, w, t) occurred in the temporal network of editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We define causal inter-event times based on the articles: we extract the time intervals between successive edits of each article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In these data, we analyze the timescales of the whole temporal network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Another dataset where we 10 dataset |V | |E| |E| Ttotal [s] Ants-1-1 89 947 1911 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='44e+03 Ants-1-2 72 862 1820 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='75e+03 Ants-2-1 71 636 975 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='44e+03 Ants-2-2 69 769 1917 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='8e+03 Ants-3-1 11 37 78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='13e+03 Ants-3-2 6 21 104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='42e+03 DNC-16 1891 5598 39264 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='49e+07 EU-email-1 309 3031 61046 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='94e+07 EU-email-2 162 1772 46772 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='94e+07 EU-email-3 89 1506 12216 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='93e+07 EU-email-4 142 1375 48141 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='94e+07 EU-email-A 986 24929 332334 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='95e+07 Gallery 10972 89034 831824 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='95e+06 HC-email 326 385 8313 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='19e+08 Hospital 75 2278 64848 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='48e+05 Hypertext 113 4392 41636 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='12e+05 OSS 5789 6888 12583 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='54e+08 Primary 242 16634 251546 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='17e+05 School-13 327 11636 377016 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='64e+05 Synthetic-1 50 500 30000 1e+05 Synthetic-2 50 500 40000 1e+05 Synthetic-3 50 500 60000 1e+05 Synthetic-4 50 898 40000 1e+05 Synthetic-5 50 500 50000 1e+05 WB-AR 1124 3334 27166 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='89e+08 WB-DE 10999 54700 464089 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='87e+08 WB-FR 9735 53606 362094 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='88e+08 Work-13 92 1510 19654 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='88e+05 Table 1: The sizes of the sets of nodes V , unique edges E, and temporal edges E of temporal networks that we analyzed in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Datasets synth-2, HC email and WB DE are in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The other datasets are shown in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' can get access to the ground truth causal structure is the public data set of Hillary Clinton’s emails (HC-email) [Kaggle, 2022], which contains the sender, the receiver, the timestamp, and the subject of each email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In this data set we analyze the timescales of node representing Hillary Clinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' While sender, receiver and the timestamp form a temporal network, email subjects allow us to obtain causal inter-event times: for each incoming email, we extract the time duration until an email with the same subject was sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We use the inter-event times between emails with the same subject and the inter-event times of articles for evaluation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' the temporal networks contain only the temporal edges and not any additional information about the ground truth timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' The details of each data-set are in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Finally, we also use empirical temporal networks where we do not know the ground truth causal path structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Datasets Ants-1-1, Ants-1-2, Ants-2-1, Ants-2-2, Ants-3-1, and Ants-3-3 [Blonder and Dornhaus, 2011] contain antenna contacts in ant colonies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Dataset DNC-16 [Rossi and Ahmed, 2015] contains emails of the US Democratic National Committee leaked in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Datasets EU- 11 email-1, EU-email-2, EU-email-3, EU-email-4, and EU-email-A [Paranjape et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2017] contain email correspondence between researchers of an EU institution from first, second, third, fourth and all departments, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Datasets Gallery [Isella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2011], Hospital [Vanhems et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2013], Hypertext [Isella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2011], Primary [Gemmetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2014, Stehl´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2011], Work-13 [G´enois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2015] and School-13 [Mastrandrea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2015] contain human face-to-face interactions in different settings measured by the SocioPatterns collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Dataset OSS [Zanetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=', 2013] contains ASSIGN relationships between members of the Open Source Software community Apache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 7 Results: Synthetic Data We present results for datasets Synthetic-1, Synthetic-3, Synthetic-4, Synthetic-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 H[ nat ] Synthetic-1 50 100 150 200 250 time [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='4 counts [103] original shuffled inter-event times Figure 3: Top: temporal path entropy as a function of the timescale τ in temporal network Synthetic- 1 and in Synthetic-1 with shuffled timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Timescale τ is represented with the x-limits of the bar, and temporal path entropy is represented as the height of the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Error bars indicate the error of the temporal path entropy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Bottom: histogram of inter-event times of synthetic causal interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 8 Results: Empirical Data with Ground Truth In this section we show results on other Wikibooks datasets that we used to test the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 7, we test temporal path entropy on the WB-AR dataset, and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 8, we test it on the WB-FR dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Similar to the WB-DE in the main paper, the bottom panel shows the yellow histogram of inter-event times of edits per article for all articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 9 Empirical data without the ground truth In this section, we show multiple datasets in which we do not have access to the ground truth temporal scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Although the lack of ground truth in these datasets makes objective evaluation 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 H[ nat ] Synthetic-3 50 100 150 200 250 time [s] 0 1 2 counts [103] original shuffled inter-event times Figure 4: Equivalent of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 3, for Synthetic-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 2 3 H[ nat ] Synthetic-4 50 100 150 200 250 time [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 counts [103] original shuffled inter-event times Figure 5: Equivalent of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 3, for Synthetic-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' of the method difficult, the results across datasets are consistent and in accordance with what one would expect: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' in the email datasets, temporal path entropy is different between the original and the shuffled network for timescales between one minute and a few days, which corresponds to what we would expect to be the interval in which emails are responded to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='2 H[ nat ] k = 2 Synthetic-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='25 H[ nat ] k = 3 50 75 100 125 150 175 200 225 250 time [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='5 counts [103] original shuffled inter-event times Figure 6: Temporal path entropy as a function of the timescale τ in temporal network Synthetic-5 and in Synthetic-5 with shuffled timestamps for orders k = 2 (top) and k = 3 (middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Timescale τ is represented with the x-limits of the bar, and temporal path entropy is represented as the height of the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Error bars indicate the error of the temporal path entropy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Bottom: histogram of inter-event times of synthetic causal interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 H[ nat ] WB-AR 100 102 104 106 time [s] 0 2 counts [103] s m h d w original shuffled inter-event times Figure 7: Top: temporal path entropy as a function of the timescale τ in WB-AR temporal network and of WB-AR temporal network with shuffled timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Timescale τ is represented with the x-limits of the bar, and temporal path entropy is represented as the height of the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Error bars indicate the error of the temporal path entropy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Bottom: histogram of inter-event times for all articles of edits of the same article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 H[ nat ] WB-FR 100 102 104 106 time [s] 0 20 counts [103] s m h d w original shuffled inter-event times Figure 8: Equivalent of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 7 for WB-FR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 15 100 101 102 103 time [s] 0 1 2 3 4 5 H[ nat ] Ants-1-1 original shuffled s m 100 101 102 103 time [s] 0 1 2 3 4 H[ nat ] Ants-1-2 original shuffled s m 100 101 102 103 time [s] 0 1 2 3 4 H[ nat ] Ants-2-1 original shuffled s m 100 101 102 103 time [s] 0 1 2 3 H[ nat ] Ants-2-2 original shuffled s m 100 101 102 103 time [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 H[ nat ] Ants-3-1 original shuffled s m 100 101 102 103 time [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 H[ nat ] Ants-3-2 original shuffled s m Figure 9: Temporal path entropy as a function of the timescale τ in temporal networks of antenna contacts in ant collonies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' For each temporal network, we show the temporal path entropy of the original and of a shuffled network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Timescale τ is represented with the x-limits of the bar, and temporal path entropy is represented as the height of the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Error bars indicate the error of the temporal path entropy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 16 100 101 102 103 104 105 106 time [s] 0 1 2 3 4 H[ nat ] DNC-16 original shuffled s m h d w 100 101 102 103 104 105 106 time [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='5 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='H[ nat ] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='Primary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='original ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='H[ nat ] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='Work-13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='original ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='shuffled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='Figure 11: Temporal path entropy as a function of the timescale τ in temporal networks of human ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='face-to-face interactions measured by the SocioPatterns collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' For each temporal network, we show the temporal path entropy of the original and of a shuffled network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Timescale τ is represented with the x-limits of the bar, and temporal path entropy is represented as the height of the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Error bars indicate the error of the temporal path entropy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 18 100 101 102 103 104 105 106 time [s] 0 1 2 3 4 H[ nat ] OSS original shuffled s m h d w Figure 12: Temporal path entropy as a function of the timescale τ in temporal networks ASSIGN relationships between members of the Open Source Software community Apache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' We show the temporal path entropy of the original and of a shuffled network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Timescale τ is represented with the x-limits of the bar, and temporal path entropy is represented as the height of the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Error bars indicate the error of the temporal path entropy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' 19 10 Conditional entropy: The chain rule For discrete random variables X and Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' the definition of the entropy (in nats) is H(X) = − � x p(X = x) ln p(X = x) and the definition of conditional entropy (in nats) H(Y |X) is: H(Y |X) = − � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='y p(X = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Y = y) ln p(X = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Y = y) p(X = x) In the following,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' we use the above definitions to derive the chain rule of conditional entropy: H(Y |X) = − � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='y p(X = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Y = y) (ln p(X = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Y = y) − ln p(X = x)) = = − � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='y p(X = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Y = y) ln p(X = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Y = y) − � − � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='y p(X = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Y = y) ln(p(X = x))) � = = H(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Y ) − � − � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content='y p(Y = y|X = x)p(X = x) ln(p(X = x))) � = = H(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Y ) − � �− � x p(X = x) ln(p(X = x))) \x18\x18\x18\x18\x18\x18\x18\x18\x18\x18\x18 :1 �� y p(Y = y|X = x) � � � = = H(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' Y ) − H(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} +page_content=' (5) 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfuy1Y/content/2301.11623v1.pdf'} diff --git a/HtE1T4oBgHgl3EQfrgVp/content/tmp_files/2301.03355v1.pdf.txt b/HtE1T4oBgHgl3EQfrgVp/content/tmp_files/2301.03355v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d00c404fd295bc52e0a1cfc69de28b427f5b25bd --- /dev/null +++ b/HtE1T4oBgHgl3EQfrgVp/content/tmp_files/2301.03355v1.pdf.txt @@ -0,0 +1,1489 @@ +Charge transfer mediated giant photo-amplification in air-stable α-CsPbI3 +nanocrystals decorated 2D-WS2 photo-FET with asymmetric contacts +Shreyasi Das1, Arup Ghorai1,2, Sourabh Pal3, Somnath Mahato1, Soumen Das4, Samit K. Ray5 * +1School of Nano Science and Technology, IIT Kharagpur, Kharagpur, West Bengal, India, 721302 +2Department of Materials Science and Engineering, Pohang University of Science and Technology, +Pohang 790-784, Korea +3Advanced Technology Development Centre, IIT Kharagpur, Kharagpur, West Bengal, India, 721302 +4School of Medical Science and Technology, IIT Kharagpur, Kharagpur, West Bengal, India, 721302 +5Department of Physics, IIT Kharagpur, Kharagpur, West Bengal, India, 721302 +Email : physkr@phy.iitkgp.ac.in + +Abstract +Hybrid heterostructure based phototransistors are attractive owing to their high gain induced +by photogating effect. However, the absence of an in-plane built-in electric field in the single +channel layer transistor results in a relatively higher dark current and require a large operating +gate voltage of the device. Here, we report novel air-stable cesium lead iodide/tungsten di- +sulfide (CsPbI3/WS2) mixed dimensional heterostructure based photo-field-effect-transistors +(photo-FETs) with asymmetric metal electrodes (Cr/WS2/Au), exhibiting extremely low dark +current (~10-12 A) with a responsivity of ~ 102 A/W at zero gate bias. The Schottky barrier +(WS2/Au interface) induced rectification characteristics in the channel accompanied by the +excellent photogating effect from solution-processed α-phase CsPbI3 NCs sensitizers, resulting +in gate-tunable broadband photodetection with a very high responsivity (~104 A/W) and +excellent sensitivity (~106). Most interestingly, the device shows superior performance even +under high humidity (50-65%) conditions owing to the formation of cubic α-phase CsPbI3 +nanocrystals with a relatively smaller lattice constant (a = 6.2315 Å) and filling of surface +vacancies (Pb2+ centres) with the sulfur atoms from WS2 layer, thus protecting it from +environmental degradation. These results emphasise a novel strategy for developing mixed +dimensional hybrid heterostructure based phototransistors for futuristic integrated nano- +optoelectronic systems. +Keywords: Two dimensional TMDs, Inorganic perovskites, Sensitizers, Asymmetric +electrodes, Mixed dimensional phototransistors + + + +Introduction +Fabrication of high performance phototransistors demands superior channel material, which +should be of high carrier mobility for high gain bandwidth product, a direct bandgap for +efficient optical absorption, a thinner layer for full depletion leading to ultralow dark current +and very low trap state density for low subthreshold swings. Layered semiconducting two +dimensional (2D) TMDs fulfil most of these requirements [1–4] making them a potential +candidate for photo-field-effect transistor (photo-FET). However, some trade-offs are found in +single semiconductor channel based phototransistors, due to the simultaneous occurrence of +both absorption and amplification processes within the same layer, compromising the device +performance [5–7]. To overcome this shortcoming as well as for fabrication simplicity, +attention has been paid on few-layer TMDs over monolayer with a wider spectral +photoresponse and relatively higher absorbance [8–10], which in turn increases the dark current +values requiring a higher gate voltage to operate the device in full depletion mode [11]. This +issue can be addressed via incorporating a Schottky barrier by judiciously selecting the metal +contacts and utilizing the developed depletion region to facilitate an unidirectional current +transport, which leads to the significant lowering of dark current in few layered TMD based +photo-FETs at zero gate bias [12–14]. The built-in electric field of the Schottky interface +further helps in the separation of photogenerated charge carriers across the channel layer [15] +leading to zero gate bias driven enhanced photosensitivity, although the responsivity of these +devices are limited owing to the lower absorption coefficient of 2D TMD material. For the +further improvement of photoresponsivity, recent works are focused on sensitizing the thin +channel material with semiconductor nanocrystals (NCs), referred to as sensitizers, having +excellent absorption characteristics and opposite doping polarity to create a vertical junction +for subsequent charge separation [16–19]. + +Recently, all-inorganic cesium lead halide (CsPbX3: X = I, Br, and Cl) perovskite NCs have +drawn tremendous interests in the field of optoelectronics [20], featuring superior emissive +(approaching photoluminescence quantum yield ~ 100%) [21] characteristics, extremely high +absorption coefficient [22], large carrier diffusion lengths (>3 mm), fast radiative +recombination rates and most importantly their improved stability [23] over organic-inorganic +hybrid perovskites [24,25]. Especially colloidal synthesised zero-dimensional (0D) α-CsPbI3 +NCs exhibit extended photoabsorption band covering the whole visible spectrum with high +absorption coefficient (~ 3 × 104 cm-1 at 680 nm) [26] and better photostability, making them +attractive as a photoactive material for high performance optoelectronic devices under ambient +condition. Weerd et al. have recently reported colloidal CsPbI3 NCs with high quantum yield +of ~ 98% prepared by hot-injection method which reveals highly efficient carrier multiplication +as well as longer build-up of free carrier concentration [27]. Among four existing phases (α- +cubic, β-tetragonal, γ-orthorhombic, and δ-orthorhombic) of CsPbI3 NCs, cubic α-phase has +high stability due to its low surface-to-volume ratio (lattice constant a = 6.23 Å) without any +octahedral inclination or lattice distortion in the [PbI6]4− octahedra as well as in the unit +cell [28–30]. These extraordinary properties of α-phase CsPbI3 NCs provoke to utilize them as +an effective sensitizer in 2D channel based hybrid phototransistor devices. However, owing to +the vulnerability towards moisture for halide perovskite NCs, recently, the surface passivation +of CsPbI3 via coordination of Pb2+ centres with sulphur donors has been explored to protect +them from environmental degradation [31] to improve the device stability under ambient +conditions. +In this work, we present a proof-of-concept for 0D/2D CsPbI3/WS2 based mixed dimensional +van der Waals heterostructure (MvWH) photo-FETs utilizing the superior photoabsorption +attributes of α-phase CsPbI3 NCs as sensitizers, along with a sub 5-nm thick 2D WS2 channel +which acts as an expressway for carrier transport. In our device configuration, the developed + +built-in electrical field at the heterostructure interface facilitates efficient transfer of +photogenerated carriers from CsPbI3 NCs into the WS2 layer resulting in an excellent +broadband visible photoabsorption in the hybrid system. Moreover, asymmetric metal contacts +(Au-Cr) as source and drain electrodes are purposefully chosen to utilize the large built-in +potential along the WS2/Au Schottky junction for a diode-like unidirectional current flow and +effective separation of photogenerated electron-hole pairs, leading to an excellent rectifying +ratio of ~ 104 with a very low dark current of the order of ~ pA under a low source-to-drain +bias (VDS) without any applied back gate voltage. Fabricated hybrid phototransistor devices +exhibit a very high photoresponsivity of ~ 104 A/W at ~ 40 V gate bias upon visible light +illumination (~ 0.1 µW) due to the photogating effect and a broad spectral bandwidth across +the entire visible range. In addition, the coordination of sulphur atoms of WS2 layer with Pb2+ +centres of CsPbI3 NCs reveals an excellent device stability under 50-65% humidity condition. +Our results not only open up new avenues for studying fundamental carrier transport and +relaxation pathways in hybrid van der Waals heterojunctions, but also pave the way for +constructing high-performance optoelectronic devices using mixed-dimensional 0D/2D hybrid +building blocks, rather than using purely 2D layered materials. +Experimental section: +Synthesis of α-phase CsPbI3 NCs: +This is a two-step reaction, where, in the first step Caesium oleate (Cs-OA) was prepared +followed by the synthesis of α-phase CsPbI3 NCs in the second step. +i) Firstly, caesium carbonate (Cs2CO3) of 814 mg and 40 ml Olive oil were added in a round +bottom two-neck flask, which was heated at 120⁰C for 1 hr under vacuum condition. +Followed by the rise in temperature to 150⁰C under N2 atmosphere for 10-15 mins, the +desired transparent solution of Cs-OA was stored for further use to synthesize CsPbI3. +ii) Next, 870 mg of PbI2 and 5 ml of olive oil (instead of 1-octadecene (ODE)) were mixed +which was heated to 120⁰C under vacuum for 1hr. Thereafter, we swiftly injected 1 ml of + +Oleyl amine (OLAm) in the reaction mixture under N2 atmosphere to get a transparent +solution. Finally, preheated (100⁰C) Cs-OA was injected to the reaction mixture and cooled +immediately in an ice bath to quench the reaction to obtain desired α-phase CsPbI3 NCs. +Finally, as-synthesized CsPbI3 was purified through centrifugation using excess hexane. The +centrifugation process (20000 rpm) was repeated for several times to remove excess +OLAm/olive oil from the product. Finally, the sedimentation was collected and re-dispersed in +hexane. The collected dispersion was stored in a sealed vial for further characterisations and +device fabrications. +CsPbI3/WS2 MvWH photo-FET device fabrication: +For the fabrication of the CsPbI3/WS2 MvWH photo-FET device, WS2 flakes were +mechanically exfoliated from the bulk WS2 crystal (2D semiconductor Inc. Scottsdale, AZ, +USA) using a Scotch tape (3M Inc. USA) on polydimethylsiloxane (PDMS) gel film (Gel-Pak +Inc. Hayward, CA, USA) and the interested few layer flakes were identified under optical +microscope, followed by the layer confirmation via Raman characteristics and AFM height +profile. Note that, mechanically exfoliated flakes were chosen for their high quality and clean +interface promising greater mobility and high performance device fabrication. Further, for the +Au contact with WS2, electrodes were patterned in advance on a pre-patterned SiO2/Si (285 nm +oxide thickness) substrate using e- beam lithography technique and then Cr (5 nm)/Au (30 nm) +were deposited via e-beam evaporation followed by lift-off with acetone. Then, the selected +few layer flakes were deterministically transferred on the targeted Au electrodes from PDMS +gel film following the dry transfer technique to make sure residue free clean transfer. After the +successful transfer of the flakes on Au electrodes, again electrodes were patterned using second +step e-beam lithography followed by metal deposition Cr (5 nm)/Au (30 nm) for Cr contacts +on WS2. To remove the resist residue and improve the contact conductance, the fabricated +devices were annealed at 150°C for 2 hrs in a high vacuum of ~ 10−3 Torr. Finally, the +synthesized diluted solution (0.1 mg mL−1) of perovskite NCs was uniformly spin-coated +several times (varying from one to four) onto WS2 layer with a speed of 2000 rpm. + +Characterisations and measurements: +X-ray diffraction (XRD, Philips MRD X-ray diffractometer) patterns were recorded using +characteristic Cu-Kα (λ = 1.5418 Å) radiation with 2.0° grazing incidence angle. For the +Transmission electron microscopy (TEM) sample preparation, CsPbI3 NCs solution was +dissolved in Hexane and then placed into a TEM grid and dried it for few minutes and did the +measurement. The TEM images were carried out using TECNAI G2 TF20-ST and JEM-2100F +Field Emission Electron Microscope operating at 200 kV equipped with Gaytan’s latest CMOS +camera. All the images were proceeding by Digital Micrograph Software for the estimation of +d-Spacing & Indexing. UV–vis–NIR absorption spectra of as synthesised CsPbI3 samples were +recorded using a fiber probe-based UV–vis–NIR spectrophotometer (Model: U-2910 +Spectrophotometer, HITACHI) and a broadband light source. Raman and PL spectra were +recorded using a semiconductor laser of excitation wavelength 532 nm, equipped with a CCD +detector, an optical microscope of 100x objective lens and a spectrometer (WITec alpha-300R). +The photogenerated carrier lifetime was measured by exciting the material with a pulsed diode +laser of wavelength 372 nm and detecting the signal using Edinburgh LifeSpec-II fluorescence +lifetime spectrometer fitted with a PMT detector. Room-temperature current−voltage +characteristics were recorded using a Keithly semiconductor parameter analyzer (4200 SCS) +in the presence of an Argon laser (514 nm) and a broadband solar simulator (AM 1.5, 100 +mW/cm2) as a visible light source. +Results and discussion + + +FIG. 1. (a) Rietveld refinements (α-phase fitting) of the XRD pattern of a film of cubic CsPbI3 +NCs. (b) 1×1 3D VESTA visualization image of α-CsPbI3 cubic crystal structure. (c) Typical +HRTEM image of CsPbI3 NCs with an energy 200 keV revealing cubic morphology. (d) FFT +patterns from a region marked by the red dotted square on the micrograph (c). (e) A magnified +view of the corresponding HRTEM image in the selected yellow square region on micrograph +(c). (f) SAED pattern of α-CsPbI3 NCs showing well defined diffraction spots indexed as (200), +(220) and (020) planes viewed along [004] zone axis. + +To study the crystal structure of as-synthesized CsPbI3 NCs, we have recorded X-ray +diffraction pattern, followed by their fitting with Rietveld refinement full proof software, which +are presented in Fig. 1(a). An excellent agreement with fitted results indicates the growth of +single-phase (α-phase) cubic CsPbI3. The crystal structure of CsPbI3 NCs visualized using +VESTA 3D software through Rietveld fitting is shown in Fig. 1(b). The VESTA 3D (1x1) +structure shows the absence of any octahedral inclination in the perovskite NCs, which is +known to be beneficial for achieving higher stability under laboratory ambient (45-50% +humidity). Typical high resolution transmission electron microscopy (HRTEM) image reveals +almost cubic shape of the synthesised NCs (15.05 nm × 18.04 nm), as shown in Fig. 1(c). +Corresponding first Fourier transform (FFT) pattern presented in Fig. 1(d) of the red squared + +(a) +C +(200) +& +Observed +(100) +米0 +Calculated +15.05nm +Intensity (arb. units) +(210) +Difference +d= 0.62 nm +Braggposition +11001 +d= 0.62 nm +[100] +10nm +10 +20 +20 (degree) +30 +40 +50 +(b) +(d) +ZA[002] +a-CsPbl3 +f) +ZA [004] +a-CsPbl +(220) +(210) +(200) +(200) +(200) +(210) +(220) +(220) ( () +2 nm +2 1/nm +(020) +Cs +Pbportion of the Fig. 1(c), shows pure cubic α-phase pattern along the zone axis [002]. Whereas, +the high-resolution fringe pattern from the yellow squared region of Fig. 1(c) shows a d-spacing +of 0.62 nm, which is in well matched with the cubic α-phase of CsPbI3 [30], as shown in Fig. +1(e). The result indicates (100) directional growth of cubic phase CsPbI3 NCs, which is in well +agreement with our previously reported results [30]. Corresponding selected area electron +diffraction (SAED) patterns shown in the Fig. 1(f), with indexed (200), (220) and (020) planes +along the zone axis [004], also corroborate the pure cubic structure of synthesized CsPbI3. +Figure 2(a) presents the optical absorption and emission properties of the as-synthesised α- +phase CsPbI3 NCs in the visible wavelength range with an absorption maxima at ~ 680 nm and +the corresponding bandgap value is ~ 1.814 eV [30], extracted from the Tauc plot shown in +Fig. S1 within the Supplimental Material. Further, the photoluminescence (PL) maxima at ~ +687 nm confirms the formation of excitons (Xα) across the direct bandgap (∼1.80 eV) of α- +phase CsPbI3 NCs represented via blue curve in Fig. 2(a). The Gaussian line shape of the PL +spectrum and the absence of any other PL peaks clearly dictate the synthesis of pure α-phase +CsPbI3 without presence of any mixed phase. To examine the charge transfer mechanism at the +CsPbI3 NCs/WS2 interface, Raman spectroscopy and micro-PL (µ-PL) measurements have +been carried out by spin-coating of a dilute solution of CsPbI3 NCs uniformly on the exfoliated +WS2 surface. For the room temperature µ-Raman-PL measurements, samples have been +excited with a CW laser having a wavelength of 532 nm with the laser power being kept at a +very low value to avoid any local heating induced sample degradation. Figure 2(b) represents +comparative Raman spectra of WS2 and mixed dimensional van der Waals heterostructure +(MvWH) samples, showing intense in-plane 2LA+E12g Raman modes at ∼ 351 cm–1 and out- +of-plane vibrational A1g peaks at ∼ 420 cm–1 [32]. The A1g vibrational Raman mode, which +preserves the symmetry of the lattice, is clearly red-shifted by ∼ 4.7 cm-1 in case of CsPbI3 +decorated WS2 layer [inset of Fig. 2(b)], revealing the interfacial charge transfer phenomena. + +The external electron doping in 2D WS2 leads to the filling-up of antibonding states of the +conduction band, mostly made up of d z2 orbitals of transition metal atoms [33]. This makes the +bonds weaker and the A1g peak of pristine WS2 is shifted towards a lower wavenumber on +significant electron doping from CsPbI3 NCs [34]. + +FIG. 2. (a) Absorption (Green) and photoluminescence (Blue) spectra of as-synthesised cubic +phase CsPbI3 NCs. (b) Comparative Raman spectra of WS2 flakes before and after CsPbI3 NCs +decoration showing characteristic E2g and A1g peaks of layered WS2. Inset shows the magnified +image of the out-of-plane A1g mode revealing a clear peak shift to lower wavenumbers due to +electron doping in WS2 flakes from CsPbI3 NCs. (c) Deconvoluted PL spectra of (i) a bare ML +WS2 flake and (ii-v) the heterostructure samples with varying number of spin coated layers of +CsPbI3 NCs on the WS2 flake. The spectra (ii), (iii), (iv) and (v) represent the PL emission from +the first, second, third and fourth spin coated layers of CsPbI3 NCs, respectively. The green +(red) peak represents the A excitonic (A- trionic) emission from ML WS2 flakes and the blue +peak represents the emission from band to band transition of cubic α-phase CsPbI3 NCs. (d) +Energy band diagram of CsPbI3/WS2 hybrid heterostructures showing effective electron doping +in WS2 from CsPbI3 sensitizers and hole trapping in the NCs giving rise to a strong photogating +effect. (e) Schematic representation of the charge transfer mechanism in 0D/2D CsPbI3/WS2 +hybrid heterostructures giving rise to trion formation in ML WS2. (f) Normalised time resolved +PL decay curves of CsPbI3 NCs (Blue curve) and CsPbI3/WS2 hybrids (Red curve), measured +using an excitation wavelength of 372 nm. + +On the other hand, the monolayer (ML) WS2 PL emission characteristics [Fig. 2c(i)] consist of +a strong A-excitonic emission at ∼ 1.995 eV, corresponding to the direct band-to-band +550 +600 +650 +700 +750 + + + + +PL intensity (arb. units) +Wavelength (nm) + + + + +(v) +(iv) +(iii) +(ii) + + +(i) +WS2 +A0 +160 +180 +200 +220 +240 +tav = 33.7 ns +PL (arb. units) +Time (ns) +CsPbI3 +CsPbI3/WS2 +tav = 40.2 ns +300 +400 +500 +410 420 430 + + + +4.7 cm-1 +400 +410 +420 +430 +440 + + + +4.7 cm-1 +CsPbI3/WS2 + + +Intensity (arb. units) +Raman shift (cm-1) +WS2 +Xα +A- +CsPbI3/WS2 +500 +600 +700 +800 +Wavelength (nm) +Nor. Abs./PL (arb. units) +Absorption +Photoluminescence +WS2 +CsPbI3 +hν +Electron +transfer +Hole +trapping +Charge +transfer +WS2 +Exciton +Trion +CsPbI3 +hν +Electrons +Holes +(d) +(c) +(a) +(f) +(b) +(e) +Increasing CsPbI3 coating layer + +transitions at the K (and/or K’) point of the Brillouin zone, and a weaker trionic A– emission at +∼ 1.965 eV with a binding energy of ∼ 30 meV, which are in good agreement with the +previously reported results [34]. It is to be noted that, ML WS2 is purposefully chosen for the +charge transfer study via PL measurements due to its extraordinary luminescence property at +room temperature owing to the direct bandgap transition. The existence of trions in the room +temperature emission spectrum indicates the unintentional doping in the un-passivated WS2 +flake from the substrate as well as the surrounding environment [35,36]. A systematic PL study +of the hybrid structure with increasing layer numbers of CsPbI3 NCs spin-coated over ML WS2 +shows a pronounced excitonic-PL quenching in MvWH as compared to both the pristine +materials (Fig. S2 within the Supplimental Material). Further, the CsPbI3/WS2 MvWHs show +relatively broad PL spectra with combined contributions from CsPbI3 NCs as well as ML WS2 +and the spectral shape changes with increasing CsPbI3 spin-coated layer numbers [Figs. 2c(ii- +v)]. To explore the effect of CsPbI3 coating over WS2, we have fitted each spectrum with three +Gaussian peaks containing the characteristics excitonic features of both the materials and +analysed their intensity variation with increasing concentration of CsPbI3 treated on WS2 +flakes, as shown in Fig. 2(c). It is noticed that the distinctive trion peak (A–) of WS2 becomes +prominent with increasing density (coating number) of CsPbI3 NCs and finally excitonic to +trionic (integrated intensity) crossover is observed above a critical concentration of CsPbI3 NCs +(Fig. S3 within the Supplimental Material). The possible explanation behind these observations +is as follows: upon illumination, photoexcited electron-hole pairs are generated in both WS2 +and CsPbI3, however, due to the type-II energy band alignment of the heterostructures, +electrons are easily transferred from CsPbI3 NCs to ML WS2, as illustrated in Fig. 2(d). On the +other hand, photogenerated holes remain trapped in the NCs, resulting in a reduced +recombination rate and giving rise to a quenched excitonic PL intensity for CsPbI3 NCs and +increased trionic emission in ML WS2. The enhanced generation rate of trions results in the + +reduced density of excitons inside the system, leading to the suppression of excitonic peak +intensity and the dominance of the trion peak in the PL spectra of MvWH, as schematically +depicted in Fig. 2(e). [34] To further confirm the charge transfer phenomena, time-resolved +photoluminescence (Tr-PL) spectra have been measured. Fig. 2(f) shows the Tr-PL decay +curves of CsPbI3 NCs (blue curve) and CsPbI3/WS2 MvWHs (red curve). The PL decay curves +have been fitted using a bi-exponential function to extract the average excitonic life time (τav). +The τav of CsPbI3 NCs decreases from 40.2 ns to 33.7 ns after hybridization with WS2, +corroborating the successful charge transfer mechanism from CsPbI3 NCs to WS2 flakes. As a +conclusion, the type-II band alignment in CsPbI3/WS2 heterostructure facilitates an efficient +electron–hole pair separation and strong electron doping into WS2 channel, making the hybrid +system ideal for fabrication of superior performance phototransistor devices [37]. + +FIG. 3. (a) Schematic 3D view of the fabricated back gated phototransistor comprising of +CsPbI3 sensitized WS2 channel with asymmetric electrodes (Au and Cr) acting as a source and +drain. An optical micrograph of the device is shown in the inset. (b) Linear IDS-VDS +characteristics of three fabricated devices with different source-drain contacts (i) Cr-Cr +(Yellow curve), (ii) Au-Au (Orange curve) and (iii) Au-Cr (Brown curve) without any back +gate bias. Inset: the corresponding semi-logarithmic IDS-VDS characteristics plots. (c) Transfer + +(a) +(b) +6 +10- +4 +10-11 +(vu) +2 +10-13 +T +Vps (M) +U +DS +-2 +Cr/Cr +Au +Au/Au +4 +Au/Cr +Watoms +Si02 +.6 +CsPbl, NCs +Satoms +-2 +-1 +0 +1 +2 +V +Ds (V) +(c) +(d) +10° +60 +20 +A0 +5V +10V +15V +(vu) +(vu) +15 +20V +'DS +10-11 +DS +10 +-20 +20 +40 +20 +WS +5 +CsPbI,/WS +4-0 +0 +-20 +0 +20 +40 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +VGs (V) +V +(V) +DS(IDS-VGS) characteristics of the CsPbI3/WS2 hybrid transistor (Green curve) and control WS2 +transistor (Blue curve) devices in linear scale and logarithmic scale (Inset). The current value +in the accumulation region decreases and the threshold voltage is shifted to a higher positive +voltage in hybrid device due to charge transfer through the CsPbI3/WS2 junction. (d) Output +characteristics of the hybrid device with varying gate voltage. + +The CsPbI3 NCs/WS2 MvWH photo-FET with asymmetric metal contacts has been +demonstrated by exploiting the Schottky barrier induced dark current suppression for the zero +gate bias driven photosensitivity of the device with a simpler fabrication technique. Figure +3(a) schematically demonstrates the as-fabricated phototransistor device structure consisting +of a few layer WS2 channel and asymmetric Cr and Au electrodes as source and drain terminals, +respectively. The inset shows the optical micrograph of the connected few layer WS2 flake +(5×2 µm2) having a thickness of ~ 4.5 nm corresponding to 4-5 atomic layers of WS2, further +corroborated by the AFM analysis (Fig. S4 within the Supplimental Material). The deposition +of a lower work function Cr (ΦCr = 4.5 eV) and higher work function Au (ΦAu = 5.1 eV) on n- +type WS2 as asymmetric contacts reveals room temperature rectifying diode characteristics +with rectification ratio up to 5.2 × 102 even at zero applied back gate bias, as depicted via the +current-voltage (IDS-VDS) characteristics in Fig. 3(b). Under an applied reverse drain voltage, +the potential barrier height between Au and WS2 becomes higher to suppress the current flow +through the junction compared to Cr, and thus exhibits the IDS-VDS characteristics of an ideal +diode [13]. The current through the Au–WS2-Cr device follows the diode behaviour and the +Schottky barrier (Φb) at Au-WS2 interface is capable of reducing the dark current significantly +making the device architecture an ideal prototype for operating in full depletion mode even at +zero gate bias, leading to a very high ON-OFF ratio of the device. On the other hand, a linear +IDS-VDS characteristics with a comparatively larger current value (100 nA) confirms the +formation of an Ohmic-like junction with a very low contact resistance for Cr-WS2-Cr +device [38]. Further, Au-WS2-Au system reveals a rectifying output characteristics with +relatively lower current than Cr contacts, confirming a typical high resistive back-to-back + +Schottky diode [15]. The energy band alignment with different metal contacts is schematically +depicted in Fig. S5 within the Supplimental Material, revealing an easy current flow through +the Ohmic Cr junction and restricted flow via built-in potential barrier in the Au Schottky +junction. Fig. 3(c) shows the transfer (IDS-VGS) characteristics of the WS2 phototransistor with +asymmetric Cr–Au contacts at a reverse drain voltage of -2V revealing excellent n-type channel +properties at room temperature with off-currents of the order of 10 pA and the transistor ON- +OFF ratio ~104. Further, to understand the effect of CsPbI3 treatment on the device +performance, the CsPbI3/WS2 hybrid transistor characteristics is compared with the pristine +WS2 one, referred to as the control device. The incorporation of sensitizing perovskite NCs on +2D-WS2 layer results in a junction formation via Fermi level alignment in equilibrium under +dark condition. In this process, the draining of electrons from the WS2 channel towards CsPbI3 +NCs results in the depletion of majority carriers in WS2 leading to the lowering of the current +flow in the channel under dark condition. Further, we have studied the output characteristics +of the hybrid phototransistor on application of gate voltage varying from 0 to +20 V, as depicted +in Fig. 3(d). For a higher positive gate voltage, more electrons are induced in the WS2 channel +and the transistor has a higher current in the saturation state. On the other hand, under a negative +gate bias a small amount of current flows through the channel owing to the depletion of carriers, +leading to the OFF state of the transistor [Fig. 3(c)]. +The performance of the CsPbI3/WS2 MvWH photo-FET has been analysed by recording the +room temperature IDS-VDS characteristics for zero gate bias under dark as well as visible +illumination using a Newport solar simulator having broadband emission with irradiance of +100 mW/cm2 under air mass (AM) 1.5G condition, as shown in Fig. 4(a). For comparison, the +characteristics of the pristine WS2 control device is also presented. The suppression of dark +current to the order of tens of pA even without any gate bias along with the significant reduction +of noise currents are attributed to the built-in electric field at the Au/WS2 Schottky junction, + +FIG. 4. (a) Comparative IDS-VDS characteristics of pristine WS2 and CsPbI3/WS2 MvWH +photo-FET under dark and illumination via broadband light source for zero gate bias. (b) +Spectral responsivity curves of MvWH photo-FET at VGS = 0V with increasing reverse VDS +from 0V to -2V, as shown via yellow, green and blue curves. The blue curve represents the +responsivity of the photo-FET at a maximum VDS of -2V, while the spectral responsivity of the +control device (orange curve) showing an order of magnitude lower device response at same +VDS. (c) COMSOL Multiphysics simulated E-field distribution at the vicinity of the hybrid +system upon excitation with an excitation wavelength of 680 nm. (d) The transfer +characteristics (IDS-VGS) of MvWH photo-FET for a range of incident powers (from 0.1 to 35 +μW) with an illumination of wavelength 514 nm at VDS = -2 V. (e) The variation of responsivity +of the device with incident illumination power for VGS varying from 0 to 40 V. (f) The shift in +threshold voltage (ΔVTh) with increasing illumination power (Pin) fitted with a power law. Blue +dots represent the extracted data points from panel (a) and green line represents the fitted curve. +Inset: The power law fit of the variation of photocurrent with incident power at VDS = -2V and +VGS = 40V showing a sublinear photocurrent dependency with incident optical power. + +which further helps in effective separation of photogenerated carriers created in WS2 channel. +On illuminating the heterojunction device, the reverse current tends to increase due to the +collection of photogenerated minority carriers (holes) at the electrodes. The photo-to-dark +current ratio of MvWH photo-FETs by illuminating with a broadband light source is estimated +to be much higher compared to the pristine WS2 one (~1000 times) under the applied reverse +bias condition, which reaches to a value of ~1.08×106 at VDS of -2V, as shown in Fig. S6 within +the Supplimental Material. The decoration of WS2 channel with superior light absorbing + +(a) +(b) +(c) +120 +(AW) +×102 +E2/E.? +6 +80 +10-8 +4 +MvWHlight +40 +MvWH dark +WS,light +2 +WS,dark +WS,1 +0 +-2 +-1 +0 +1 +2 +300400500600700800900 +Vps (V) +Wavelength(nm) +(d) +(e) +(f) +0.3 +-10 +35μW +(A/W) +- 40V +20V +200 +10-7 +104 +-15 +(nA) +40V +30V +-10V +150 +(μA) +0.2 ++ +-OV +20 +DS +100 +10-1 +0μw +Tocp0.48 +10 +50 +10-13 +AV +-25 +-20 +Vcs (M) +40 +0 +10 +20 +30 +30 +Pin (μW) +Resi +10 +-35 +ocp0.17 +0.0 +40 +-20 +20 +40 +0.1 +1 +10 +0 +7 +14 +0 +21 +28 +35 +(V)perovskite CsPbI3 NCs facilitates enhancement in the photocurrent by elevating the +photogenerated carriers in the channel via efficient charge transfer from CsPbI3 to WS2 due to +type-II energy band alignment [7,19]. This explains the significant enhancement (~103 times) +of response of MvWH transistor over the control one, revealing the role of photoabsorbing +CsPbI3 NCs in boosting the performance of the phototransistor. Further, the spectral +responsivity of the fabricated MvWH photo-FET, the most important figure of merit to evaluate +a detector performance, has been studied displaying a broadband spectral photoresponse +covering the entire visible wavelength range, as shown in Fig. 4(b). It may be noted that a peak +responsivity of ~1.05×102 A/W at ~ 460 nm at an applied bias (VDS) of -2 V is achieved, which +is close to the C-exciton absorption edge of WS2. Two other peaks at ~ 620 nm (R ~ 0.97×102 +A/W) and ~ 720 nm (R ~ 0.77×102 A/W) correspond to the direct bandgap absorption of few +layer WS2 and α-phase CsPbI3 NCs, respectively. Further, the spectral responsivity increases +with increasing reverse VDS that assists in the efficient extraction of photogenerated carriers. +On the other hand, the control WS2 based device also exhibits a similar trend with increasing +bias showing a maximum peak responsivity of ~ 10 A/W at ~ 460 nm at -2 V applied VDS (Fig. +S7 within the Supplimental Material). It is to be noted that the decoration of CsPbI3 NCs on +WS2 flakes not only improves the detector responsivity by more than 10-fold but also extends +the spectral responsivity window up to 800 nm, as shown comparatively in Fig. 4(b). Hence, +the decoration of WS2 active channel layer with excellent photoabsorbing CsPbI3 NCs appear +to be a promising approach for next generation high performance optoelectronic applications. +To further investigate the role of CsPbI3 in photocarrier generation and efficient charge +transfer, the electromagnetic simulations have been performed using the COMSOL +Multiphysics software. Figure 4(c) shows the electric field distribution of the hybrid +CsPbI3/WS2 device, illuminated with an electromagnetic plane wave of wavelength λ = 680 +nm from top, which propagates through air and the nanostructure. The distribution clearly + +depicts that the electric field is trapped along the edges of the nano-cubes of CsPbI3 with the +maximum confinement occurring near the base (as demonstrated by the colour index profile), +resulting in strong charge transport in CsPbI3/WS2 hybrid heterostructure. +Further to explore the impact of gate bias on transistor performance, the photo-induced transfer +characteristics (IDS−VGS) of the WS2/CsPbI3 MvWH photo-FET is recorded under the dark +(black markers) and 514 nm illumination with a range of optical powers (from 0.1 μW to 35 +μW) at a constant VDS of -2V, as illustrated in Fig. 4(d). The corresponding logarithmic current +representation is depicted in the inset. Under illumination of a fixed power of 35 μW, the drain +current of the MvWH photo-FET is enhanced by ~ 13 times (from ~ 20 nA to ~ 0.26 μA) at a +constant gate voltage of ~ 40V, manifested by the strong photoabsorption in CsPbI3 and +subsequent transfer of photoexcited electrons to the WS2 channel. Further, with the increase of +laser power, the photocurrent significantly increases in the accumulation region (i.e. VGS>VTh) +and the transfer curves are gradually shifted to a negative gate voltage. As illustrated in Fig. +2(d), the favourable energy band alignment rules out the possibility of hole injection from +CsPbI3 into WS2, leading to the trapped holes induced strong photogating effect in the hybrid +system. This leads to significant photocurrent increment in the accumulation region and +negative threshold voltage shift (ΔVTh) with increasing incident power density of +illumination [39]. To investigate in greater detail, the calculated responsivity as a function of +illumination power has been plotted for different gate bias voltages in Fig. 4(e). Here, the +responsivity value increases with increasing positive gate bias in case of MvWH photo-FETs +and reaches to a high value of ~ 1.1 × 104 A W−1 at a back gate voltage of ~ 40 V under an +illumination power of 0.1 μW, which is quite remarkable compared to those previously +reported 0D/2D hybrid phototransistors [37,39]. Note that, for all the gate voltages, the +measured responsivity dropped with increasing power because of the saturation of sensitizing +traps in CsPbI3 NCs, which is a characteristic footprint of trap-dominated photoresponse [40– + +42] . Further, we have extracted the threshold voltage via extrapolating the linear region of +each transfer curve under different incident laser powers and the shift in threshold voltage is +plotted as a function of incident power. The variation is fitted with the power law function +𝑉𝑇ℎ ∝ 𝑃𝑏, as depicted in Fig. 4(f), to understand the possible photoconduction mechanism. The +extracted fitting exponent, b ~ 0.17 clearly indicates a sublinear dependency on laser power +confirming the existence of photogating dominant carrier conduction in MvWH photo- +FETs [43]. Further, it is also observed that the change in VTh is large in the lower power region +and starts to saturate gradually at a higher power owing to the saturated trap states present in +sensitizer interface leading to the saturation of the photogating effect. The photocurrent IPh = +IPhoto − IDark versus gate voltage for different illumination intensity [Fig. 5(a)] shows a strong +modulation with VGS, and a clear maximum in response can be identified around +35 V. The +strongest response of the FET device corresponds to the region with highest transconductance, +due to the favourable Fermi level alignment, for low-contact resistance operation leading to +many cycles of electron circulation to produce maximum gain. Hitherto, in this region the FET +device operates at a relatively higher dark current, compromising the signal-to-noise ratio +(SNR) of the device, which is also a very important figure of merit of photo-FETs. The SNR +defined as IPhoto/IDark is illustrated in the same panel, Fig. 5(a), which reveals the potential of +0D/2D hybrid phototransistors for highest sensitivity detection in its depletion regime with VGS +from 0 to 5V. In this region, a lowest dark current and a maximum sensitivity are achieved, +despite the devices’ concurrent drop in the photocurrent. So the maximum sensitivity of the +device can be achieved via contact engineering where the transistor is operated in the depletion +region, even without applying any gate bias, hitherto unreported for photo-FET devices. While +the peak responsivity of our device is comparable or superior to the reported 2D materials based +hybrid phototransistor devices with perovskite sensitizers, the sensitivity is found to be +significantly higher without application of any external gate bias (see Table 1). These results + +illustrate the superior performance of broadband phototransistor, with ultrahigh sensitivity and +responsivity, using CsPbI3 NCs sensitized 2D WS2 layer. + +FIG. 5. (a) Back-gate bias dependent photocurrent (right axis) and photo-to-dark current ratio, +(left axis) of the phototransistor device under five different illumination intensities (from 0.1 +µW to 10 µW) for 514 nm. Despite the strongest photoresponse at higher gate bias (VGS ≈ 40 +V), highest sensitivity of the device is achieved in the depletion regime (VGS ≈ 0V). The +schematic representation of channel current transport mechanism and energy band diagram of +the asymmetric contact hybrid phototransistor under reverse drain-source voltage with (b) zero +and (c-d) different gate bias conditions. + +On the other hand, a remarkable photoresponse of CsPbI3/WS2 MvWH photo-FET is explained +by considering the influence of positive gate voltage on energy band alignment at the contact +interfaces and heterostuctures leading to efficient charge injection into n-type WS2 channel +Table 1. Comparison of device performances with reported 2D material based hybrid photo- +FETs with perovskite sensitizers + +(a) +(b) +hy +High sen sitivity +High photoresponse +104 +90 +0.1 μW +0.5 μW +(vu) +5 μW +10μW +CsPbI3 +103 +0.00 +60 +Photo +10 +Au +WS2 +Cr +30 +10 +-ve ++ve +100 +00 +20 +0 +20 +40 +Vcs (V) +Underillumination +(c) +(p) +hy +CsPbl +CsPbI3 +Au +Au +Cr +WS2 +-ve +-ve +Cr +WS2 ++ve ++ve +Depletion region +Accumulation regionDevice +structure +Sensitizer +Operational +spectral +range +Idark +w/o +applied +VGS +Iphoto/Idark +@ +VGS=0V +Responsivity +for different +values of VGs +Ref. +Au / ML +WS2 / Au +CH3NH3PbI3 +450-700 nm +5nA +104 +2.5 A/W @ 0V [44] +Au / ML +MoS2 / Au +CsPbBr3 +350-550nm +0.2 nA +103 +4.4 A/W @ 0V [45] +Au / Ti / ML +MoS2 / Ti / +Au +Ch3NH3PbBr3 +/ CsPbI3-xBrx +532 and 355 +nm +4 nA +104 +7 × 104 A/W +@ 60V + [46] +Au / Ti / FL +MoS2 / Ti / +Au +CsPbBr3 +405 nm +10 nA +10 +4.7 × 104 A/W +@ 20V + [47] +Au / FL BP / +Au +CsPbBr3 +405 nm +2 nA +102 +357.2 mA/W +@ 0V + [48] +Au / ML +MoS2 / Au +CsPbI3-xBrx +532 nm +0.2 µA +103 +1.13 × 105 +A/W @ 60V + [49] +Au / FL BP / +Al / Au +MAPbI3−xClx +400-900 nm +0.1 µA +/µm +102 +4 × 106 A/W +@ 40V + [50] +Au / Ti / FL +MoSe2(WSe2 +) / Ti / Au +CsPb(Cl/Br)3 +455 nm +0.8 nA +10 +102 A/W @ +50V + [51] +Au / Cr / FL +Ta2NiSe5 / Cr +/ Au +CH3NH3PbI3 +800 nm +3.5 µA +10 +2.4 × 102 A/W +@ 0V + [37] +Au / Cr / FL +WS2 / Au / +Cr +α-phase +CsPbI3 +400-800 nm +2 pA +106 +104 A/W @ +40V +Our +work + +layer from photoabsorbing CsPbI3 NCs. As illustrated in Fig. 5(b), the Schottky barrier at the +Au/WS2 interface is high enough to inhibit the charge conduction mechanism across the WS2 +channel layer at reverse drain bias without any gate electric field under dark condition. Hence, +the transistor immediately goes to the OFF state with very low dark current in the order of pA. +At this condition, when the visible light is illuminated on the 0D/2D heterostructure, the +photogeneration takes place in both CsPbI3 NCs as well as WS2 channel layer, as depicted in +Fig. 5(b). The effective photogenerated carrier separation takes place by the built-in electric +field at the Schottky junction (WS2/Au) as well as at CsPbI3/WS2 interfaces. The subsequent +transition of photoexcited electrons from CsPbI3 to WS2 starts to populate the active channel + +layer which are collected by the external electrodes under an applied reverse VDS, leading to +the photoresponsivity of ~ 102 A/W at VDS = ‒2V and VGS = 0V. Further, the application of a +back gate voltage (VGS) to the device modulates the Schottky barrier height at Au/WS2 interface +as shown in Figs. 5(c)-(d) [52,53]. An application of negative gate bias (VGS < VTh) increases +the barrier height leading to the transistor operation in the depletion region, where the +photosensitivity (IPhoto/IDark) of the device is maximum. On the other hand, on increasing the +VGS beyond VTh initiates the lowering of the Schottky barrier at Au/WS2 interface, resulting in +a higher magnitude of charge carrier injection from the Au electrode to WS2 channel layer +through thermoionic as well as tunnelling mechanisms, as illustrated in Fig. 5(d). Thus, the +cumulative effects of CsPbI3 NCs decoration mediated strong photogating phenomena as well +as the gate voltage induced Schottky barrier lowering result in a drastic enhancement of the +photocurrent (IPhoto − IDark) through the transistor channel at ON state (VGS > VTh). This leads +to an ultrahigh photoresponsivity of the order of ~ 104 A/W at VGS = 40 V. Such gate modulated +responsivity and sensitivity of MvWH photo-FET devices via interface engineering offers a +novel pathway for next generation high performance and low power integrated photonic +technology. +Temporal photoresponse is also an important parameter for the phototransistors performance +in terms of switching speed and device stability. The transient photoresponse of the as- +fabricated CsPbI3/WS2 MvWH photo-FET upon visible illumination (𝜆 = 514 nm) at VGS = 0V +with varying reverse VDS is demonstrated in Fig. 6(a). Upon illumination of four periodic +pulses of the Argon-ion laser, relatively fast and consistent photocurrent modulation +characteristics of the device reveals the stability and reproducibility of the as-fabricated MvWH +photo-FET. The device exhibits a much stronger photoresponse characteristics revealing ratio +of ~106 as compared to the control device with pristine WS2 with the value ~103 (Fig. S8 within +the Supplimental Material), which is attributed to the injection of high density + + +FIG. 6. (a) Transient response of the MvWH photo-FET device under illumination of a 514 nm +laser at different applied VDS. (b) Temporal photocurrent response of the MvWH device for a +wavelength of 514 nm with and without any applied gate bias. The temporal response indicates +a significant decrease in rise time (from 43.8 to 34 ms) as well as fall time (from 32.7 to 24 +ms), measured at a relatively higher power of 35 µW. (c) Operational stability of the fabricated +MvWH photo-FET device under visible illumination for more than half an hour. (d) The +transient photocurrent response of the fabricated transistor over a span of seven days from the +beginning and end of the stability test. (e) Stability of the device tested under extreme humid +conditions (varying from 50 to 65% RH). The last four cycle is the response under 65% of +humidity showing around 5% decay in the photoresponse. + + photogenerated charge carriers into the WS2 channel from strong light absorbing CsPbI3 NCs. +With the increment of reverse VDS, a consistent photocurrent enhancement is distinctly noticed +from the switching characteristics owing to the increase of depletion region width at the +Schottky barrier interface and subsequent separation of photogenerated charge carriers. +Further, the rise and fall times of the fabricated device in the absence of gate bias have been +estimated using an enlarged single cycle response [Fig. 6(b)] and are found to be around ∼ +43.8 ms and ∼ 32.7 ms, respectively, which are further reduced to 34 ms and 24 ms, +respectively on applying a gate voltage of 40 V. The response speed of these devices are found +to be relatively slower, which is attributed to the trapping of charge carriers in various structural + +(a) +(c) +3 +21 +-1.5V +-1V +2.5 +2.0 +HA +1.5 +DS +DS +1.0 +0.5 +ON +ON +0 +OFF +0.0 +0 +20 +40 +60 +80 +0 +20 +40 +0 +602 +1205 +1807 +18901920 +Time (sec) +Time (sec) +Time (sec) +Time (sec) +(b) +(d) +(e) +1.2 +1.2 +oV +3 +VDs = -2V +2=514nm +50% humidity +65% humidity +GS +Dav 1 +Day3 +Day 5 +Dav 7 +Normalized +0.8 +Normalized +0.8 +(vn) +2 +0.4 +DS +0.4 +0.0 +0.0 +5% +decay +6.75 +6.80 +6.85 +6.90 +6.95 +7.00 +Time (sec) +Time (sec) +Time (sec)and surface defect states present in WS2 as well as CsPbI3 NCs and their local junction +interfaces. These interface traps present in WS2 layer are mostly empty when biased under +depletion condition, i.e. VGS < VTh owing to lack of enough mobile carriers in the channel. This +allows a large number of photogenerated electrons to get trapped by the defect states while +some of the gate induced electrons, although small in number, can be trapped as well. This +results in a relatively slow rise of current, as depicted in the photocurrent dynamic response. +On the other hand, the interface traps are nearly filled up with gate-induced electrons in +accumulation condition, when VGS > VTh, as shown in Fig. 6(b). Hence, the trapping +probability of photogenerated carriers is lower and a relatively faster response (~34 ms) is +observed in MvWH photo-FET devices. Further, owing to the fact that the perovskite materials +are prone to environmental degradation via oxygen diffusion through iodide vacancies upon +illumination, the long term operation stability of the fabricated devices have been tested in this +study upon visible light illumination at zero gate bias for prolonged duration (more than 60 +min). From the I–t curves for the first 100 s [Fig. 6(c), left] and the last 100 s [Fig. 6(c), right], +it is observed that the photocurrent has almost no attenuation, indicating that these devices +show an excellent light stability under ambient condition, even without the use of a glovebox +or encapsulation. The device stability has also been tested via recording the photocurrent under +illumination over a period of one week, as illustrated in Fig. 6(d). Here, the phototransistor +sustains under laboratory ambient conditions (relative humidity (RH) ~ 45-50%, temperature +~ 22oC) for one week with negligible change in the photocurrent via degradation after storing. +Further, as CsPbI3 NCs are vulnerable to environmental humidity, to explore the device +performance in the extreme humid condition, we have performed the temporal response under +65% RH showing an insignificant degradation (5% decay) in terms of device response [Fig. +6(e)]. This superior performance stability is due to the surface defect passivation of CsPbI3 +through the interaction with the sulphur of WS2 ensuring the outstanding environmental + +stability of as-fabricated CsPbI3/WS2 MvWH photo-FETs. The sulfur atoms present on the top +layer of WS2 may have stronger coordination to the Pb2+ centers of CsPbI3 NCs leads to reduced +defect states in perovskites enabling higher reluctance to the degradation [31]. It may be noted +that the performance of the devices could be further improved by process optimization, device +encapsulation and incorporation of buffer layers. This work reveals the significant potential of +colloidal synthesized air-stable α-CsPbI3 NCs on 2D materials in fabricating 0D/2D mixed- +dimensional heterostructure photo-FETs for applications in next generation optoelectronic +devices. +Conclusion: +To summarize, significant improvements in performance have been realized in CsPbI3/WS2 +0D/2D mixed-dimensional phototransistors with asymmetric metal electrodes leading to +combinatorial effect of Schottky barrier induced suppression of dark current and efficient +charge transfer from photoabsorbing CsPbI3 nanocrystals, resulting in enhanced +photosensitivity and spectral responsivity. The WS2 channel with asymmetric contacts +(Cr/WS2/Au) shows a rectifying I-V characteristics under an applied VDS with the dark current +in the order of pA. Further, by combining the channel sensitization via decorating the WS2 with +photosensitive air-stable α-phase CsPbI3 NCs, a responsivity of ~102 A/W has been achieved +at low VDS (~ -2V) for an incident optical power of 0.1 µW even without any external gate +bias. The device exhibits a broad spectral photoresponsivity between 400 and 800 nm due to +the extended visible light absorption features of CsPbI3 NCs. Using gate-controlled carrier +modulation in the transistor channel, a peak responsivity ~104 A/W (VGS = +40 V) has been +achieved owing to the photogating effect mediated charge conduction whereas the maximum +sensitivity (~ 106 at ~ VDS = -2 V) in terms of signal-to-noise ratio is observed by depleting the +channel carries (VGS = 0 to 5 V). These devices show superior performance in terms of +environment stability, owing to the filling of surface trap states present in CsPbI3 NCs via + +conjugation with sulfur atoms of 2D WS2 layer. 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Joo, Photosensitive N- +Type Doping Using Perovskite CsPbX 3 Quantum Dots for Two-Dimensional MSe 2 +(M = Mo and W) Field-Effect Transistors, ACS Appl. Mater. Interfaces 12, 25159 +(2020). +[52] S. Aftab, M. W. Iqbal, A. M. Afzal, M. F. Khan, G. Hussain, H. S. Waheed, and M. A. +Kamran, Formation of an MoTe 2 Based Schottky Junction Employing Ultra-Low and +High Resistive Metal Contacts, RSC Adv. 9, 10017 (2019). +[53] M. H. Yang, K. B. K. Teo, W. I. Milne, and D. G. Hasko, Carbon Nanotube Schottky +Diode and Directionally Dependent Field-Effect Transistor Using Asymmetrical +Contacts, Appl. Phys. Lett. 87, 253116 (2005). + + + + +Supplemental Material + +Charge transfer mediated giant photo-amplification in air-stable α-CsPbI3 +nanocrystals decorated 2D-WS2 photo-FET with asymmetric contacts +Shreyasi Das1, Arup Ghorai1,2, Sourabh Pal3, Somnath Mahato1, Soumen Das4, Samit K. Ray5 * +1School of Nano Science and Technology, IIT Kharagpur, Kharagpur, West Bengal, India, 721302 +2Department of Materials Science and Engineering, Pohang University of Science and Technology, +Pohang 790-784, Korea +3Advanced Technology Development Centre, IIT Kharagpur, Kharagpur, West Bengal, India, 721302 +4School of Medical Science and Technology, IIT Kharagpur, Kharagpur, West Bengal, India, 721302 +5Department of Physics, IIT Kharagpur, Kharagpur, West Bengal, India, 721302 +Email : physkr@phy.iitkgp.ac.in + + + + +S1: Tauc plot of CsPbI3 NCs + +Fig. S1. Tauc plot of as synthesised α-phase CsPbI3 NCs + +S2: Photoluminescence spectra of WS2, CsPbI3 and CsPbI3/WS2 hybrid + +Fig. S2. Comparative PL spectrum of bare ML WS2 flake, CsPbI3 NCs and their +heterostructures where three different concentrations of CsPbI3 NCs (different steps of spin +coating) incorporated on WS2 flakes. After formation of heterostructures, the PL intensity of +both bare ML WS2 as well as CsPbI3 NCs get reduced. + +(αhv)? (a.u.) +E. = 1.814 eV +1.6 +1.8 +2.0 +2.2 +2.4 +2.6 +Energy (eV)PL Intensity (a.u.) +Ws? +CsPbI3 +Step 1 +Step 2 +Step 3 +550 +600 +650 +700 +750 +Wavelength (nm)S3: PL integrated intensity ratio vs coating step + +Fig. S3. Variation of PL integrated intensity ratio of WS2 trion peak (A-) to excitonic peak +(A) with increasing concentration of CsPbI3 decoration (increasing spin coating step) on WS2 +flakes. + +S4: Thickness of the exfoliated flake analysis using AFM + +Fig. S4. Atomic force microscopy image of the few layer WS2 flakes. Inset shows the height +profile along the yellow dashed line confirming the thickness of the flakes around 4.5 nm. + +7 +Increasing CsPbI, +6 +concentration +5 +2 +1 +0 +Step 3 +Step 2 +Step 1 +Ws.0 +5 +10 +15 +20 +25 +30 +35 +40 +45μm +nm +30 +Height (nm) +5 +27.5 +25 +10- +22.5 +15 +-20 +20 +0 +17.5 +25 +0.0 +0.4 +0.8 +1.2 +-15 +Distance (um) +30- +12.5 +-10 +35 +7.5 +40 +-5 +45 +2.5 +50 +umS5: Energy band structures of WS2 at the contacts + +Fig. S5. The corresponding energy band structures with different combination of metal +electrodes before contact and after contact condition under applied reverse bias. + +S6: Photo to dark current ratio + +Fig. S6. Photo to dark current ratio for control device (WS2 FET) and MvWH photo-FET +with varying drain to source voltage. +Cr +Cr +WS2 +Ohmic contact ++ve +-ve +Au +Au +WS2 +Symmetric contact ++ve +-ve +Au +-ve +Cr ++ve +WS2 +Asymmetric contact +Schottky diode +Au +Cr +WS2 +4.6 eV +4.5 eV +5.1 eV +Evac +EF +Before contact + +1.2x10 +Ws +9.0x1( +sPbI. +rk +Dal +6.0x10 +3.0x10 +0.0 +2.0 +-1.5 +-1.0 +-0.5 +0.0 +Vps (V)S7: Spectral responsivity of the control WS2 FET device + +Fig. S7. Spectral responsivity curves of control WS2 FET device at VGS = 0V with increasing +reverse VDS from 0V to -2V + +S8: Transient response of the control WS2 FET device + + +Fig. S8. Transient response of the control WS2 FET device under illumination of a 514 nm +laser at different applied VDS. + +300 400 500 600 700 800 900 +0 +3 +5 +8 +10 +Responsivity (A/W) +WS2 +Wavelength (nm) + -2V + -1V + 0V + +8 +-1.5V +-1V +40 +6 +(nA) +4 +2 +ON +OFF ON +0 +0 +20 +40 +60 +80 +Time (sec) \ No newline at end of file diff --git a/HtE1T4oBgHgl3EQfrgVp/content/tmp_files/load_file.txt b/HtE1T4oBgHgl3EQfrgVp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe18fc11fe90f516ad807446e9f9320ab58dba8c --- /dev/null +++ b/HtE1T4oBgHgl3EQfrgVp/content/tmp_files/load_file.txt @@ -0,0 +1,1189 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf,len=1188 +page_content='Charge transfer mediated giant photo-amplification in air-stable α-CsPbI3 nanocrystals decorated 2D-WS2 photo-FET with asymmetric contacts Shreyasi Das1, Arup Ghorai1,2, Sourabh Pal3, Somnath Mahato1, Soumen Das4, Samit K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Ray5 * 1School of Nano Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' IIT Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' West Bengal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' India,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 721302 2Department of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Pohang University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Pohang 790-784,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Korea 3Advanced Technology Development Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' IIT Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' West Bengal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' India,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 721302 4School of Medical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' IIT Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' West Bengal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' India,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 721302 5Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' IIT Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' West Bengal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' India,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 721302 Email : physkr@phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='iitkgp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='in Abstract Hybrid heterostructure based phototransistors are attractive owing to their high gain induced by photogating effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' However, the absence of an in-plane built-in electric field in the single channel layer transistor results in a relatively higher dark current and require a large operating gate voltage of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Here, we report novel air-stable cesium lead iodide/tungsten di- sulfide (CsPbI3/WS2) mixed dimensional heterostructure based photo-field-effect-transistors (photo-FETs) with asymmetric metal electrodes (Cr/WS2/Au), exhibiting extremely low dark current (~10-12 A) with a responsivity of ~ 102 A/W at zero gate bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The Schottky barrier (WS2/Au interface) induced rectification characteristics in the channel accompanied by the excellent photogating effect from solution-processed α-phase CsPbI3 NCs sensitizers, resulting in gate-tunable broadband photodetection with a very high responsivity (~104 A/W) and excellent sensitivity (~106).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Most interestingly, the device shows superior performance even under high humidity (50-65%) conditions owing to the formation of cubic α-phase CsPbI3 nanocrystals with a relatively smaller lattice constant (a = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='2315 Å) and filling of surface vacancies (Pb2+ centres) with the sulfur atoms from WS2 layer, thus protecting it from environmental degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' These results emphasise a novel strategy for developing mixed dimensional hybrid heterostructure based phototransistors for futuristic integrated nano- optoelectronic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Keywords: Two dimensional TMDs, Inorganic perovskites, Sensitizers, Asymmetric electrodes, Mixed dimensional phototransistors Introduction Fabrication of high performance phototransistors demands superior channel material, which should be of high carrier mobility for high gain bandwidth product, a direct bandgap for efficient optical absorption, a thinner layer for full depletion leading to ultralow dark current and very low trap state density for low subthreshold swings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Layered semiconducting two dimensional (2D) TMDs fulfil most of these requirements [1–4] making them a potential candidate for photo-field-effect transistor (photo-FET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' However, some trade-offs are found in single semiconductor channel based phototransistors, due to the simultaneous occurrence of both absorption and amplification processes within the same layer, compromising the device performance [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' To overcome this shortcoming as well as for fabrication simplicity, attention has been paid on few-layer TMDs over monolayer with a wider spectral photoresponse and relatively higher absorbance [8–10], which in turn increases the dark current values requiring a higher gate voltage to operate the device in full depletion mode [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' This issue can be addressed via incorporating a Schottky barrier by judiciously selecting the metal contacts and utilizing the developed depletion region to facilitate an unidirectional current transport, which leads to the significant lowering of dark current in few layered TMD based photo-FETs at zero gate bias [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The built-in electric field of the Schottky interface further helps in the separation of photogenerated charge carriers across the channel layer [15] leading to zero gate bias driven enhanced photosensitivity, although the responsivity of these devices are limited owing to the lower absorption coefficient of 2D TMD material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' For the further improvement of photoresponsivity, recent works are focused on sensitizing the thin channel material with semiconductor nanocrystals (NCs), referred to as sensitizers, having excellent absorption characteristics and opposite doping polarity to create a vertical junction for subsequent charge separation [16–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Recently, all-inorganic cesium lead halide (CsPbX3: X = I, Br, and Cl) perovskite NCs have drawn tremendous interests in the field of optoelectronics [20], featuring superior emissive (approaching photoluminescence quantum yield ~ 100%) [21] characteristics, extremely high absorption coefficient [22], large carrier diffusion lengths (>3 mm), fast radiative recombination rates and most importantly their improved stability [23] over organic-inorganic hybrid perovskites [24,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Especially colloidal synthesised zero-dimensional (0D) α-CsPbI3 NCs exhibit extended photoabsorption band covering the whole visible spectrum with high absorption coefficient (~ 3 × 104 cm-1 at 680 nm) [26] and better photostability, making them attractive as a photoactive material for high performance optoelectronic devices under ambient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Weerd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' have recently reported colloidal CsPbI3 NCs with high quantum yield of ~ 98% prepared by hot-injection method which reveals highly efficient carrier multiplication as well as longer build-up of free carrier concentration [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Among four existing phases (α- cubic, β-tetragonal, γ-orthorhombic, and δ-orthorhombic) of CsPbI3 NCs, cubic α-phase has high stability due to its low surface-to-volume ratio (lattice constant a = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='23 Å) without any octahedral inclination or lattice distortion in the [PbI6]4− octahedra as well as in the unit cell [28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' These extraordinary properties of α-phase CsPbI3 NCs provoke to utilize them as an effective sensitizer in 2D channel based hybrid phototransistor devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' However, owing to the vulnerability towards moisture for halide perovskite NCs, recently, the surface passivation of CsPbI3 via coordination of Pb2+ centres with sulphur donors has been explored to protect them from environmental degradation [31] to improve the device stability under ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' In this work, we present a proof-of-concept for 0D/2D CsPbI3/WS2 based mixed dimensional van der Waals heterostructure (MvWH) photo-FETs utilizing the superior photoabsorption attributes of α-phase CsPbI3 NCs as sensitizers, along with a sub 5-nm thick 2D WS2 channel which acts as an expressway for carrier transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' In our device configuration, the developed built-in electrical field at the heterostructure interface facilitates efficient transfer of photogenerated carriers from CsPbI3 NCs into the WS2 layer resulting in an excellent broadband visible photoabsorption in the hybrid system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Moreover, asymmetric metal contacts (Au-Cr) as source and drain electrodes are purposefully chosen to utilize the large built-in potential along the WS2/Au Schottky junction for a diode-like unidirectional current flow and effective separation of photogenerated electron-hole pairs, leading to an excellent rectifying ratio of ~ 104 with a very low dark current of the order of ~ pA under a low source-to-drain bias (VDS) without any applied back gate voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Fabricated hybrid phototransistor devices exhibit a very high photoresponsivity of ~ 104 A/W at ~ 40 V gate bias upon visible light illumination (~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='1 µW) due to the photogating effect and a broad spectral bandwidth across the entire visible range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' In addition, the coordination of sulphur atoms of WS2 layer with Pb2+ centres of CsPbI3 NCs reveals an excellent device stability under 50-65% humidity condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Our results not only open up new avenues for studying fundamental carrier transport and relaxation pathways in hybrid van der Waals heterojunctions, but also pave the way for constructing high-performance optoelectronic devices using mixed-dimensional 0D/2D hybrid building blocks, rather than using purely 2D layered materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Experimental section: Synthesis of α-phase CsPbI3 NCs: This is a two-step reaction, where, in the first step Caesium oleate (Cs-OA) was prepared followed by the synthesis of α-phase CsPbI3 NCs in the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' i) Firstly, caesium carbonate (Cs2CO3) of 814 mg and 40 ml Olive oil were added in a round bottom two-neck flask, which was heated at 120⁰C for 1 hr under vacuum condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Followed by the rise in temperature to 150⁰C under N2 atmosphere for 10-15 mins, the desired transparent solution of Cs-OA was stored for further use to synthesize CsPbI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' ii) Next, 870 mg of PbI2 and 5 ml of olive oil (instead of 1-octadecene (ODE)) were mixed which was heated to 120⁰C under vacuum for 1hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Thereafter, we swiftly injected 1 ml of Oleyl amine (OLAm) in the reaction mixture under N2 atmosphere to get a transparent solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Finally, preheated (100⁰C) Cs-OA was injected to the reaction mixture and cooled immediately in an ice bath to quench the reaction to obtain desired α-phase CsPbI3 NCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Finally, as-synthesized CsPbI3 was purified through centrifugation using excess hexane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The centrifugation process (20000 rpm) was repeated for several times to remove excess OLAm/olive oil from the product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Finally, the sedimentation was collected and re-dispersed in hexane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The collected dispersion was stored in a sealed vial for further characterisations and device fabrications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' CsPbI3/WS2 MvWH photo-FET device fabrication: For the fabrication of the CsPbI3/WS2 MvWH photo-FET device, WS2 flakes were mechanically exfoliated from the bulk WS2 crystal (2D semiconductor Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Scottsdale, AZ, USA) using a Scotch tape (3M Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' USA) on polydimethylsiloxane (PDMS) gel film (Gel-Pak Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Hayward, CA, USA) and the interested few layer flakes were identified under optical microscope, followed by the layer confirmation via Raman characteristics and AFM height profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Note that, mechanically exfoliated flakes were chosen for their high quality and clean interface promising greater mobility and high performance device fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further, for the Au contact with WS2, electrodes were patterned in advance on a pre-patterned SiO2/Si (285 nm oxide thickness) substrate using e- beam lithography technique and then Cr (5 nm)/Au (30 nm) were deposited via e-beam evaporation followed by lift-off with acetone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Then, the selected few layer flakes were deterministically transferred on the targeted Au electrodes from PDMS gel film following the dry transfer technique to make sure residue free clean transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' After the successful transfer of the flakes on Au electrodes, again electrodes were patterned using second step e-beam lithography followed by metal deposition Cr (5 nm)/Au (30 nm) for Cr contacts on WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' To remove the resist residue and improve the contact conductance, the fabricated devices were annealed at 150°C for 2 hrs in a high vacuum of ~ 10−3 Torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Finally, the synthesized diluted solution (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='1 mg mL−1) of perovskite NCs was uniformly spin-coated several times (varying from one to four) onto WS2 layer with a speed of 2000 rpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Characterisations and measurements: X-ray diffraction (XRD, Philips MRD X-ray diffractometer) patterns were recorded using characteristic Cu-Kα (λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5418 Å) radiation with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0° grazing incidence angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' For the Transmission electron microscopy (TEM) sample preparation, CsPbI3 NCs solution was dissolved in Hexane and then placed into a TEM grid and dried it for few minutes and did the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The TEM images were carried out using TECNAI G2 TF20-ST and JEM-2100F Field Emission Electron Microscope operating at 200 kV equipped with Gaytan’s latest CMOS camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' All the images were proceeding by Digital Micrograph Software for the estimation of d-Spacing & Indexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' UV–vis–NIR absorption spectra of as synthesised CsPbI3 samples were recorded using a fiber probe-based UV–vis–NIR spectrophotometer (Model: U-2910 Spectrophotometer, HITACHI) and a broadband light source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Raman and PL spectra were recorded using a semiconductor laser of excitation wavelength 532 nm, equipped with a CCD detector, an optical microscope of 100x objective lens and a spectrometer (WITec alpha-300R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The photogenerated carrier lifetime was measured by exciting the material with a pulsed diode laser of wavelength 372 nm and detecting the signal using Edinburgh LifeSpec-II fluorescence lifetime spectrometer fitted with a PMT detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Room-temperature current−voltage characteristics were recorded using a Keithly semiconductor parameter analyzer (4200 SCS) in the presence of an Argon laser (514 nm) and a broadband solar simulator (AM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5, 100 mW/cm2) as a visible light source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Results and discussion FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (a) Rietveld refinements (α-phase fitting) of the XRD pattern of a film of cubic CsPbI3 NCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (b) 1×1 3D VESTA visualization image of α-CsPbI3 cubic crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (c) Typical HRTEM image of CsPbI3 NCs with an energy 200 keV revealing cubic morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (d) FFT patterns from a region marked by the red dotted square on the micrograph (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (e) A magnified view of the corresponding HRTEM image in the selected yellow square region on micrograph (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (f) SAED pattern of α-CsPbI3 NCs showing well defined diffraction spots indexed as (200), (220) and (020) planes viewed along [004] zone axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' To study the crystal structure of as-synthesized CsPbI3 NCs, we have recorded X-ray diffraction pattern, followed by their fitting with Rietveld refinement full proof software, which are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' An excellent agreement with fitted results indicates the growth of single-phase (α-phase) cubic CsPbI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The crystal structure of CsPbI3 NCs visualized using VESTA 3D software through Rietveld fitting is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The VESTA 3D (1x1) structure shows the absence of any octahedral inclination in the perovskite NCs, which is known to be beneficial for achieving higher stability under laboratory ambient (45-50% humidity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Typical high resolution transmission electron microscopy (HRTEM) image reveals almost cubic shape of the synthesised NCs (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='05 nm × 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='04 nm), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Corresponding first Fourier transform (FFT) pattern presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 1(d) of the red squared (a) C (200) & Observed (100) 米0 Calculated 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='05nm Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' units) (210) Difference d= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='62 nm Braggposition 11001 d= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='62 nm [100] 10nm 10 20 20 (degree) 30 40 50 (b) (d) ZA[002] a-CsPbl3 f) ZA [004] a-CsPbl (220) (210) (200) (200) (200) (210) (220) (220) ( () 2 nm 2 1/nm (020) Cs Pbportion of the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 1(c), shows pure cubic α-phase pattern along the zone axis [002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Whereas, the high-resolution fringe pattern from the yellow squared region of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 1(c) shows a d-spacing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='62 nm, which is in well matched with the cubic α-phase of CsPbI3 [30], as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 1(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The result indicates (100) directional growth of cubic phase CsPbI3 NCs, which is in well agreement with our previously reported results [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Corresponding selected area electron diffraction (SAED) patterns shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 1(f), with indexed (200), (220) and (020) planes along the zone axis [004], also corroborate the pure cubic structure of synthesized CsPbI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Figure 2(a) presents the optical absorption and emission properties of the as-synthesised α- phase CsPbI3 NCs in the visible wavelength range with an absorption maxima at ~ 680 nm and the corresponding bandgap value is ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='814 eV [30], extracted from the Tauc plot shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S1 within the Supplimental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further, the photoluminescence (PL) maxima at ~ 687 nm confirms the formation of excitons (Xα) across the direct bandgap (∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='80 eV) of α- phase CsPbI3 NCs represented via blue curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The Gaussian line shape of the PL spectrum and the absence of any other PL peaks clearly dictate the synthesis of pure α-phase CsPbI3 without presence of any mixed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' To examine the charge transfer mechanism at the CsPbI3 NCs/WS2 interface, Raman spectroscopy and micro-PL (µ-PL) measurements have been carried out by spin-coating of a dilute solution of CsPbI3 NCs uniformly on the exfoliated WS2 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' For the room temperature µ-Raman-PL measurements, samples have been excited with a CW laser having a wavelength of 532 nm with the laser power being kept at a very low value to avoid any local heating induced sample degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Figure 2(b) represents comparative Raman spectra of WS2 and mixed dimensional van der Waals heterostructure (MvWH) samples, showing intense in-plane 2LA+E12g Raman modes at ∼ 351 cm–1 and out- of-plane vibrational A1g peaks at ∼ 420 cm–1 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The A1g vibrational Raman mode, which preserves the symmetry of the lattice, is clearly red-shifted by ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='7 cm-1 in case of CsPbI3 decorated WS2 layer [inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 2(b)], revealing the interfacial charge transfer phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The external electron doping in 2D WS2 leads to the filling-up of antibonding states of the conduction band, mostly made up of d z2 orbitals of transition metal atoms [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' This makes the bonds weaker and the A1g peak of pristine WS2 is shifted towards a lower wavenumber on significant electron doping from CsPbI3 NCs [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (a) Absorption (Green) and photoluminescence (Blue) spectra of as-synthesised cubic phase CsPbI3 NCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (b) Comparative Raman spectra of WS2 flakes before and after CsPbI3 NCs decoration showing characteristic E2g and A1g peaks of layered WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Inset shows the magnified image of the out-of-plane A1g mode revealing a clear peak shift to lower wavenumbers due to electron doping in WS2 flakes from CsPbI3 NCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (c) Deconvoluted PL spectra of (i) a bare ML WS2 flake and (ii-v) the heterostructure samples with varying number of spin coated layers of CsPbI3 NCs on the WS2 flake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The spectra (ii), (iii), (iv) and (v) represent the PL emission from the first, second, third and fourth spin coated layers of CsPbI3 NCs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The green (red) peak represents the A excitonic (A- trionic) emission from ML WS2 flakes and the blue peak represents the emission from band to band transition of cubic α-phase CsPbI3 NCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (d) Energy band diagram of CsPbI3/WS2 hybrid heterostructures showing effective electron doping in WS2 from CsPbI3 sensitizers and hole trapping in the NCs giving rise to a strong photogating effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (e) Schematic representation of the charge transfer mechanism in 0D/2D CsPbI3/WS2 hybrid heterostructures giving rise to trion formation in ML WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (f) Normalised time resolved PL decay curves of CsPbI3 NCs (Blue curve) and CsPbI3/WS2 hybrids (Red curve), measured using an excitation wavelength of 372 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' On the other hand, the monolayer (ML) WS2 PL emission characteristics [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 2c(i)] consist of a strong A-excitonic emission at ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='995 eV, corresponding to the direct band-to-band 550 600 650 700 750 PL intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' units) Wavelength (nm) (v) (iv) (iii) (ii) (i) WS2 A0 160 180 200 220 240 tav = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='7 ns PL (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' units) Time (ns) CsPbI3 CsPbI3/WS2 tav = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='2 ns 300 400 500 410 420 430 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='7 cm-1 400 410 420 430 440 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='7 cm-1 CsPbI3/WS2 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' units) Raman shift (cm-1) WS2 Xα A- CsPbI3/WS2 500 600 700 800 Wavelength (nm) Nor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='/PL (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' units) Absorption Photoluminescence WS2 CsPbI3 hν Electron transfer Hole trapping Charge transfer WS2 Exciton Trion CsPbI3 hν Electrons Holes (d) (c) (a) (f) (b) (e) Increasing CsPbI3 coating layer transitions at the K (and/or K’) point of the Brillouin zone, and a weaker trionic A– emission at ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='965 eV with a binding energy of ∼ 30 meV, which are in good agreement with the previously reported results [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' It is to be noted that, ML WS2 is purposefully chosen for the charge transfer study via PL measurements due to its extraordinary luminescence property at room temperature owing to the direct bandgap transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The existence of trions in the room temperature emission spectrum indicates the unintentional doping in the un-passivated WS2 flake from the substrate as well as the surrounding environment [35,36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' A systematic PL study of the hybrid structure with increasing layer numbers of CsPbI3 NCs spin-coated over ML WS2 shows a pronounced excitonic-PL quenching in MvWH as compared to both the pristine materials (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S2 within the Supplimental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further, the CsPbI3/WS2 MvWHs show relatively broad PL spectra with combined contributions from CsPbI3 NCs as well as ML WS2 and the spectral shape changes with increasing CsPbI3 spin-coated layer numbers [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 2c(ii- v)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' To explore the effect of CsPbI3 coating over WS2, we have fitted each spectrum with three Gaussian peaks containing the characteristics excitonic features of both the materials and analysed their intensity variation with increasing concentration of CsPbI3 treated on WS2 flakes, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' It is noticed that the distinctive trion peak (A–) of WS2 becomes prominent with increasing density (coating number) of CsPbI3 NCs and finally excitonic to trionic (integrated intensity) crossover is observed above a critical concentration of CsPbI3 NCs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S3 within the Supplimental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The possible explanation behind these observations is as follows: upon illumination, photoexcited electron-hole pairs are generated in both WS2 and CsPbI3, however, due to the type-II energy band alignment of the heterostructures, electrons are easily transferred from CsPbI3 NCs to ML WS2, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' On the other hand, photogenerated holes remain trapped in the NCs, resulting in a reduced recombination rate and giving rise to a quenched excitonic PL intensity for CsPbI3 NCs and increased trionic emission in ML WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The enhanced generation rate of trions results in the reduced density of excitons inside the system, leading to the suppression of excitonic peak intensity and the dominance of the trion peak in the PL spectra of MvWH, as schematically depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 2(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' [34] To further confirm the charge transfer phenomena, time-resolved photoluminescence (Tr-PL) spectra have been measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 2(f) shows the Tr-PL decay curves of CsPbI3 NCs (blue curve) and CsPbI3/WS2 MvWHs (red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The PL decay curves have been fitted using a bi-exponential function to extract the average excitonic life time (τav).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The τav of CsPbI3 NCs decreases from 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='2 ns to 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='7 ns after hybridization with WS2, corroborating the successful charge transfer mechanism from CsPbI3 NCs to WS2 flakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' As a conclusion, the type-II band alignment in CsPbI3/WS2 heterostructure facilitates an efficient electron–hole pair separation and strong electron doping into WS2 channel, making the hybrid system ideal for fabrication of superior performance phototransistor devices [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (a) Schematic 3D view of the fabricated back gated phototransistor comprising of CsPbI3 sensitized WS2 channel with asymmetric electrodes (Au and Cr) acting as a source and drain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' An optical micrograph of the device is shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (b) Linear IDS-VDS characteristics of three fabricated devices with different source-drain contacts (i) Cr-Cr (Yellow curve), (ii) Au-Au (Orange curve) and (iii) Au-Cr (Brown curve) without any back gate bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Inset: the corresponding semi-logarithmic IDS-VDS characteristics plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (c) Transfer (a) (b) 6 10- 4 10-11 (vu) 2 10-13 T Vps (M) U DS 2 Cr/Cr Au Au/Au 4 Au/Cr Watoms Si02 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content="6 CsPbl, NCs Satoms 2 1 0 1 2 V Ds (V) (c) (d) 10° 60 20 A0 5V 10V 15V (vu) (vu) 15 20V 'DS 10-11 DS 10 20 20 40 20 WS 5 CsPbI,/WS 4-0 0 20 0 20 40 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 VGs (V) V (V) DS(IDS-VGS) characteristics of the CsPbI3/WS2 hybrid transistor (Green curve) and control WS2 transistor (Blue curve) devices in linear scale and logarithmic scale (Inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The current value in the accumulation region decreases and the threshold voltage is shifted to a higher positive voltage in hybrid device due to charge transfer through the CsPbI3/WS2 junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (d) Output characteristics of the hybrid device with varying gate voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The CsPbI3 NCs/WS2 MvWH photo-FET with asymmetric metal contacts has been demonstrated by exploiting the Schottky barrier induced dark current suppression for the zero gate bias driven photosensitivity of the device with a simpler fabrication technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Figure 3(a) schematically demonstrates the as-fabricated phototransistor device structure consisting of a few layer WS2 channel and asymmetric Cr and Au electrodes as source and drain terminals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The inset shows the optical micrograph of the connected few layer WS2 flake (5×2 µm2) having a thickness of ~ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 nm corresponding to 4-5 atomic layers of WS2, further corroborated by the AFM analysis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S4 within the Supplimental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The deposition of a lower work function Cr (ΦCr = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 eV) and higher work function Au (ΦAu = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='1 eV) on n- type WS2 as asymmetric contacts reveals room temperature rectifying diode characteristics with rectification ratio up to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='2 × 102 even at zero applied back gate bias, as depicted via the current-voltage (IDS-VDS) characteristics in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Under an applied reverse drain voltage, the potential barrier height between Au and WS2 becomes higher to suppress the current flow through the junction compared to Cr, and thus exhibits the IDS-VDS characteristics of an ideal diode [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The current through the Au–WS2-Cr device follows the diode behaviour and the Schottky barrier (Φb) at Au-WS2 interface is capable of reducing the dark current significantly making the device architecture an ideal prototype for operating in full depletion mode even at zero gate bias, leading to a very high ON-OFF ratio of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' On the other hand, a linear IDS-VDS characteristics with a comparatively larger current value (100 nA) confirms the formation of an Ohmic-like junction with a very low contact resistance for Cr-WS2-Cr device [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further, Au-WS2-Au system reveals a rectifying output characteristics with relatively lower current than Cr contacts, confirming a typical high resistive back-to-back Schottky diode [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The energy band alignment with different metal contacts is schematically depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S5 within the Supplimental Material, revealing an easy current flow through the Ohmic Cr junction and restricted flow via built-in potential barrier in the Au Schottky junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 3(c) shows the transfer (IDS-VGS) characteristics of the WS2 phototransistor with asymmetric Cr–Au contacts at a reverse drain voltage of -2V revealing excellent n-type channel properties at room temperature with off-currents of the order of 10 pA and the transistor ON- OFF ratio ~104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further, to understand the effect of CsPbI3 treatment on the device performance, the CsPbI3/WS2 hybrid transistor characteristics is compared with the pristine WS2 one, referred to as the control device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The incorporation of sensitizing perovskite NCs on 2D-WS2 layer results in a junction formation via Fermi level alignment in equilibrium under dark condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' In this process, the draining of electrons from the WS2 channel towards CsPbI3 NCs results in the depletion of majority carriers in WS2 leading to the lowering of the current flow in the channel under dark condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further, we have studied the output characteristics of the hybrid phototransistor on application of gate voltage varying from 0 to +20 V, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' For a higher positive gate voltage, more electrons are induced in the WS2 channel and the transistor has a higher current in the saturation state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' On the other hand, under a negative gate bias a small amount of current flows through the channel owing to the depletion of carriers, leading to the OFF state of the transistor [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 3(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The performance of the CsPbI3/WS2 MvWH photo-FET has been analysed by recording the room temperature IDS-VDS characteristics for zero gate bias under dark as well as visible illumination using a Newport solar simulator having broadband emission with irradiance of 100 mW/cm2 under air mass (AM) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5G condition, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' For comparison, the characteristics of the pristine WS2 control device is also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The suppression of dark current to the order of tens of pA even without any gate bias along with the significant reduction of noise currents are attributed to the built-in electric field at the Au/WS2 Schottky junction, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (a) Comparative IDS-VDS characteristics of pristine WS2 and CsPbI3/WS2 MvWH photo-FET under dark and illumination via broadband light source for zero gate bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (b) Spectral responsivity curves of MvWH photo-FET at VGS = 0V with increasing reverse VDS from 0V to -2V, as shown via yellow, green and blue curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The blue curve represents the responsivity of the photo-FET at a maximum VDS of -2V, while the spectral responsivity of the control device (orange curve) showing an order of magnitude lower device response at same VDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (c) COMSOL Multiphysics simulated E-field distribution at the vicinity of the hybrid system upon excitation with an excitation wavelength of 680 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (d) The transfer characteristics (IDS-VGS) of MvWH photo-FET for a range of incident powers (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='1 to 35 μW) with an illumination of wavelength 514 nm at VDS = -2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (e) The variation of responsivity of the device with incident illumination power for VGS varying from 0 to 40 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (f) The shift in threshold voltage (ΔVTh) with increasing illumination power (Pin) fitted with a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Blue dots represent the extracted data points from panel (a) and green line represents the fitted curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Inset: The power law fit of the variation of photocurrent with incident power at VDS = -2V and VGS = 40V showing a sublinear photocurrent dependency with incident optical power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' which further helps in effective separation of photogenerated carriers created in WS2 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' On illuminating the heterojunction device, the reverse current tends to increase due to the collection of photogenerated minority carriers (holes) at the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The photo-to-dark current ratio of MvWH photo-FETs by illuminating with a broadband light source is estimated to be much higher compared to the pristine WS2 one (~1000 times) under the applied reverse bias condition, which reaches to a value of ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='08×106 at VDS of -2V, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S6 within the Supplimental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The decoration of WS2 channel with superior light absorbing (a) (b) (c) 120 (AW) ×102 E2/E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 6 80 10-8 4 MvWHlight 40 MvWH dark WS,light 2 WS,dark WS,1 0 2 1 0 1 2 300400500600700800900 Vps (V) Wavelength(nm) (d) (e) (f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='3 10 35μW (A/W) 40V 20V 200 10-7 104 15 (nA) 40V 30V 10V 150 (μA) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='2 + OV 20 DS 100 10-1 0μw Tocp0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='48 10 50 10-13 AV 25 20 Vcs (M) 40 0 10 20 30 30 Pin (μW) Resi 10 35 ocp0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0 40 20 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='1 1 10 0 7 14 0 21 28 35 (V)perovskite CsPbI3 NCs facilitates enhancement in the photocurrent by elevating the photogenerated carriers in the channel via efficient charge transfer from CsPbI3 to WS2 due to type-II energy band alignment [7,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' This explains the significant enhancement (~103 times) of response of MvWH transistor over the control one, revealing the role of photoabsorbing CsPbI3 NCs in boosting the performance of the phototransistor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further, the spectral responsivity of the fabricated MvWH photo-FET, the most important figure of merit to evaluate a detector performance, has been studied displaying a broadband spectral photoresponse covering the entire visible wavelength range, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' It may be noted that a peak responsivity of ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='05×102 A/W at ~ 460 nm at an applied bias (VDS) of -2 V is achieved, which is close to the C-exciton absorption edge of WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Two other peaks at ~ 620 nm (R ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='97×102 A/W) and ~ 720 nm (R ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='77×102 A/W) correspond to the direct bandgap absorption of few layer WS2 and α-phase CsPbI3 NCs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further, the spectral responsivity increases with increasing reverse VDS that assists in the efficient extraction of photogenerated carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' On the other hand, the control WS2 based device also exhibits a similar trend with increasing bias showing a maximum peak responsivity of ~ 10 A/W at ~ 460 nm at -2 V applied VDS (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S7 within the Supplimental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' It is to be noted that the decoration of CsPbI3 NCs on WS2 flakes not only improves the detector responsivity by more than 10-fold but also extends the spectral responsivity window up to 800 nm, as shown comparatively in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Hence, the decoration of WS2 active channel layer with excellent photoabsorbing CsPbI3 NCs appear to be a promising approach for next generation high performance optoelectronic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' To further investigate the role of CsPbI3 in photocarrier generation and efficient charge transfer, the electromagnetic simulations have been performed using the COMSOL Multiphysics software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Figure 4(c) shows the electric field distribution of the hybrid CsPbI3/WS2 device, illuminated with an electromagnetic plane wave of wavelength λ = 680 nm from top, which propagates through air and the nanostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The distribution clearly depicts that the electric field is trapped along the edges of the nano-cubes of CsPbI3 with the maximum confinement occurring near the base (as demonstrated by the colour index profile), resulting in strong charge transport in CsPbI3/WS2 hybrid heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further to explore the impact of gate bias on transistor performance, the photo-induced transfer characteristics (IDS−VGS) of the WS2/CsPbI3 MvWH photo-FET is recorded under the dark (black markers) and 514 nm illumination with a range of optical powers (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='1 μW to 35 μW) at a constant VDS of -2V, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The corresponding logarithmic current representation is depicted in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Under illumination of a fixed power of 35 μW, the drain current of the MvWH photo-FET is enhanced by ~ 13 times (from ~ 20 nA to ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='26 μA) at a constant gate voltage of ~ 40V, manifested by the strong photoabsorption in CsPbI3 and subsequent transfer of photoexcited electrons to the WS2 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further, with the increase of laser power, the photocurrent significantly increases in the accumulation region (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' VGS>VTh) and the transfer curves are gradually shifted to a negative gate voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 2(d), the favourable energy band alignment rules out the possibility of hole injection from CsPbI3 into WS2, leading to the trapped holes induced strong photogating effect in the hybrid system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' This leads to significant photocurrent increment in the accumulation region and negative threshold voltage shift (ΔVTh) with increasing incident power density of illumination [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' To investigate in greater detail, the calculated responsivity as a function of illumination power has been plotted for different gate bias voltages in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 4(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Here, the responsivity value increases with increasing positive gate bias in case of MvWH photo-FETs and reaches to a high value of ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='1 × 104 A W−1 at a back gate voltage of ~ 40 V under an illumination power of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='1 μW, which is quite remarkable compared to those previously reported 0D/2D hybrid phototransistors [37,39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Note that, for all the gate voltages, the measured responsivity dropped with increasing power because of the saturation of sensitizing traps in CsPbI3 NCs, which is a characteristic footprint of trap-dominated photoresponse [40– 42] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further, we have extracted the threshold voltage via extrapolating the linear region of each transfer curve under different incident laser powers and the shift in threshold voltage is plotted as a function of incident power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The variation is fitted with the power law function 𝑉𝑇ℎ ∝ 𝑃𝑏, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 4(f), to understand the possible photoconduction mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The extracted fitting exponent, b ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='17 clearly indicates a sublinear dependency on laser power confirming the existence of photogating dominant carrier conduction in MvWH photo- FETs [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further, it is also observed that the change in VTh is large in the lower power region and starts to saturate gradually at a higher power owing to the saturated trap states present in sensitizer interface leading to the saturation of the photogating effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The photocurrent IPh = IPhoto − IDark versus gate voltage for different illumination intensity [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 5(a)] shows a strong modulation with VGS, and a clear maximum in response can be identified around +35 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The strongest response of the FET device corresponds to the region with highest transconductance, due to the favourable Fermi level alignment, for low-contact resistance operation leading to many cycles of electron circulation to produce maximum gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Hitherto, in this region the FET device operates at a relatively higher dark current, compromising the signal-to-noise ratio (SNR) of the device, which is also a very important figure of merit of photo-FETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The SNR defined as IPhoto/IDark is illustrated in the same panel, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 5(a), which reveals the potential of 0D/2D hybrid phototransistors for highest sensitivity detection in its depletion regime with VGS from 0 to 5V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' In this region, a lowest dark current and a maximum sensitivity are achieved, despite the devices’ concurrent drop in the photocurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' So the maximum sensitivity of the device can be achieved via contact engineering where the transistor is operated in the depletion region, even without applying any gate bias, hitherto unreported for photo-FET devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' While the peak responsivity of our device is comparable or superior to the reported 2D materials based hybrid phototransistor devices with perovskite sensitizers, the sensitivity is found to be significantly higher without application of any external gate bias (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' These results illustrate the superior performance of broadband phototransistor, with ultrahigh sensitivity and responsivity, using CsPbI3 NCs sensitized 2D WS2 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (a) Back-gate bias dependent photocurrent (right axis) and photo-to-dark current ratio, (left axis) of the phototransistor device under five different illumination intensities (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='1 µW to 10 µW) for 514 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Despite the strongest photoresponse at higher gate bias (VGS ≈ 40 V), highest sensitivity of the device is achieved in the depletion regime (VGS ≈ 0V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The schematic representation of channel current transport mechanism and energy band diagram of the asymmetric contact hybrid phototransistor under reverse drain-source voltage with (b) zero and (c-d) different gate bias conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' On the other hand, a remarkable photoresponse of CsPbI3/WS2 MvWH photo-FET is explained by considering the influence of positive gate voltage on energy band alignment at the contact interfaces and heterostuctures leading to efficient charge injection into n-type WS2 channel Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Comparison of device performances with reported 2D material based hybrid photo- FETs with perovskite sensitizers (a) (b) hy High sen sitivity High photoresponse 104 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='1 μW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 μW (vu) 5 μW 10μW CsPbI3 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='00 60 Photo 10 Au WS2 Cr 30 10 ve +ve 100 00 20 0 20 40 Vcs (V) Underillumination (c) (p) hy CsPbl CsPbI3 Au Au Cr WS2 ve ve Cr WS2 +ve +ve Depletion region Accumulation regionDevice structure Sensitizer Operational spectral range Idark w/o applied VGS Iphoto/Idark @ VGS=0V Responsivity for different values of VGs Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Au / ML WS2 / Au CH3NH3PbI3 450-700 nm 5nA 104 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 A/W @ 0V [44] Au / ML MoS2 / Au CsPbBr3 350-550nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='2 nA 103 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='4 A/W @ 0V [45] Au / Ti / ML MoS2 / Ti / Au Ch3NH3PbBr3 / CsPbI3-xBrx 532 and 355 nm 4 nA 104 7 × 104 A/W @ 60V [46] Au / Ti / FL MoS2 / Ti / Au CsPbBr3 405 nm 10 nA 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='7 × 104 A/W @ 20V [47] Au / FL BP / Au CsPbBr3 405 nm 2 nA 102 357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='2 mA/W @ 0V [48] Au / ML MoS2 / Au CsPbI3-xBrx 532 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='2 µA 103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='13 × 105 A/W @ 60V [49] Au / FL BP / Al / Au MAPbI3−xClx 400-900 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='1 µA /µm 102 4 × 106 A/W @ 40V [50] Au / Ti / FL MoSe2(WSe2 ) / Ti / Au CsPb(Cl/Br)3 455 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='8 nA 10 102 A/W @ 50V [51] Au / Cr / FL Ta2NiSe5 / Cr / Au CH3NH3PbI3 800 nm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 µA 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='4 × 102 A/W @ 0V [37] Au / Cr / FL WS2 / Au / Cr α-phase CsPbI3 400-800 nm 2 pA 106 104 A/W @ 40V Our work layer from photoabsorbing CsPbI3 NCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 5(b), the Schottky barrier at the Au/WS2 interface is high enough to inhibit the charge conduction mechanism across the WS2 channel layer at reverse drain bias without any gate electric field under dark condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Hence, the transistor immediately goes to the OFF state with very low dark current in the order of pA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' At this condition, when the visible light is illuminated on the 0D/2D heterostructure, the photogeneration takes place in both CsPbI3 NCs as well as WS2 channel layer, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The effective photogenerated carrier separation takes place by the built-in electric field at the Schottky junction (WS2/Au) as well as at CsPbI3/WS2 interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The subsequent transition of photoexcited electrons from CsPbI3 to WS2 starts to populate the active channel layer which are collected by the external electrodes under an applied reverse VDS, leading to the photoresponsivity of ~ 102 A/W at VDS = ‒2V and VGS = 0V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further, the application of a back gate voltage (VGS) to the device modulates the Schottky barrier height at Au/WS2 interface as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 5(c)-(d) [52,53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' An application of negative gate bias (VGS < VTh) increases the barrier height leading to the transistor operation in the depletion region, where the photosensitivity (IPhoto/IDark) of the device is maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' On the other hand, on increasing the VGS beyond VTh initiates the lowering of the Schottky barrier at Au/WS2 interface, resulting in a higher magnitude of charge carrier injection from the Au electrode to WS2 channel layer through thermoionic as well as tunnelling mechanisms, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Thus, the cumulative effects of CsPbI3 NCs decoration mediated strong photogating phenomena as well as the gate voltage induced Schottky barrier lowering result in a drastic enhancement of the photocurrent (IPhoto − IDark) through the transistor channel at ON state (VGS > VTh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' This leads to an ultrahigh photoresponsivity of the order of ~ 104 A/W at VGS = 40 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Such gate modulated responsivity and sensitivity of MvWH photo-FET devices via interface engineering offers a novel pathway for next generation high performance and low power integrated photonic technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Temporal photoresponse is also an important parameter for the phototransistors performance in terms of switching speed and device stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The transient photoresponse of the as- fabricated CsPbI3/WS2 MvWH photo-FET upon visible illumination (𝜆 = 514 nm) at VGS = 0V with varying reverse VDS is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Upon illumination of four periodic pulses of the Argon-ion laser, relatively fast and consistent photocurrent modulation characteristics of the device reveals the stability and reproducibility of the as-fabricated MvWH photo-FET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The device exhibits a much stronger photoresponse characteristics revealing ratio of ~106 as compared to the control device with pristine WS2 with the value ~103 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S8 within the Supplimental Material), which is attributed to the injection of high density FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (a) Transient response of the MvWH photo-FET device under illumination of a 514 nm laser at different applied VDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (b) Temporal photocurrent response of the MvWH device for a wavelength of 514 nm with and without any applied gate bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The temporal response indicates a significant decrease in rise time (from 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='8 to 34 ms) as well as fall time (from 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='7 to 24 ms), measured at a relatively higher power of 35 µW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (c) Operational stability of the fabricated MvWH photo-FET device under visible illumination for more than half an hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (d) The transient photocurrent response of the fabricated transistor over a span of seven days from the beginning and end of the stability test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (e) Stability of the device tested under extreme humid conditions (varying from 50 to 65% RH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The last four cycle is the response under 65% of humidity showing around 5% decay in the photoresponse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' photogenerated charge carriers into the WS2 channel from strong light absorbing CsPbI3 NCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' With the increment of reverse VDS, a consistent photocurrent enhancement is distinctly noticed from the switching characteristics owing to the increase of depletion region width at the Schottky barrier interface and subsequent separation of photogenerated charge carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further, the rise and fall times of the fabricated device in the absence of gate bias have been estimated using an enlarged single cycle response [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 6(b)] and are found to be around ∼ 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='8 ms and ∼ 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='7 ms, respectively, which are further reduced to 34 ms and 24 ms, respectively on applying a gate voltage of 40 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The response speed of these devices are found to be relatively slower, which is attributed to the trapping of charge carriers in various structural (a) (c) 3 21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5V 1V 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0 HA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 DS DS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 ON ON 0 OFF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0 0 20 40 60 80 0 20 40 0 602 1205 1807 18901920 Time (sec) Time (sec) Time (sec) Time (sec) (b) (d) (e) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='2 oV 3 VDs = -2V 2=514nm 50% humidity 65% humidity GS Dav 1 Day3 Day 5 Dav 7 Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='8 Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='8 (vn) 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='4 DS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0 5% decay 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='75 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='80 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='85 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='90 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='95 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='00 Time (sec) Time (sec) Time (sec)and surface defect states present in WS2 as well as CsPbI3 NCs and their local junction interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' These interface traps present in WS2 layer are mostly empty when biased under depletion condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' VGS < VTh owing to lack of enough mobile carriers in the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' This allows a large number of photogenerated electrons to get trapped by the defect states while some of the gate induced electrons, although small in number, can be trapped as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' This results in a relatively slow rise of current, as depicted in the photocurrent dynamic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' On the other hand, the interface traps are nearly filled up with gate-induced electrons in accumulation condition, when VGS > VTh, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Hence, the trapping probability of photogenerated carriers is lower and a relatively faster response (~34 ms) is observed in MvWH photo-FET devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further, owing to the fact that the perovskite materials are prone to environmental degradation via oxygen diffusion through iodide vacancies upon illumination, the long term operation stability of the fabricated devices have been tested in this study upon visible light illumination at zero gate bias for prolonged duration (more than 60 min).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' From the I–t curves for the first 100 s [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 6(c), left] and the last 100 s [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 6(c), right], it is observed that the photocurrent has almost no attenuation, indicating that these devices show an excellent light stability under ambient condition, even without the use of a glovebox or encapsulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The device stability has also been tested via recording the photocurrent under illumination over a period of one week, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 6(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Here, the phototransistor sustains under laboratory ambient conditions (relative humidity (RH) ~ 45-50%, temperature ~ 22oC) for one week with negligible change in the photocurrent via degradation after storing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further, as CsPbI3 NCs are vulnerable to environmental humidity, to explore the device performance in the extreme humid condition, we have performed the temporal response under 65% RH showing an insignificant degradation (5% decay) in terms of device response [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 6(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' This superior performance stability is due to the surface defect passivation of CsPbI3 through the interaction with the sulphur of WS2 ensuring the outstanding environmental stability of as-fabricated CsPbI3/WS2 MvWH photo-FETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The sulfur atoms present on the top layer of WS2 may have stronger coordination to the Pb2+ centers of CsPbI3 NCs leads to reduced defect states in perovskites enabling higher reluctance to the degradation [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' It may be noted that the performance of the devices could be further improved by process optimization, device encapsulation and incorporation of buffer layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' This work reveals the significant potential of colloidal synthesized air-stable α-CsPbI3 NCs on 2D materials in fabricating 0D/2D mixed- dimensional heterostructure photo-FETs for applications in next generation optoelectronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Conclusion: To summarize, significant improvements in performance have been realized in CsPbI3/WS2 0D/2D mixed-dimensional phototransistors with asymmetric metal electrodes leading to combinatorial effect of Schottky barrier induced suppression of dark current and efficient charge transfer from photoabsorbing CsPbI3 nanocrystals, resulting in enhanced photosensitivity and spectral responsivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The WS2 channel with asymmetric contacts (Cr/WS2/Au) shows a rectifying I-V characteristics under an applied VDS with the dark current in the order of pA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Further, by combining the channel sensitization via decorating the WS2 with photosensitive air-stable α-phase CsPbI3 NCs, a responsivity of ~102 A/W has been achieved at low VDS (~ -2V) for an incident optical power of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='1 µW even without any external gate bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The device exhibits a broad spectral photoresponsivity between 400 and 800 nm due to the extended visible light absorption features of CsPbI3 NCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Using gate-controlled carrier modulation in the transistor channel, a peak responsivity ~104 A/W (VGS = +40 V) has been achieved owing to the photogating effect mediated charge conduction whereas the maximum sensitivity (~ 106 at ~ VDS = -2 V) in terms of signal-to-noise ratio is observed by depleting the channel carries (VGS = 0 to 5 V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' These devices show superior performance in terms of environment stability, owing to the filling of surface trap states present in CsPbI3 NCs via conjugation with sulfur atoms of 2D WS2 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The fabricated hybrid heterostructure devices combining 2D TMDs and superior light absorbing 0D perovskite nanocrsytals, through proper interface engineering, would open up new pathways for novel optoelectronic functionalities and energy-harvesting applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Acknowledgement: SKR 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H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Zhang, In Situ Preparation of a CsPbBr 3 /Black Phosphorus Heterostructure with an Optimized Interface and Photodetector Application, Nanoscale 11, 16852 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' [49] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Kang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Waheed, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Kamran, Formation of an MoTe 2 Based Schottky Junction Employing Ultra-Low and High Resistive Metal Contacts, RSC Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 9, 10017 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' [53] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Yang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Teo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Milne, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Hasko, Carbon Nanotube Schottky Diode and Directionally Dependent Field-Effect Transistor Using Asymmetrical Contacts, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 87, 253116 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Supplemental Material Charge transfer mediated giant photo-amplification in air-stable α-CsPbI3 nanocrystals decorated 2D-WS2 photo-FET with asymmetric contacts Shreyasi Das1, Arup Ghorai1,2, Sourabh Pal3, Somnath Mahato1, Soumen Das4, Samit K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Ray5 * 1School of Nano Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' IIT Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' West Bengal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' India,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 721302 2Department of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Pohang University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Pohang 790-784,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Korea 3Advanced Technology Development Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' IIT Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' West Bengal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' India,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 721302 4School of Medical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' IIT Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' West Bengal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' India,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 721302 5Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' IIT Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' West Bengal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' India,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 721302 Email : physkr@phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='iitkgp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='in S1: Tauc plot of CsPbI3 NCs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Tauc plot of as synthesised α-phase CsPbI3 NCs S2: Photoluminescence spectra of WS2, CsPbI3 and CsPbI3/WS2 hybrid Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Comparative PL spectrum of bare ML WS2 flake, CsPbI3 NCs and their heterostructures where three different concentrations of CsPbI3 NCs (different steps of spin coating) incorporated on WS2 flakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' After formation of heterostructures, the PL intensity of both bare ML WS2 as well as CsPbI3 NCs get reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (αhv)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=') E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='814 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='6 Energy (eV)PL Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=') Ws?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' CsPbI3 Step 1 Step 2 Step 3 550 600 650 700 750 Wavelength (nm)S3: PL integrated intensity ratio vs coating step Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Variation of PL integrated intensity ratio of WS2 trion peak (A-) to excitonic peak (A) with increasing concentration of CsPbI3 decoration (increasing spin coating step) on WS2 flakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S4: Thickness of the exfoliated flake analysis using AFM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Atomic force microscopy image of the few layer WS2 flakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Inset shows the height profile along the yellow dashed line confirming the thickness of the flakes around 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 7 Increasing CsPbI, 6 concentration 5 2 1 0 Step 3 Step 2 Step 1 Ws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0 5 10 15 20 25 30 35 40 45μm nm 30 Height (nm) 5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 25 10- 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 15 20 20 0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='2 15 Distance (um) 30- 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 10 35 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 40 5 45 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 50 umS5: Energy band structures of WS2 at the contacts Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' The corresponding energy band structures with different combination of metal electrodes before contact and after contact condition under applied reverse bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S6: Photo to dark current ratio Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Photo to dark current ratio for control device (WS2 FET) and MvWH photo-FET with varying drain to source voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Cr Cr WS2 Ohmic contact +ve ve Au Au WS2 Symmetric contact +ve ve Au ve Cr +ve WS2 Asymmetric contact Schottky diode Au Cr WS2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='6 eV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 eV 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='1 eV Evac EF Before contact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='2x10 Ws 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0x1( sPbI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' rk Dal 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0x10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0x10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='0 Vps (V)S7: Spectral responsivity of the control WS2 FET device Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Spectral responsivity curves of control WS2 FET device at VGS = 0V with increasing reverse VDS from 0V to -2V S8: Transient response of the control WS2 FET device Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' Transient response of the control WS2 FET device under illumination of a 514 nm laser at different applied VDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content=' 300 400 500 600 700 800 900 0 3 5 8 10 Responsivity (A/W) WS2 Wavelength (nm) 2V 1V 0V 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} +page_content='5V 1V 40 6 (nA) 4 2 ON OFF ON 0 0 20 40 60 80 Time (sec)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfrgVp/content/2301.03355v1.pdf'} diff --git 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b/KNE2T4oBgHgl3EQfpghg/content/tmp_files/2301.04029v1.pdf.txt @@ -0,0 +1,1098 @@ +arXiv:2301.04029v1 [math.CO] 10 Jan 2023 +On the set of stable matchings of a bipartite graph +Alexander V. Karzanov ∗ +Abstract. The topic of stable matchings (marriages) in a bipartite graph +has become widely popular, starting with the appearance of the classical +work by Gale and Shapley. We give a detailed survey on selected known +results in this field that demonstrate structural, polyhedral and algorithmic +properties of such matchings and their sets, providing our description with +relatively short proofs. (The paper is written in Russian.) +1 +Введение +Начиная с классической работы Гейла и Шепли [3], область задач о стабильных +марьяжах и их обобщений привлекала внимание многочисленных исследователей, +и в этой области был собран богатый урожай интересных результатов, как теоре- +тического, так и прикладного характера. +В задаче о стабильном марьяже (или, более определенно, о стабильном матчин- +ге в двудольном графе) рассматривается двудольный граф G = (V, E), в котором +для каждой вершины v задан линейный порядок u e′, то e′ >v e′′. +Действительно, пусть для определенности e′ ∈ M. Тогда e, e′′ ∈ L. Если бы +было e′ v1 e2. +Ввиду этого, последовательно применяя (2.1) к тройкам соседних ребер в P, +получаем, что ek−1 >vk−1 ek. Но тогда при нечетном k ребро ek является блокиру- +ющим для L (так как ek−1 ∈ L, ek /∈ L и vk /∈ VL), а при четном k ребро ek является +блокирующим для M (так как ek−1 ∈ M, ek /∈ M и vk /∈ VM); противоречие. +□ +Будем обозначать множество вершин в G, покрытых стабильным матчингом, +через �V . Очевидно, при удалении вершин в V −�V каждый стабильный матчинг для +G остается стабильным и для результирующего графа �G. Однако в �G могут по- +явиться новые стабильные матчинги (и, следовательно, включение M(G) ⊆ M( �G) +может быть строгим), как показывает простой пример в левом фрагменте на Рис. 1 +Здесь отношения предпочтений устроены так: a < b, b < c, c < d, d < e < a, +и можно проверить, что имеется единственный стабильный матчинг, а именно, +M = {b, d}. Тогда вершина v не покрыта M, однако при ее удалении (вместе с +ребром e) появляется новый стабильный матчинг {a, c}. +Таким образом, при исследовании множества стабильных матчингов для G +мы, вообще говоря, не можем выкидывать из рассмотрения непокрытые верши- +ны, оставляя только подграф �G = (�V = (�I ⊔ �J), �E) (где доли �I = I ∩ �V и �J = J ∩ �V +5 + +a +b +c +d +e +a +c +b +d +e +b' +a' +d' +c' +v +Рис. 1: Два примера +имеют одинаковый размер, и все стабильные матчинги совершенные). По похожим +причинам в случае отсутствия непокрытых вершин удаление ребра, не входяще- +го ни в один стабильный матчинг, может повлечь появление нового стабильного +матчинга. (В правом фрагменте на Рис. 1 изображено расширение предыдущего +графа с аналогичными предпочтениями для новых ребер, а именно, a′ < b′, b′ < c′, +c′ < d′ и d′ < e < a′. Здесь имеются три стабильных матчинга, а именно, {a, c, b′d′}, +{b, d, a′, c′} и {b, d, b′, d′}, однако при удалении непокрытого ребра e появляется чет- +вертый стабильный матчинг {a, c, a′, c′}, для которого ранее имелось блокирующее +ребро e.) +II. Далее рассмотрим упорядоченную пару (M, L) стабильных матчингов в G. +Из Предложения 2.2 следует, что граф ∆M,L, индуцированный M△L, распа- +дается на некоторое множество C = C(M, L) непересекающихся чередующих- +ся циклов. Считая G ориентированным от I к J, будем полагать каждый цикл +C = (v0, e1, v1, . . . , ek, vk = v0) ∈ C направленным согласно ориентации ребер в L, +т.е. прямые ребра в C принадлежат L, а обратные принадлежат M. Ввиду (2.1), +все предпочтения вдоль C “направлены в одну сторону”, а именно, +(2.2) для цикла C = e1e2 · · · ek ∈ C(M, L) (применяя обозначение C через ребра), +если ei < ei+1 выполнено для некоторого i, то это выполняется для всех +i = 1, . . . , k (полагая ek+1 := e1). +В этом случае скажем, что цикл C повышающий, или правый, относительно +M. Иначе C считается понижающим, или левым. Обозначим C+(M, L) и C−(M, L) +множества правых и левых циклов для (M, L), соответственно. Если матчинг M′ +получается из M заменой ребер вдоль цикла C будем писать M +C +−→ M′ или M′ = +Repl(M, C), и аналогично, будем писать L +C +−→ L′ или L′ = Repl(L, C) для замены +вдоль C в L. Заметим, что если цикл C правый относительно M, то по сравнению +с M матчинг M′ является менее предпочтительным для всех вершин в I ∩ C +(“men”) и более предпочтительным для всех вершин в J ∩ C (“women”), а для L +и L′ поведение противоположное. (Иначе говоря, при замене вдоль повышающего +цикла мы как бы удаляемся от Mmin и приближаемся к Mmax.) Повышающий цикл +показан на Рис. 2. +Можно было бы ожидать, что матчинг M′ = Repl(M, C) должен быть стабиль- +ным. Однако это может быть неверным. Например, рассмотрим граф как в правом +фрагменте на Рис. 1 (с указанными выше предпочтениями) и возьмем стабильные +матчинги M = {b, d, a′, c′} и L = {a, c, b′, d′}. Тогда ребра a, b, c, d порождают цикл +6 + +m1 +m2 +m3 +mk +w1 +w2 +w3 +wk +< +< +< +< +< +< +< +< +. . . +Рис. 2: Повышающий цикл для (M, L). Матчинг M изображен жирным, а L – +тонким. Вершины mi принадлежат доле I. +C в ∆M,L, но матчинг M′ = {a, c, a′, c′}, получающийся в результате замены ребер +в M вдоль C, не является стабильным. +Тем не менее, стабильность сохраняется при замене сразу во всех циклах в +C+(M, L) или в C−(M, L). В этих случаях будем писать M′ = Repl(M, C+(M, L)) и +M′ = Repl(M, C−(M, L)), соответственно. +Лемма 2.3 [10] Если M, L – стабильные матчинги, то M′ = Repl(M, C+(M, L)) +– тоже стабильный матчинг. Аналогично для Repl(M, C−(M, L)). +Доказательство Очевидно, M′ является матчингом. Предположим, M′ имеет +блокирующее ребро e = mw (где m ∈ I). Легко видеть, что это в принципе воз- +можно только если одна из вершин m, w принадлежит правому циклу, а другая – +левому циклу в C(M, L). Без ограничения общности можно считать, что m при- +надлежит циклу C ∈ C+(M, L), а w – циклу C′ ∈ C−(M, L). Пусть вершина m +инцидентна ребрам a ∈ M и b ∈ L в C, а w инцидентна ребрам c ∈ M и d ∈ L в C′. +Тогда a 0} и V − := {v ∈ VQ : ζ(v) < 0}. Граф �Q +получается из Q добавлением двух вершин: “источника” s и “стока” t, а также +множества ребер E+ из s в v для всех v ∈ V + и множества ребер E− из u в t для +всех u ∈ V −. Пропускные способности ребер e ∈ �E задаются так: +h(e) := + + + +ζ(v), +если e = sv ∈ E+; +|ζ(u)|, +если e = ut ∈ E−; +∞, +если e ∈ EQ. +Для подмножества S ⊆ �V такого, что s ∈ X ̸∋ t, обозначим через δ(S) мно- +жество ребер в �Q, идущих из S в �V − S, называемое s–t разрезом; величина +h(δ(S)) := �(h(e): e ∈ δ(S)) считается пропускной способностью этого разреза. +Лемма 5.1 [13] Подмножество X ⊆ VQ является замкнутым множеством ми- +нимального веса в (Q, ζ) тогда и только тогда, когда δ((VQ − X) ∪ {s}) является +s–t разрезом минимальной пропускной способности в ( �Q, h). +Доказательство Заметим, что X ⊆ VQ – замкнутое множество тогда и только +тогда, когда разрез E′ = δ((VQ − X) ∪ {s}) не содержит ребер бесконечной про- +пускной способности; эквивалентно, E′ содержится в E+∪E−. Для такого разреза, +состоящего из ребер вида sv для v ∈ X и ребер вида ut для u ∈ VQ−X, пропускная +способность выглядит так: +h(E′) = ζ(X ∩ V +) + +� +(|ζ(u)|: u ∈ (VQ − X) ∩ V −) += ζ(X ∩ V +) + ζ(X ∩ V −) − ζ(V −) = ζ(X) − ζ(V −). +Следовательно, вес замкнутого множества отличается от пропускной способно- +сти соответствующего разреза на постоянную величину −ζ(V −), откуда получаем +требуемое утверждение. +□ +Таким образом, задача (5.1) сводится к задаче о минимильном двухполюсном +разрезе в сети с N = O(|E|) вершинами и A = O(|E|) ребрами (учитывая то, +что вместо всего посета (RG, ⋖) достаточно рассматривать порождающий граф +H, имеющий (O(|E|) ребер (см. Предложение 4.2). Применяя быстрые алгоритмы +для задачи о максимальном потоке и минимальном разрезе (см., например, обзор +в [18, Sec. 10.8]), можно получить временную оценку O(NA log N) ≃ O(|V |4 log |V |). +В [8] приведена имплементация, решающая (5.1) за время O(|V |4). +Замечание. В [9] доказывалась труднорешаемость некоторых вариантов задачи +о замкнутых множествах. Используя один из них, а также тот факт, что любой +транзитивно замкнутый граф реализуется как посет ротаций (о чем будет сказано +в разделе 7), можно показать NP-трудность следующего усиления задачи (5.1): +для двудольного G = (V, E, <), функций c, g : E → R+ и числа K ∈ R+ найти +18 + +стабильный матчинг M ∈ M(G), минимизирующий c(M) при условии g(M) ≥ +K. (Здесь g(M) можно интерпретировать как прибыль, а c(M) как затраты при +организации союзов (или контрактов) в M.) +6 +Полиэдральные аспекты и медианные стабильные мат- +чинги +Как мы упоминали ранее, для двудольного графа G = (V = I ⊔ J, E, <) имеет- +ся полиэдральная характеризация многогранника стабильных матчингов Pst(G), +задаваемая линейным от |V |, |E| числом неравенств. Здесь Pst(G) – выпуклая обо- +лочка множества характеристических векторов χM стабильных матчингов M в +пространстве RE. Первоначальное описание Pst(G) (в случае G ≃ Kn,n) было дано +в работе Ванде Вейта [23] (и повторена в работе [16]). Ниже мы приводим описание +(несколько отличающееся по виду, но близкое к тому, что в [23]) и доказательство, +основываясь на изложении в [18, Sec. 18.5g]. +Для ребер e, f ∈ E будем писать f ≺ e, если они имеют общую вершину v, и +выполняется f 0} и обозначим V + множество вершин в G, +покрытых E+. Для v ∈ V + обозначим ev наилучшее ребро в δ(v)∩E+ относительно +порядка u e) ≤ 1 − +� +(x(f): f >u e) (применяя (6.2) к u). +19 + +Здесь все неравенства должны обращаться в равенства. Это дает �(x(f): f >u +e) = 0 и x(δ(u)) = 1, откуда следует (6.4). +Образуем множества M := {e ∈ E+ : e = ev для v ∈ I} и L := {e ∈ E+ : e = +ev для v ∈ J}. Для любой вершины v ∈ I ∩ V + наилучшее ребро в δ(v) ∩ E+ +принадлежит M, и наихудшее ребро, и только оно, принадлежит L (ввиду (6.4)); +для вершин в J ∩V + поведение обратное. Отсюда следует, что оба M и L являются +матчингами. Каждое ребро e ∈ M ∩ L образует компоненту в подграфе (V +, E+); +это дает x(e) = 1 (ввиду (6.2),(6.3)). В частности, x целочисленный, если M = L. +Пусть теперь M ̸= L. Очевидно, для любого ребра e в M′ := M − L или в +L′ := L−M выполняется 0 < x(e) < 1. Поэтому можно выбрать достаточно малое +число ε > 0 так, чтобы вектора x′ := x + εχM′ − εχL′ удовлетворяли (6.1) и (6.2). +Проверим, что оба x′ и x′′ удовлетворяют также и (6.3). +Для этого рассмотрим ребро e = mw с x(e) < 1 и предположим, что x′(γ(e)) < +x(γ(e)). Это возможно только если a ≤w e 0 следует +x(γ(e)) = +� +(x(f): f ≤w e) ≤ x(δ(w)) − x(b) < 1, +что невозможно. Аналогично, (6.3) верно для x′′. +Таким образом, x′, x′′ ∈ P′. Но x′ ̸= x′′ и (x′ + x′′)/2 = x. Это противоречит +тому, что x – вершина в P′. +□ +Имеется еще одно полезное свойство, показанное в [17]; мы приводим его в +несколько иной, но эквивалентной, форме. +Лемма 6.2 +Пусть x ∈ Pst(G), и пусть e = mw – ребро в G, для которого +x(e) > 0. Тогда x(δ(m)) = x(δ(w)) = x(γ(e)) = 1. +Доказательство Представим x как α1χM1 + · · · + αkχMk, где Mi – стабильный +матчинг, αi > 0, и α1 + · · · + αk = 1. Обозначим xi := χMi. Ввиду x(e) > 0, ребро e +принадлежит некоторому Mi. Тогда xi(δ(m)) = xi(δ(w)) = xi(γ(e)) = 1. Аналогич- +ные равенства имеют место и для матчинга Mj, не содержащего e. Действительно, +так как Mi и Mj покрывают одно и то же множество вершин (ввиду Предложе- +ния 2.2), в Mj есть ребро e′, инцидентное m, и ребро e′′, инцидентное w. Более +того, рассматривая пару Mi, Mj и применяя (2.1), получаем либо e′ m e >w e′′. В обоих случаях в γ(e) попадает ровно одно ребро из e′, e′′; +поэтому xj(γ(e)) = 1. Теперь утверждение следует из α1 + · · · + αk = 1. +□ +Эта лемма помогает получить интересный результат Тео и Сетарамана. +Предложение 6.3 [21] Пусть M1, . . . , Mℓ – стабильные матчинги в G. Для +каждого m ∈ �I обозначим Em список (с возможными повторениями) ребер в +δ(m), принадлежащих этим матчингам, упорядоченный согласно 0 на ℓx(e) =: r +параллельных ребер e1, . . . , er. При этом продолжение порядка w · · · >w er. Требуемое утверждение для этого общего случая получается +повторением (с незначительными уточнениями) рассуждений для случая непере- +секающихся матчингов выше. +□ +Следствие 6.4 Если ℓ нечетно, то для k := (ℓ+1)/2 множество, состоящее из +k-х по порядку элементов в списках ребер Ev, инцидентных v и принадлежащих +M1, . . . , Mℓ, для всех вершин v ∈ �V , является стабильным матчингом. +Такой матчинг называют медианным для M1, . . . , Mℓ. В [21] был поставлен во- +прос о возможности эффективного нахождения медианного стабильного матчинга +среди всех стабильных матчингов в G (или “почти медианного”, когда число ста- +бильных матчингов четное). Нам это представляется маловероятным, в свете того, +что задача вычисления числа |M(G)| является труднорешаемой, о чем будет ска- +зано ниже. +21 + +7 +Дополнительные замечания +Кнут [10] указал примеры, когда число стабильных матчингов двудольного гра- +фа экспоненциально велико по сравнению с размером графа, и задал вопрос о +сложности точного вычисления этого числа. Ответ был дан Ирвингом и Лейте- +ром [7], которые показали труднорешаемость такой задачи (рассматривая графы +вида (Kn,n, <)). Ниже мы кратко поясним идею их доказательства. +Напомним некоторые понятия (отсылая за точными определениями к [22] +или [5, Разд. 7.3]). Не вдаваясь в полную логическую строгость, можно пони- +мать, что описание той или иной перечислительной задачи (enumeration problem) +P состоит из бесконечного семейства S конечных множеств, и для каждого S ∈ S +имеется семейство F(S) подмножеств в S (“объектов”). В задаче P требуется для +заданного S ∈ S определить число |F(S)|. Говорят, что P является #P-задачей +(или KP-задачей, в терминологии некоторых авторов), если распознавание объ- +екта проводится за полиномиальное время, т.е. имеется алгоритм, который для +любых S ∈ S и X ⊆ S за время, полиномиальное от |S|, определяет принад- +лежит или нет данное множество X семейству F(S). Говорят, что #P-задача +P = {F(S), S ∈ S} является #P-полной (или универсальной в классе #P), если +любая другая #P-задача P′ = {F ′(S′), S′ ∈ S′} сводится к ней за полиномиальное +время (т.е. есть отображение ω : S′ → S такое, что для каждого S′ ∈ S число +|F ′(S′)| определяется из |F(ω(S′))| за время, полиномиальное от |S′|). +В рассматриваемой нами задаче роль cемейства S играет совокупность ребер- +ных множеств E двудольных графов G = (V, E, <), а роль семейства F(S), S ∈ S, +– соответствующее множество стабильных матчингов в G. Задача определения +|M(G)| – это действительно #P-задача, так как для любого подмножества M ⊆ E +можно определить, является ли M стабильным матчингом, за время O(|E|). +Заметим, что #P-аналог любой NP-полной задачи является “труднорешае- +мым” (поскольку в последней задаче требуется “всего лишь” определить, является +ли непустым соответствующее семейство объектов F(S) (например, содержит ли +заданный граф хотя бы один гамильтонов цикл), а в первой нужно найти количе- +ство объектов). Однако есть P-задачи, чьи перечислительные аналоги являются +#P-полными. Именно таковой и является задача определения |M(G)|. Это выте- +кает из следующих двух результатов. +Предложение 7.1 Задача определения числа анти-цепей в конечном посете яв- +ляется #P-полной. +Предложение 7.2 Пусть (P, <′) – посет на n элементах. Существует и +может быть построен за время полиномиальное от n двудольный граф +G = (V, E, <) такой, что его ротационный посет (RG, ⋖) изоморфен (P, <′). Сле- +довательно (ввиду Предложения 4.1), число |M(G)| стабильных матчингов в G +равно числу анти-цепей (или числу идеалов) в посете (P, <′). +Предложение 7.1 было установлено Прованом и Боллом [15]. Предложение 7.2 +было доказано в [7, Sec. 5] путем явной конструкции требуемого графа G для +заданного посета (P, <′). +22 + +Список литературы +[1] M. Baiou and M. Balinski, Erratum: the stable allocation (or ordinal transportation) +problem. Math. Oper. Res. 27 (4) (2002) 662–680. +[2] B.C. Dean and S. Munshi, Faster algorithms for stable allocation problems. Algorithmica +58 (1) (2010) 59–81. +[3] D. Gale and L.S. Shapley, College admissions and the stability of marriage. Amer. Math. +Monthly 69 (1) (1962) 9–15. +[4] D. Gusfield, Three fast algorithms for four problems in stable marriage. SIAM J. Comput. +16 (1987) 111–128. +[5] М. Гэри, Д. Джонсон, Вычислительные машины и труднорешаемые задачи, М.: +Мир, 1982 +[6] R.W. Irving, An efficient algorithm for the “stable roommates” problem. 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Ball, The complexity of counting cuts and of computing the probability +that a graph is connected. SIAM J. Comput. 12 (1983) 777–788. +[16] U.G. Rothblum, Characterization of stable matchings as extreme points of a polytope. +Math. Programming 54 (1992) 57–67. +[17] A.E. Roth, U.G. Rothblum, J.H. Vande Vate, Stable matching, optimal assignments and +linear programming. Math. Oper. Res. 18 (1993) 808–828. +[18] A. Schrijver, Combinatorial Optimization, Vol. A, Springer, 2003. +23 + +[19] J. Tan, A necessary and sufficient condition for the existence of a complete stable +matching. J. Algorithms 12 (1991) 154–178. +[20] `E. Tardos, A strongly polynomial algorithm to solve combinatorial linear problems. Oper. +Research 34 (1986) 250–256. +[21] C.P. Teo, J. Sethuraman. The geometry of fractional stable matchings and its +applications. Math. Oper. Res. 23 (4) (1998) 874–891. +[22] L.G. Valiant, The complexity of computing the permanent. Theoret. Comp. Sci. 8 (1979) +189–201. +[23] J.H. Vande Vate, Linear programming brings marital bliss. Oper. Res. Lett. 8 (1989) +147–153. +24 + diff --git a/KNE2T4oBgHgl3EQfpghg/content/tmp_files/load_file.txt b/KNE2T4oBgHgl3EQfpghg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c09b935e934070de1c4626635b9cb75b6b906af --- /dev/null +++ b/KNE2T4oBgHgl3EQfpghg/content/tmp_files/load_file.txt @@ -0,0 +1,905 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfpghg/content/2301.04029v1.pdf,len=904 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfpghg/content/2301.04029v1.pdf'} +page_content='04029v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfpghg/content/2301.04029v1.pdf'} +page_content='CO] 10 Jan 2023 On the set of stable matchings of a bipartite graph Alexander V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfpghg/content/2301.04029v1.pdf'} +page_content=' Karzanov ∗ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfpghg/content/2301.04029v1.pdf'} +page_content=' The topic of stable matchings (marriages) in a bipartite graph has become widely popular, starting with the appearance of the classical work by Gale and Shapley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfpghg/content/2301.04029v1.pdf'} +page_content=' We give a detailed survey on selected known results in this field that demonstrate structural, polyhedral and algorithmic properties of such matchings and their sets, providing our description with relatively short proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfpghg/content/2301.04029v1.pdf'} +page_content=' (The paper is written in Russian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfpghg/content/2301.04029v1.pdf'} +page_content=') 1 Введение Начиная с классической работы Гейла и Шепли [3], область задач о стабильных марьяжах и их обобщений привлекала внимание многочисленных исследователей, и в этой области был собран богатый урожай интересных результатов, как теоре- тического, так и прикладного характера.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfpghg/content/2301.04029v1.pdf'} +page_content=' В задаче о стабильном марьяже (или, более определенно, о стабильном матчин- ге в двудольном графе) рассматривается двудольный граф G = (V, E), в котором для каждой вершины v задан линейный порядок 0, x ∈ R}, we consider a nonlinear parabolic equation +ut + v(u)ux − t(a2(u)ux)x = 0, +(1) +where v(u), a(u) ∈ L∞(R), a(u) ≥ 0. Since the diffusion coefficient a(u) may take zero +value, equation (1) is degenerate. In the case when a(u) ≡ 0 it reduces to a first order +conservation law +ut + ϕ(u)x = 0, +(2) +where ϕ′(u) = v(u). Similarly, a general equation (1) can be written in the conservative +form +ut + ϕ(u)x − tA(u)xx = 0 +with A′(u) = a2(u), which allows to define weak solutions of this equation. Unfortunately, +weak solutions to a Cauchy problem for equation (1) are not unique in general, and some +additional entropy conditions are required. We consider the Cauchy problem with initial +data +u(0, x) = u0(x), +(3) +where u0(x) ∈ L∞(R). Recall the notion of entropy solution (e.s. for short) in the sense of +Carrillo [1]. +Definition 1. A function u = u(t, x) ∈ L∞(Π) is called an e.s. of (1), (3) if +(i) the distribution A(u) ∈ L2 +loc(Π); +(ii) for all k ∈ R +|u − k|t + (sign(u − k)(ϕ(u) − ϕ(k)))x − (t sign(u − k)(A(u) − A(k)))xx ≤ 0 +(4) +in the sense of distributions (in D′(Π)); +(iii) ess lim +t→0 +u(t, ·) = u0 in L1 +loc(R). +1 +arXiv:2301.13292v1 [math.AP] 30 Jan 2023 + +Entropy condition (4) means that for each nonnegative test function f = f(t, x) ∈ +C2 +0(Π) +� +Π +[|u − k|ft + sign(u − k)((ϕ(u) − ϕ(k))fx + t(A(u) − A(k))fxx)]dtdx ≥ 0. +(5) +In the case of conservation laws (2) the notion of e.s. reduces to the notion of generalized +e.s. in the sense of Kruzhkov [2]. Taking in (4) k = ±M, M ≥ ∥u∥∞, we derive that +ut + ϕ(u)x − tA(u)xx = 0 in D′(Π), +that is, an e.s. u of (1), (3) is a weak solution of this problem. It is known that an e.s. of (1), +(3) always exists and is unique. In general multidimensional setting this was demonstrated +in [2] for conservation laws and in [1] for the general case. If to be precise, in [1] the case of +usual diffusion term A(u)xx was studied but the proofs can be readily adapted to the case +of the self-similar diffusion tA(u)xx. +If u = u(t, x) is a piecewise C2-smooth e.s. of equation (1) then it must satisfy this +equation in classic sense in each smoothness domain. +Applying relation (1) to a test +function f = f(t, x) ∈ C2 +0(Π) supported in a neighborhood of a discontinuity line x = x(t) +and integrating by parts, we then obtain the identity +(−x′(t)[u] + [ϕ(u)] − t[A(u)x])f + t[A(u)]fx = 0 +(6) +a.e. on the line x = x(t). Here we denote by [w] the jump of a function w = w(t, x) on the +line x = x(t) so that +[w] = w(t, x(t)+) − w(t, x(t)−), +where w(t, x(t)±) = +lim +y→x(t)± w(t, y). +Since the functions f, fx are arbitrary and independent on the line x = x(t), identity (6) +implies the following two relations of Rankine-Hugoniot type +[A(u)] = 0, +(7) +−x′(t)[u] + [ϕ(u)] − t[A(u)x] = 0. +(8) +Similarly, it follows from entropy relation (5), after integration by parts, that +(−x′(t)[|u − k|] + [sign(u − k)(ϕ(u) − ϕ(k))] − t[sign(u − k)A(u)x])f+ +t[sign(u − k)(A(u) − A(k))]fx ≤ 0. +(9) +Since the function A(u) increases, it follows from (7) that A(u) = const when u lies between +the values u(t, x(t)−) and u(t, x(t)+). This implies that [sign(u − k)(A(u) − A(k))] = 0 +and in view of arbitrariness of f ≥ 0 it follows from (9) that +− x′(t)[|u − k|] + [sign(u − k)(ϕ(u) − ϕ(k))] − t[sign(u − k)A(u)x] ≤ 0. +(10) +In the case when k lies out of the interval with the endpoints u± .= u(t, x(t)±) relation (10) +follows from (8) and fulfils with equality sign. When u− < k < u+ this relation reads +−x′(t)(u+ + u− − 2k) + ϕ(u+) + ϕ(u−) − 2ϕ(k) − t(A(u)+ +x + A(u)− +x ) ≤ 0, +2 + +where A(u)± +x = A(u)x(t, x(t)±). Adding (8) to this relation and dividing the result by 2, +we arrive at the following analogue of the famous Oleinik condition (see [3]) known for +conservation laws. +− x′(t)(u+ − k) + ϕ(u+) − tA(u)+ +x − ϕ(k) ≤ 0 +∀k ∈ [u−, u+]. +(11) +In the case u+ < u− this condition has the form +− x′(t)(u+ − k) + ϕ(u+) − tA(u)+ +x − ϕ(k) ≥ 0 +∀k ∈ [u+, u−] +(12) +and can be derived similarly. +Geometric interpretation of these conditions is that the +graph of the flux function ϕ(u) lies not below (not above) of the segment connecting the +points (u−, ϕ(u−) − tA(u)− +x ), (u+, ϕ(u+) − tA(u)+ +x ) when u− ≤ u ≤ u+ (respectively, when +u+ ≤ u ≤ u−), see Figure 1. We take here into account that in view of condition (8) +the vector (−x′(t), 1) is a normal to the indicated segment. We also notice that it follows +from relations (11), (12) with k = u± and from the Rankine-Hugoniot condition (8) that +A(u)± +x ≥ 0 (A(u)± +x ≤ 0) whenever u+ > u− (u+ < u−). +u +u- +u+ +y=φ(u) +y +tA(u)x +- +tA(u)x ++ +(-x',1) +k +Figure 1: Oleinik condition. +2 +The case of piecewise constant coefficients. +Below we will assume that the functions v(u), a(u) are piecewise constant, v(u) = vk, +a(u) = ak when uk < u < uk+1, k = 0, . . . , n − 1, where +α = u0 < u1 < · · · < un−1 < un = β. +We will study problem (1), (3) with the Riemann data u0(x) = +� α, +x < 0, +β, +x > 0. +Since this +problem is invariant under the scaling transformations t → λt, x → λx, λ > 0 then, by +the uniqueness, the e.s. u = u(t, x) is self-similar: u(t, x) = u(λt, λx). This implies that +u = u(x/t). Suppose that ak > 0. Then in a domain where uk < u(ξ) < uk+1 with ξ = x/t +equation (1) reduces to the second order ODE +(vk − ξ)u′ − a2 +ku′′ = 0, +3 + +the general solution of which is u = C1F((ξ − vk)/ak) + C2; C1, C2 = const, where +F(z) = +1 +√ +2π +� z +−∞ +e−s2/2ds +is the error function. Therefore, it is natural to seek the e.s. of our problem in the following +form +u(ξ) = +� uk + +uk+1−uk +F((ξk+1−vk)/ak)−F((ξk−vk)/ak)(F((ξ − vk)/ak) − F((ξk − vk)/ak)) +, +ak > 0, +uk +, +ak = 0, +(13) +ξk < ξ < ξk+1, k = 0, . . . , d, +where +d = +� n − 1 +, +an−1 > 0, +n +, +an−1 = 0, +− ∞ = ξ0 < ξ1 ≤ · · · ≤ ξd < ξd+1 = +∞, +and we agree that an = 0, F(−∞) = 0, F(+∞) = 1. We also assume that ξk+1 > ξk +whenever ak > 0. The rays x = ξkt for finite ξk are (weak or strong) discontinuity lines of +u, they correspond to discontinuity points ξk of the function u(ξ). Observe that conditions +(7), (8) turns into the following relations at points ξk +[A(u)] = A(u(ξk+)) − A(u(ξk−)) = 0, +(14) +−ξk[u] + [ϕ(u)] − [A(u)′] = −ξk(u(ξk+) − u(ξk−)) + ϕ(u(ξk+))− +ϕ(u(ξk−)) − A(u)′(ξk+) + A(u)′(ξk−) = 0. +(15) +Here w(ξk±) denotes unilateral limits of a function w(ξ) at the point ξk. Similarly, the +Oleinik condition (11) reads +− ξk(u(ξk+) − k) + ϕ(u(ξk+)) − A(u)′(ξk+) − ϕ(k) ≤ 0 +∀k ∈ [u(ξk−), u(ξk+)]. +(16) +Notice that our solution (13) is a nonstrictly increasing function of the self-similar variable +ξ and, therefore, u(ξk−) ≤ u(ξk+). +Let us firstly analyze the solution (13) in the case ξk−1 < ξk < ξk+1. If ak−1, ak > 0 +then u(ξk−) = u(ξk+) = uk so that condition (14) fulfils while (15) reduces to the equality +[A(u)′] = 0, which is revealed as +ak(uk+1 − uk) +F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)F ′((ξk − vk)/ak) = +ak−1(uk − uk−1) +F((ξk − vk−1)/ak−1) − F((ξk−1 − vk−1)/ak−1)F ′((ξk − vk−1)/ak−1). +(17) +In this situation ξ = ξk is a weak discontinuity point, the function u(ξ) itself is continuous, +only its derivative u′(ξ) may be discontinuous. Moreover, it follows from (17) that both +functions u(ξ) and u′(ξ) are continuous at point ξk if ak = ak−1 > 0. +If ak−1 > ak = 0 then again u(ξ) is continuous at uk and (15) reduces to the relation +ak−1(uk − uk−1) +F((ξk − vk−1)/ak−1) − F((ξk−1 − vk−1)/ak−1)F ′((ξk − vk−1)/ak−1) = 0, +(18) +4 + +which is impossible. If ak > ak−1 = 0 then u(ξk−) = uk−1 < uk = u(ξk+), that is, ξk is +a strong discontinuity point. Condition (14) holds because A(u) is constant on [uk−1, uk] +(A′(u) = a2 +k−1 = 0) while (15) turns into +(vk−1 − ξk)(uk − uk−1) − +ak(uk+1 − uk) +F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)F ′((ξk − vk)/ak) = 0, (19) +where we use the fact that ϕ(uk) − ϕ(uk−1) = vk−1(uk − uk−1). It remains to analyze the +situation when ak = ak−1 = 0. In this case again u(ξk−) = uk−1 < uk = u(ξk+) and +A(uk−1) = A(uk) while condition (15) turns into the simple relation +ξk = vk−1. +(20) +Finally, since the function ϕ(u) is affine on the segment [uk−1, uk] and A(u)′(ξk±) ≥ 0, then +entropy relation (16) is always satisfied. +Now we consider the case when there exists a nontrivial family of mutually equaled +values ξi, ξi = ξk for i = k, . . . , l, where l > k. We can assume that this family is maximal, +that is, +−∞ ≤ ξk−1 < ξk = · · · = ξl < ξl+1 ≤ +∞. +Then ai = 0 for i = k, . . . , l − 1, and the point ξ = c .= ξk is a discontinuity point of u(ξ) +with the unilateral limits +u(c+) = ul, u(c−) = uk′, +where k′ = +� k +, +ak−1 > 0, +k − 1 +, +ak−1 = 0 . +Since a(u) = 0 for u(c−) < u < u(c+), we find that A(u(c−)) = A(u(c+)) and condition +(14) is satisfied. Further, we notice that +l−1 +� +i=k′ +(−ξi+1(ui+1 − ui)) = −c +l−1 +� +i=k′ +(ui+1 − ui) = −c(ul − uk′), +l−1 +� +i=k′ +vi(ui+1 − ui) = +l−1 +� +i=k′ +(ϕ(ui+1) − ϕ(ui)) = ϕ(ul) − ϕ(uk′). +Therefore, condition (15) can be written in the form +l−1 +� +i=k′ +(vi − ξi+1)(ui+1 − ui) − (A(u)′(c+) − A(u)′(c−)) = 0, +(21) +where, as is easy to verify, +A(u)′(c−) = +� +0 +, +ak−1 = 0, +ak−1(uk−uk−1) +F((ξk−vk−1)/ak−1)−F((ξk−1−vk−1)/ak−1)F ′((ξk − vk−1)/ak−1) +, +ak−1 > 0, +(22) +A(u)′(c+) = +� +0 +, +al = 0, +al(ul+1−ul) +F((ξl+1−vl)/al)−F((ξl−vl)/al)F ′((ξl − vl)/al) +, +al > 0. +(23) +5 + +In the similar way we can write the Oleinik condition (16) as follows +l−1 +� +i=j +(vi − ξi+1)(ui+1 − ui) − A(u)′(c+) ≤ 0 +k′ < j < l. +(24) +We use here the fact the function ϕ(u) is piecewise affine and, therefore, it is enough to +verify the Oleinik condition (16) only at the nodal points k = uj. +The above reasoning remains valid also in the case when l = k. In this case, relation +(21) reduces to one of conditions (17), (18), (19), (20) while (24) is trivial. +3 +The entropy function +We introduce the convex cone Ω ⊂ Rd consisting of points ¯ξ = (ξ1, . . . , ξd) with increasing +coordinates, ξ1 ≤ ξ2 ≤ · · · ≤ ξd such that ξk+1 > ξk whenever ak > 0, k = 1, . . . , d − 1. +Each point ¯ξ ∈ Ω determines a function u(ξ) in correspondence with formula (13). Assume +firstly that ¯ξ ∈ Int Ω, that is, the values ξk are strictly increasing. Then conditions (17), +(18), (19), (20) coincides with the equality +∂ +∂ξk E(¯ξ) = 0, where +E(¯ξ) = − +� +k=0,...,n−1,ak>0 +(ak)2(uk+1 − uk) ln(F((ξk+1 − vk)/ak) − F((ξk − vk)/ak))+ +1 +2 +� +k=0,...,n−1,ak=0 +(uk+1 − uk)(ξk+1 − vk)2, +¯ξ = (ξ1, . . . , ξd) ∈ Ω. +(25) +We will call this function the entropy because it depends only on the discontinuities of a +solution. Thus, for ¯ξ ∈ Int Ω the e.s. (13) corresponds to a critical point of the entropy. We +are going to demonstrate that the entropy is strictly convex and coercive in Ω. Therefore, +it has a unique global minimum point in Ω. In the case when this minimum point lies in +Int Ω it is a unique critical point. +Obviously, E(¯ξ) ∈ C∞(Ω). Notice that for all k = 0, . . . , n − 1, such that ak > 0 +ln(F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)) < 0. +Therefore, all terms in expression (25) are nonnegative and, in particular, E(¯ξ) ≥ 0. +Proposition 1 (coercivity). The sets E(¯ξ) ≤ c are compact for each constant c ≥ 0. +Proof. If E(¯ξ) ≤ c then it follows from nonnegativity of all terms in (25) that for all +k = 0, . . . , n − 1 +−(ak)2(uk+1 − uk) ln(F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)) ≤ E(¯ξ) ≤ c if ak > 0, +(26) +(uk+1 − uk)(ξk+1 − vk)2/2 ≤ c if ak = 0. +(27) +Relation (26) implies the estimate +F((ξk+1 − vk)/ak) − F((ξk − vk)/ak) ≥ δ .= exp(−c/m) > 0, +(28) +where m = +min +k=0,...,n−1,ak>0(ak)2(uk+1 − uk) > 0. If a0 > 0 relation (28) with k = 0 reads +F((ξ1 − v0)/a0) > δ (notice that F((ξ0 − v0)/a0) = F(−∞) = 0), which implies that +ξ1 ≥ v0 + a0F −1(δ). +6 + +On the other hand, if a0 = 0 then (u1 − u0)(ξ1 − v0)2 ≤ 2c, in view of (27) with k = 0, and +ξ1 ≥ v0 − (2c/(u1 − u0))1/2. +In any case, +ξ1 ≥ r1 .= v0 + min(a0F −1(δ), −(2c/(u1 − u0))1/2). +(29) +To get an upper bound, we remark that in the case an−1 > 0 it follows from (28) with +k = d = n − 1 that F(−(ξn−1 − vn−1)/an−1) = 1 − F((ξn−1 − vn−1)/an−1) ≥ δ (observe that +F((ξn − vn−1)/an−1) = F(+∞) = 1), which implies the estimate +ξd ≤ vn−1 − an−1F −1(δ). +If an−1 = 0 then d = n and in view of inequality (27) with k = n − 1 we find (un − +un−1)(ξn − vn−1)2/2 ≤ c, that is, +ξd ≤ vn−1 + (2c/(un − un−1))1/2. +In both cases +ξd ≤ r2 .= vn−1 + max(−an−1F −1(δ), (2c/(un − un−1))1/2). +(30) +Since all coordinates of ¯ξ lie between ξ1 and ξd, estimates (29), (30) imply the bound +|¯ξ|∞ = max +k=1,...,d |ξk| ≤ r .= max(|r1|, |r2|). +Further, since F ′(x) = +1 +√ +2πe−x2/2 < 1, the function F(x) is Lipschitz with constant 1 and +it follows from (28) that +(ξk+1 − ξk)/ak ≥ F((ξk+1 − vk)/ak) − F((ξk − vk)/ak) ≥ δ, +k = 1, . . . , d − 1, ak > 0. +We find that +ξk+1 − ξk ≥ akδ +(this also includes the case ak = 0). We conclude tat the set E(¯ξ) ≤ c lies in the compact +set +K = { ¯ξ = (ξ1, . . . , ξd) ∈ Rd | |¯ξ|∞ ≤ r, ξk+1 − ξk ≥ akδ ∀k = 1, . . . , d − 1 }. +By the continuity of E(¯ξ) the set E(¯ξ) ≤ c is a closed subset of K and therefore is +compact. +We take c > N .= inf E(¯ξ). Then the set E(¯ξ) ≤ c is not empty. By Proposition 1 +this set is compact and therefore the continuous function E(¯ξ) reaches the minimal value +on it, which is evidently equal to N. We proved the existence of global minimum E(¯ξ0) = +min E(¯ξ). The uniqueness of the minimum point is a consequence of strict convexity of the +entropy, which is stated in Proposition 2 below. The following lemma plays a key role. +Lemma 1. The function P(x, y) = − ln(F(x) − F(y)) is strictly convex in the half-plane +x > y. +7 + +Proof. The function P(x, y) is infinitely differentiable in the domain x > y. To prove the +lemma, we need to establish that the Hessian D2P is positive definite at every point. By +the direct computation we find +∂2 +∂x2P(x, y) = (F ′(x))2 − F ′′(x)(F(x) − F(y)) +(F(x) − F(y))2 +, +∂2 +∂y2P(x, y) = (F ′(y))2 − F ′′(y)(F(y) − F(x)) +(F(x) − F(y))2 +, +∂2 +∂x∂yP(x, y) = − +F ′(x)F ′(y) +(F(x) − F(y))2. +We have to prove positive definiteness of the matrix Q = (F(x) − F(y))2D2P(x, y) with +the components +Q11 = (F ′(x))2 − F ′′(x)(F(x) − F(y)), +Q22 = (F ′(y))2 − F ′′(y)(F(y) − F(x)), Q12 = Q21 = −F ′(x)F ′(y). +Since F ′(x) = e−x2/2, then F ′′(x) = −xF ′(x) and the diagonal elements of this matrix can +be written in the form +Q11 = F ′(x)(x(F(x) − F(y)) + F ′(x)) = +F ′(x)(x(F(x) − F(y)) + (F ′(x) − F ′(y))) + F ′(x)F ′(y), +Q22 = F ′(y)(y(F(y) − F(x)) + (F ′(y) − F ′(x))) + F ′(x)F ′(y). +By Cauchy mean value theorem there exists such a value z ∈ (y, x) that +F ′(x) − F ′(y) +F(x) − F(y) = F ′′(z) +F ′(z) = −z. +Therefore, +Q11 = F ′(x)(F(x) − F(y))(x − z) + F ′(x)F ′(y), +Q22 = F ′(y)(F(x) − F(y))(z − y) + F ′(x)F ′(y), +and it follows that Q = R1 +F ′(x)F ′(y)R2, where R1 is a diagonal matrix with the positive +diagonal elements F ′(x)(F(x)−F(y))(x−z), F ′(y)(F(x)−F(y))(z−y) while R2 = +� 1 +−1 +−1 +1 +� +. +Since R1 > 0, R2 ≥ 0, then the matrix Q > 0, as was to be proved. +Corollary 1. The functions P(x, −∞) = − ln F(x), P(+∞, x) = − ln(1 − F(x)) of single +variable are strictly convex. +Proof. Since 1 − F(x) = F(−x), we see that P(+∞, x) = P(−x, −∞), and it is sufficient +to prove the strict convexity of the function P(x, −∞) = − ln F(x). By Lemma 1 in the +limit as y → −∞ we obtain that this function is convex, moreover, +0 ≤ (F(x))2 d2 +dx2P(x, −∞) = lim +y→−∞ Q11 = F ′(x)(xF(x) + F ′(x)). +If +d2 +dx2P(x, −∞) = 0 at some point x = x0 then 0 = x0F(x0)+F ′(x0) is the minimum of the +nonnegative function xF(x) + F ′(x). Therefore, its derivative (xF + F ′)′(x0) = 0. Since +F ′′(x) = −xF ′(x), this derivative +(xF + F ′)′(x0) = F(x0) + x0F ′(x0) + F ′′(x0) = F(x0) > 0. +But this contradicts our assumption. We conclude that +d2 +dx2P(x, −∞) > 0 and the function +P(x, −∞) is strictly convex. +8 + +Proposition 2 (convexity). The entropy function E(¯ξ) is strictly convex on Ω. +Proof. For k = 0, . . . , n − 1 we denote Pk(¯ξ) = − ln(F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)) +if ak > 0, and Pk(¯ξ) = (ξk+1 − vk)2 if ak = 0. In view of (25) the entropy E(¯ξ) is a linear +combination of the functions Pk with positive coefficients, and convexity of the entropy +readily follows from the statements of Lemma 1 and Corollary 1. To establish the strict +convexity, we have to demonstrate that the Hessian matrix D2E(¯ξ) is strictly positive. +Assume that for some ζ = (ζ1, . . . , ζd) ∈ Rd +D2E(¯ξ)ζ · ζ = +d +� +i,j=1 +∂2E(¯ξ) +∂ξi∂ξj +ζiζj = 0. +(31) +Since E(¯ξ) is a linear combination of convex functions Pk(¯ξ) with positive coefficients, we +find that +D2Pk(¯ξ)ζ · ζ = 0 +∀k = 0, . . . , n − 1. +This can be written in the form +� +i,j=k,k+1 +∂2Pk(¯ξ) +∂ξi∂ξj +ζiζj = 0 if 0 < k < n − 1, ak > 0; +∂2Pk(¯ξ) +∂ξ2 +k+1 +ζ2 +k+1 if k = 0 or ak = 0. +In view of Lemma 1 and Corollary 1 the functions Pk in above equalities are strictly convex +as functions of either two variables (ξk, ξk+1) or single variable ξk+1. +Therefore, these +equalities imply that in any case ζk+1 = 0, k = 0, . . . , n − 2, and ζn = 0 if an−1 = 0 (when +d = n). We conclude that all coordinates ζi = 0, i = 1, . . . , d. Hence, equality (31) can +hold only for ζ = 0 and the matrix D2P(¯ξ) > 0 for all ¯ξ ∈ Ω. This completes the proof. +4 +The variational formulation +Let ¯ξ0 = (ξ1, . . . , ξd) ∈ Ω be the unique minimum point of E(¯ξ). +The necessary and +sufficient condition for ¯ξ0 to be a minimum point is the following one +∇E(¯ξ0) · p ≥ 0 +∀p ∈ T(¯ξ0) = { p ∈ Rd | ∃h > 0 ¯ξ0 + hp ∈ Ω }, +(32) +so that T(¯ξ0) is the tangent cone to Ω at the point ¯ξ0. If ¯ξ0 ∈ Int Ω then T(¯ξ0) = Rd +and (32) reduces to the requirement ∇E(¯ξ0) = 0. As we have already demonstrated, this +requirement coincides with jump conditions (17), (18), (19), (20) for all k = 1, . . . , d. But +these conditions are equivalent to the statement that the function (13) is an e.s. of (1), +(3). In the general situation when ¯ξ0 can belong to the boundary of Ω, the coordinates of +¯ξ0 may coincides. Let ξk = · · · = ξl = c be a maximal family of coinciding coordinates, +that is, ξk−1 < ξk = ξl < ξl+1 (it is possible here that k = l). Then, as is easy to realize, +the vector p = (p1, . . . , pd), with arbitrary increasing coordinates pk ≤ · · · ≤ pl and with +zero remaining coordinates, belong to the tangent cone T(¯ξ0). In view of (32) +l +� +i=k +∂ +∂ξi +E(¯ξ0)pi ≥ 0 +9 + +for any such a vector. Using the summation by parts formula, we realize that the above +condition is equivalent to the following requirements +l +� +i=k +∂ +∂ξi +E(¯ξ0) = 0, +(33) +l +� +i=j +∂ +∂ξi +E(¯ξ0) ≥ 0, k < j ≤ l. +(34) +Recall that ai = 0 for k ≤ i < l. By the direct computation we find +∂ +∂ξi +E(¯ξ0) = (ui − ui−1)(ξi − vi−1), +k < i < l, +∂ +∂ξk +E(¯ξ0) = +� (uk − uk−1)(ξk − vk−1) +, +ak−1 = 0, +−A(u)′(c−) +, +ak−1 > 0; +∂ +∂ξl +E(¯ξ0) = A(u)′(c+), +where A(u)′(c±) are given by (22), (23). Putting these expressions into (33), (34), we +obtain exactly the jump conditions (21), (24). Therefore, the function (13) corresponding +to the point ¯ξ0 is an e.s. of (1), (3). Conversely, if (13) is an e.s. then relations (33), (34) +holds for all groups of coinciding coordinates. As is easy to verify, this is equivalent to the +criterion (32). We have proved our main result. +Theorem 1. The function (13) is an e.s. of (1), (3) if and only if ¯ξ0 = (ξ1, . . . , ξd) is the +minimum point of the entropy E(¯ξ). +Remark 1. Adding to the entropy (25) the constant +� +k=0,...,n−1,ak>0 +(ak)2(uk+1 − uk) ln((uk+1 − uk)/ak), +we obtain the alternative variant of the entropy +E1(¯ξ) = − +� +k=0,...,n−1,ak>0 +(ak)2(uk+1 − uk) ln +�F((ξk+1 − vk)/ak) − F((ξk − vk)/ak) +(uk+1 − uk)/ak +� ++1 +2 +� +k=0,...,n−1,ak=0 +(uk+1 − uk)(ξk+1 − vk)2. (35) +If we consider the values vk, ak as a piecewise constant approximation of an arbitrary +velocity function v(u) and, respectively, a diffusion function a(u) ≥ 0 then, passing in +(35) to the limit as max(uk+1 − uk) → 0, we find that the entropy E1(¯ξ) turns into the +variational functional +J(ξ) = − +� +{u∈[α,β],a(u)>0} +(a(u))2 ln(F ′((ξ(u) − v(u))/a(u))ξ′(u))du+ +1 +2 +� +{u∈[α,β],a(u)=0} +(ξ(u) − v(u))2du, +10 + +where ξ(u) is an increasing function on [α, β], which is expected to be the inverse function +to a self-similar solution u = u(ξ) of the problem (1), (3). Taking into account that +ln(F ′((ξ(u) − v(u))/a(u))ξ′(u)) = ln F ′((ξ(u) − v(u))/a(u)) + ln ξ′(u) = +−(ξ(u) − v(u))2 +2a2(u) ++ ln ξ′(u), +we may simplify the expression for the functional J(ξ) +J(ξ) = +� β +α +[(ξ(u) − v(u))2/2 − (a(u))2 ln(ξ′(u))]du. +(36) +We see that this functional is strictly convex. The corresponding Euler-Lagrange equation +has the form +ξ(u) − v(u) + ((a(u))2/ξ′(u))′ = 0. +(37) +Since u′(ξ) = 1/ξ′(u), u = u(ξ), we can transform (37) as follows +ξ(u) − v(u) + ((a(u))2u′(ξ))′ +u = 0. +Multiplying this equation by u′(ξ), we obtain the equation +(a2u′)′ = (v − ξ)u′, +u = u(ξ), +which is exactly our equation (1) written in the self-similar variable. +Remark 2. In the case of conservation laws (2) the e.s. u = u(ξ) of (2), (3) is piecewise +constant, and, by expression (13), +u(ξ) = uk, +ξk < ξ < ξk+1, k = 0, . . . , n, +where −∞ = ξ0 < ξ1 ≤ · · · ≤ ξn < ξn+1 = +∞. In this case the entropy function is +particularly simple, it is the quadratic function +E(¯ξ) = 1 +2 +n +� +k=1 +(uk − uk−1)(ξk − vk−1)2, +defined on the closed polyhedral cone +Ω = { ¯ξ = (ξ1, . . . , ξn) ∈ Rn | ξk+1 ≥ ξk ∀k = 1, . . . , n − 1 }. +Existence and uniqueness of a minimal point in this case is trivial. By Theorem 1 and +Remark 1 we obtain new, variational formulation of the entropy solution. +Acknowledgments +The research was supported by the Russian Science Foundation, grant 22-21-00344. +11 + +References +[1] J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. +Anal., 147 (1999), 269–361. +[2] S. N. Kruzhkov, First order quasilinear equations in several independent variables, Mat. +Sb. (N.S.), 81 (1970), 228–255. +[3] O. A. Oleinik, Uniqueness and stability of the generalized solution of the Cauchy prob- +lem for a quasi-linear equation, Uspekhi Mat. Nauk, 14:2(86) (1959), 165–170. +12 + diff --git a/KdFQT4oBgHgl3EQfTTZa/content/tmp_files/load_file.txt b/KdFQT4oBgHgl3EQfTTZa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e100172c72b74674b1d8397d01d25397e182f669 --- /dev/null +++ b/KdFQT4oBgHgl3EQfTTZa/content/tmp_files/load_file.txt @@ -0,0 +1,335 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf,len=334 +page_content='On the structure of entropy solutions to the Riemann problem for a degenerate nonlinear parabolic equation Evgeny Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Panov Yaroslav-the-Wise Novgorod State University, Veliky Novgorod, Russian Federation, Research and Development Center, Veliky Novgorod, Russian Federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Abstract We find an explicit form of entropy solutions to a Riemann problem for a de- generate nonlinear parabolic equation with piecewise constant velocity and diffusion coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' It is demonstrated that this solution corresponds to the minimum point of some strictly convex function of a finite number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' 1 Introduction In a half-plane Π = {(t, x) | t > 0, x ∈ R}, we consider a nonlinear parabolic equation ut + v(u)ux − t(a2(u)ux)x = 0, (1) where v(u), a(u) ∈ L∞(R), a(u) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Since the diffusion coefficient a(u) may take zero value, equation (1) is degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' In the case when a(u) ≡ 0 it reduces to a first order conservation law ut + ϕ(u)x = 0, (2) where ϕ′(u) = v(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Similarly, a general equation (1) can be written in the conservative form ut + ϕ(u)x − tA(u)xx = 0 with A′(u) = a2(u), which allows to define weak solutions of this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Unfortunately, weak solutions to a Cauchy problem for equation (1) are not unique in general, and some additional entropy conditions are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' We consider the Cauchy problem with initial data u(0, x) = u0(x), (3) where u0(x) ∈ L∞(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Recall the notion of entropy solution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' for short) in the sense of Carrillo [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' A function u = u(t, x) ∈ L∞(Π) is called an e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' of (1), (3) if (i) the distribution A(u) ∈ L2 loc(Π);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (ii) for all k ∈ R |u − k|t + (sign(u − k)(ϕ(u) − ϕ(k)))x − (t sign(u − k)(A(u) − A(k)))xx ≤ 0 (4) in the sense of distributions (in D′(Π));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (iii) ess lim t→0 u(t, ·) = u0 in L1 loc(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='13292v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='AP] 30 Jan 2023 Entropy condition (4) means that for each nonnegative test function f = f(t, x) ∈ C2 0(Π) � Π [|u − k|ft + sign(u − k)((ϕ(u) − ϕ(k))fx + t(A(u) − A(k))fxx)]dtdx ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (5) In the case of conservation laws (2) the notion of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' reduces to the notion of generalized e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' in the sense of Kruzhkov [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Taking in (4) k = ±M, M ≥ ∥u∥∞, we derive that ut + ϕ(u)x − tA(u)xx = 0 in D′(Π), that is, an e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' u of (1), (3) is a weak solution of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' It is known that an e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' of (1), (3) always exists and is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' In general multidimensional setting this was demonstrated in [2] for conservation laws and in [1] for the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' If to be precise, in [1] the case of usual diffusion term A(u)xx was studied but the proofs can be readily adapted to the case of the self-similar diffusion tA(u)xx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' If u = u(t, x) is a piecewise C2-smooth e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' of equation (1) then it must satisfy this equation in classic sense in each smoothness domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Applying relation (1) to a test function f = f(t, x) ∈ C2 0(Π) supported in a neighborhood of a discontinuity line x = x(t) and integrating by parts, we then obtain the identity (−x′(t)[u] + [ϕ(u)] − t[A(u)x])f + t[A(u)]fx = 0 (6) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' on the line x = x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Here we denote by [w] the jump of a function w = w(t, x) on the line x = x(t) so that [w] = w(t, x(t)+) − w(t, x(t)−), where w(t, x(t)±) = lim y→x(t)± w(t, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Since the functions f, fx are arbitrary and independent on the line x = x(t), identity (6) implies the following two relations of Rankine-Hugoniot type [A(u)] = 0, (7) −x′(t)[u] + [ϕ(u)] − t[A(u)x] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (8) Similarly, it follows from entropy relation (5), after integration by parts, that (−x′(t)[|u − k|] + [sign(u − k)(ϕ(u) − ϕ(k))] − t[sign(u − k)A(u)x])f+ t[sign(u − k)(A(u) − A(k))]fx ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (9) Since the function A(u) increases, it follows from (7) that A(u) = const when u lies between the values u(t, x(t)−) and u(t, x(t)+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' This implies that [sign(u − k)(A(u) − A(k))] = 0 and in view of arbitrariness of f ≥ 0 it follows from (9) that − x′(t)[|u − k|] + [sign(u − k)(ϕ(u) − ϕ(k))] − t[sign(u − k)A(u)x] ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (10) In the case when k lies out of the interval with the endpoints u± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='= u(t, x(t)±) relation (10) follows from (8) and fulfils with equality sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' When u− < k < u+ this relation reads −x′(t)(u+ + u− − 2k) + ϕ(u+) + ϕ(u−) − 2ϕ(k) − t(A(u)+ x + A(u)− x ) ≤ 0, 2 where A(u)± x = A(u)x(t, x(t)±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Adding (8) to this relation and dividing the result by 2, we arrive at the following analogue of the famous Oleinik condition (see [3]) known for conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' − x′(t)(u+ − k) + ϕ(u+) − tA(u)+ x − ϕ(k) ≤ 0 ∀k ∈ [u−, u+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (11) In the case u+ < u− this condition has the form − x′(t)(u+ − k) + ϕ(u+) − tA(u)+ x − ϕ(k) ≥ 0 ∀k ∈ [u+, u−] (12) and can be derived similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Geometric interpretation of these conditions is that the graph of the flux function ϕ(u) lies not below (not above) of the segment connecting the points (u−, ϕ(u−) − tA(u)− x ), (u+, ϕ(u+) − tA(u)+ x ) when u− ≤ u ≤ u+ (respectively, when u+ ≤ u ≤ u−), see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' We take here into account that in view of condition (8) the vector (−x′(t), 1) is a normal to the indicated segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' We also notice that it follows from relations (11), (12) with k = u± and from the Rankine-Hugoniot condition (8) that A(u)± x ≥ 0 (A(u)± x ≤ 0) whenever u+ > u− (u+ < u−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=" u u- u+ y=φ(u) y tA(u)x tA(u)x + (-x',1) k Figure 1: Oleinik condition." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' 2 The case of piecewise constant coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Below we will assume that the functions v(u), a(u) are piecewise constant, v(u) = vk, a(u) = ak when uk < u < uk+1, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , n − 1, where α = u0 < u1 < · · · < un−1 < un = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' We will study problem (1), (3) with the Riemann data u0(x) = � α, x < 0, β, x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Since this problem is invariant under the scaling transformations t → λt, x → λx, λ > 0 then, by the uniqueness, the e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' u = u(t, x) is self-similar: u(t, x) = u(λt, λx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' This implies that u = u(x/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Suppose that ak > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Then in a domain where uk < u(ξ) < uk+1 with ξ = x/t equation (1) reduces to the second order ODE (vk − ξ)u′ − a2 ku′′ = 0, 3 the general solution of which is u = C1F((ξ − vk)/ak) + C2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' C1, C2 = const, where F(z) = 1 √ 2π � z −∞ e−s2/2ds is the error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Therefore, it is natural to seek the e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' of our problem in the following form u(ξ) = � uk + uk+1−uk F((ξk+1−vk)/ak)−F((ξk−vk)/ak)(F((ξ − vk)/ak) − F((ξk − vk)/ak)) , ak > 0, uk , ak = 0, (13) ξk < ξ < ξk+1, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , d, where d = � n − 1 , an−1 > 0, n , an−1 = 0, − ∞ = ξ0 < ξ1 ≤ · · · ≤ ξd < ξd+1 = +∞, and we agree that an = 0, F(−∞) = 0, F(+∞) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' We also assume that ξk+1 > ξk whenever ak > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' The rays x = ξkt for finite ξk are (weak or strong) discontinuity lines of u, they correspond to discontinuity points ξk of the function u(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Observe that conditions (7), (8) turns into the following relations at points ξk [A(u)] = A(u(ξk+)) − A(u(ξk−)) = 0, (14) −ξk[u] + [ϕ(u)] − [A(u)′] = −ξk(u(ξk+) − u(ξk−)) + ϕ(u(ξk+))− ϕ(u(ξk−)) − A(u)′(ξk+) + A(u)′(ξk−) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (15) Here w(ξk±) denotes unilateral limits of a function w(ξ) at the point ξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Similarly, the Oleinik condition (11) reads − ξk(u(ξk+) − k) + ϕ(u(ξk+)) − A(u)′(ξk+) − ϕ(k) ≤ 0 ∀k ∈ [u(ξk−), u(ξk+)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (16) Notice that our solution (13) is a nonstrictly increasing function of the self-similar variable ξ and, therefore, u(ξk−) ≤ u(ξk+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Let us firstly analyze the solution (13) in the case ξk−1 < ξk < ξk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' If ak−1, ak > 0 then u(ξk−) = u(ξk+) = uk so that condition (14) fulfils while (15) reduces to the equality [A(u)′] = 0, which is revealed as ak(uk+1 − uk) F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)F ′((ξk − vk)/ak) = ak−1(uk − uk−1) F((ξk − vk−1)/ak−1) − F((ξk−1 − vk−1)/ak−1)F ′((ξk − vk−1)/ak−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (17) In this situation ξ = ξk is a weak discontinuity point, the function u(ξ) itself is continuous, only its derivative u′(ξ) may be discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Moreover, it follows from (17) that both functions u(ξ) and u′(ξ) are continuous at point ξk if ak = ak−1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' If ak−1 > ak = 0 then again u(ξ) is continuous at uk and (15) reduces to the relation ak−1(uk − uk−1) F((ξk − vk−1)/ak−1) − F((ξk−1 − vk−1)/ak−1)F ′((ξk − vk−1)/ak−1) = 0, (18) 4 which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' If ak > ak−1 = 0 then u(ξk−) = uk−1 < uk = u(ξk+), that is, ξk is a strong discontinuity point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Condition (14) holds because A(u) is constant on [uk−1, uk] (A′(u) = a2 k−1 = 0) while (15) turns into (vk−1 − ξk)(uk − uk−1) − ak(uk+1 − uk) F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)F ′((ξk − vk)/ak) = 0, (19) where we use the fact that ϕ(uk) − ϕ(uk−1) = vk−1(uk − uk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' It remains to analyze the situation when ak = ak−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' In this case again u(ξk−) = uk−1 < uk = u(ξk+) and A(uk−1) = A(uk) while condition (15) turns into the simple relation ξk = vk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (20) Finally, since the function ϕ(u) is affine on the segment [uk−1, uk] and A(u)′(ξk±) ≥ 0, then entropy relation (16) is always satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Now we consider the case when there exists a nontrivial family of mutually equaled values ξi, ξi = ξk for i = k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , l, where l > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' We can assume that this family is maximal, that is, −∞ ≤ ξk−1 < ξk = · · · = ξl < ξl+1 ≤ +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Then ai = 0 for i = k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , l − 1, and the point ξ = c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='= ξk is a discontinuity point of u(ξ) with the unilateral limits u(c+) = ul, u(c−) = uk′, where k′ = � k , ak−1 > 0, k − 1 , ak−1 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Since a(u) = 0 for u(c−) < u < u(c+), we find that A(u(c−)) = A(u(c+)) and condition (14) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Further, we notice that l−1 � i=k′ (−ξi+1(ui+1 − ui)) = −c l−1 � i=k′ (ui+1 − ui) = −c(ul − uk′), l−1 � i=k′ vi(ui+1 − ui) = l−1 � i=k′ (ϕ(ui+1) − ϕ(ui)) = ϕ(ul) − ϕ(uk′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Therefore, condition (15) can be written in the form l−1 � i=k′ (vi − ξi+1)(ui+1 − ui) − (A(u)′(c+) − A(u)′(c−)) = 0, (21) where, as is easy to verify, A(u)′(c−) = � 0 , ak−1 = 0, ak−1(uk−uk−1) F((ξk−vk−1)/ak−1)−F((ξk−1−vk−1)/ak−1)F ′((ξk − vk−1)/ak−1) , ak−1 > 0, (22) A(u)′(c+) = � 0 , al = 0, al(ul+1−ul) F((ξl+1−vl)/al)−F((ξl−vl)/al)F ′((ξl − vl)/al) , al > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (23) 5 In the similar way we can write the Oleinik condition (16) as follows l−1 � i=j (vi − ξi+1)(ui+1 − ui) − A(u)′(c+) ≤ 0 k′ < j < l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (24) We use here the fact the function ϕ(u) is piecewise affine and, therefore, it is enough to verify the Oleinik condition (16) only at the nodal points k = uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' The above reasoning remains valid also in the case when l = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' In this case, relation (21) reduces to one of conditions (17), (18), (19), (20) while (24) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' 3 The entropy function We introduce the convex cone Ω ⊂ Rd consisting of points ¯ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , ξd) with increasing coordinates, ξ1 ≤ ξ2 ≤ · · · ≤ ξd such that ξk+1 > ξk whenever ak > 0, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Each point ¯ξ ∈ Ω determines a function u(ξ) in correspondence with formula (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Assume firstly that ¯ξ ∈ Int Ω, that is, the values ξk are strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Then conditions (17), (18), (19), (20) coincides with the equality ∂ ∂ξk E(¯ξ) = 0, where E(¯ξ) = − � k=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=',n−1,ak>0 (ak)2(uk+1 − uk) ln(F((ξk+1 − vk)/ak) − F((ξk − vk)/ak))+ 1 2 � k=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=',n−1,ak=0 (uk+1 − uk)(ξk+1 − vk)2, ¯ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , ξd) ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (25) We will call this function the entropy because it depends only on the discontinuities of a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Thus, for ¯ξ ∈ Int Ω the e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (13) corresponds to a critical point of the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' We are going to demonstrate that the entropy is strictly convex and coercive in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Therefore, it has a unique global minimum point in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' In the case when this minimum point lies in Int Ω it is a unique critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Obviously, E(¯ξ) ∈ C∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Notice that for all k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , n − 1, such that ak > 0 ln(F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Therefore, all terms in expression (25) are nonnegative and, in particular, E(¯ξ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Proposition 1 (coercivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' The sets E(¯ξ) ≤ c are compact for each constant c ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' If E(¯ξ) ≤ c then it follows from nonnegativity of all terms in (25) that for all k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , n − 1 −(ak)2(uk+1 − uk) ln(F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)) ≤ E(¯ξ) ≤ c if ak > 0, (26) (uk+1 − uk)(ξk+1 − vk)2/2 ≤ c if ak = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (27) Relation (26) implies the estimate F((ξk+1 − vk)/ak) − F((ξk − vk)/ak) ≥ δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='= exp(−c/m) > 0, (28) where m = min k=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=',n−1,ak>0(ak)2(uk+1 − uk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' If a0 > 0 relation (28) with k = 0 reads F((ξ1 − v0)/a0) > δ (notice that F((ξ0 − v0)/a0) = F(−∞) = 0), which implies that ξ1 ≥ v0 + a0F −1(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' 6 On the other hand, if a0 = 0 then (u1 − u0)(ξ1 − v0)2 ≤ 2c, in view of (27) with k = 0, and ξ1 ≥ v0 − (2c/(u1 − u0))1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' In any case, ξ1 ≥ r1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='= v0 + min(a0F −1(δ), −(2c/(u1 − u0))1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (29) To get an upper bound, we remark that in the case an−1 > 0 it follows from (28) with k = d = n − 1 that F(−(ξn−1 − vn−1)/an−1) = 1 − F((ξn−1 − vn−1)/an−1) ≥ δ (observe that F((ξn − vn−1)/an−1) = F(+∞) = 1), which implies the estimate ξd ≤ vn−1 − an−1F −1(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' If an−1 = 0 then d = n and in view of inequality (27) with k = n − 1 we find (un − un−1)(ξn − vn−1)2/2 ≤ c, that is, ξd ≤ vn−1 + (2c/(un − un−1))1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' In both cases ξd ≤ r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='= vn−1 + max(−an−1F −1(δ), (2c/(un − un−1))1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (30) Since all coordinates of ¯ξ lie between ξ1 and ξd, estimates (29), (30) imply the bound |¯ξ|∞ = max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=',d |ξk| ≤ r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='= max(|r1|, |r2|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Further, since F ′(x) = 1 √ 2πe−x2/2 < 1, the function F(x) is Lipschitz with constant 1 and it follows from (28) that (ξk+1 − ξk)/ak ≥ F((ξk+1 − vk)/ak) − F((ξk − vk)/ak) ≥ δ, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , d − 1, ak > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' We find that ξk+1 − ξk ≥ akδ (this also includes the case ak = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' We conclude tat the set E(¯ξ) ≤ c lies in the compact set K = { ¯ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , ξd) ∈ Rd | |¯ξ|∞ ≤ r, ξk+1 − ξk ≥ akδ ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , d − 1 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' By the continuity of E(¯ξ) the set E(¯ξ) ≤ c is a closed subset of K and therefore is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' We take c > N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='= inf E(¯ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Then the set E(¯ξ) ≤ c is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' By Proposition 1 this set is compact and therefore the continuous function E(¯ξ) reaches the minimal value on it, which is evidently equal to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' We proved the existence of global minimum E(¯ξ0) = min E(¯ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' The uniqueness of the minimum point is a consequence of strict convexity of the entropy, which is stated in Proposition 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' The following lemma plays a key role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' The function P(x, y) = − ln(F(x) − F(y)) is strictly convex in the half-plane x > y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' The function P(x, y) is infinitely differentiable in the domain x > y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' To prove the lemma, we need to establish that the Hessian D2P is positive definite at every point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' By the direct computation we find ∂2 ∂x2P(x, y) = (F ′(x))2 − F ′′(x)(F(x) − F(y)) (F(x) − F(y))2 , ∂2 ∂y2P(x, y) = (F ′(y))2 − F ′′(y)(F(y) − F(x)) (F(x) − F(y))2 , ∂2 ∂x∂yP(x, y) = − F ′(x)F ′(y) (F(x) − F(y))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' We have to prove positive definiteness of the matrix Q = (F(x) − F(y))2D2P(x, y) with the components Q11 = (F ′(x))2 − F ′′(x)(F(x) − F(y)), Q22 = (F ′(y))2 − F ′′(y)(F(y) − F(x)), Q12 = Q21 = −F ′(x)F ′(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Since F ′(x) = e−x2/2, then F ′′(x) = −xF ′(x) and the diagonal elements of this matrix can be written in the form Q11 = F ′(x)(x(F(x) − F(y)) + F ′(x)) = F ′(x)(x(F(x) − F(y)) + (F ′(x) − F ′(y))) + F ′(x)F ′(y), Q22 = F ′(y)(y(F(y) − F(x)) + (F ′(y) − F ′(x))) + F ′(x)F ′(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' By Cauchy mean value theorem there exists such a value z ∈ (y, x) that F ′(x) − F ′(y) F(x) − F(y) = F ′′(z) F ′(z) = −z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Therefore, Q11 = F ′(x)(F(x) − F(y))(x − z) + F ′(x)F ′(y), Q22 = F ′(y)(F(x) − F(y))(z − y) + F ′(x)F ′(y), and it follows that Q = R1 +F ′(x)F ′(y)R2, where R1 is a diagonal matrix with the positive diagonal elements F ′(x)(F(x)−F(y))(x−z), F ′(y)(F(x)−F(y))(z−y) while R2 = � 1 −1 −1 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Since R1 > 0, R2 ≥ 0, then the matrix Q > 0, as was to be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' The functions P(x, −∞) = − ln F(x), P(+∞, x) = − ln(1 − F(x)) of single variable are strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Since 1 − F(x) = F(−x), we see that P(+∞, x) = P(−x, −∞), and it is sufficient to prove the strict convexity of the function P(x, −∞) = − ln F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' By Lemma 1 in the limit as y → −∞ we obtain that this function is convex, moreover, 0 ≤ (F(x))2 d2 dx2P(x, −∞) = lim y→−∞ Q11 = F ′(x)(xF(x) + F ′(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' If d2 dx2P(x, −∞) = 0 at some point x = x0 then 0 = x0F(x0)+F ′(x0) is the minimum of the nonnegative function xF(x) + F ′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Therefore, its derivative (xF + F ′)′(x0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Since F ′′(x) = −xF ′(x), this derivative (xF + F ′)′(x0) = F(x0) + x0F ′(x0) + F ′′(x0) = F(x0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' But this contradicts our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' We conclude that d2 dx2P(x, −∞) > 0 and the function P(x, −∞) is strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' 8 Proposition 2 (convexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' The entropy function E(¯ξ) is strictly convex on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' For k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , n − 1 we denote Pk(¯ξ) = − ln(F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)) if ak > 0, and Pk(¯ξ) = (ξk+1 − vk)2 if ak = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' In view of (25) the entropy E(¯ξ) is a linear combination of the functions Pk with positive coefficients, and convexity of the entropy readily follows from the statements of Lemma 1 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' To establish the strict convexity, we have to demonstrate that the Hessian matrix D2E(¯ξ) is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Assume that for some ζ = (ζ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , ζd) ∈ Rd D2E(¯ξ)ζ · ζ = d � i,j=1 ∂2E(¯ξ) ∂ξi∂ξj ζiζj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (31) Since E(¯ξ) is a linear combination of convex functions Pk(¯ξ) with positive coefficients, we find that D2Pk(¯ξ)ζ · ζ = 0 ∀k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' This can be written in the form � i,j=k,k+1 ∂2Pk(¯ξ) ∂ξi∂ξj ζiζj = 0 if 0 < k < n − 1, ak > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' ∂2Pk(¯ξ) ∂ξ2 k+1 ζ2 k+1 if k = 0 or ak = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' In view of Lemma 1 and Corollary 1 the functions Pk in above equalities are strictly convex as functions of either two variables (ξk, ξk+1) or single variable ξk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Therefore, these equalities imply that in any case ζk+1 = 0, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , n − 2, and ζn = 0 if an−1 = 0 (when d = n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' We conclude that all coordinates ζi = 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Hence, equality (31) can hold only for ζ = 0 and the matrix D2P(¯ξ) > 0 for all ¯ξ ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' 4 The variational formulation Let ¯ξ0 = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , ξd) ∈ Ω be the unique minimum point of E(¯ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' The necessary and sufficient condition for ¯ξ0 to be a minimum point is the following one ∇E(¯ξ0) · p ≥ 0 ∀p ∈ T(¯ξ0) = { p ∈ Rd | ∃h > 0 ¯ξ0 + hp ∈ Ω }, (32) so that T(¯ξ0) is the tangent cone to Ω at the point ¯ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' If ¯ξ0 ∈ Int Ω then T(¯ξ0) = Rd and (32) reduces to the requirement ∇E(¯ξ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' As we have already demonstrated, this requirement coincides with jump conditions (17), (18), (19), (20) for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' But these conditions are equivalent to the statement that the function (13) is an e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' of (1), (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' In the general situation when ¯ξ0 can belong to the boundary of Ω, the coordinates of ¯ξ0 may coincides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Let ξk = · · · = ξl = c be a maximal family of coinciding coordinates, that is, ξk−1 < ξk = ξl < ξl+1 (it is possible here that k = l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Then, as is easy to realize, the vector p = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , pd), with arbitrary increasing coordinates pk ≤ · · · ≤ pl and with zero remaining coordinates, belong to the tangent cone T(¯ξ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' In view of (32) l � i=k ∂ ∂ξi E(¯ξ0)pi ≥ 0 9 for any such a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Using the summation by parts formula, we realize that the above condition is equivalent to the following requirements l � i=k ∂ ∂ξi E(¯ξ0) = 0, (33) l � i=j ∂ ∂ξi E(¯ξ0) ≥ 0, k < j ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (34) Recall that ai = 0 for k ≤ i < l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' By the direct computation we find ∂ ∂ξi E(¯ξ0) = (ui − ui−1)(ξi − vi−1), k < i < l, ∂ ∂ξk E(¯ξ0) = � (uk − uk−1)(ξk − vk−1) , ak−1 = 0, −A(u)′(c−) , ak−1 > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' ∂ ∂ξl E(¯ξ0) = A(u)′(c+), where A(u)′(c±) are given by (22), (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Putting these expressions into (33), (34), we obtain exactly the jump conditions (21), (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Therefore, the function (13) corresponding to the point ¯ξ0 is an e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' of (1), (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Conversely, if (13) is an e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' then relations (33), (34) holds for all groups of coinciding coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' As is easy to verify, this is equivalent to the criterion (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' We have proved our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' The function (13) is an e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' of (1), (3) if and only if ¯ξ0 = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , ξd) is the minimum point of the entropy E(¯ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Adding to the entropy (25) the constant � k=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=',n−1,ak>0 (ak)2(uk+1 − uk) ln((uk+1 − uk)/ak), we obtain the alternative variant of the entropy E1(¯ξ) = − � k=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=',n−1,ak>0 (ak)2(uk+1 − uk) ln �F((ξk+1 − vk)/ak) − F((ξk − vk)/ak) (uk+1 − uk)/ak � +1 2 � k=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=',n−1,ak=0 (uk+1 − uk)(ξk+1 − vk)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (35) If we consider the values vk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' ak as a piecewise constant approximation of an arbitrary velocity function v(u) and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' a diffusion function a(u) ≥ 0 then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' passing in (35) to the limit as max(uk+1 − uk) → 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' we find that the entropy E1(¯ξ) turns into the variational functional J(ξ) = − � {u∈[α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='β],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='a(u)>0} (a(u))2 ln(F ′((ξ(u) − v(u))/a(u))ξ′(u))du+ 1 2 � {u∈[α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='β],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='a(u)=0} (ξ(u) − v(u))2du,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' 10 where ξ(u) is an increasing function on [α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' β],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' which is expected to be the inverse function to a self-similar solution u = u(ξ) of the problem (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Taking into account that ln(F ′((ξ(u) − v(u))/a(u))ξ′(u)) = ln F ′((ξ(u) − v(u))/a(u)) + ln ξ′(u) = −(ξ(u) − v(u))2 2a2(u) + ln ξ′(u), we may simplify the expression for the functional J(ξ) J(ξ) = � β α [(ξ(u) − v(u))2/2 − (a(u))2 ln(ξ′(u))]du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (36) We see that this functional is strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' The corresponding Euler-Lagrange equation has the form ξ(u) − v(u) + ((a(u))2/ξ′(u))′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (37) Since u′(ξ) = 1/ξ′(u), u = u(ξ), we can transform (37) as follows ξ(u) − v(u) + ((a(u))2u′(ξ))′ u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Multiplying this equation by u′(ξ), we obtain the equation (a2u′)′ = (v − ξ)u′, u = u(ξ), which is exactly our equation (1) written in the self-similar variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' In the case of conservation laws (2) the e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' u = u(ξ) of (2), (3) is piecewise constant, and, by expression (13), u(ξ) = uk, ξk < ξ < ξk+1, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , n, where −∞ = ξ0 < ξ1 ≤ · · · ≤ ξn < ξn+1 = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' In this case the entropy function is particularly simple, it is the quadratic function E(¯ξ) = 1 2 n � k=1 (uk − uk−1)(ξk − vk−1)2, defined on the closed polyhedral cone Ω = { ¯ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , ξn) ∈ Rn | ξk+1 ≥ ξk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' , n − 1 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Existence and uniqueness of a minimal point in this case is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' By Theorem 1 and Remark 1 we obtain new, variational formulation of the entropy solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Acknowledgments The research was supported by the Russian Science Foundation, grant 22-21-00344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' 11 References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Carrillo, Entropy solutions for nonlinear degenerate problems, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=', 147 (1999), 269–361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Kruzhkov, First order quasilinear equations in several independent variables, Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' ), 81 (1970), 228–255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' [3] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Oleinik, Uniqueness and stability of the generalized solution of the Cauchy prob- lem for a quasi-linear equation, Uspekhi Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' Nauk, 14:2(86) (1959), 165–170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} +page_content=' 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'} diff --git a/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf b/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..19bdeae58982b100a2c6bfb99da8db5cd6596861 --- /dev/null +++ b/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5fc369ae1d7c8a45f049a5f204a0ef73ee652ff0084a2d9ad6e15198f8620215 +size 17207322 diff --git a/KtAzT4oBgHgl3EQfyf5z/content/tmp_files/2301.01754v1.pdf.txt b/KtAzT4oBgHgl3EQfyf5z/content/tmp_files/2301.01754v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9fc6cb597c11340c561253f8bb22394bc3009a89 --- /dev/null +++ b/KtAzT4oBgHgl3EQfyf5z/content/tmp_files/2301.01754v1.pdf.txt @@ -0,0 +1,1195 @@ +Status of leptoquark models after LHC Run-2 and discovery +prospects at future colliders +Nishita Desai∗ +Department of Theoretical Physics, +Tata Institute of Fundamental Research, +Mumbai, India 400005 +Amartya Sengupta +Meghnad Saha Pally, Burdwan, India, 713104 † +We study limits from dilepton searches on leptoquark completions to the Standard +Model in the parameter space motivated by anomalies in the b → s sector. After a full +Run-2 analysis by LHCb, the disparity in lepton flavour violation has disappeared. +However, the mismatch in angular distributions as well as in Bs → µ+µ− partial +width is still unresolved and still implies a possible new physics contribution. We +probe three models of leptoquarks — scalar models S3 and R2 as well as vector +leptoquark model U1 using non-resonant dilepton searches to place limit on both the +mass and couplings to SM fermions. The exclusions of leptoquarks coupling either +non-uniformly to different lepton flavours or uniformly is examined. Interestingly, if +leptoquark couplings to electrons and muons are indeed universal, then the U1 model +parameter space that corresponds to the anomalous contribution should already +accessible with Run-2 data in the non-resonant eµ channel. In the non-universal +case, there is a significant exclusion in couplings, but not enough to reach regions +that explain observed anomalies. We, therefore, examine the prospective sensitivity +at the HL-LHC as well as of a 3 TeV future muon collider. For the vector leptoquark +model, we find that a muon collider can probe all of the relevant parameter space at +95% confidence with just 1 fb−1 data whereas R2 and S3 models can be excluded at +95% with 5 fb−1 and 6.5 fb−1 luminosity respectively. +∗ nishita.desai@tifr.res.in +† amartya.sengupta@studenti.unipd.it +arXiv:2301.01754v1 [hep-ph] 4 Jan 2023 + +2 +I. +INTRODUCTION +An exciting development in recent years has been the measurement of ratios of de- +cay widths in the semileptonic rare decays of B-mesons [1–4], hinting at lepton flavour- +universality violation (LFV). The latest of these [1, 2] showed a measurement consistent +with the SM for certain lepton universality, however, there remains a mismatch with the +measured branching fraction of Bs → µ+µ− [3, 4] and in the angular distribution in the +decay B → K∗µ+µ− [5, 6]. Unsurprisingly, this has led to a spirited effort to understand +the source of the mismatch with the predictions of the Standard Model (SM) and to provide +new physics explanations for it. In particular, there have been several dedicated studies that +determine global fits to data in terms of effective field theoretic operators (see e.g. [7–11]). +There has also been some effort to explain the anomalies in terms of new particles, notably +with new vector bosons or leptoquarks [12–15]. The effects of the presence of such new +particles can generally be seen in other observables besides the LFV ratios, and in partic- +ular, in the high energy tails of certain distributions observable at the LHC. In this paper, +we examine the expected effects of leptoquarks with minimally required properties to cause +observed anomalies in the B-sector and report on current constraints and future prospects +of their detection. +We start by providing a bare-bones introduction to how the Effective Field Theoretic +(EFT) framework is used and translated to the measurement of the high-energy observables +that we examine in this paper. EFT provides a useful method to describe the low-energy +physics processes in which the short-distance (i.e. high-energy or UV) physics is encapsulated +in the Wilson coefficients whilst the rest of the long-distance physics is expressed in terms of +effective operators with those having dimensions higher than four being suppressed by powers +of an energy scale to maintain the mass dimension of each term in the Lagrangian. The +analytic form of the Wilson coefficient can then be calculated by “matching” the expressions +calculated from the EFT with the expressions from the full UV theory. We can use the +published value by one of the multiple groups to translate the B-meson observations into +best-fit values of the appropriate Wilson coefficients [7–11, 16]. We then match these values +to the expressions derived from the leptoquark model under study and study the consequence +of what that means on other production mechanisms at the LHC. +The anomalies seen in the data fall into two categories — (1) in the neutral current sector + +3 +with b → s transitions, and (2) in the charged current sector with b → c transitions. In +this work, we concentrate mainly on models that explain the first of these [17], however, it +is known that one of the models we study viz. the U1 vector leptoquark can explain both +simultaneously(see e.g. table 2. of [12]) +The relevant observations that motivate this work based on the full Run 1 and 2 dataset +are shown in table II in the appendix. For completeness, we show both the pre-December +2022 LHCb announcement [1, 2] numbers, as well as the latest measurements. +The low-energy effective theory for the b → s flavour changing neutral current sector is +described in terms of an effective Hamiltonian which can be written as +Heff = −4GF +√ +2 VtbV ∗ +ts +� � +Ci(µ)Oi(µ) +� +where Ci(µ) are the Wilson coefficients. The effective operators relevant to our study are +Ol1l2 +9 += +e2 +(4π)2(¯sγµPLb)(¯l1γµl2), +Ol1l2 +10 = +e2 +(4π)2(¯sγµPLb)(¯l1γµγ5l2) +(I.1) +Multiple fitting studies have found that the operator whose Wilson coefficient shows sig- +nificant deviation from the predicted SM value is the C9 and that the most likely discrepancy +seems to be in the Cµ+µ− +9 +coefficient. To stay consistent with the latest data, we use the +Author (Year) +Model Dependent Data Driven +Ciuchini et al (2022) [7] +[−1.25, −0.72] +[−1.10, 1.05] +Ciuchini et al (2019) [8] +[−1.37, −1.05] +[−1.47, −0.93] +Alguer´o et al (2019) [9] +[−1.15, −0.81] +Alok et al (2019) [10] +[−1.27, −0.91] +Mahmoudi et al (2021) [11] +[−1.07, −0.83] +TABLE I. Best Fit values for the new physics contribution to the operator C9. The first of these +contains the updated 2022 results. The fits taking into account angular distributions still favour +a similar range as before the 2022 LHCb data release even though the overall best fit 1σ range is +now consistent with the SM value of zero. + +4 +most recent best-fit results as reported in [7]. We shall use the best fit values that correctly +give the angular correlations as well (the so called “model-dependent” fit). However, later +in the paper when we examine future prospects, we also show the overlap with the fully +agnostic data-driven fits. For an overview of the best-fit C9 values see table I. Currently, we +proceed by using the value +Cµ+µ− +9 += −0.98 ± 0.27, +Multiple studies have also examined the leptoquark UV completion and calculated explicit +expressions for Cµ+µ− +9 +from each model. In this work, we use these expressions to investigate +the LHC constraints on the couplings and mass of the leptoquarks. +We make only the +minimal assumptions, i.e. only the couplings that are necessary to give a contribution to +the b → s anomalies is assumed to be non-zero. As we shall see, in each leptoquark model, +the Wilson coefficients C9(10) depend on three parameters roughly as +C9 ∼ +�y22 y32 +M +�2 +where y22 is the sµ coupling, y32 is the bµ coupling and M is the mass of the leptoquark. We +start by constraining (y22, y32, M) in other production modes without any further assump- +tions on other leptoquark couplings. This results in the most conservative limits. In the case +where there is no LFV, one would expect identical couplings of the leptoquark to electrons, +i.e. y22 = y21 and y32 = y31. This would also lead to signatures with different flavored +dileptons which often have much stronger constraints. These constraints are examined in +section III. In the flavour universal case, the strongest limits on leptoquark masses will come +from µ → e processes including µ → eγ [18] and µ → 3e[19] measurements. However, it +might be possible that the effects of leptoquarks could be cancelled in loop-induced processes +by the presence of other new particles. Studying direct leptoquark production at the LHC +allows us to directly probe the lepton-universal case because the observed number of events +in µµ, ee and µe channels will be correlated. +Our paper is structured as follows: we start by listing out the model Lagrangian and +the resulting Wilson coefficients for C9 in section II. We then examine the current LHC +constraints in various search channels in section III and expected detection prospects of +future colliders are calculated in section IV. + +5 +II. +LEPTOQUARK MODELS +Leptoquarks are bosons which carry both SU(2)L and colour SU(3) charges and therefore +couple to both leptons and quarks. Given that we need to get the right contribution to +Cµ+µ− +9 +, this corresponds to a leptoquark that at a minimum couples to muons and to b and +s quarks. There are three known leptoquark models that give the right kind of contribu- +tion [12–14, 20], which we describe below. We use the standard names for the fields, viz. S3, +R2 and U1 and the numbers in brackets that follow correspond to (n-plet of SU(3), n-plet of +SU(2), U(1)Y hypercharge). Of these, S3 and R2 are scalar fields and U1 is a vector field. +A. +Scalar Leptoquark S3 (¯3, 3, 1/3) +The first leptoquark model we consider is S3(¯3, 3, 1/3) which is a SU(2)L triplet of scalar +leptoquark states with hypercharge 1/3. S3 is the only scalar leptoquark model that can +simultaneously predict Rexp +K∗ < RSM +K∗ and Rexp +K∗ < RSM +K∗ at tree level [21–24]. The Lagrangian +for the S3 model is +LS3 = yij +L ¯QC +i iτ2(τkSk +3)Lj + h.c., +(II.1) +where Qi and Lj are SU(2)L doublet fermion fields corresponding to quarks and leptons of +the ith( jth) generation respectively, τk are the generators of SU(2)L, and yij +L stands for a +Yukawa matrix for the left-handed fermions. The three triplet component states of S3 carry +charges Q = −2/3, 1/3 and 4/3 respectively. Expanding out the SU(2)L components and +referring to the leptoquarks as SQ +3 , we get +LS3 = −yij +L ¯dC +LiνLjS1/3 +3 +− +√ +2yij +L ¯dC +LiℓLjS4/3 +3 ++ +√ +2(V ∗yL)ij¯uC +LiνLjS−2/3 +3 +− (V ∗yL)ij¯uC +LiℓLjS1/3 +3 ++ h.c., +(II.2) +of which only the ¯dC +LiℓLjS4/3 +3 +term contributes to O9. One can extract the Wilson coefficients +for the b → sl−l+ decay [12–14, 20], +Cℓ1ℓ2 +9 += −Cℓ1ℓ2 +10 += +πv2 +VtbV ∗ +tsαem +ybℓ1 +L (ysℓ2 +L )∗ +m2 +S3 +, +(II.3) + +6 +B. +Scalar Leptoquark R2 (3, 2, 7/6) +The second case we consider is a weak doublet of scalar leptoquarks with hypercharge Y = +7/6, i.e. R2 (3, 2, 7/6).[25] The most general Lagrangian describing the Yukawa interactions +with R2 can be written as, +LR2 = yij +R ¯QilRjR2 − yij +L ¯uRiR2iτ2Lj + h.c., +(II.4) +where yL and yR are the Yukawa matrices corresponding to left- and right-handed lepton +fields respectively. In terms of the components with RQ +2 denoting each leptoquark state with +charge Q, the Lagrangian can be written as +LR2 = (V yR)ij¯uLiℓRjR5/3 +2 ++ (yR)ij ¯dLiℓRjR2/3 +2 ++ (yL)ij¯uRiνLjR2/3 +2 +− (yL)ij¯uRiℓLjR5/3 +2 ++ h.c. +(II.5) +The tree-level contribution to the Wilson coefficients C9 through the term (yR)ij ¯dLiℓRjR2/3 +2 +amounts to +Cℓ1ℓ2 +9 += Cℓ1ℓ2 +10 += − +πv2 +2VtbV ∗ +tsαem +ysℓ1 +R (ybℓ2 +R )∗ +m2 +R2 +, +(II.6) +C. +Vector Leptoquark U1 (3, 1, 2/3) +Finally, we describe the only vector leptoquark model considered in this paper, mainly +because it has been the only model that could simultaneously explain both charged current +and neutral current anomalies [12]. We consider the U1 (3, 1, 2/3) model which gives a single +leptoquark state with charge 2/3. The most general Lagrangian consistent with the SM +gauge symmetry allows couplings to both left-handed and right-handed fermions, namely +LU1 = βij +L ¯QiγµLjU µ +1 + βij +R ¯dRiγµℓRjU µ +1 + h.c., +(II.7) +with couplings βij +L and βij +R. The contributions to the left-handed couplings to the effective +Lagrangian amount to +Cℓ1ℓ2 +9 += −Cℓ1ℓ2 +10 += − +πv2 +VtbV ∗ +tsαem +βsℓ1 +L (βbℓ2 +L )∗ +m2 +U1 +, +(II.8) + +7 +III. +LHC LIMITS +Our goal is to use published LHC data to simultaneously constrain the mass and Yukawa +couplings of the leptoquarks. The Wilson coefficient C9 depends on three parameters roughly +as +Cℓ,ℓ +9 +∼ +�y2ℓ y3ℓ +M +�2 +where yij refers to the leptoquark coupling between the ith generation of quark and jth +generation lepton. This corresponds to Yukawa couplings for S3 and R2 models and the gauge +coupling for the U1 model. Therefore, its possible to find a surface in the 3D parameter space +that gives the required value of C9. However, most LHC search constraints are in principle +only 2D — one coupling that determines the cross section of the final state and one mass. +We, therefore, have several options in which to view the full constraints. +Let us start with ℓ = 2 (i.e. µ) which contributes to Cµµ +9 . To be able to independently +constrain the two Yukawa couplings y22 and y32, we study three different cases — first +setting only y22 non-zero (see figure 1, second setting only y32 non-zero (see figure 4) and +third, setting both equal (see figure 5). Using the upper limits from the non-resonant dimuon +search gives us an upper limit on y22 at each mass value. It is possible to also determine the +minimal allowed value of y22 that is consistent with C9 by requiring y32 ≤ 1. +Since the latest LHCb data seem to indicate that electrons and muons have identical +behaviour, we can indeed also do a similar exercise with y21 and y31 which would contribute +to Cee +9 . Besides these, non-zero values of all four couplings (or even a single electron and +a single muon coupling) — y21, y31, y22 and y32 can give signatures that have differently +flavoured leptons in the final state, but without missing energy and therefore with no SM +background. +It should be noted that in the case where a single leptoquark state can couple to both +electrons and muons, the strongest constraints on couplings and mass of course come from +low energy processes in the µ → e sector [18, 19, 26]. However, it can still be an interesting +exercise to directly probe the case where both yk1 and yk2 are non-zero. As we see in figure 2, +this case is strongly constrained by the LHC, with the U1 model likely to be ruled out already +with full run-2 data of 139 fb−1. +Since multiple leptoquark states come from the same multiplet, they have identical mass +and switching on a single coupling allows the production of multiple states. For calculating + +8 +the LHC limits, we allow the production of all leptoquark states and select only that fraction +that decays into the final state selected for by the analysis being reinterpreted. For example, +in the S3 case, if we look for pair production of leptoquark followed by decay of each into +a muon and a jet by turning on y22 ̸= 0 alone, we allow both the production of pairs of +S4/3 +3 +→ ¯sµ+ as well as pairs of S1/3 +3 +→ ¯cµ+. Our limits, therefore, are not identical to the +simplified model limits that the experimental analysis publishes by producing only one state +at a time, with 100% branching fraction into a certain channel. Similarly, when looking at +dilepton distributions, we take into account, with interference, all leptoquark states in the +t-channel that are allowed by non-zero couplings. +A. +Computational setup +Since we examine the limits from dilepton searches which have been presented in the form +of upper limits on generator-level cross sections with fiducial cuts, our computational setup +is much simplified. We generate events using +Madgraph5 amc@NLO [27] with the required +fiducial cuts and do not need to perform further detector simulation. This approach has +been proven to work well [28] and reproduces expected limits. For the UV models, we use +the scalar leptoquark models for S3 and R2 described in [29] and for the vector leptoquark +model for the U1 case, we use the model described in [30–32]. When more complicated +functionality is required, we use Pythia8 [33] to shower, hadronize and apply the required +kinematic cuts on events. +B. +Limits from resonant and non-resonant dilepton searches +We re-interpreted both the dilepton resonance search with 139 fb−1 [34] and the non- +resonant dilepton search at 139 fb−1 [35] from ATLAS. We find that the non-resonant search +results in much stronger limits and we continue with this search for the rest of our study. +The exclusive dilepton state can only be seen with a t-channel leptoquark exchange. It is +possible to have a dilepton plus two jets from strong production of leptoquarks, however, +this process does not depend on the leptoquark-fermion couplings and results in only a mass +limit which we deal with in the next subsection. With the interference of SM Drell-Yan +production of leptons with the t-channel leptoquark mediated production, one expects to + +9 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +MLQ [GeV] +y2k +Allowed +S3 +C9 ⇒ y3 k > 1 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +MLQ [GeV] +y2k +Allowed +R2 +C9 ⇒ y3 k > 1 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +MLQ [GeV] +β2k +Allowed +U1 +C9 ⇒ β3 k > 1 +FIG. 1. +Exclusion plots y2ℓ versus Mass of leptoquark for the S3 (top-left), R2 (top-right) and +U1 models (bottom). The bright red regions at the top are disallowed from dimuon searches. The +corresponding di-electron limit is the lighter line inside the red region. The solid regions at the +bottom are from requiring perturbative couplings consistent with allowed C9. The vertical lines +are mass limits from direct leptoquark pair production with the solid line corresponding to second +generation leptons and the dotted corresponding to first generation. The limits correspond to 139 +fb−1 data. +see a change in the shape of the dilepton invariant mass distribution mℓℓ where ℓ = µ or e. +We apply the limits from the ATLAS non-resonant dilepton search by generating events +using +Madgraph5 amc@NLO according to fiducial cuts listed in [35] and using the 95% upper +limits for the most conservative signal region called the “µ+µ− constructive signal region” (or + +10 +analogously the e+e− constructive signal region). The constructive signal region corresponds +to the case where you expect signal events above the EW expectation, which is similar to +our case. The experimental analysis uses LO signal shape to model the expected number of +events and we therefore also do not use any NLO corrections. The upper limits are provided +on the additional cross section above the expected SM Electro-Weak (EW) prediction in the +cumulative signal region where mµ+µ− ≥ 2070 GeV (or me+e− ≥ 2200 GeV). +As expected, the effect of having heavy new leptoquarks in t-channel dies down when +either the leptoquark mass is too high or the Yukawa coupling is too small. To account for +the interference correctly, we use the difference of the cross-section pp → ℓ+ℓ− with both +leptoquark and EW bosons, and with only EW gauge bosons as our new physics contribution. +The result is an excluded region near high Yukawa coupling values, with a larger range ruled +out for smaller leptoquark masses. This is shown as a bright red region in figure 1. The +highest allowed value of y2k is referred to as y2k max and can be used to further restrict what +values of y3k are consistent with C9. +Currently, there is one different flavour dilepton search [36] performed at 13 TeV, but +with only 3.2 fb data analysed. Aside from cuts on pT of 65 and 50 GeV on electrons and +muons respectively, there are requirements that missing energy be less than 25 GeV and +mT < 50 GeV to remove contamination from W-boson production which we apply using +Pythia 8.3 [33]. The expected background for meµ > 2 TeV is 0.02 ± 0.02. They see one +event and interpreting it as a statistical fluctuation, set a limit on new physics cross section. +We extrapolate the expected limits from this search at 139 fb−1. The limits on the eµ case +for the U1 model can be seen in figure 2. The expected background at 139 fb−1 is 2.78 +events, resulting in an expected 95% upper limit of 0.0185 fb on production cross section +times branching. As can be seen, the U1 model should be completely ruled out with 139 +fb−1 data. For results in the eµ channel for S3 and R2 models, refer to appendix C. +C. +Limits from leptoquark-pair production +Direct limits on the mass of the leptoquark based on strong pair-production mode followed +by the decay of each leptoquark into a lepton and a jet are presented in [37]. The limits are +also presented on generator-level cross-section times branching fraction and can be applied +directly to our model. The resulting limit is shown as a solid black vertical line. Since there + +11 +is no significant improvement in the limit from b-tagging, we use the general lepton+jet +limits in all cases. When only yk2 is non-zero, i.e. the leptoquark decays to a muon and a +jet, we obtain a mass limit for S3 leptoquark at 1774 GeV, for the R2 leptoquark at 1720 +GeV and the U1 leptoquark at 2309 GeV. For the case where the leptoquark decays into +electron alone, we get a mass limit for S3 leptoquark at 1828 GeV, for the R2 leptoquark at +1773 GeV and the U1 leptoquark at 2419 GeV. +There is no direct limit on the case with an eµ final state in the published search, which +if it existed, would give a far better exclusion simply because there is no irreducible SM +background and the dominant background would be from mis-identification of leptons. +D. +Missing search: top FCNC decay +Given the need for non-zero leptoquark coupling to the third generation of quarks, this +also implies a coupling between the top quark and second generation leptons for both the +S3 and U1 models. In the R2 case, the coupling is either CKM suppressed (in the case of +left-handed) or entirely independent and therefore set to zero (in the right-handed case). It +would therefore be possible to search directly for FCNC top decay via t → cµµ. +Currently, there are no searches for t → cµ+µ− except for a t → cZ search which requires +the dimuon mass to be within 15 GeV of the Z mass [38] and therefore is not directly +applicable to our model. A similar measurement from CMS [39] is available from the 8 TeV +run. +The main background for a t → cµ+µ− search is from the SM production of t¯tµ+µ− +via an off-shell Z or γ produced in association with t¯t. To remove contamination from on- +shell Z, we apply a cut instead Mℓℓ > 105 which is outside the Z-mass window selected for +by the t → cZ searches. Assuming the identification acceptances do not change, we can +estimate the background for our proposed search using the data driven estimate presented +in [38] (denoted by σBG,ATLAS). +Since we have identical SM production modes for t¯tZ +and t¯tµ+µ−, we assume that the generator level transfer factor between these processes is +transmitted all the way to the final selection. The kinematic effect of changing the mℓℓ cut +from |Mℓℓ − MZ| < 15 to Mℓℓ > 105 can be estimated at generator level and is encapsulated +in a single number fℓℓ Also, we assume that the enhancement in production of t¯tZ in going + +12 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +MLQ [GeV] +β2k +Allowed +U1 +C9 ⇒ β3 K > 1 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +MLQ [GeV] +β3k +C9 ⇒ β2 k > 1 +U1 +C9 ⇒ β2 k > β2 k max +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +MLQ [GeV] +β2k,3k +C9 best fit +U1 +FIG. 2. +Limits for the Leptoquark Couplings versus mass for the U1 Model. The dilepton process, +in this case, is pp → µe which does not exist in the SM. We, therefore, have strong limits even +with 3.2 fb−1 data as published in [36]. The top-left panel shows limits on the coupling to second +generation quarks with y22 = y21, the top-right panel on the coupling to third generation quarks +with y32 = y31 and the bottom panel shows the case where all four couplings are equal. The green +band shows the values corresponding to the best fit values of C9 The dotted line in this figure +shows the expected limit after analysing full 139 fb−1 of run-2 data by ATLAS (only partial result +is published so far). We see clearly that the universal scenario is likely already ruled out by run-2 +data. + +13 +from 13 TeV to 13.6 TeV (fE = σ13.6 +σ13 ) remains the same also for t¯tµ+µ−. Thus we have +σBG(√s = 13.6) = +σBG,ATLAS +× fE × fℓℓ +× σ(pp → t¯tµ+µ−; √s = 13) +σ(pp → t¯tZ; √s = 13) +(III.1) +Using this, and the expected background cross section from ATLAS, we calculate an +expected background of 7±2 events. Given that with the Z-window, the background is esti- +mated at 119±10 events, this would correspond to over an order of magnitude improvement +in the sensitivity to FCNC branching fraction of the top quark. +IV. +FUTURE PROSPECTS +The best-fit value of the Wilson coefficients for operators that explain the b → s anomalies +suggests a high suppression scale. Using equations (II.6), (II.3) and (II.8), we find that the +required scale for both couplings set to one is 16183 GeV for the R2 case and 22887 GeV +for the S3 and U1 cases. Naturally, resonantly producing a leptoquark of this mass scale is +out of the question at the LHC. We, therefore, investigate both the expected reach of the +LHC after the planned high-luminosity run and estimate a conservative reach for a muon +collider with CM energy of 3 TeV [40–43]. To illustrate the highest sensitivity case, we +choose y22 = y32 for this calculation. This also allows us to make a comment on the ability +of the collider to explore the entire parameter space of interest. A summary of the expected +reach of future colliders can be seen in figure 3 +A. +LHC High-Lumi expected limits +Projections for the HL-LHC are made with the luminosity of 3000 fb−1. From previous +experience, we know that the improvements in limits scale with about the square root of +luminosity. Using the expected number of signal and background events for the non-resonant +dilepton search, we can probe effects of leptoquarks up to mass 5 TeV for the S3, 3 TeV for +the R2 and 9.5 TeV for the U1 model. Conversely, we can probe coupling values as small as +0.4 for S3, 0.55 for R2 and 0.15 for U1 models respectively at 1 TeV leptoquark mass. For + +14 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +MLQ [GeV] +y22,32 +LHC13 +MuonC, 1/fb +5σ +2σ +C9 +fit +S3 +LHC13-Mass +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +MLQ [GeV] +y22,32 +LHC13 +MuonC, 1/fb +5σ +2σ +C9 +fit +R2 +LHC13-Mass +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +MLQ [GeV] +β22,32 +LHC13 +MuonC, 1/fb +5σ +2σ +C9 +fit +U1 +LHC13-Mass +FIG. 3. Current and future reach in leptoquark coupling to muons with leptoquark mass for the S3 +Model (top-left), R2 Model (top-right) and U1 Model (bottom). The green region corresponds to +the 1σ region given by global fit C9 values in the model-dependent case whereas the yellow is the +data-driven 1σ region ([7], also see table I). The solid red region is the current 139 fb−1 limits with +the dotted red line the expected reach after 3 ab−1 at the HL-LHC. The solid and dotted vertical +lines correspond to mass limits from pair production again corresponding to the 139 fb−1 and 3 +ab−1 luminosity respectively. The blue region corresponds to the parameter space that can be +discovered with a 5σ significance at a 3 TeV muon collider with 1 fb−1 whereas the orange region +corresponds to the further region that can be probed at 95% confidence at the same collider. The +U1 model can be fully excluded with just 1 fb−1 data. The S3 and R2 models can also be fully +probed with 6.5fb−1 and 5fb−1 respectively. +comparison, C9 best fit predicts a minimum value of coupling at 0.04, 0.06 and 0.04 for the +three models when we set both couplings equal. +The direct search limits from strong production are calculated in a similar way using the + +15 +published upper limits at 139/fb. We find that the HL-LHC can exclude leptoquark masses +of 2.2 TeV for both the S3 and R2 case and 2.8 TeV for the U1 case for the leptoquark +decaying into a muon and a jet and 2.3 TeV for both the S3 and R2 case and 2.9 TeV for +the U1 case for the leptoquark decaying into an electron and a jet. +B. +Reach of a Future Muon Collider +Estimating the reach of a future muon collider is more difficult since we do not currently +have a detector configuration to be able to simulate a realistic analysis. However, taking +lessons from the dilepton and dijet searches at the LHC, we know that a single-bin analysis +with a high enough cut on the invariant mass provides a very reliable estimate of reach. We +look at µ+µ− → jj as our signal. Obviously using b-tagging will be a further improvement +that can pinpoint the underlying scenario. However, for this estimate, we just use untagged +jets. Given that acceptance efficiencies of jets are expected to be similar for both signal and +background events for a simple dijet search, we proceed with using just generator-level cross +sections. A further advantage is the much reduced probability of extra initial state radiation +jets from initial state muons (in sharp contrast to a pp machine). +The main background from the SM comes from s-channel photon or Z exchange. In the +presence of the leptoquark, another Feynman diagram with a t-channel leptoquark exchange +needs to be taken into account. We look only at events with Mjj > 500 GeV. The SM-only +cross section at LO is 5.96×10−2 pb which corresponds to a statistical error of about 8 events +at a luminosity of 1 fb−1. Using this, we can calculate the parameter space corresponding to +a 5σ discovery as well as regions that can be excluded at 2σ. They are shown in figure 3 as +blue and orange regions respectively. In the U1 case, we see that a muon collider is capable +of excluding the entire viable parameter space with 1 fb−1. To exclude the R2 and S3 models +would need a luminosity of 6.5 fb−1 for S3 and 5 fb−1 for R2. +V. +SUMMARY AND CONCLUSIONS +We examine the limits from direct collider searches on leptoquark models that are capable +of explaining the anomalous measurements in the decays of B-mesons. We focus on three +specific models — two scalar leptoquark models S3 and R2 and one vector leptoquark model + +16 +U1. Aside from limits on the mass of the leptoquarks (which can be pair-produced by strong +interactions), it is possible to also constrain the couplings to fermions by looking at changes +to the shape of the dilepton mass spectrum. Reinterpreting full Run-2 limits from the pair +production and non-resonant dilepton searches by ATLAS experiment, we find that current +mass limits are 1.77 TeV, 1.72 TeV and 2.3 TeV respectively for the three models. We can +expect to reach up to 2.2 TeV for S3 and R2 and 2.8 TeV for the U1 respectively with the +High-Luminosity LHC run. +Effects of leptoquarks with couplings to muons can potentially be probed in a muon +collider. Since there has been considerable interest in a future muon collider recently, we +also estimate what the reach of the proposed 3 TeV muon collider would be for the three +models in question. 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Virto, Optimizing the basis of B → K∗ll +observables in the full kinematic range, JHEP 05 (2013) 137 [1303.5794]. + +21 +Appendix A: Relevant observables in the b → s sector +Observable +Experiment +Theory (SM) +RK[0.1,1.1] +0.994 +0.090 +−0.082 (stat) +0.029 +−0.027 (syst) [2022] [1, 2] +1.00 ± 0.01 [44]) +RK∗[0.1,1.1] +0.927 +0.093 +−0.087 (stat) +0.036 +−0.035 (syst) [2022] [1, 2] +1.00 ± 0.01 [44]) +RK[1.1,6] +0.949 +0.042 +−0.041 (stat) +0.022 +−0.022 (syst) [2022] [1, 2] +1.00 ± 0.01 [44]) +RK∗[1.1,6] +1.027 +0.072 +−0.068 (stat) +0.027 +−0.026 (syst) [2022] [1, 2] +1.00 ± 0.01 [44]) +R[0.045,1.1] +K∗ +0.66+0.11 +−0.07 ± 0.03 [2021] [45] +0.906 ± 0.028 [44] +R[1.1,6.0] +K∗ +0.69+0.11 +−0.07 ± 0.05 [2021] [45] +1.00 ± 0.01 [44] +R[1.1,6.0] +K +0.846+0.042+0.013 +−0.039−0.012 [2021] [46] +1.00 ± 0.01 [44] +B(Bs → µ+µ−) +(2.85+0.32 +−0.31) × 10−9 [3, 4] +(3.66 ± 0.14) × 10−9[47]) +P ′ +5 in B → K(∗) l+ l− +[5, 48, 49] +[6, 50] +TABLE II. A summary of the most relevant experimental results and SM predictions for the +observables in b → s sector. + +22 +Appendix B: Limits on leptoquark couplings to third generation quarks y3k. +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +MLQ [GeV] +y3k +C9 ⇒ y2 k > 1 +S3 +C9 ⇒ y2 k > y2 k max +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +MLQ [GeV] +y3k +C9 ⇒ y2 K > 1 +R2 +C9 ⇒ y2 K > y2 K max +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +MLQ [GeV] +β3k +C9 ⇒ β2 k > β2 k max +U1 +FIG. 4. +Exclusion plots y3ℓ versus Mass of leptoquark for the S3 (top-left), R2 (top-right) and +U1 models (bottom). The solid regions at the bottom are from requiring perturbative couplings +consistent with allowed C9. The darker region is inconsistent with the observed upper limits on y2k +in figure 1. The vertical lines are mass limits from direct leptoquark pair production with the solid +line corresponding to second generation leptons and the dotted corresponding to first generation. +The limits correspond to 139 fb−1 data. + +23 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +MLQ [GeV] +y2k,3k +C9 best fit +S3 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +MLQ [GeV] +y2k,3k +C9 best fit +R2 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +MLQ [GeV] +β2k,3k +C9 best fit +U1 +FIG. 5. +Exclusion plots in the limited case of y2ℓ = y3ℓ versus Mass of leptoquark for the S3 +(top-left), R2 (top-right) and U1 models (bottom). +The solid red region at the top are limits +from non-resonant dilepton searches in µ+µ−. The lighter lines inside this region correspond to +subleading limits from the similar e+e− search. +The vertical lines are mass limits from direct +leptoquark pair production with the solid line corresponding to second generation leptons and the +dotted corresponding to first generation. The limits correspond to 139 fb−1 data. The green band +is the region that corresponds to the coefficient C9 within one sigma of best fit to data. +– + +24 +Appendix C: Limits on S3 and R2 model parameters in the Lepton Flavour Universal +case +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +MLQ [GeV] +y2k +Allowed +S3 +C9 ⇒ y3 K > 1 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +MLQ [GeV] +y3k +C9 ⇒ y2 k > 1 +S3 +C9 ⇒ y2 k > y2 k max +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +MLQ [GeV] +y2k,3k +C9 best fit +S3 +FIG. 6. Limits on the leptoquark couplings via the process p p → µ e in the case of flavour universal +couplings to electrons and muons for the S3 Model. + +25 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +MLQ [GeV] +y2k +Allowed +R2 +C9 ⇒ y3 K > 1 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +MLQ [GeV] +y3k +C9 ⇒ y2 k > 1 +R2 +C9 ⇒ y2 k > y2 k max +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +MLQ [GeV] +y2k,3k +C9 best fit +R2 +FIG. 7. Limits on the leptoquark couplings via the process p p → µ e in the case of flavour universal +couplings to electrons and muons for the R2 Model. + diff --git a/KtAzT4oBgHgl3EQfyf5z/content/tmp_files/load_file.txt b/KtAzT4oBgHgl3EQfyf5z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..41c46115978c2b6504ab36e871f1a1fc009fbe69 --- /dev/null +++ b/KtAzT4oBgHgl3EQfyf5z/content/tmp_files/load_file.txt @@ -0,0 +1,786 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf,len=785 +page_content='Status of leptoquark models after LHC Run-2 and discovery prospects at future colliders Nishita Desai∗ Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai, India 400005 Amartya Sengupta Meghnad Saha Pally, Burdwan, India, 713104 † We study limits from dilepton searches on leptoquark completions to the Standard Model in the parameter space motivated by anomalies in the b → s sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' After a full Run-2 analysis by LHCb, the disparity in lepton flavour violation has disappeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' However, the mismatch in angular distributions as well as in Bs → µ+µ− partial width is still unresolved and still implies a possible new physics contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We probe three models of leptoquarks — scalar models S3 and R2 as well as vector leptoquark model U1 using non-resonant dilepton searches to place limit on both the mass and couplings to SM fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The exclusions of leptoquarks coupling either non-uniformly to different lepton flavours or uniformly is examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Interestingly, if leptoquark couplings to electrons and muons are indeed universal, then the U1 model parameter space that corresponds to the anomalous contribution should already accessible with Run-2 data in the non-resonant eµ channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' In the non-universal case, there is a significant exclusion in couplings, but not enough to reach regions that explain observed anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We, therefore, examine the prospective sensitivity at the HL-LHC as well as of a 3 TeV future muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' For the vector leptoquark model, we find that a muon collider can probe all of the relevant parameter space at 95% confidence with just 1 fb−1 data whereas R2 and S3 models can be excluded at 95% with 5 fb−1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='5 fb−1 luminosity respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' ∗ nishita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='desai@tifr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='in † amartya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='sengupta@studenti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='it arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='01754v1 [hep-ph] 4 Jan 2023 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' INTRODUCTION An exciting development in recent years has been the measurement of ratios of de- cay widths in the semileptonic rare decays of B-mesons [1–4], hinting at lepton flavour- universality violation (LFV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The latest of these [1, 2] showed a measurement consistent with the SM for certain lepton universality, however, there remains a mismatch with the measured branching fraction of Bs → µ+µ− [3, 4] and in the angular distribution in the decay B → K∗µ+µ− [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Unsurprisingly, this has led to a spirited effort to understand the source of the mismatch with the predictions of the Standard Model (SM) and to provide new physics explanations for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' In particular, there have been several dedicated studies that determine global fits to data in terms of effective field theoretic operators (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' [7–11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' There has also been some effort to explain the anomalies in terms of new particles, notably with new vector bosons or leptoquarks [12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The effects of the presence of such new particles can generally be seen in other observables besides the LFV ratios, and in partic- ular, in the high energy tails of certain distributions observable at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' In this paper, we examine the expected effects of leptoquarks with minimally required properties to cause observed anomalies in the B-sector and report on current constraints and future prospects of their detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We start by providing a bare-bones introduction to how the Effective Field Theoretic (EFT) framework is used and translated to the measurement of the high-energy observables that we examine in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' EFT provides a useful method to describe the low-energy physics processes in which the short-distance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' high-energy or UV) physics is encapsulated in the Wilson coefficients whilst the rest of the long-distance physics is expressed in terms of effective operators with those having dimensions higher than four being suppressed by powers of an energy scale to maintain the mass dimension of each term in the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The analytic form of the Wilson coefficient can then be calculated by “matching” the expressions calculated from the EFT with the expressions from the full UV theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We can use the published value by one of the multiple groups to translate the B-meson observations into best-fit values of the appropriate Wilson coefficients [7–11, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We then match these values to the expressions derived from the leptoquark model under study and study the consequence of what that means on other production mechanisms at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The anomalies seen in the data fall into two categories — (1) in the neutral current sector 3 with b → s transitions, and (2) in the charged current sector with b → c transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' In this work, we concentrate mainly on models that explain the first of these [17], however, it is known that one of the models we study viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' the U1 vector leptoquark can explain both simultaneously(see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' of [12]) The relevant observations that motivate this work based on the full Run 1 and 2 dataset are shown in table II in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' For completeness, we show both the pre-December 2022 LHCb announcement [1, 2] numbers, as well as the latest measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The low-energy effective theory for the b → s flavour changing neutral current sector is described in terms of an effective Hamiltonian which can be written as Heff = −4GF √ 2 VtbV ∗ ts � � Ci(µ)Oi(µ) � where Ci(µ) are the Wilson coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The effective operators relevant to our study are Ol1l2 9 = e2 (4π)2(¯sγµPLb)(¯l1γµl2), Ol1l2 10 = e2 (4π)2(¯sγµPLb)(¯l1γµγ5l2) (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='1) Multiple fitting studies have found that the operator whose Wilson coefficient shows sig- nificant deviation from the predicted SM value is the C9 and that the most likely discrepancy seems to be in the Cµ+µ− 9 coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' To stay consistent with the latest data, we use the Author (Year) Model Dependent Data Driven Ciuchini et al (2022) [7] [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='25, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='72] [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='10, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='05] Ciuchini et al (2019) [8] [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='37, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='05] [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='47, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='93] Alguer´o et al (2019) [9] [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='15, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='81] Alok et al (2019) [10] [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='27, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='91] Mahmoudi et al (2021) [11] [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='07, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='83] TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Best Fit values for the new physics contribution to the operator C9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The first of these contains the updated 2022 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The fits taking into account angular distributions still favour a similar range as before the 2022 LHCb data release even though the overall best fit 1σ range is now consistent with the SM value of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' 4 most recent best-fit results as reported in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We shall use the best fit values that correctly give the angular correlations as well (the so called “model-dependent” fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' However, later in the paper when we examine future prospects, we also show the overlap with the fully agnostic data-driven fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' For an overview of the best-fit C9 values see table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Currently, we proceed by using the value Cµ+µ− 9 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='27, Multiple studies have also examined the leptoquark UV completion and calculated explicit expressions for Cµ+µ− 9 from each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' In this work, we use these expressions to investigate the LHC constraints on the couplings and mass of the leptoquarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We make only the minimal assumptions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' only the couplings that are necessary to give a contribution to the b → s anomalies is assumed to be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' As we shall see, in each leptoquark model, the Wilson coefficients C9(10) depend on three parameters roughly as C9 ∼ �y22 y32 M �2 where y22 is the sµ coupling, y32 is the bµ coupling and M is the mass of the leptoquark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We start by constraining (y22, y32, M) in other production modes without any further assump- tions on other leptoquark couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' This results in the most conservative limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' In the case where there is no LFV, one would expect identical couplings of the leptoquark to electrons, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' y22 = y21 and y32 = y31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' This would also lead to signatures with different flavored dileptons which often have much stronger constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' These constraints are examined in section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' In the flavour universal case, the strongest limits on leptoquark masses will come from µ → e processes including µ → eγ [18] and µ → 3e[19] measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' However, it might be possible that the effects of leptoquarks could be cancelled in loop-induced processes by the presence of other new particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Studying direct leptoquark production at the LHC allows us to directly probe the lepton-universal case because the observed number of events in µµ, ee and µe channels will be correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Our paper is structured as follows: we start by listing out the model Lagrangian and the resulting Wilson coefficients for C9 in section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We then examine the current LHC constraints in various search channels in section III and expected detection prospects of future colliders are calculated in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' 5 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' LEPTOQUARK MODELS Leptoquarks are bosons which carry both SU(2)L and colour SU(3) charges and therefore couple to both leptons and quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Given that we need to get the right contribution to Cµ+µ− 9 , this corresponds to a leptoquark that at a minimum couples to muons and to b and s quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' There are three known leptoquark models that give the right kind of contribu- tion [12–14, 20], which we describe below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We use the standard names for the fields, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' S3, R2 and U1 and the numbers in brackets that follow correspond to (n-plet of SU(3), n-plet of SU(2), U(1)Y hypercharge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Of these, S3 and R2 are scalar fields and U1 is a vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Scalar Leptoquark S3 (¯3, 3, 1/3) The first leptoquark model we consider is S3(¯3, 3, 1/3) which is a SU(2)L triplet of scalar leptoquark states with hypercharge 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' S3 is the only scalar leptoquark model that can simultaneously predict Rexp K∗ < RSM K∗ and Rexp K∗ < RSM K∗ at tree level [21–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The Lagrangian for the S3 model is LS3 = yij L ¯QC i iτ2(τkSk 3)Lj + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=', (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='1) where Qi and Lj are SU(2)L doublet fermion fields corresponding to quarks and leptons of the ith( jth) generation respectively, τk are the generators of SU(2)L, and yij L stands for a Yukawa matrix for the left-handed fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The three triplet component states of S3 carry charges Q = −2/3, 1/3 and 4/3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Expanding out the SU(2)L components and referring to the leptoquarks as SQ 3 , we get LS3 = −yij L ¯dC LiνLjS1/3 3 − √ 2yij L ¯dC LiℓLjS4/3 3 + √ 2(V ∗yL)ij¯uC LiνLjS−2/3 3 − (V ∗yL)ij¯uC LiℓLjS1/3 3 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=', (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='2) of which only the ¯dC LiℓLjS4/3 3 term contributes to O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' One can extract the Wilson coefficients for the b → sl−l+ decay [12–14, 20], Cℓ1ℓ2 9 = −Cℓ1ℓ2 10 = πv2 VtbV ∗ tsαem ybℓ1 L (ysℓ2 L )∗ m2 S3 , (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='3) 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Scalar Leptoquark R2 (3, 2, 7/6) The second case we consider is a weak doublet of scalar leptoquarks with hypercharge Y = 7/6, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' R2 (3, 2, 7/6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' [25] The most general Lagrangian describing the Yukawa interactions with R2 can be written as, LR2 = yij R ¯QilRjR2 − yij L ¯uRiR2iτ2Lj + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=', (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='4) where yL and yR are the Yukawa matrices corresponding to left- and right-handed lepton fields respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' In terms of the components with RQ 2 denoting each leptoquark state with charge Q, the Lagrangian can be written as LR2 = (V yR)ij¯uLiℓRjR5/3 2 + (yR)ij ¯dLiℓRjR2/3 2 + (yL)ij¯uRiνLjR2/3 2 − (yL)ij¯uRiℓLjR5/3 2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='5) The tree-level contribution to the Wilson coefficients C9 through the term (yR)ij ¯dLiℓRjR2/3 2 amounts to Cℓ1ℓ2 9 = Cℓ1ℓ2 10 = − πv2 2VtbV ∗ tsαem ysℓ1 R (ybℓ2 R )∗ m2 R2 , (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='6) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Vector Leptoquark U1 (3, 1, 2/3) Finally, we describe the only vector leptoquark model considered in this paper, mainly because it has been the only model that could simultaneously explain both charged current and neutral current anomalies [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We consider the U1 (3, 1, 2/3) model which gives a single leptoquark state with charge 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The most general Lagrangian consistent with the SM gauge symmetry allows couplings to both left-handed and right-handed fermions, namely LU1 = βij L ¯QiγµLjU µ 1 + βij R ¯dRiγµℓRjU µ 1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=', (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='7) with couplings βij L and βij R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The contributions to the left-handed couplings to the effective Lagrangian amount to Cℓ1ℓ2 9 = −Cℓ1ℓ2 10 = − πv2 VtbV ∗ tsαem βsℓ1 L (βbℓ2 L )∗ m2 U1 , (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='8) 7 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' LHC LIMITS Our goal is to use published LHC data to simultaneously constrain the mass and Yukawa couplings of the leptoquarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The Wilson coefficient C9 depends on three parameters roughly as Cℓ,ℓ 9 ∼ �y2ℓ y3ℓ M �2 where yij refers to the leptoquark coupling between the ith generation of quark and jth generation lepton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' This corresponds to Yukawa couplings for S3 and R2 models and the gauge coupling for the U1 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Therefore, its possible to find a surface in the 3D parameter space that gives the required value of C9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' However, most LHC search constraints are in principle only 2D — one coupling that determines the cross section of the final state and one mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We, therefore, have several options in which to view the full constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Let us start with ℓ = 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' µ) which contributes to Cµµ 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' To be able to independently constrain the two Yukawa couplings y22 and y32, we study three different cases — first setting only y22 non-zero (see figure 1, second setting only y32 non-zero (see figure 4) and third, setting both equal (see figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Using the upper limits from the non-resonant dimuon search gives us an upper limit on y22 at each mass value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' It is possible to also determine the minimal allowed value of y22 that is consistent with C9 by requiring y32 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Since the latest LHCb data seem to indicate that electrons and muons have identical behaviour, we can indeed also do a similar exercise with y21 and y31 which would contribute to Cee 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Besides these, non-zero values of all four couplings (or even a single electron and a single muon coupling) — y21, y31, y22 and y32 can give signatures that have differently flavoured leptons in the final state, but without missing energy and therefore with no SM background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' It should be noted that in the case where a single leptoquark state can couple to both electrons and muons, the strongest constraints on couplings and mass of course come from low energy processes in the µ → e sector [18, 19, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' However, it can still be an interesting exercise to directly probe the case where both yk1 and yk2 are non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' As we see in figure 2, this case is strongly constrained by the LHC, with the U1 model likely to be ruled out already with full run-2 data of 139 fb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Since multiple leptoquark states come from the same multiplet, they have identical mass and switching on a single coupling allows the production of multiple states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' For calculating 8 the LHC limits, we allow the production of all leptoquark states and select only that fraction that decays into the final state selected for by the analysis being reinterpreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' For example, in the S3 case, if we look for pair production of leptoquark followed by decay of each into a muon and a jet by turning on y22 ̸= 0 alone, we allow both the production of pairs of S4/3 3 → ¯sµ+ as well as pairs of S1/3 3 → ¯cµ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Our limits, therefore, are not identical to the simplified model limits that the experimental analysis publishes by producing only one state at a time, with 100% branching fraction into a certain channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Similarly, when looking at dilepton distributions, we take into account, with interference, all leptoquark states in the t-channel that are allowed by non-zero couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Computational setup Since we examine the limits from dilepton searches which have been presented in the form of upper limits on generator-level cross sections with fiducial cuts, our computational setup is much simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We generate events using Madgraph5 amc@NLO [27] with the required fiducial cuts and do not need to perform further detector simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' This approach has been proven to work well [28] and reproduces expected limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' For the UV models, we use the scalar leptoquark models for S3 and R2 described in [29] and for the vector leptoquark model for the U1 case, we use the model described in [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' When more complicated functionality is required, we use Pythia8 [33] to shower, hadronize and apply the required kinematic cuts on events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Limits from resonant and non-resonant dilepton searches We re-interpreted both the dilepton resonance search with 139 fb−1 [34] and the non- resonant dilepton search at 139 fb−1 [35] from ATLAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We find that the non-resonant search results in much stronger limits and we continue with this search for the rest of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The exclusive dilepton state can only be seen with a t-channel leptoquark exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' It is possible to have a dilepton plus two jets from strong production of leptoquarks, however, this process does not depend on the leptoquark-fermion couplings and results in only a mass limit which we deal with in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' With the interference of SM Drell-Yan production of leptons with the t-channel leptoquark mediated production, one expects to 9 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='500 1 MLQ [GeV] y2k Allowed S3 C9 ⇒ y3 k > 1 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='500 1 MLQ [GeV] y2k Allowed R2 C9 ⇒ y3 k > 1 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='500 1 MLQ [GeV] β2k Allowed U1 C9 ⇒ β3 k > 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Exclusion plots y2ℓ versus Mass of leptoquark for the S3 (top-left), R2 (top-right) and U1 models (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The bright red regions at the top are disallowed from dimuon searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The corresponding di-electron limit is the lighter line inside the red region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The solid regions at the bottom are from requiring perturbative couplings consistent with allowed C9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The vertical lines are mass limits from direct leptoquark pair production with the solid line corresponding to second generation leptons and the dotted corresponding to first generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The limits correspond to 139 fb−1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' see a change in the shape of the dilepton invariant mass distribution mℓℓ where ℓ = µ or e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We apply the limits from the ATLAS non-resonant dilepton search by generating events using Madgraph5 amc@NLO according to fiducial cuts listed in [35] and using the 95% upper limits for the most conservative signal region called the “µ+µ− constructive signal region” (or 10 analogously the e+e− constructive signal region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The constructive signal region corresponds to the case where you expect signal events above the EW expectation, which is similar to our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The experimental analysis uses LO signal shape to model the expected number of events and we therefore also do not use any NLO corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The upper limits are provided on the additional cross section above the expected SM Electro-Weak (EW) prediction in the cumulative signal region where mµ+µ− ≥ 2070 GeV (or me+e− ≥ 2200 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' As expected, the effect of having heavy new leptoquarks in t-channel dies down when either the leptoquark mass is too high or the Yukawa coupling is too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' To account for the interference correctly, we use the difference of the cross-section pp → ℓ+ℓ− with both leptoquark and EW bosons, and with only EW gauge bosons as our new physics contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The result is an excluded region near high Yukawa coupling values, with a larger range ruled out for smaller leptoquark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' This is shown as a bright red region in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The highest allowed value of y2k is referred to as y2k max and can be used to further restrict what values of y3k are consistent with C9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Currently, there is one different flavour dilepton search [36] performed at 13 TeV, but with only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='2 fb data analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Aside from cuts on pT of 65 and 50 GeV on electrons and muons respectively, there are requirements that missing energy be less than 25 GeV and mT < 50 GeV to remove contamination from W-boson production which we apply using Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='3 [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The expected background for meµ > 2 TeV is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' They see one event and interpreting it as a statistical fluctuation, set a limit on new physics cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We extrapolate the expected limits from this search at 139 fb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The limits on the eµ case for the U1 model can be seen in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The expected background at 139 fb−1 is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='78 events, resulting in an expected 95% upper limit of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='0185 fb on production cross section times branching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' As can be seen, the U1 model should be completely ruled out with 139 fb−1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' For results in the eµ channel for S3 and R2 models, refer to appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Limits from leptoquark-pair production Direct limits on the mass of the leptoquark based on strong pair-production mode followed by the decay of each leptoquark into a lepton and a jet are presented in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The limits are also presented on generator-level cross-section times branching fraction and can be applied directly to our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The resulting limit is shown as a solid black vertical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Since there 11 is no significant improvement in the limit from b-tagging, we use the general lepton+jet limits in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' When only yk2 is non-zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' the leptoquark decays to a muon and a jet, we obtain a mass limit for S3 leptoquark at 1774 GeV, for the R2 leptoquark at 1720 GeV and the U1 leptoquark at 2309 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' For the case where the leptoquark decays into electron alone, we get a mass limit for S3 leptoquark at 1828 GeV, for the R2 leptoquark at 1773 GeV and the U1 leptoquark at 2419 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' There is no direct limit on the case with an eµ final state in the published search, which if it existed, would give a far better exclusion simply because there is no irreducible SM background and the dominant background would be from mis-identification of leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Missing search: top FCNC decay Given the need for non-zero leptoquark coupling to the third generation of quarks, this also implies a coupling between the top quark and second generation leptons for both the S3 and U1 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' In the R2 case, the coupling is either CKM suppressed (in the case of left-handed) or entirely independent and therefore set to zero (in the right-handed case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' It would therefore be possible to search directly for FCNC top decay via t → cµµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Currently, there are no searches for t → cµ+µ− except for a t → cZ search which requires the dimuon mass to be within 15 GeV of the Z mass [38] and therefore is not directly applicable to our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' A similar measurement from CMS [39] is available from the 8 TeV run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The main background for a t → cµ+µ− search is from the SM production of t¯tµ+µ− via an off-shell Z or γ produced in association with t¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' To remove contamination from on- shell Z, we apply a cut instead Mℓℓ > 105 which is outside the Z-mass window selected for by the t → cZ searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Assuming the identification acceptances do not change, we can estimate the background for our proposed search using the data driven estimate presented in [38] (denoted by σBG,ATLAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Since we have identical SM production modes for t¯tZ and t¯tµ+µ−, we assume that the generator level transfer factor between these processes is transmitted all the way to the final selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The kinematic effect of changing the mℓℓ cut from |Mℓℓ − MZ| < 15 to Mℓℓ > 105 can be estimated at generator level and is encapsulated in a single number fℓℓ Also, we assume that the enhancement in production of t¯tZ in going 12 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='500 MLQ [GeV] β2k Allowed U1 C9 ⇒ β3 K > 1 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='500 MLQ [GeV] β3k C9 ⇒ β2 k > 1 U1 C9 ⇒ β2 k > β2 k max 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='500 MLQ [GeV] β2k,3k C9 best fit U1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Limits for the Leptoquark Couplings versus mass for the U1 Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The dilepton process, in this case, is pp → µe which does not exist in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We, therefore, have strong limits even with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='2 fb−1 data as published in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The top-left panel shows limits on the coupling to second generation quarks with y22 = y21, the top-right panel on the coupling to third generation quarks with y32 = y31 and the bottom panel shows the case where all four couplings are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The green band shows the values corresponding to the best fit values of C9 The dotted line in this figure shows the expected limit after analysing full 139 fb−1 of run-2 data by ATLAS (only partial result is published so far).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We see clearly that the universal scenario is likely already ruled out by run-2 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' 13 from 13 TeV to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='6 TeV (fE = σ13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='6 σ13 ) remains the same also for t¯tµ+µ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Thus we have σBG(√s = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='6) = σBG,ATLAS × fE × fℓℓ × σ(pp → t¯tµ+µ−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' √s = 13) σ(pp → t¯tZ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' √s = 13) (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='1) Using this, and the expected background cross section from ATLAS, we calculate an expected background of 7±2 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Given that with the Z-window, the background is esti- mated at 119±10 events, this would correspond to over an order of magnitude improvement in the sensitivity to FCNC branching fraction of the top quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' FUTURE PROSPECTS The best-fit value of the Wilson coefficients for operators that explain the b → s anomalies suggests a high suppression scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Using equations (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='6), (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='3) and (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='8), we find that the required scale for both couplings set to one is 16183 GeV for the R2 case and 22887 GeV for the S3 and U1 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Naturally, resonantly producing a leptoquark of this mass scale is out of the question at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We, therefore, investigate both the expected reach of the LHC after the planned high-luminosity run and estimate a conservative reach for a muon collider with CM energy of 3 TeV [40–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' To illustrate the highest sensitivity case, we choose y22 = y32 for this calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' This also allows us to make a comment on the ability of the collider to explore the entire parameter space of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' A summary of the expected reach of future colliders can be seen in figure 3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' LHC High-Lumi expected limits Projections for the HL-LHC are made with the luminosity of 3000 fb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' From previous experience, we know that the improvements in limits scale with about the square root of luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Using the expected number of signal and background events for the non-resonant dilepton search, we can probe effects of leptoquarks up to mass 5 TeV for the S3, 3 TeV for the R2 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='5 TeV for the U1 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Conversely, we can probe coupling values as small as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='4 for S3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='55 for R2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='15 for U1 models respectively at 1 TeV leptoquark mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' For 14 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='0 MLQ [GeV] y22,32 LHC13 MuonC, 1/fb 5σ 2σ C9 fit S3 LHC13-Mass 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='0 MLQ [GeV] y22,32 LHC13 MuonC, 1/fb 5σ 2σ C9 fit R2 LHC13-Mass 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='0 MLQ [GeV] β22,32 LHC13 MuonC, 1/fb 5σ 2σ C9 fit U1 LHC13-Mass FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Current and future reach in leptoquark coupling to muons with leptoquark mass for the S3 Model (top-left), R2 Model (top-right) and U1 Model (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The green region corresponds to the 1σ region given by global fit C9 values in the model-dependent case whereas the yellow is the data-driven 1σ region ([7], also see table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The solid red region is the current 139 fb−1 limits with the dotted red line the expected reach after 3 ab−1 at the HL-LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The solid and dotted vertical lines correspond to mass limits from pair production again corresponding to the 139 fb−1 and 3 ab−1 luminosity respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The blue region corresponds to the parameter space that can be discovered with a 5σ significance at a 3 TeV muon collider with 1 fb−1 whereas the orange region corresponds to the further region that can be probed at 95% confidence at the same collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The U1 model can be fully excluded with just 1 fb−1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The S3 and R2 models can also be fully probed with 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='5fb−1 and 5fb−1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' comparison, C9 best fit predicts a minimum value of coupling at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='06 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='04 for the three models when we set both couplings equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The direct search limits from strong production are calculated in a similar way using the 15 published upper limits at 139/fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We find that the HL-LHC can exclude leptoquark masses of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='2 TeV for both the S3 and R2 case and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='8 TeV for the U1 case for the leptoquark decaying into a muon and a jet and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='3 TeV for both the S3 and R2 case and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='9 TeV for the U1 case for the leptoquark decaying into an electron and a jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Reach of a Future Muon Collider Estimating the reach of a future muon collider is more difficult since we do not currently have a detector configuration to be able to simulate a realistic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' However, taking lessons from the dilepton and dijet searches at the LHC, we know that a single-bin analysis with a high enough cut on the invariant mass provides a very reliable estimate of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We look at µ+µ− → jj as our signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Obviously using b-tagging will be a further improvement that can pinpoint the underlying scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' However, for this estimate, we just use untagged jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Given that acceptance efficiencies of jets are expected to be similar for both signal and background events for a simple dijet search, we proceed with using just generator-level cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' A further advantage is the much reduced probability of extra initial state radiation jets from initial state muons (in sharp contrast to a pp machine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The main background from the SM comes from s-channel photon or Z exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' In the presence of the leptoquark, another Feynman diagram with a t-channel leptoquark exchange needs to be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We look only at events with Mjj > 500 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The SM-only cross section at LO is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='96×10−2 pb which corresponds to a statistical error of about 8 events at a luminosity of 1 fb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Using this, we can calculate the parameter space corresponding to a 5σ discovery as well as regions that can be excluded at 2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' They are shown in figure 3 as blue and orange regions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' In the U1 case, we see that a muon collider is capable of excluding the entire viable parameter space with 1 fb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' To exclude the R2 and S3 models would need a luminosity of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='5 fb−1 for S3 and 5 fb−1 for R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' SUMMARY AND CONCLUSIONS We examine the limits from direct collider searches on leptoquark models that are capable of explaining the anomalous measurements in the decays of B-mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We focus on three specific models — two scalar leptoquark models S3 and R2 and one vector leptoquark model 16 U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Aside from limits on the mass of the leptoquarks (which can be pair-produced by strong interactions), it is possible to also constrain the couplings to fermions by looking at changes to the shape of the dilepton mass spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Reinterpreting full Run-2 limits from the pair production and non-resonant dilepton searches by ATLAS experiment, we find that current mass limits are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='77 TeV, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='72 TeV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='3 TeV respectively for the three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We can expect to reach up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='2 TeV for S3 and R2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='8 TeV for the U1 respectively with the High-Luminosity LHC run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Effects of leptoquarks with couplings to muons can potentially be probed in a muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Since there has been considerable interest in a future muon collider recently, we also estimate what the reach of the proposed 3 TeV muon collider would be for the three models in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' We find that with very minimal assumptions, S3, R2 and U1 models show significant deviation in dijet distributions that can be observable for the entire range of interest with less than 6 fb−1 data for all three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' ACKNOWLEDGEMENTS ND is supported by the Ramanujan Fellowship grant SB/S2/RJN-070 from the Department of Science and Technology of the Government of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' [1] LHCb collaboration, Test of lepton universality in b → sℓ+ℓ− decays, 2212.' metadata={'source': 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Observable Experiment Theory (SM) RK[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='994 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='090 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='082 (stat) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='029 −0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='087 (stat) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='036 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='035 (syst) [2022] [1, 2] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='01 [44]) RK[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='1,6] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='949 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='042 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='041 (stat) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='022 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='022 (syst) [2022] [1, 2] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='01 [44]) RK∗[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='1,6] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='027 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='072 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='068 (stat) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='027 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='026 (syst) [2022] [1, 2] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='01 [44]) R[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='045,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='1] K∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='66+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='03 [2021] [45] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='906 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='028 [44] R[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='1,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='0] K∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='69+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='05 [2021] [45] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='01 [44] R[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='1,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='0] K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='846+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='042+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='013 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='039−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='012 [2021] [46] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='01 [44] B(Bs → µ+µ−) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='85+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='31) × 10−9 [3, 4] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='14) × 10−9[47]) P ′ 5 in B → K(∗) l+ l− [5, 48, 49] [6, 50] TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' A summary of the most relevant experimental results and SM predictions for the observables in b → s sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' 22 Appendix B: Limits on leptoquark couplings to third generation quarks y3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='500 1 MLQ [GeV] y3k C9 ⇒ y2 k > 1 S3 C9 ⇒ y2 k > y2 k max 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='500 1 MLQ [GeV] y3k C9 ⇒ y2 K > 1 R2 C9 ⇒ y2 K > y2 K max 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='500 1 MLQ [GeV] β3k C9 ⇒ β2 k > β2 k max U1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Exclusion plots y3ℓ versus Mass of leptoquark for the S3 (top-left), R2 (top-right) and U1 models (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The solid regions at the bottom are from requiring perturbative couplings consistent with allowed C9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The darker region is inconsistent with the observed upper limits on y2k in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The vertical lines are mass limits from direct leptoquark pair production with the solid line corresponding to second generation leptons and the dotted corresponding to first generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The limits correspond to 139 fb−1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' 23 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='500 1 MLQ [GeV] y2k,3k C9 best fit S3 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='500 1 MLQ [GeV] y2k,3k C9 best fit R2 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='500 1 MLQ [GeV] β2k,3k C9 best fit U1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Exclusion plots in the limited case of y2ℓ = y3ℓ versus Mass of leptoquark for the S3 (top-left), R2 (top-right) and U1 models (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The solid red region at the top are limits from non-resonant dilepton searches in µ+µ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The lighter lines inside this region correspond to subleading limits from the similar e+e− search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The vertical lines are mass limits from direct leptoquark pair production with the solid line corresponding to second generation leptons and the dotted corresponding to first generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The limits correspond to 139 fb−1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' The green band is the region that corresponds to the coefficient C9 within one sigma of best fit to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' – 24 Appendix C: Limits on S3 and R2 model parameters in the Lepton Flavour Universal case 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='500 1 MLQ [GeV] y2k Allowed S3 C9 ⇒ y3 K > 1 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='500 1 MLQ [GeV] y3k C9 ⇒ y2 k > 1 S3 C9 ⇒ y2 k > y2 k max 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='500 1 MLQ [GeV] y2k,3k C9 best fit S3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' Limits on the leptoquark couplings via the process p p → µ e in the case of flavour universal couplings to electrons and muons for the S3 Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content=' 25 1000 2000 3000 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfyf5z/content/2301.01754v1.pdf'} +page_content='010 0.' metadata={'source': 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S¨aubert,1, 2, ∗ C. Franz,1, 2, 3 J.K. Jochum,1, 2 G. Benka,1 A. Bauer,1, 4 S.M. Shapiro,5 P. B¨oni,1 and C. Pfleiderer1, 4, 6 +1Physik-Department, Technische Universit¨at M¨unchen, D-85748 Garching, Germany +2Heinz Maier-Leibnitz Zentrum, Technische Universit¨at M¨unchen, D-85748 Garching, Germany +3J¨ulich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), +Forschungszentrum J¨ulich GmbH, Garching, Germany +4Zentrum f¨ur QuantumEngineering (ZQE), Technische Universit¨at M¨unchen, D-85748 Garching, Germany +5Brookhaven National Laboratory, Department of Physics, Upton, NY 11973, USA +6Munich Center for Quantum Science and Technology (MCQST), +Technische Universit¨at M¨unchen, D-85748 Garching, Germany +(Dated: January 12, 2023) +In the iron–chromium system, FexCr1−x, a wide dome of spin-glass behavior emerges when the +ferromagnetism of iron is suppressed and the antiferromagnetism of chromium emerges as a function +of increasing iron content x. As both, the high-temperature state and the characteristic cluster size +vary as a function of x, different regimes of spin-glass behavior may be compared in a single, +isostructural material system. Here, we report a study of the spin dynamics across the freezing +process into the spin-glass state for different iron concentrations (x = 0.145, 0.175, 0.21) using +Modulation of IntEnsity with Zero Effort (MIEZE) spectroscopy. In the parameter range studied, the +relaxation process observed experimentally may be described well in terms of a stretched exponential. +In the reentrant cluster-glass regime, x = 0.145, this behavior persists up to high temperatures. In +comparison, in the superparamagnetic regime, x = 0.175 and x = 0.21, a single relaxation time at +elevated temperatures is observed. For all samples studied, the spin relaxation exhibits a momentum +dependence consistent with a power law, providing evidence of a dispersive character of the spin +relaxation. +A. +Introduction +In spin-glasses, the combination of random site oc- +cupation and disorder with competing interactions, +anisotropy, and frustration leads to a collective freezing +of the spins in random orientations at the glass temper- +ature Tg1,2. The freezing process may involve a broad +distribution of relaxation times, resulting from varying +correlation lengths of the individual magnetic clusters. +These clusters may vary strongly in size, ranging from in- +dividual spins in canonical spin-glasses, over interacting +clusters of spins in cluster glasses, to collectively behav- +ing domains in superparamagnetic systems3. In dilute +spin-glasses, the perhaps most comprehensively studied +class of glassy magnetic systems4–12, the relaxation times +of the spin freezing exhibit no dispersion due to the short +range of the interactions. +Seminal +studies +of +the +spin-glass +behavior +in +Cu1–xMnx +(x += +0.1, 0.16, and 0.35) as well as +Au1–xFex (x = 0.14) using neutron spin echo (NSE) +spectroscopy11,13 suggested non-exponential relaxation. +To go beyond an account of the spin relaxation in terms +of a stretched exponential, which cannot distinguish be- +tween a distribution of parallel relaxation channels and +hierarchical relaxation comprising intercluster and intra- +cluster processes, the Weron model14 was found to pro- +vide a universal description. +This raises the question +for key characteristics of the spin relaxation when the +concentration of magnetic atoms is increased to form +cluster-glass or superparamagnetic systems. +Moreover, +reports of a momentum independent quasielastic peak +in the spin-glass regime15–20 contrast dispersive behavior +that has been attributed to either the coexistence of fer- +romagnetism and spin-glass behavior21–23 or momentum- +dependent dynamics of the spin-glass24. +To advance these questions, we report a study of the +spin-glass dynamics in FexCr1–x. When combining the +itinerant-electron ferromagnet iron and the spin-density +wave antiferromagnet chromium in the isostructural al- +loy FexCr1–x, the ferromagnetic transition temperature +is suppressed and long-range spin-density wave order +emerges with increasing x for xc = 0.15 − 0.1725–30. +A dome of spin-glass behavior is located at low temper- +atures in the vicinity of xc16,31–37. The dome extends +from x ∼ 0.10 to x ∼ 0.25, reaching well into concentra- +tion regimes which exhibit ferromagnetic and antiferro- +magnetic order at higher temperatures, respectively38–41. +Thus, for different x, the spin-glass behavior in FexCr1–x +may emerge with decreasing temperature from ferromag- +netic or antiferromagnetic order due to spin freezing42–45. +Moreover, the ferromagnetically ordered clusters initially +grow in size as a function of increasing x, changing the +character of the spin-glass state from a cluster-glass to +a superparamagnet41,46,47. As both iron and chromium +display spin wave dispersions that are prototypical for +ferro- and antiferromagnetism, respectively, a question +concerns the existence and character of dispersive behav- +ior in the spin-glass regime48–55. +Quasielastic neutron scattering has been established as +an indispensable tool in the study of spin relaxation pro- +cesses. For a wide range of relaxation times expected in +spin-glasses, neutron spin-echo spectroscopy appears to +be ideally suited, since the associated measurements of +the intermediate scattering function S(q, τ), as opposed +to the dynamic structure factor S(q, ℏω), allows to sep- +arXiv:2301.04495v1 [cond-mat.str-el] 11 Jan 2023 + +2 +arate dynamic processes on very different time scales. +However, in conventional neutron spin-echo spectroscopy +depolarizing samples or samples under depolarizing sam- +ple environment may only be measured at a high penalty +in neutron flux56,57. On this note, Modulation of IntEn- +sity with Zero Effort58,59 (MIEZE) is a spin-echo tech- +nique which permits measurements under depolarizing +conditions for small momentum transfers. It is therefore +ideally suited for the study of spin relaxation dynamics +in nearly ferromagnetic and ferromagnetic systems. +In the study reported in this paper, we used the lon- +gitudinal MIEZE technique to determine the spin dy- +namics in FexCr1–x as a function of temperature for +three different compositions. For the reentrant cluster- +glass (x = 0.145), a broad distribution of relaxation +times is observed over the entire temperature range that +may be described well in terms of a stretched exponen- +tial. For compositions in the superparamagnetic regime +(x = 0.175 and x = 0.21), a broad distribution of relax- +ation times at low temperatures that may be described +well in terms of a stretched exponential is contrasted by +a single relaxation time at elevated temperatures. Our +data recorded in the sample with x = 0.175 are con- +sistent with an earlier study using transverse NRSE and +conventional NSE in the same sample12, which, however +exhibited more scatter and covered a smaller dynamical +range. For all three compositions we find dispersive be- +havior of the spin relaxation that follows a power-law de- +pendence of the momentum, q, consistent with Γ ∝ qz, +where the dynamical exponent z decreases from z ∼ 1.5 +to z ∼ 1.0 with increasing x. This behavior is loosely +reminiscent of the quasielastic linewidth in Fe0.7Al0.324. +However, for FexCr1–x the origin of the dispersive behav- +ior may, in principle, comprise a combination of different +contributions. +Our paper is organized as follows. In Section B, spe- +cific aspects of the neutron scattering experiments and +the crystal growth techniques are presented. +In Sec- +tions C 1 and C 2 the elastic and quasielastic neutron +scattering experiments are described. In Section D, the +implications of the experiments are discussed, followed +by a summary of the findings of this study in Section E. +B. +Experimental Methods +Our experiments were conducted at the beamline +RESEDA60,61 at the Heinz Maier-Leibnitz Zentrum using +the longitudinal MIEZE option62. The MIEZE method +is particularly well suited to study dynamics close to +the [0, 0, 0] Bragg peak, corresponding to small-angle +neutron scattering (SANS)61,62. +In Fig. 1 the experi- +mental setup used for our experiments is shown. +The +sample-detector distance, LSD += 2.335 m, was maxi- +mized to ensure highest q resolution. To provide high +neutron flux and to cover the desired dynamic range, +the wavelength was set to 6 �A with a wavelength spread +∆λ/λ = 0.12. In this configuration, a dynamic range +(c) +n +(b) +detector +sample +aperture +beam +stop +sample +(a) +detector +sample +polarizer +analyzer +π/2 +flipper +π/2 +flipper +NSE +solenoid +NRSE +RF flipper +n +LSD +direct +beam +beam stop +11 +FIG. +1. +Longitudinal +MIEZE +setup +used +in +our +experiments51,62. (a) Schematic depiction of the spectrom- +eter. All spin manipulations are performed upstream to the +sample, rendering the method insensitive to depolarizing con- +ditions at the sample position. The red arrows indicate the di- +rection of the magnetic guide fields. (b) Schematic depiction +of the neutron flight path through the spectrometer show- +ing the sample aperture, sample, beam stop, and detector. +(c) Detector segments used in the evaluation of the MIEZE +scans. Segments represent parts of circular rings centered at +the direct beam with an opening angle of 60° and a width of +10 pixels. The grey shaded area represents the location of the +beam stop. +from ∼ 6 · 10−6 ns to 2 ns was accessible. Fig. 1(b) shows +a schematic of the neutron flight path through the spec- +trometer. Data were recorded with a 20 cm × 20 cm 2D +CASCADE detector63, covering a q range from 0.016 �A +−1 +to 0.085 �A +−1 at λ = 6 �A and LSD = 2.335 m. Recent de- +velopments at RESEDA made it possible to increase LSD +up to 3.43 m improving the spatial resolution further61. +The grouping of the detector segments for evaluating +the quasielastic data are shown in Fig. 1(c). For the elas- +tic measurements, the identical setup was used with nar- +rower grouped detector segments, i.e., data were eval- +uated in the same area as in Fig. 1(c) but divided in +25 instead of 11 detector segments. +The sample was +cooled to temperatures between 4 K and 300 K using a +top-loading closed-cycle refrigerator and the temperature +was controlled with two sensors close to the sample. The +temperature stability was ∼0.05 K. +No hysteresis was +observed in temperature scans. +Data were normalized +using data recorded at the base temperature of ∼4 K, as- +suming that the spin dynamics in FexCr1–x are frozen at +temperatures well below Tg11. This procedure minimizes +systematic errors since the experimental setup remains +unchanged throughout the entire experiment on a given +sample. +We investigated three textured polycrystalline samples +of FexCr1–x containing large grains with iron concentra- +tions of x = 0.145, 0.175, and 0.21. The samples were +prepared by means of arc melting from pure starting ma- +terials and annealed for 4 days at 1100 °C before quench- +ing in water16. To remove strain, the samples were sub- + +3 +sequently annealed at 1000 °C for 1 day16. Following this +process ingots of cylindrical shape with a height of ap- +proximately 20 mm and a diameter of about 10 mm were +obtained. +To optimize the signal-to-noise ratio, slabs with a +thickness of 8 mm were cut from the ingots. +A circu- +lar aperture made of cadmium with a diameter of 10 mm +was attached directly to the samples. Thus, the neutron +beam effectively illuminated samples of cylindrical shape +with a diameter of 10 mm and a length of 8 mm, where +the cylindrical axis was parallel to the incident neutron +beam. Measurements on a sample with x = 0.17 from +Benka et al.41 (not shown) were in excellent agreement +with the results presented in the following. +Parts of the sample with x = 0.145 were recently +used to investigate the influence of concentration fluc- +tuations on relaxation processes in spin-glasses12. Using +atom probe tomography, a high-resolution local probe, +together with neutron resonant spin-echo spectroscopy it +was shown that small-scale inhomogenieties in the mi- +crostructure influence the relaxation processes of a spin- +glass material12. Using the Weron model which had pre- +viously been applied to dilute spin-glasses13, the relax- +ation processes were described in this study. +Tg +TC +Tg +TC +Tg +TC +Burke Refs. [36, +38-40] +TN +Tg +TC +Benka Ref. [41] +TN +Tg +TX +TC +this study +Tg +TC +FexCr1 +x +x = 0.145 +FexCr1 +x +x = 0.175 +FexCr1 +x +x = 0.210 +FM +AFM +SG +AFM/FM +14.5% +17.5% +21.0% +0 +125 +250 +T (K) +0 +1 +2 +3 +Int.×10 +5 (a.u.) +0 +125 +250 +T (K) +0 +125 +250 +T (K) +0.10 +0.15 +0.20 +0.25 +x in FexCr1 +x +0 +30 +60 +90 +Temperature (K) +(b) +(c) +(d) +(a) +FIG. 2. Magnetic properties of the FexCr1–x system. An- +tiferromagnetic (AFM, green), ferromagnetic (FM, blue), +spin-glass (SG, orange), and antiferromagnetic-ferromagnetic +(AFM/FM, purple) regimes are distinguished. +(a) Tem- +perature vs concentration phase diagram combining data +from neutron scattering and low-field magnetization by Burke +et al. (squares)36,38–40, ac susceptibility and magnetization +by Benka et al. (triangles)41, and neutron scattering (present +work, stars). (b)-(d) Integrated SANS intensity for the three +samples measured at RESEDA. The shaded areas indicate +magnetic regimes as inferred from the phase diagram. +C. +Experimental Results +The phase diagram of FexCr1–x as a function of tem- +perature and iron concentration x is shown in Fig. 2(a) as +reproduced from literature36,38–41. For comparison, the +temperature dependence of the integrated SANS inten- +sity of the three samples measured in this study is shown +in Figs. 2(b) to 2(d). At the border between ferromag- +netic and antiferromagnetic order, a dome of spin-glass +behavior emerges, covering the regime of putative quan- +tum phase transitions. At low temperatures, this spin- +glass regime includes concentrations for which at high +temperatures long-range ferromagnetic or antiferromag- +netic order are observed. +In our study, three compositions were investigated: (i) +x = 0.145 which exhibits transitions from paramag- +netism (PM) to antiferromagnetism (AFM) to a spin- +glass (SG) with a glass temperature Tg = 11 K ± 2 K41, +(ii) x += +0.175 which exhibits transitions from PM +via a small region reminiscent of FM order41 to a SG +with a glass temperature Tg = 20 K ± 2 K41, and (iii) +x = 0.21 which exhibits transitions from PM to fer- +romagnetism (FM) to a SG with a glass temperature +Tg = 14 K ± 2 K41. +1. +Elastic Scattering +The temperature dependence of the integrated SANS +intensity is shown in Figs. 2(b) to 2(d). As no magnetic +scattering was observed at high temperatures, the data +recorded at ∼300 K were used for background subtrac- +tion. +The scattered intensities as a function of temperature +for x = 0.145 and x = 0.175 behave similarly. With de- +creasing temperature the intensity increases as expected +for a transition into a ferromagnetically ordered state. +When entering the spin-glass state, the system becomes +static on the time scales probed by SANS, and the inten- +sity increases with a change in slope, forming a plateau +that starts at the onset of the spin-glass regime64. +The bulk properties and phase diagram show that the +sample with x = 0.145 enters the spin-glass regime via +an antiferromagnetic state, which may not be identified +microscopically for the parameter range probed in SANS. +Interestingly, the increase in intensity is, analogous to +the sample with x = 0.175, reminiscent of ferromagnetic +order in agreement with Ref. [41]. +For the sample with x = 0.21, a broad feature with +a maximum at ∼75 K defines TC, followed by a sharp +increase in intensity. +A change of slope with decreas- +ing temperature, close to the transition temperature +reported previously in Ref. [41], defines Tg. +The sig- +natures defined in neutron scattering are denoted by +stars in Fig. 2(a). +Discrepancies as compared to the +phase boundaries inferred from the ac susceptibility and +magnetization41 may reflect the different time scales +probed by the different methods. + +4 +The q dependence of the SANS data, cf. Section B for +details on data analysis, is shown in Figs. 3(a) to 3(c) for +different temperatures. Following the approach taken in +related SANS studies on other materials65,66 we consider +a single power law form +I ∝ q−n. +(1) +Fitting the experimental data yields exponents as a func- +tion of temperature as shown in Figs. 3(d) to 3(f). +10 +6 +10 +5 +10 +4 +Intensity (a.u.) +10 +6 +10 +5 +10 +4 +Intensity (a.u.) +0.02 +0.04 +0.06 +0.1 +q (Å +1) +10 +6 +10 +5 +10 +4 +Intensity (a.u.) +0 +1 +2 +n +0 +1 +2 +n +n +2 +1 +0 +T (K) +200 +100 +0 +FM +SG +AFM +AFM/FM +T (K) +5 +15 +10 +20 +12 +25 +30 +70 +34 +80 +50 +100 +150 +200 +x = 0.145 +x = 0.175 +x = 0.210 +FexCr1 +x +x = 0.145 +x = 0.175 +x = 0.210 +Tg +TC +TC +Tg +Tg +TC +(a) +(b) +(c) +(d) +(e) +(f) +FIG. 3. Momentum dependence of SANS of FexCr1–x as mea- +sured at RESEDA. (a)-(c) Intensity as a function of scattering +vector q. The accessible q range was 0.02 �A +−1 < q < 0.08 �A +−1. +Solid lines are fits using Eq. (1). (d)-(f) Temperature depen- +dence of the exponent n obtained from the fits in (a)-(c). +Solid lines are guides to the eye. The shaded areas indicate +the regimes according to the phase diagram, namely AFM +(green), AFM/FM (purple), FM (blue), and SG (orange). +Samples with iron concentrations of x = 0.145 (top row), +x = 0.175 (middle row), and x = 0.21 (bottom row) were +evaluated at different temperatures. +With decreasing temperature, the exponent n increases +from ∼ 0.3, reaching low-temperature values of n ∼ 1.3 +for x = 0.145, n ∼ 1.8 for x = 0.175, and n ∼ 2.1 for +x = 0.21. Even though the data are well-described by +the power law in Eq. (1), unambiguous interpretation of +the exponent n proves difficult. Assuming a two-phase +system where smooth clusters are isolated, n = 4 would +be expected. Deviations from this behavior would indi- +cate more complex phases including fractal surfaces. An +exponent of n = 2, for instance, was reported in the +perovskite manganite Pr1–xCaxMnO3 and attributed to +sheets of inter-penetrating ferromagnetic and antiferro- +magnetic phases65. In the same material, Viret et al.66 +found n = 1.6 − 1.7 which was attributed to filamentary +ferromagnetic chains in analogy to polymers. An increase +in n with decreasing temperature suggests a coarsening of +the interfaces as the system enters the spin-glass regime, +consistent with an increase in cluster size. +To address the question of the nature of the low- +temperature magnetic structure as a function of iron con- +centration, it may be necessary to collect data at even +smaller q values, which is beyond the technical limits of +the work reported here. Alternatively, more complex de- +pendencies, e.g., comprising several different power-law +contributions may be possible. Such descriptions would +require theoretical modelling beyond the scope of our +study. +2. +Quasielastic Measurements +The normalized intermediate scattering function, +I(q, τ) = S(q, τ)/S(q, 0), +(2) +in the temperature range 4 K ≤ T ≤ 150 K for q = +0.044 �A +−1 and all FexCr1–x compositions investigated is +shown in Fig. 4. Following careful comparison of the data +with different relaxation models models4,6–8,10,11,13,67 we +find that the data are already described well by a +stretched exponential characteristic of a distribution of +relaxation rates, namely +I(q, τ) = Ielastic + (1 − Ielastic) exp +� +− (Γτ)β� +. +(3) +Here, Γ is the decay rate, corresponding to the inverse +spin relaxation time, 0 < β ≤ 1 stretches the clas- +sic exponential decay (β = 1), and the prefactor Ielastic +describes the elastic contribution to the signal. In the +light of the microscopic complexity of the FexCr1−x sys- +tem, which may support both parallel as well as hierar- +chical relaxation, we refrain from an analysis in terms of +more sophisticated mechanisms as assumed in the Weron +model12–14,67. +The normalized intermediate scattering functions in +Fig. 4 have been shifted vertically in steps of 0.25 for bet- +ter visibility. Fig. 4(a) shows the sample with x = 0.145. +The intermediate scattering function is constant below +∼4 K, confirming that the spin dynamics are frozen11 +with respect to the normalizing data at 4 K. For tem- +peratures around the glass temperature, Tg ∼ 15 K, the +sample starts to show signatures of dynamic behavior, +notably spin relaxation as indicated by a decrease in the +intermediate scattering function. The relaxation time of +the spins is inferred from a fit to the data using τ = ℏ +Γ. +Increasing the temperature shifts the spin relaxation to +shorter spin-echo times. Ielastic describes the elastic con- +tribution to the intermediate scattering function, corre- +sponding to the fraction of the sample that is static on +the time scales probed here, i.e., clusters fluctuating on +much longer time scales. +The elastic signal decreases with increasing tempera- +ture, reaching a minimum of Ielastic ≈ 0.25 for tempera- +tures above 34 K. For all temperatures, the system may + +5 +(b) +(c) +(a) +4 K +12 K +15 K +20 K +25 K +30 K +34 K +80 K +150 K +FexCr1 +x +x = 0.145 +0.044 Å +1 +10 +4 +10 +3 +10 +2 +10 +1 +10 +0 +(ns) +0 +1 +2 +3 +I(q, ) +4 K +10 K +15 K +20 K +25 K +30 K +34 K +80 K +150 K +FexCr1 +x +x = 0.175 +0.044 Å +1 +10 +4 +10 +3 +10 +2 +10 +1 +10 +0 +(ns) +4 K +10 K +15 K +20 K +25 K +34 K +50 K +80 K +150 K +FexCr1 +x +x = 0.210 +0.044 Å +1 +10 +4 +10 +3 +10 +2 +10 +1 +10 +0 +(ns) +FIG. 4. Normalized intermediate scattering function as measured for different temperatures in FexCr1–x. Data were recorded +using neutrons with a mean wavelength of λ = 6.0 �A and are shown at q = 0.044 �A +−1 for (a) x = 0.145, (b) x = 0.175, (c) +x = 0.21. For better visibility, data are shifted vertically. The solid lines are fits to the data using the stretched exponential in +Eq. (3). The dashed lines represent fits using a single exponential relaxation, corresponding to β = 1 in Eq. (3). +be described by a stretched exponential decay, β < 1. +To highlight the need for the stretching parameter β, ex- +ponential decays (β = 1) are shown for comparison as +dashed lines in Figs. 4(a) to 4(c). +The deviation from +simple exponential behavior gets more pronounced with +decreasing temperature. +The intermediate scattering functions for the sample +with x = 0.175 are shown in Fig. 4(b). +For tempera- +tures below Tg, the data resemble the behavior observed +for x = 0.145. Under increasing temperature the spin re- +laxation is similarly shifted to shorter times but remains +larger than for x = 0.145. The minimum of the elastic +background is reduced to Ielastic ≈ 0.15. In contrast to +the sample with x = 0.145, at higher temperatures the +spin relaxation can be described by a simple exponential +decay. +For x = 0.21, shown in Fig. 4(c), the behavior again +is highly reminiscent of the other two compositions. Ac- +cordingly, we observe a shift of the relaxation time to- +wards shorter times as the temperature increases. How- +ever, for temperatures above 50 K the relaxation time no +longer decreases. The elastic contribution reaches a min- +imum of Ielastic ≈ 0.1. In a paramagnetic sample, where +all scattering is dynamic, the spin-echo curve would de- +cay to Ielastic = 0 at high temperatures or long spin-echo +times. The observed sample dependence of Ielastic may +be attributed empirically to the different iron contents +of the samples, which lead to different high-temperature +magnetic phases. +Since the measurements were performed in a SANS ge- +ometry, close to the ferromagnetic Bragg peak at q = 0, +ferromagnetic fluctuations will contribute strongly to the +measured intensity. However, as possible antiferromag- +netic scattering intensity cannot be observed in the vicin- +ity of q = 0, antiferromagnetic fluctuations will not con- +tribute to the exponential decay, decreasing the dynamic +contribution to the intensity in samples with a larger an- +tiferromagnetic fraction. +This observation is in accor- +dance with the elastic measurements shown in Figs. 2(b) +to 2(d), which indicate that the intensity increases with +increasing iron content. +The temperature dependences of the fit parameters +Ielastic and β are summarized in Fig. 5 for q = 0.044 �A +−1. +For all samples the elastic contribution Ielastic decreases +linearly with increasing temperature reaching a constant +value above ∼40 K, i.e., far above the glass temperature. +A shrinking elastic contribution with increasing temper- +ature suggests that parts of the sample slowly unfreeze +on the time scales studied in our experiments. +The exponent β provides an estimate of the broaden- +ing of the spectrum of relaxation times. For all samples, +β decreases drastically when the temperature decreases +towards the spin-glass regime. This evolution suggests +an increase of the distribution of relaxation times, as ex- +pected of the formation of different-sized domains fluctu- +ating on different time scales before freezing at the lowest + +6 +FM +SG +AFM +AFM/FM +x = 0.210 +Ielastic +0 +50 +100 +150 +T (K) +0 +0.5 +1.0 +(c) +x = 0.145 +0 +0.5 +1.0 +(d) +x = 0.175 +0 +0.5 +1.0 +(e) +(f) +x = 0.210 +0 +50 +100 +150 +T (K) +0 +0.5 +1.0 +(b) +x = 0.175 +0 +0.5 +1.0 +Ielastic +(a) +FexCr1 +x +x = 0.145 +0.044 Å +1 +0 +0.5 +1.0 +Ielastic +FIG. 5. Temperature dependence of the fit parameters Ielastic +and β in Eq. (3) for the three FexCr1–x samples with (a),(d) +x = 0.145, (b),(e) x = 0.175, and (c),(f) x = 0.21. Data +are shown for q = 0.044 �A +−1. +The shaded areas indicate +the phases according to the phase diagram: AFM (green), +AFM/FM (purple), FM (blue), and SG (orange). +(a)-(c) +Elastic contribution Ielastic as a function of temperature, +showing a linear increase for temperatures below ∼40 K. Solid +lines are guides to the eye. The elastic background, i.e., the +constant value at temperatures above 40 K, is connected to +the magnetic signal-to-noise ratio, and thus different for the +different samples, see main text for details. (d)-(f) Stretched +exponent β as function of temperature showing a drastic de- +crease of β close to Tg. Solid lines are guides to the eye. +temperatures. For x = 0.145, β stays below 1 over the +entire temperature range, while for the other two com- +positions β is close to 1 at temperatures above Tg. This +means that the dynamics may be described with a single +relaxation time. +Numerical calculations of the non-exponential relax- +ation in spin-glasses and glassy systems68,69 have shown +that β = 1/3 is approached at Tg when the system is +close to its percolation limit. The decrease of β we ob- +serve as the spin-glass regime is approached is in quali- +tative agreement with these calculations. +D. +Discussion +The area detector used in our MIEZE measurements +allowed studying the normalized intermediate scattering +function I(q, τ) simultaneously over a wide range of mo- +mentum transfers q. +The q dependence of the decay +rate Γ and therefore the spin relaxation time is shown +in Figs. 6(a) to 6(c). Within experimental accuracy, our +data it are consistent with +Γ = Aqz, +(4) +where q is the momentum transfer, z the dynamical expo- +nent, and A the energy scale of the exchange interactions. +The decay rate Γ as a function of q, shown in Fig. 6, was +fitted using Eq. (4) with fixed values of z, i.e., z = 1.0, +z = 1.5, and z = 2.0, as well as with z as an independent +fitting parameter. Additionally, data were fitted for each +sample and for all temperatures independently, as well as +simultaneously for all temperatures. The fit results are +summarized in Tab. I. A χ2-analysis of the fits with fixed +parameter z shows that with decreasing iron content z +decreases from ∼ 1.5 to ∼ 1.0, which is supported by +the fits with z as a free parameter. The resulting values +of z and A as a function of temperature are depicted in +Figs. 6(d) to 6(i). +For small β, the distribution of relaxation times is very +broad and the data cannot be described with a single +τ. +Therefore a meaningful linewidth Γ cannot be ex- +tracted for any of the three samples below and around +the glass temperature Tg. For x = 0.145, Γ could only +be extracted for a small temperature window in the fer- +romagnetic regime, 25 K ≤ T ≤ 34 K. Even at these +temperatures, the exponential decay is already stretched +and the values of Γ determined experimentally represent +a mean relaxation time rather than a single relaxation +time. +At higher temperatures, the exponential decay +is strongly stretched such that a single dominant relax- +ation time could not be determined. The sample with +x = 0.175 allowed us to analyze Γ for temperatures be- +tween 25 K ≤ T ≤ 80 K. For x = 0.21, Γ could be +extracted for all temperatures above the freezing tem- +perature Tg. +For all concentrations, Γ depends on q according to +the relation given in Eq. (4). The exponent z increases +for increasing iron concentration from z ≈ 1 to z ≈ 2. +According to dynamic scaling theory the critical expo- +nent at Tc for pure ferromagnets in the limit q → 0 +corresponds to z = 2.5 while antiferromagnetic correla- +tions for q → QAFM lead to z = 1.5. Heuristically, one +might hence explain the exponents in FexCr1–x in terms +of a competition of ferromagnetic and antiferromagnetic +correlations. +Tajima et al. used a spin diffusion model with Γ ∝ q2.0 +to describe the q dependence of Γ in the Invar alloy +Fe65Ni35 over a rather wide q range70. This model as- +sumes a hydrodynamic behavior of uncorrelated spins. +The authors argued that the system never reaches criti- +cality, i.e., z = 2.5 for ferromagnetically correlated spins, +due to impurity scattering of the electrons. +Along this line, the presence of chromium atoms acting +as impurities in ferromagnetic iron clusters could prevent +the system from reaching ferromagnetic critical dynam- +ics, reducing z to 2.0. A large fraction of chromium, and +therefore an increase in antiferromagnetic correlations in +FexCr1–x could reduce the exponent z further towards +1.5 leading to an effective range of exponents between +z = 1.5 for antiferromagnetic correlations and z = 2.0 +as expected in the spin diffusion model. Values of z below +1 may reflect the large number of different time scales in +spin-glasses. Unusually small values of z found in spin- +glasses were previously attributed to disorder, the prox- + +7 +TABLE I. Summary of the fitting procedure of the spin dynamics in FexCr1–x using Eq. (4) with different approaches, namely: +(i) fixing the exponent z = 1.0, (ii) fixing the exponent z = 1.5, (iii) fixing the exponent z = 2.0, (iv) leaving the exponent +z as a free fit parameter. Additionally, data were also fitted with (I) all temperatures independently and (II) all temperatures +simultaneously. The χ2 value of each fit is used as an indicator for the goodness of the fits. +FexCr1–x with x = 0.145; all temperatures fitted independently +z = 1.0 +z = 1.5 +z = 2.0 +z = free +T (K) A +Aerr z +zerr χ2 +A +Aerr z +zerr χ2 +A +Aerr z +zerr χ2 +A +Aerr +z +zerr χ2 +25 +273 10 +1.0 − +1.20 +1127 84 +1.5 − +4.95 +4394 506 +2.0 − +11.19 +223 +61 +0.9 0.1 1.13 +30 +333 48 +1.0 − +18.63 +1536 225 1.5 − +18.84 +6576 1042 2.0 − +21.47 +629 +745 +1.2 0.4 18.22 +34 +400 40 +1.0 − +20.21 +1723 184 1.5 − +22.79 +6953 914 +2.0 − +32.99 +589 +507 +1.1 0.3 19.74 +FexCr1–x with x = 0.145; all temperatures fitted simultaneously +z = 1.0 +z = 1.5 +z = 2.0 +z = free +T (K) A +Aerr z +zerr χ2 +A +Aerr z +zerr χ2 +A +Aerr z +zerr χ2 +A +Aerr +z +zerr χ2 +all +336 22 +1.0 − +51.27 +1443 104 1.5 − +60.63 +5794 498 +2.0 − +82.4 +377.0 190.0 1.0 0.2 51.18 +FexCr1–x with x = 0.175; all temperatures fitted independently +z = 1.0 +z = 1.5 +z = 2.0 +z = free +T (K) A +Aerr z +zerr χ2 +A +Aerr z +zerr χ2 +A +Aerr z +zerr χ2 +A +Aerr +z +zerr χ2 +25 +233 17 +1.0 − +7.60 +1073 88 +1.5 − +9.88 +4663 537 +2.0 − +18.63 +293 +249 +1.1 0.3 7.54 +30 +212 12 +1.0 − +12.51 +950 +29 +1.5 − +3.94 +3912 306 +2.0 − +24.21 +656 +158 +1.4 0.1 2.94 +34 +196 7 +1.0 − +5.55 +891 +31 +1.5 − +5.03 +3877 273 +2.0 − +20.56 +445 +178 +1.3 0.1 3.34 +80 +239 13 +1.0 − +15.82 +1060 22 +1.5 − +2.30 +4484 213 +2.0 − +12.26 +1145 298 +1.5 0.1 2.26 +FexCr1–x with x = 0.175; all temperatures fitted simultaneously +z = 1.0 +z = 1.5 +z = 2.0 +z = free +T (K) A +Aerr z +zerr χ2 +A +Aerr z +zerr χ2 +A +Aerr z +zerr χ2 +A +Aerr +z +zerr χ2 +all +216 6 +1.0 − +55.92 +973 +22 +1.5 − +33.46 +4138 149 +2.0 − +86.81 +700 +163 +1.4 0.1 31.52 +FexCr1–x with x = 0.21; all temperatures fitted independently +z = 1.0 +z = 1.5 +z = 2.0 +z = free +T (K) A +Aerr z +zerr χ2 +A +Aerr z +zerr χ2 +A +Aerr z +zerr χ2 +A +Aerr +z +zerr χ2 +15 +303 37 +1.0 − +9.90 +1495 173 1.5 − +9.14 +6761 928 +2.0 − +12.33 +849 +889 +1.3 0.3 8.84 +20 +266 7 +1.0 − +2.05 +1204 61 +1.5 − +6.80 +5096 479 +2.0 − +22.29 +360 +83 +1.1 0.1 1.68 +25 +310 26 +1.0 − +23.64 +1504 83 +1.5 − +10.65 +6646 519 +2.0 − +20.54 +1462 742 +1.5 0.2 10.64 +30 +311 15 +1.0 − +11.96 +1469 45 +1.5 − +4.97 +6321 511 +2.0 − +33.39 +844 +149 +1.3 0.1 2.26 +34 +338 24 +1.0 − +29.58 +1641 52 +1.5 − +6.28 +7232 457 +2.0 − +24.47 +1578 448 +1.5 0.1 6.27 +50 +430 33 +1.0 − +43.43 +2011 66 +1.5 − +8.49 +8643 446 +2.0 − +20.47 +2393 764 +1.6 0.1 8.20 +70 +418 37 +1.0 − +43.72 +2041 64 +1.5 − +5.76 +9099 392 +2.0 − +10.77 +3350 838 +1.7 0.1 3.76 +80 +423 32 +1.0 − +66.27 +1952 48 +1.5 − +7.52 +8283 387 +2.0 − +26.58 +2506 565 +1.6 0.1 6.48 +100 +383 25 +1.0 − +54.38 +1763 18 +1.5 − +1.30 +7499 327 +2.0 − +24.07 +1984 181 +1.5 0.1 1.07 +150 +291 37 +1.0 − +137.06 1457 78 +1.5 − +27.54 +6468 214 +2.0 − +10.63 +4169 1032 1.8 0.1 7.52 +FexCr1–x with x = 0.21; all temperatures fitted simultaneously +z = 1.0 +z = 1.5 +z = 2.0 +z = free +T (K) A +Aerr z +zerr χ2 +A +Aerr z +zerr χ2 +A +Aerr z +zerr χ2 +A +Aerr +z +zerr χ2 +all +348 10 +1.0 − +610.23 1659 31 +1.5 − +256.94 7203 163 +2.0 − +363.71 2205 364 +1.6 0.1 249.2 +imity to the spin-glass transition or a reentrant phase, or +the complex character of the interactions71,72. +A q dependence of Γ has, finally, been reported by Bao +et al., who found that the spin dynamics in Fe0.7Al0.3 +can be approximated with Γ ∝ q2.5 for q values from +0.05 ˚A−1 − 0.5 ˚A−1. This finding is in contrast to studies +of dilute spin-glasses which reported that the relaxation +process does not depend on q. Such a result is expected +for dilute systems that are homogeneous over the q scale +studied. +E. +Conclusions +In summary, we reported a MIEZE spectroscopy study +of the spin dynamics in FexCr1–x samples with x = + +8 +FM +SG +AFM +AFM/FM +0 +0.04 +0.08 +q (Å +1) +0 +50 +100 +150 +(meV) +0 +0.04 +0.08 +q (Å +1) +0 +0.04 +0.08 +q (Å +1) +0 +0.5 +1.0 +1.5 +2.0 +0 +50 +100 +150 +T (K) +0 +1.5 +3.0 +4.5 +z +eVÅz) +A ( +0 +50 +100 +150 +T (K) +0 +50 +100 +150 +T (K) +x = 0.175 +x = 0.21 +FexCr1 +x +x = 0.145 +(a) +(b) +(c) += Aqz += Aqz += Aqz +(d) +(e) +(f) +x = 0.145 +x = 0.175 +x = 0.21 +(g) +(h) +(i) +x = 0.145 +x = 0.175 +x = 0.21 +15 K +20 K +25 K +30 K +34 K +50 K +70 K +80 K +100 K +150 K +T (K) +FIG. 6. Momentum dependence of the decay rate Γ. Data are +shown for (a) x = 0.145, (b) x = 0.21, and (c) x = 0.175 +and temperatures where a meaningful linewidth Γ could be +extracted, see main text for details. The solid lines are fits to +the data using Eq. (4). The temperature dependence of the fit +parameters z and A are shown for x = 0.145 (left column), +x = 0.175 (middle column), and x = 0.21 (right column). +The shaded areas indicate the phases according to the phase +diagram: AFM (green), AFM/FM (purple), FM (blue), and +SG (orange). +0.145, x += +0.175, and x += +0.21. +These composi- +tions were chosen to compare the spin relaxation for dif- +ferent spin-glass classifications and reentrant tempera- +ture dependencies from different high-temperature prop- +erties, namely antiferromagnetic, paramagnetic, and fer- +romagnetic states. The dynamic properties of all sam- +ples studied are consistent with a comprehensive study of +polycrystaline samples41. Properties of the sample with +x = 0.175 have been reported in Ref. [12] using trans- +verse NRSE and conventional NSE featuring more scatter +and a smaller dynamic range. All samples show a broad +distribution of relaxation times close to the spin-glass +regime which, in the simplest approach, can be described +by a stretched exponential. The stretching exponent β +approaches 1/3 as the spin-glass regime is approached, +suggesting proximity to the percolation limit. +For in- +creasing iron content and hence increasing tendency to +ferromagnetic order, the spin relaxation can be described +with a simple exponential (Debye) decay at high temper- +atures, where a single relaxation time can be extracted, +indicating a smaller spread in relaxation times and mag- +netic cluster sizes. As explained above we refrain from +an analysis in terms of more specific mechanisms as as- +sumed, e.g., in the Weron model. +The spin relaxation dynamics of all three compositions +x investigated in our study depends on q. Such a depen- +dence contrasts dilute spin-glasses in which no disper- +sive behavior has been observed9. Within the scatter of +our data the dispersive behavior may be described by a +power-law dependence where the exponent z decreases +with increasing iron concentration. We note, however, +that the dispersive character may be the result of a com- +bination of different contributions with different values +of z, a scenario that we cannot disentangle further. +On a technical note, the present study highlights that +the MIEZE technique allows to perform high-resolution +neutron spin-echo spectroscopy over a large dynamic +range in materials hosting ferromagnetic domains that +may depolarize the neutron beam, posing a major lim- +itation in conventional neutron spin-echo spectroscopy. +Samples with ferromagnetic, antiferromagnetic, or para- +magnetic high-temperature properties may therefore be +investigated using MIEZE with the same instrument and +sample environment. +ACKNOWLEDGMENTS +We wish to thank F. Haslbeck, M. Mantwill, S. Mayr, +S. M¨uhlbauer, and A. Wendl for fruitful discussions +and assistance with the experiments. +This work has +been funded by the Deutsche Forschungsgemeinschaft +(DFG, German Research Foundation) under TRR80 +(From Electronic Correlations to Functionality, Project +No. 107745057, Project E1) and the excellence clus- +ter MCQST under Germany’s Excellence Strategy EXC- +2111 (Project No. 390814868). Financial support by the +Bundesministerium f¨ur Bildung und Forschung (BMBF) +through Project No. 05K16WO6 as well as by the Euro- +pean Research Council (ERC) through Advanced Grants +No. 291079 (TOPFIT) and No. 788031 (ExQuiSid) is +gratefully acknowledged. G.B. and S.S. acknowledge fi- +nancial support through the TUM Graduate School. +∗ steffen.saeubert@gmail.com +1 P. Nordblad, Competing Interaction in Magnets: The Root +of Ordered Disorder or only Frustration? Physica Scripta +88, 058301 (2013). +2 J. A. Mydosh, Spin Glasses: +Redux: +An Updated Ex- +perimental/Materials Survey, Rep. Prog. 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Mezei, Critical +Dynamics of AuFe Reentrant Ferromagnets Close to the +Percolation Threshold, Physica B 165-166, 191 (1990). + diff --git a/NtE3T4oBgHgl3EQfZQpQ/content/tmp_files/load_file.txt b/NtE3T4oBgHgl3EQfZQpQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d41386bfc3e1d1f2b61effee77e6b67bf7673ef --- /dev/null +++ b/NtE3T4oBgHgl3EQfZQpQ/content/tmp_files/load_file.txt @@ -0,0 +1,1345 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf,len=1344 +page_content='Evolution of the spin dynamics during freezing in the spin-glass FexCr1–x S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' S¨aubert,1, 2, ∗ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Franz,1, 2, 3 J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Pfleiderer1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 6 1Physik-Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Technische Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' D-85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Germany 2Heinz Maier-Leibnitz Zentrum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Technische Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' D-85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Germany 3J¨ulich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Forschungszentrum J¨ulich GmbH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Germany 4Zentrum f¨ur QuantumEngineering (ZQE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Technische Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' D-85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Germany 5Brookhaven National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Upton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' NY 11973,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' USA 6Munich Center for Quantum Science and Technology (MCQST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Technische Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' D-85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Germany (Dated: January 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 2023) In the iron–chromium system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' FexCr1−x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' a wide dome of spin-glass behavior emerges when the ferromagnetism of iron is suppressed and the antiferromagnetism of chromium emerges as a function of increasing iron content x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' As both, the high-temperature state and the characteristic cluster size vary as a function of x, different regimes of spin-glass behavior may be compared in a single, isostructural material system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Here, we report a study of the spin dynamics across the freezing process into the spin-glass state for different iron concentrations (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21) using Modulation of IntEnsity with Zero Effort (MIEZE) spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' In the parameter range studied, the relaxation process observed experimentally may be described well in terms of a stretched exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' In the reentrant cluster-glass regime, x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145, this behavior persists up to high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' In comparison, in the superparamagnetic regime, x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 and x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21, a single relaxation time at elevated temperatures is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For all samples studied, the spin relaxation exhibits a momentum dependence consistent with a power law, providing evidence of a dispersive character of the spin relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Introduction In spin-glasses, the combination of random site oc- cupation and disorder with competing interactions, anisotropy, and frustration leads to a collective freezing of the spins in random orientations at the glass temper- ature Tg1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The freezing process may involve a broad distribution of relaxation times, resulting from varying correlation lengths of the individual magnetic clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' These clusters may vary strongly in size, ranging from in- dividual spins in canonical spin-glasses, over interacting clusters of spins in cluster glasses, to collectively behav- ing domains in superparamagnetic systems3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' In dilute spin-glasses, the perhaps most comprehensively studied class of glassy magnetic systems4–12, the relaxation times of the spin freezing exhibit no dispersion due to the short range of the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Seminal studies of the spin-glass behavior in Cu1–xMnx (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='16, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='35) as well as Au1–xFex (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='14) using neutron spin echo (NSE) spectroscopy11,13 suggested non-exponential relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' To go beyond an account of the spin relaxation in terms of a stretched exponential, which cannot distinguish be- tween a distribution of parallel relaxation channels and hierarchical relaxation comprising intercluster and intra- cluster processes, the Weron model14 was found to pro- vide a universal description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' This raises the question for key characteristics of the spin relaxation when the concentration of magnetic atoms is increased to form cluster-glass or superparamagnetic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Moreover, reports of a momentum independent quasielastic peak in the spin-glass regime15–20 contrast dispersive behavior that has been attributed to either the coexistence of fer- romagnetism and spin-glass behavior21–23 or momentum- dependent dynamics of the spin-glass24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' To advance these questions, we report a study of the spin-glass dynamics in FexCr1–x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' When combining the itinerant-electron ferromagnet iron and the spin-density wave antiferromagnet chromium in the isostructural al- loy FexCr1–x, the ferromagnetic transition temperature is suppressed and long-range spin-density wave order emerges with increasing x for xc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='15 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='1725–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' A dome of spin-glass behavior is located at low temper- atures in the vicinity of xc16,31–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The dome extends from x ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='10 to x ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='25, reaching well into concentra- tion regimes which exhibit ferromagnetic and antiferro- magnetic order at higher temperatures, respectively38–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Thus, for different x, the spin-glass behavior in FexCr1–x may emerge with decreasing temperature from ferromag- netic or antiferromagnetic order due to spin freezing42–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Moreover, the ferromagnetically ordered clusters initially grow in size as a function of increasing x, changing the character of the spin-glass state from a cluster-glass to a superparamagnet41,46,47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' As both iron and chromium display spin wave dispersions that are prototypical for ferro- and antiferromagnetism, respectively, a question concerns the existence and character of dispersive behav- ior in the spin-glass regime48–55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Quasielastic neutron scattering has been established as an indispensable tool in the study of spin relaxation pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For a wide range of relaxation times expected in spin-glasses, neutron spin-echo spectroscopy appears to be ideally suited, since the associated measurements of the intermediate scattering function S(q, τ), as opposed to the dynamic structure factor S(q, ℏω), allows to sep- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='04495v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='str-el] 11 Jan 2023 2 arate dynamic processes on very different time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' However, in conventional neutron spin-echo spectroscopy depolarizing samples or samples under depolarizing sam- ple environment may only be measured at a high penalty in neutron flux56,57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' On this note, Modulation of IntEn- sity with Zero Effort58,59 (MIEZE) is a spin-echo tech- nique which permits measurements under depolarizing conditions for small momentum transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' It is therefore ideally suited for the study of spin relaxation dynamics in nearly ferromagnetic and ferromagnetic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' In the study reported in this paper, we used the lon- gitudinal MIEZE technique to determine the spin dy- namics in FexCr1–x as a function of temperature for three different compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For the reentrant cluster- glass (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145), a broad distribution of relaxation times is observed over the entire temperature range that may be described well in terms of a stretched exponen- tial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For compositions in the superparamagnetic regime (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 and x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21), a broad distribution of relax- ation times at low temperatures that may be described well in terms of a stretched exponential is contrasted by a single relaxation time at elevated temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Our data recorded in the sample with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 are con- sistent with an earlier study using transverse NRSE and conventional NSE in the same sample12, which, however exhibited more scatter and covered a smaller dynamical range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For all three compositions we find dispersive be- havior of the spin relaxation that follows a power-law de- pendence of the momentum, q, consistent with Γ ∝ qz, where the dynamical exponent z decreases from z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 to z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 with increasing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' This behavior is loosely reminiscent of the quasielastic linewidth in Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='7Al0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' However, for FexCr1–x the origin of the dispersive behav- ior may, in principle, comprise a combination of different contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' In Section B, spe- cific aspects of the neutron scattering experiments and the crystal growth techniques are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' In Sec- tions C 1 and C 2 the elastic and quasielastic neutron scattering experiments are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' In Section D, the implications of the experiments are discussed, followed by a summary of the findings of this study in Section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Experimental Methods Our experiments were conducted at the beamline RESEDA60,61 at the Heinz Maier-Leibnitz Zentrum using the longitudinal MIEZE option62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The MIEZE method is particularly well suited to study dynamics close to the [0, 0, 0] Bragg peak, corresponding to small-angle neutron scattering (SANS)61,62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 1 the experi- mental setup used for our experiments is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The sample-detector distance, LSD = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='335 m, was maxi- mized to ensure highest q resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' To provide high neutron flux and to cover the desired dynamic range, the wavelength was set to 6 �A with a wavelength spread ∆λ/λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' In this configuration, a dynamic range (c) n (b) detector sample aperture beam stop sample (a) detector sample polarizer analyzer π/2 flipper π/2 flipper NSE solenoid NRSE RF flipper n LSD direct beam beam stop 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Longitudinal MIEZE setup used in our experiments51,62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (a) Schematic depiction of the spectrom- eter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' All spin manipulations are performed upstream to the sample, rendering the method insensitive to depolarizing con- ditions at the sample position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The red arrows indicate the di- rection of the magnetic guide fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (b) Schematic depiction of the neutron flight path through the spectrometer show- ing the sample aperture, sample, beam stop, and detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (c) Detector segments used in the evaluation of the MIEZE scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Segments represent parts of circular rings centered at the direct beam with an opening angle of 60° and a width of 10 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The grey shaded area represents the location of the beam stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' from ∼ 6 · 10−6 ns to 2 ns was accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 1(b) shows a schematic of the neutron flight path through the spec- trometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Data were recorded with a 20 cm × 20 cm 2D CASCADE detector63, covering a q range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='016 �A −1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='085 �A −1 at λ = 6 �A and LSD = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='335 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Recent de- velopments at RESEDA made it possible to increase LSD up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='43 m improving the spatial resolution further61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The grouping of the detector segments for evaluating the quasielastic data are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For the elas- tic measurements, the identical setup was used with nar- rower grouped detector segments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=', data were eval- uated in the same area as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 1(c) but divided in 25 instead of 11 detector segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The sample was cooled to temperatures between 4 K and 300 K using a top-loading closed-cycle refrigerator and the temperature was controlled with two sensors close to the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The temperature stability was ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='05 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' No hysteresis was observed in temperature scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Data were normalized using data recorded at the base temperature of ∼4 K, as- suming that the spin dynamics in FexCr1–x are frozen at temperatures well below Tg11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' This procedure minimizes systematic errors since the experimental setup remains unchanged throughout the entire experiment on a given sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' We investigated three textured polycrystalline samples of FexCr1–x containing large grains with iron concentra- tions of x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The samples were prepared by means of arc melting from pure starting ma- terials and annealed for 4 days at 1100 °C before quench- ing in water16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' To remove strain, the samples were sub- 3 sequently annealed at 1000 °C for 1 day16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Following this process ingots of cylindrical shape with a height of ap- proximately 20 mm and a diameter of about 10 mm were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' To optimize the signal-to-noise ratio, slabs with a thickness of 8 mm were cut from the ingots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' A circu- lar aperture made of cadmium with a diameter of 10 mm was attached directly to the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Thus, the neutron beam effectively illuminated samples of cylindrical shape with a diameter of 10 mm and a length of 8 mm, where the cylindrical axis was parallel to the incident neutron beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Measurements on a sample with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='17 from Benka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='41 (not shown) were in excellent agreement with the results presented in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Parts of the sample with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145 were recently used to investigate the influence of concentration fluc- tuations on relaxation processes in spin-glasses12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Using atom probe tomography, a high-resolution local probe, together with neutron resonant spin-echo spectroscopy it was shown that small-scale inhomogenieties in the mi- crostructure influence the relaxation processes of a spin- glass material12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Using the Weron model which had pre- viously been applied to dilute spin-glasses13, the relax- ation processes were described in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Tg TC Tg TC Tg TC Burke Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' [36, 38-40] TN Tg TC Benka Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' [41] TN Tg TX TC this study Tg TC FexCr1 x x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145 FexCr1 x x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 FexCr1 x x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='210 FM AFM SG AFM/FM 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0% 0 125 250 T (K) 0 1 2 3 Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='×10 5 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=') 0 125 250 T (K) 0 125 250 T (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='25 x in FexCr1 x 0 30 60 90 Temperature (K) (b) (c) (d) (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Magnetic properties of the FexCr1–x system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' An- tiferromagnetic (AFM, green), ferromagnetic (FM, blue), spin-glass (SG, orange), and antiferromagnetic-ferromagnetic (AFM/FM, purple) regimes are distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (a) Tem- perature vs concentration phase diagram combining data from neutron scattering and low-field magnetization by Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (squares)36,38–40, ac susceptibility and magnetization by Benka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (triangles)41, and neutron scattering (present work, stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (b)-(d) Integrated SANS intensity for the three samples measured at RESEDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The shaded areas indicate magnetic regimes as inferred from the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Experimental Results The phase diagram of FexCr1–x as a function of tem- perature and iron concentration x is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 2(a) as reproduced from literature36,38–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For comparison, the temperature dependence of the integrated SANS inten- sity of the three samples measured in this study is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 2(b) to 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' At the border between ferromag- netic and antiferromagnetic order, a dome of spin-glass behavior emerges, covering the regime of putative quan- tum phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' At low temperatures, this spin- glass regime includes concentrations for which at high temperatures long-range ferromagnetic or antiferromag- netic order are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' In our study, three compositions were investigated: (i) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145 which exhibits transitions from paramag- netism (PM) to antiferromagnetism (AFM) to a spin- glass (SG) with a glass temperature Tg = 11 K ± 2 K41, (ii) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 which exhibits transitions from PM via a small region reminiscent of FM order41 to a SG with a glass temperature Tg = 20 K ± 2 K41, and (iii) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21 which exhibits transitions from PM to fer- romagnetism (FM) to a SG with a glass temperature Tg = 14 K ± 2 K41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Elastic Scattering The temperature dependence of the integrated SANS intensity is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 2(b) to 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' As no magnetic scattering was observed at high temperatures, the data recorded at ∼300 K were used for background subtrac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The scattered intensities as a function of temperature for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145 and x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 behave similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' With de- creasing temperature the intensity increases as expected for a transition into a ferromagnetically ordered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' When entering the spin-glass state, the system becomes static on the time scales probed by SANS, and the inten- sity increases with a change in slope, forming a plateau that starts at the onset of the spin-glass regime64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The bulk properties and phase diagram show that the sample with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145 enters the spin-glass regime via an antiferromagnetic state, which may not be identified microscopically for the parameter range probed in SANS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Interestingly, the increase in intensity is, analogous to the sample with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175, reminiscent of ferromagnetic order in agreement with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For the sample with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21, a broad feature with a maximum at ∼75 K defines TC, followed by a sharp increase in intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' A change of slope with decreas- ing temperature, close to the transition temperature reported previously in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' [41], defines Tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The sig- natures defined in neutron scattering are denoted by stars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Discrepancies as compared to the phase boundaries inferred from the ac susceptibility and magnetization41 may reflect the different time scales probed by the different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 4 The q dependence of the SANS data, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Section B for details on data analysis, is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 3(a) to 3(c) for different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Following the approach taken in related SANS studies on other materials65,66 we consider a single power law form I ∝ q−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (1) Fitting the experimental data yields exponents as a func- tion of temperature as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 3(d) to 3(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 10 6 10 5 10 4 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=') 10 6 10 5 10 4 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='1 q (Å 1) 10 6 10 5 10 4 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=') 0 1 2 n 0 1 2 n n 2 1 0 T (K) 200 100 0 FM SG AFM AFM/FM T (K) 5 15 10 20 12 25 30 70 34 80 50 100 150 200 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='210 FexCr1 x x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='210 Tg TC TC Tg Tg TC (a) (b) (c) (d) (e) (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Momentum dependence of SANS of FexCr1–x as mea- sured at RESEDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (a)-(c) Intensity as a function of scattering vector q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The accessible q range was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='02 �A −1 < q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='08 �A −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Solid lines are fits using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (d)-(f) Temperature depen- dence of the exponent n obtained from the fits in (a)-(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Solid lines are guides to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The shaded areas indicate the regimes according to the phase diagram, namely AFM (green), AFM/FM (purple), FM (blue), and SG (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Samples with iron concentrations of x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145 (top row), x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 (middle row), and x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21 (bottom row) were evaluated at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' With decreasing temperature, the exponent n increases from ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='3, reaching low-temperature values of n ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='3 for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145, n ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='8 for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175, and n ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='1 for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Even though the data are well-described by the power law in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (1), unambiguous interpretation of the exponent n proves difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Assuming a two-phase system where smooth clusters are isolated, n = 4 would be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Deviations from this behavior would indi- cate more complex phases including fractal surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' An exponent of n = 2, for instance, was reported in the perovskite manganite Pr1–xCaxMnO3 and attributed to sheets of inter-penetrating ferromagnetic and antiferro- magnetic phases65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' In the same material, Viret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='66 found n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='6 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='7 which was attributed to filamentary ferromagnetic chains in analogy to polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' An increase in n with decreasing temperature suggests a coarsening of the interfaces as the system enters the spin-glass regime, consistent with an increase in cluster size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' To address the question of the nature of the low- temperature magnetic structure as a function of iron con- centration, it may be necessary to collect data at even smaller q values, which is beyond the technical limits of the work reported here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Alternatively, more complex de- pendencies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=', comprising several different power-law contributions may be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Such descriptions would require theoretical modelling beyond the scope of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Quasielastic Measurements The normalized intermediate scattering function, I(q, τ) = S(q, τ)/S(q, 0), (2) in the temperature range 4 K ≤ T ≤ 150 K for q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='044 �A −1 and all FexCr1–x compositions investigated is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Following careful comparison of the data with different relaxation models models4,6–8,10,11,13,67 we find that the data are already described well by a stretched exponential characteristic of a distribution of relaxation rates, namely I(q, τ) = Ielastic + (1 − Ielastic) exp � − (Γτ)β� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (3) Here, Γ is the decay rate, corresponding to the inverse spin relaxation time, 0 < β ≤ 1 stretches the clas- sic exponential decay (β = 1), and the prefactor Ielastic describes the elastic contribution to the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' In the light of the microscopic complexity of the FexCr1−x sys- tem, which may support both parallel as well as hierar- chical relaxation, we refrain from an analysis in terms of more sophisticated mechanisms as assumed in the Weron model12–14,67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The normalized intermediate scattering functions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 4 have been shifted vertically in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='25 for bet- ter visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 4(a) shows the sample with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The intermediate scattering function is constant below ∼4 K, confirming that the spin dynamics are frozen11 with respect to the normalizing data at 4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For tem- peratures around the glass temperature, Tg ∼ 15 K, the sample starts to show signatures of dynamic behavior, notably spin relaxation as indicated by a decrease in the intermediate scattering function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The relaxation time of the spins is inferred from a fit to the data using τ = ℏ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Increasing the temperature shifts the spin relaxation to shorter spin-echo times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Ielastic describes the elastic con- tribution to the intermediate scattering function, corre- sponding to the fraction of the sample that is static on the time scales probed here, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=', clusters fluctuating on much longer time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The elastic signal decreases with increasing tempera- ture, reaching a minimum of Ielastic ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='25 for tempera- tures above 34 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For all temperatures, the system may 5 (b) (c) (a) 4 K 12 K 15 K 20 K 25 K 30 K 34 K 80 K 150 K FexCr1 x x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='044 Å 1 10 4 10 3 10 2 10 1 10 0 (ns) 0 1 2 3 I(q, ) 4 K 10 K 15 K 20 K 25 K 30 K 34 K 80 K 150 K FexCr1 x x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='044 Å 1 10 4 10 3 10 2 10 1 10 0 (ns) 4 K 10 K 15 K 20 K 25 K 34 K 50 K 80 K 150 K FexCr1 x x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='044 Å 1 10 4 10 3 10 2 10 1 10 0 (ns) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Normalized intermediate scattering function as measured for different temperatures in FexCr1–x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Data were recorded using neutrons with a mean wavelength of λ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 �A and are shown at q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='044 �A −1 for (a) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145, (b) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175, (c) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For better visibility, data are shifted vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The solid lines are fits to the data using the stretched exponential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The dashed lines represent fits using a single exponential relaxation, corresponding to β = 1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' be described by a stretched exponential decay, β < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' To highlight the need for the stretching parameter β, ex- ponential decays (β = 1) are shown for comparison as dashed lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 4(a) to 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The deviation from simple exponential behavior gets more pronounced with decreasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The intermediate scattering functions for the sample with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For tempera- tures below Tg, the data resemble the behavior observed for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Under increasing temperature the spin re- laxation is similarly shifted to shorter times but remains larger than for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The minimum of the elastic background is reduced to Ielastic ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' In contrast to the sample with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145, at higher temperatures the spin relaxation can be described by a simple exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 4(c), the behavior again is highly reminiscent of the other two compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Ac- cordingly, we observe a shift of the relaxation time to- wards shorter times as the temperature increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' How- ever, for temperatures above 50 K the relaxation time no longer decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The elastic contribution reaches a min- imum of Ielastic ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' In a paramagnetic sample, where all scattering is dynamic, the spin-echo curve would de- cay to Ielastic = 0 at high temperatures or long spin-echo times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The observed sample dependence of Ielastic may be attributed empirically to the different iron contents of the samples, which lead to different high-temperature magnetic phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Since the measurements were performed in a SANS ge- ometry, close to the ferromagnetic Bragg peak at q = 0, ferromagnetic fluctuations will contribute strongly to the measured intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' However, as possible antiferromag- netic scattering intensity cannot be observed in the vicin- ity of q = 0, antiferromagnetic fluctuations will not con- tribute to the exponential decay, decreasing the dynamic contribution to the intensity in samples with a larger an- tiferromagnetic fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' This observation is in accor- dance with the elastic measurements shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 2(b) to 2(d), which indicate that the intensity increases with increasing iron content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The temperature dependences of the fit parameters Ielastic and β are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 5 for q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='044 �A −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For all samples the elastic contribution Ielastic decreases linearly with increasing temperature reaching a constant value above ∼40 K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=', far above the glass temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' A shrinking elastic contribution with increasing temper- ature suggests that parts of the sample slowly unfreeze on the time scales studied in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The exponent β provides an estimate of the broaden- ing of the spectrum of relaxation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For all samples, β decreases drastically when the temperature decreases towards the spin-glass regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' This evolution suggests an increase of the distribution of relaxation times, as ex- pected of the formation of different-sized domains fluctu- ating on different time scales before freezing at the lowest 6 FM SG AFM AFM/FM x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='210 Ielastic 0 50 100 150 T (K) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 (c) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 (d) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 (e) (f) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='210 0 50 100 150 T (K) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 (b) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 Ielastic (a) FexCr1 x x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='044 Å 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 Ielastic FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Temperature dependence of the fit parameters Ielastic and β in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (3) for the three FexCr1–x samples with (a),(d) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145, (b),(e) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175, and (c),(f) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Data are shown for q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='044 �A −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The shaded areas indicate the phases according to the phase diagram: AFM (green), AFM/FM (purple), FM (blue), and SG (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (a)-(c) Elastic contribution Ielastic as a function of temperature, showing a linear increase for temperatures below ∼40 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Solid lines are guides to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The elastic background, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=', the constant value at temperatures above 40 K, is connected to the magnetic signal-to-noise ratio, and thus different for the different samples, see main text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (d)-(f) Stretched exponent β as function of temperature showing a drastic de- crease of β close to Tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Solid lines are guides to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145, β stays below 1 over the entire temperature range, while for the other two com- positions β is close to 1 at temperatures above Tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' This means that the dynamics may be described with a single relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Numerical calculations of the non-exponential relax- ation in spin-glasses and glassy systems68,69 have shown that β = 1/3 is approached at Tg when the system is close to its percolation limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The decrease of β we ob- serve as the spin-glass regime is approached is in quali- tative agreement with these calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Discussion The area detector used in our MIEZE measurements allowed studying the normalized intermediate scattering function I(q, τ) simultaneously over a wide range of mo- mentum transfers q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The q dependence of the decay rate Γ and therefore the spin relaxation time is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 6(a) to 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Within experimental accuracy, our data it are consistent with Γ = Aqz, (4) where q is the momentum transfer, z the dynamical expo- nent, and A the energy scale of the exchange interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The decay rate Γ as a function of q, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 6, was fitted using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (4) with fixed values of z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=', z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0, z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5, and z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0, as well as with z as an independent fitting parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Additionally, data were fitted for each sample and for all temperatures independently, as well as simultaneously for all temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The fit results are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' A χ2-analysis of the fits with fixed parameter z shows that with decreasing iron content z decreases from ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 to ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0, which is supported by the fits with z as a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The resulting values of z and A as a function of temperature are depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 6(d) to 6(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For small β, the distribution of relaxation times is very broad and the data cannot be described with a single τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Therefore a meaningful linewidth Γ cannot be ex- tracted for any of the three samples below and around the glass temperature Tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145, Γ could only be extracted for a small temperature window in the fer- romagnetic regime, 25 K ≤ T ≤ 34 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Even at these temperatures, the exponential decay is already stretched and the values of Γ determined experimentally represent a mean relaxation time rather than a single relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' At higher temperatures, the exponential decay is strongly stretched such that a single dominant relax- ation time could not be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The sample with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 allowed us to analyze Γ for temperatures be- tween 25 K ≤ T ≤ 80 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21, Γ could be extracted for all temperatures above the freezing tem- perature Tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For all concentrations, Γ depends on q according to the relation given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The exponent z increases for increasing iron concentration from z ≈ 1 to z ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' According to dynamic scaling theory the critical expo- nent at Tc for pure ferromagnets in the limit q → 0 corresponds to z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 while antiferromagnetic correla- tions for q → QAFM lead to z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Heuristically, one might hence explain the exponents in FexCr1–x in terms of a competition of ferromagnetic and antiferromagnetic correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Tajima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' used a spin diffusion model with Γ ∝ q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 to describe the q dependence of Γ in the Invar alloy Fe65Ni35 over a rather wide q range70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' This model as- sumes a hydrodynamic behavior of uncorrelated spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The authors argued that the system never reaches criti- cality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=', z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 for ferromagnetically correlated spins, due to impurity scattering of the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Along this line, the presence of chromium atoms acting as impurities in ferromagnetic iron clusters could prevent the system from reaching ferromagnetic critical dynam- ics, reducing z to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' A large fraction of chromium, and therefore an increase in antiferromagnetic correlations in FexCr1–x could reduce the exponent z further towards 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 leading to an effective range of exponents between z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 for antiferromagnetic correlations and z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 as expected in the spin diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Values of z below 1 may reflect the large number of different time scales in spin-glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Unusually small values of z found in spin- glasses were previously attributed to disorder, the prox- 7 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Summary of the fitting procedure of the spin dynamics in FexCr1–x using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (4) with different approaches, namely: (i) fixing the exponent z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0, (ii) fixing the exponent z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5, (iii) fixing the exponent z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0, (iv) leaving the exponent z as a free fit parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Additionally, data were also fitted with (I) all temperatures independently and (II) all temperatures simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The χ2 value of each fit is used as an indicator for the goodness of the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' FexCr1–x with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' all temperatures fitted independently z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 z = free T (K) A Aerr z zerr χ2 A Aerr z zerr χ2 A Aerr z zerr χ2 A Aerr z zerr χ2 25 273 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='20 1127 84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='95 4394 506 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='19 223 61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='13 30 333 48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='63 1536 225 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 − 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='84 6576 1042 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='47 629 745 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='22 34 400 40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21 1723 184 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 − 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='79 6953 914 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='99 589 507 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='3 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='74 FexCr1–x with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' all temperatures fitted simultaneously z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 z = free T (K) A Aerr z zerr χ2 A Aerr z zerr χ2 A Aerr z zerr χ2 A Aerr z zerr χ2 all 336 22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='27 1443 104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 − 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='63 5794 498 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='4 377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='2 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='18 FexCr1–x with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' all temperatures fitted independently z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 z = free T (K) A Aerr z zerr χ2 A Aerr z zerr χ2 A Aerr z zerr χ2 A Aerr z zerr χ2 25 233 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='60 1073 88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='88 4663 537 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='63 293 249 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='54 30 212 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='51 950 29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='94 3912 306 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21 656 158 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='94 34 196 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='55 891 31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='03 3877 273 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='56 445 178 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='34 80 239 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='82 1060 22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='30 4484 213 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='26 1145 298 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='26 FexCr1–x with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' all temperatures fitted simultaneously z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 z = free T (K) A Aerr z zerr χ2 A Aerr z zerr χ2 A Aerr z zerr χ2 A Aerr z zerr χ2 all 216 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='92 973 22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 − 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='46 4138 149 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='81 700 163 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='1 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='52 FexCr1–x with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' all temperatures fitted independently z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 z = free T (K) A Aerr z zerr χ2 A Aerr z zerr χ2 A Aerr z zerr χ2 A Aerr z zerr χ2 15 303 37 1.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='06 1457 78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 − 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='54 6468 214 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='63 4169 1032 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='52 FexCr1–x with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' all temperatures fitted simultaneously z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 z = free T (K) A Aerr z zerr χ2 A Aerr z zerr χ2 A Aerr z zerr χ2 A Aerr z zerr χ2 all 348 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='23 1659 31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 − 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='94 7203 163 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 − 363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='71 2205 364 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='1 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='2 imity to the spin-glass transition or a reentrant phase, or the complex character of the interactions71,72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' A q dependence of Γ has, finally, been reported by Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=', who found that the spin dynamics in Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='7Al0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='3 can be approximated with Γ ∝ q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 for q values from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='05 ˚A−1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 ˚A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' This finding is in contrast to studies of dilute spin-glasses which reported that the relaxation process does not depend on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Such a result is expected for dilute systems that are homogeneous over the q scale studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Conclusions In summary, we reported a MIEZE spectroscopy study of the spin dynamics in FexCr1–x samples with x = 8 FM SG AFM AFM/FM 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='08 q (Å 1) 0 50 100 150 (meV) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='08 q (Å 1) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='08 q (Å 1) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 0 50 100 150 T (K) 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='5 z eVÅz) A ( 0 50 100 150 T (K) 0 50 100 150 T (K) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21 FexCr1 x x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145 (a) (b) (c) = Aqz = Aqz = Aqz (d) (e) (f) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21 (g) (h) (i) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21 15 K 20 K 25 K 30 K 34 K 50 K 70 K 80 K 100 K 150 K T (K) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Momentum dependence of the decay rate Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Data are shown for (a) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145, (b) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21, and (c) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 and temperatures where a meaningful linewidth Γ could be extracted, see main text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The solid lines are fits to the data using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The temperature dependence of the fit parameters z and A are shown for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145 (left column), x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 (middle column), and x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21 (right column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The shaded areas indicate the phases according to the phase diagram: AFM (green), AFM/FM (purple), FM (blue), and SG (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='145, x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175, and x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' These composi- tions were chosen to compare the spin relaxation for dif- ferent spin-glass classifications and reentrant tempera- ture dependencies from different high-temperature prop- erties, namely antiferromagnetic, paramagnetic, and fer- romagnetic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The dynamic properties of all sam- ples studied are consistent with a comprehensive study of polycrystaline samples41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Properties of the sample with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='175 have been reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' [12] using trans- verse NRSE and conventional NSE featuring more scatter and a smaller dynamic range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' All samples show a broad distribution of relaxation times close to the spin-glass regime which, in the simplest approach, can be described by a stretched exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The stretching exponent β approaches 1/3 as the spin-glass regime is approached, suggesting proximity to the percolation limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' For in- creasing iron content and hence increasing tendency to ferromagnetic order, the spin relaxation can be described with a simple exponential (Debye) decay at high temper- atures, where a single relaxation time can be extracted, indicating a smaller spread in relaxation times and mag- netic cluster sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' As explained above we refrain from an analysis in terms of more specific mechanisms as as- sumed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=', in the Weron model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' The spin relaxation dynamics of all three compositions x investigated in our study depends on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Such a depen- dence contrasts dilute spin-glasses in which no disper- sive behavior has been observed9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Within the scatter of our data the dispersive behavior may be described by a power-law dependence where the exponent z decreases with increasing iron concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' We note, however, that the dispersive character may be the result of a com- bination of different contributions with different values of z, a scenario that we cannot disentangle further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' On a technical note, the present study highlights that the MIEZE technique allows to perform high-resolution neutron spin-echo spectroscopy over a large dynamic range in materials hosting ferromagnetic domains that may depolarize the neutron beam, posing a major lim- itation in conventional neutron spin-echo spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Samples with ferromagnetic, antiferromagnetic, or para- magnetic high-temperature properties may therefore be investigated using MIEZE with the same instrument and sample environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' ACKNOWLEDGMENTS We wish to thank F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Haslbeck, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Mantwill, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Mayr, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' M¨uhlbauer, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Wendl for fruitful discussions and assistance with the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under TRR80 (From Electronic Correlations to Functionality, Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 107745057, Project E1) and the excellence clus- ter MCQST under Germany’s Excellence Strategy EXC- 2111 (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 390814868).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Financial support by the Bundesministerium f¨ur Bildung und Forschung (BMBF) through Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 05K16WO6 as well as by the Euro- pean Research Council (ERC) through Advanced Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 291079 (TOPFIT) and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 788031 (ExQuiSid) is gratefully acknowledged.' metadata={'source': 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Magnetic Order in CrFe Alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Antiferromagnetic Alloys Close to the Critical Concentration, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' F: Met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 13, 441 (1983).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Rainford, The Evolution of Magnetic Order in CrFe Alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Onset of Ferromagnetism, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' F: Met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 13, 451 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' 40 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Burke and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfZQpQ/content/2301.04495v1.pdf'} +page_content=' Rainford, The Evolution of Mag- netic Order in CrFe Alloys.' metadata={'source': 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sha256:d5aa62814da2d175e9156de817f7580cde21c9e2f43c429afbca253f6faa0d30 +size 118018 diff --git a/O9E0T4oBgHgl3EQfTgDH/content/tmp_files/2301.02238v1.pdf.txt b/O9E0T4oBgHgl3EQfTgDH/content/tmp_files/2301.02238v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a906028b489898a4412dade87b410d83a5de5369 --- /dev/null +++ b/O9E0T4oBgHgl3EQfTgDH/content/tmp_files/2301.02238v1.pdf.txt @@ -0,0 +1,1542 @@ +HyperReel: High-Fidelity 6-DoF Video with Ray-Conditioned Sampling +Benjamin Attal∗ +Carnegie Mellon University +Jia-Bin Huang +Meta & UMD +Christian Richardt +Reality Labs Research +Michael Zollh¨ofer +Reality Labs Research +Johannes Kopf +Meta +Matthew O’Toole +Carnegie Mellon University +Changil Kim +Meta +https://hyperreel.github.io +Dynamic 6-DoF rendering +t = 0 s +t = 0.25 s +t = 0.5 s +t = 0.75 s +t = 1 s +t = 1.25 s +t = 1.5 s +Static 6-DoF rendering +Figure 1. HyperReel: A novel 6-DoF video representation. HyperReel converts synchronized multi-view video streams into a high-fidelity, +memory efficient scene representation that can be rendered from novel views and time steps at interactive rates. HyperReel’s combination of +high rendering quality, speed, and compactness sets it apart from existing 6-DoF video representations. The upper two rows show 6-DoF +(i.e., varying viewpoint and viewing orientation) rendering of dynamic scenes [9, 44]; the lower two of static scenes [57, 58]. +Abstract +Volumetric scene representations enable photorealistic view +synthesis for static scenes and form the basis of several ex- +isting 6-DoF video techniques. However, the volume render- +ing procedures that drive these representations necessitate +careful trade-offs in terms of quality, rendering speed, and +memory efficiency. In particular, existing methods fail to +simultaneously achieve real-time performance, small mem- +ory footprint, and high-quality rendering for challenging +∗ This work was done while Benjamin was an intern at Meta. +real-world scenes. To address these issues, we present Hy- +perReel — a novel 6-DoF video representation. The two core +components of HyperReel are: (1) a ray-conditioned sample +prediction network that enables high-fidelity, high frame rate +rendering at high resolutions and (2) a compact and memory- +efficient dynamic volume representation. Our 6-DoF video +pipeline achieves the best performance compared to prior +and contemporary approaches in terms of visual quality with +small memory requirements, while also rendering at up to +18 frames-per-second at megapixel resolution without any +custom CUDA code. +1 +arXiv:2301.02238v1 [cs.CV] 5 Jan 2023 + +WDGREEN +WD +7mmS +ALCOHOL +ALSOFFALCOHOL +ALSOFF +mmonsongobcuALCOHOL +ALSOFF +allens +onels!mmt +ALCOHOL +ALSOFFWD GREEN +WD +2.5/7mmSd +5"/7mmDi +ALCOHOL +ALSOFFWDGREEN +WD +/7mmSc +7mmD +ALCOHOL +ALSOFFWDGREEN +zmmS +ALCOHOL +ALSOFFPACE FS +HEHANGEDMA +工PACES +HEHANGEDMAPACES +QUEEN +CHANGEDMAPACE S +QUEEN +HANGEDMAPACES +QUEEN +IANGEDMAPACE S +QUEENF +HANGEDMAPACE S +QUEEN +EHANGEDMA1. Introduction +Six–Degrees-of-Freedom (6-DoF) videos allow for free ex- +ploration of an environment by giving the users the ability +to change their head position (3 degrees of freedom) and +orientation (3 degrees of freedom). As such, 6-DoF videos +offer immersive experiences with many exciting applications +in AR/VR. The underlying methodology that drives 6-DoF +video is view synthesis: the process of rendering new, unob- +served views of an environment—static or dynamic—from a +set of posed images or videos. Volumetric scene representa- +tions such as neural radiance fields [29] and instant neural +graphics primitives [30] have recently made great strides +toward photorealistic view synthesis for static scenes. +While several recent works build dynamic view synthe- +sis pipelines on top of these volumetric representations +[14, 23, 24, 33, 61], it remains a challenging task to cre- +ate a 6-DoF video format that can achieve high quality, +fast rendering, and a small memory footprint (even given +many synchronized video streams from multi-view camera +rigs [9, 35, 44]). Existing approaches that attempt to create +memory-efficient 6-DoF video can take nearly a minute to +render a single megapixel image [23]. Works that target ren- +dering speed and represent dynamic volumes directly with +3D textures require gigabytes of storage even for short video +clips [56]. While other volumetric methods achieve memory +efficiency and speed by leveraging sparse or compressed vol- +ume storage for static scenes [11, 30], only contemporary +work [22, 49] addresses the extension of these approaches to +dynamic scenes. Moreover, all of the above representations +struggle to capture highly view-dependent appearance, such +as reflections and refractions caused by non-planar surfaces. +In this paper, we present HyperReel, a novel 6-DoF video +representation that achieves state-of-the-art quality while +being memory efficient and real-time renderable at high res- +olution. The first ingredient of our approach is a novel ray- +conditioned sample prediction network that predicts sparse +point samples for volume rendering. In contrast to exist- +ing static view synthesis methods that use sample networks +[20, 31], our design is unique in that it both (1) acceler- +ates volume rendering and at the same time (2) improves +rendering quality for challenging view-dependent scenes. +Second, we introduce a memory-efficient dynamic vol- +ume representation that achieves a high compression rate +by exploiting the spatio-temporal redundancy of a dynamic +scene. Specifically, we extend Tensorial Radiance Fields [11] +to compactly represent a set of volumetric keyframes, and +capture intermediate frames with trainable scene flow. +The combination of these two techniques comprises our +high-fidelity 6-DoF video representation, HyperReel. We +validate the individual components of our approach and our +representation as a whole with comparisons to state-of-the- +art sampling network-based approaches for static scenes as +well as 6-DoF video representations for dynamic scenes. Not +only does HyperReel outperform these existing works, but it +also provides high-quality renderings for scenes with chal- +lenging non-Lambertian appearances. Our system renders at +up to 18 frames-per-second at megapixel resolution without +using any custom CUDA code. +In summary, the contributions of our work include the +following: +1. A novel sample prediction network for volumetric view +synthesis that accelerates volume rendering and accu- +rately represents complex view-dependent effects. +2. A memory-efficient dynamic volume representation +that compactly represents a dynamic scene. +3. HyperReel, a 6-DoF video representation that achieves +a desirable trade-off between speed, quality, and mem- +ory, while rendering in real time at megapixel resolu- +tion. +2. Related Work +Novel View Synthesis. +Novel-view synthesis is the pro- +cess of rendering new views of a scene given a set of input +posed images. Classical image-based rendering techniques +use approximate scene geometry to reproject and blend +source image content onto novel views [10, 37, 46]. Recent +works leverage the power of deep learning and neural fields +[62] to improve image-based rendering from both structured +(e.g., light fields [16, 21]) and unstructured data [7, 50]. +Rather than performing image-based rendering, which re- +quires storing the input images, another approach is to op- +timize some 3D scene representation augmented with ap- +pearance information [41]. Examples of such representa- +tions include point clouds [1, 40], voxel grids [27, 32, 47], +meshes [42, 43], or layered representations like multi-plane +[13, 28, 65] or multi-sphere images [9]. +Neural Radiance Fields. +NeRFs are one such 3D scene +representation for view synthesis [29] that parameterize the +appearance and density of every point in 3D space with a +multilayer perceptron (MLP). While NeRFs enable high- +quality view synthesis at a small memory cost, they do not +lend themselves to real-time rendering. To render the color +of a ray from a NeRF, one must evaluate and integrate the +color and opacity of many points along a ray—necessitating, +in the case of NeRF, many hundreds of MLP evaluations +per pixel. Still, due to its impressive performance for static +view synthesis, many recent methods build on NeRFs in the +quest for higher visual quality, more efficient training, and +faster rendering speed [15, 53]. Several works improve the +quality of NeRFs by accounting for finite pixels and aper- +tures [5, 60], by enabling application to unbounded scenes +[6, 63, 64], or by modifying the representation to allow for +better reproduction of challenging view-dependent appear- +ances like distorted reflections and refractions [8, 17, 19, 54]. +2 + +One can achieve significant training and inference speed im- +provements by replacing the deep multilayer perceptron with +a feature voxel grid in combination with a small neural net- +work [11, 30, 51] or no network at all [18, 63]. Several other +works achieve both fast rendering and memory-efficient stor- +age with tensor factorizations [11], learned appearance code- +books, or quantized volumetric features [52]. +Adaptive Sampling for Neural Volume Rendering. +Other works aim to improve the speed of volumetric repre- +sentations by reducing the number of volume queries re- +quired to render a single ray. Approaches like DoNeRF +[31], TermiNeRF [38], and AdaNeRF [20] learn weights +for each segment along a ray as a function of the ray itself, +and use these weights for adaptive evaluation of the under- +lying NeRF. In doing so, they can achieve near-real-time +rendering. NeuSample [12] replaces the NeRF coarse net- +work with a module that directly predicts the distance to each +sample point along a ray. Methods like AutoInt [26], DIVeR +[59], and neural light fields [4, 25, 48] learn integrated opac- +ity and color along a small set of ray segments (or just one +segment), requiring only a single network evaluation per +segment. A key component of our framework is a flexible +sampling network, which is among one of the few schemes +that both accelerates volume rendering, and also improves +volume rendering quality for challenging scenes. +6–Degrees-of-Freedom Video. +6-DoF video is an emer- +gent technology that allows users to explore new views +within videos [41]. Systems for 6-DoF video [35] use multi- +view camera rigs that capture a full 360-degree field of view +and use variants of depth-based reprojection [45] for view +synthesis at each frame of the video. Other methods optimize +time-varying multi-sphere images (MSIs) [2, 9], which can +provide better visual quality but at a higher training cost. +6-DoF from Monocular Captures. +Due to the success of +neural radiance fields for static view synthesis, many recent +approaches attempt to extend volumetric scene representa- +tions to dynamic scenes. Several such works reconstruct +6-DoF video from single-view (i.e. monocular) RGB se- +quences [24, 33]. This is a highly under-constrained setting, +which requires decoupling camera and object motion. The +natural signal priors provided by neural radiance fields help +during reconstruction. However, most methods typically rely +on additional priors, such as off-the-shelf networks for pre- +dicting scene flow and geometry or depth from ToF cameras +[3, 61]. Still, other approaches model the scene at different +time steps as smoothly “warped” copies of some canoni- +cal frame [33, 39], which works best for small temporal +windows and smooth object motion. +6-DoF from Multi-View Captures. +Other methods, like +ours, aim to produce 6-DoF video from multi-view camera +rigs [9, 23, 27]. Despite the additional constraints provided +by multiple cameras, this remains a challenging task; an ideal +6-DoF video format must simultaneously achieve high visual +quality, rendering speed, and memory efficiency. Directly +extending recent volumetric methods to dynamic scenes can +achieve high quality and rendering speed [56], but at the +cost of substantial memory requirements, potentially giga- +bytes of memory [63] for each video frame. Contemporary +works such as StreamRF [22] and NeRFPlayer [49] design +volumetric 6-DoF video representations that mitigate storage +requirements but sacrifice either rendering speed or visual +quality. On the other hand, our approach achieves both fast +and high-quality 6-DoF video rendering while maintaining a +small memory footprint. +3. Method +We start by considering the problem of optimizing a volu- +metric representation for static view synthesis. Volume repre- +sentations like NeRF [29] model the density and appearance +of a static scene at every point in the 3D space. More specif- +ically, a function Fθ : (x, ⃗ω) → (Le(x, ⃗ω), σ(x)) maps +position x and direction ⃗ω along a ray to a color Le(x, ⃗ω) +and density σ(x). Here, the trainable parameters θ may be +neural network weights, N-dimensional array entries, or a +combination of both. +We can then render new views of a static scene with +C(o, ⃗ω) = +� tf +tn +T(o, xt) +� +�� +� +Transmittance +σ(xt) +� �� � +Density +Le(xt, ⃗ω) +� +�� +� +Radiance +dt, +(1) +where T (o, xt) denotes the transmittance from o to xt. +In practice, we can evaluate Equation 1 using numerical +quadrature by taking many sample points along a given ray: +C(o, ⃗ω) ≈ +N +� +k=1 +wk Le(xk, ⃗ω) , +(2) +where the weights wk = ˆT (o, xk) (1−e−σ(xk)∆xk) specify +the contribution of each sample point’s color to the output. +3.1. Sample Networks for Volume Rendering +Most scenes consist of solid objects whose surfaces lie on a +2D manifold within the 3D scene volume. In this case, only +a small set of sample points contributes to the rendered color +for each ray. To accelerate volume rendering, we would like +to query color and opacity only for points with non-zero wk. +While most volume representations use importance sampling +and pruning schemes that help reduce sample counts, they +often require hundreds or even thousands of queries per ray +to produce accurate renderings [11, 30]. +As shown in Figure 2, we use a feed-forward network +to predict a set of sample locations xk. Specifically, we use +a sample prediction network Eφ : (o, ⃗ω) → (x1, . . . , xn) +that maps a ray (o, ⃗ω) to the sample points xk for volume +3 + +𝝎 +𝝎×𝒐 +𝒐 +(𝒐, 𝝎) +𝒓 = 𝝎, 𝝎×𝒐 +Compute sample points 𝐱! by +(i) intersecting ray 𝒓 with primitive 𝐺" +(ii) adding displacement vector 𝐝𝐤 +𝐱% = 𝑖𝑛𝑡𝑒𝑟 𝐺&; 𝒐, 𝝎 + 𝐝% +Input ray +Plücker coordinates +Plücker coordinate illustration +𝐶 𝒐, 𝝎 = 4 +& +𝑤& 𝐿'(𝐱%, 𝝎) +Compute volume rendering integral +with numerical quadrature +Rendered image +{𝐝(, 𝐝), ⋯ , 𝐝𝐧} +{𝐺(, 𝐺), ⋯ , 𝐺+} +Displacement vectors +Geometric primitives +(e.g., planes, spheres) +Sample Prediction +Network 𝐸$ 𝑟 +𝒓 +(a) Ray parameterization +(b) Sample prediction network +(c) Sample generation +(d) Volume rendering +Figure 2. Overview of HyperReel for static scenes. Given a set of images and camera poses, the training objective is to reconstruct the +measured color associated with every ray. (a) Given an input ray originating at the camera origin o and traveling in direction ⃗ω, we first +reparameterize the ray using Pl¨ucker coordinates. (b) A network Eφ takes this ray as input and outputs the parameters for a set of geometric +primitives {Gk} (such as axis-aligned planes and spheres) and displacement vectors {dk}. (c) To generate sample points {xk} for volume +rendering, we compute the intersections between the ray and the geometric primitives, and add the displacement vectors to the results. +Predicting geometric primitives has the advantage of making the sample signal smooth and easy to interpolate (see Section 3.1). The +displacement vectors grant additional flexibility to the sample points, enabling better capture of complex view-dependent appearance. (d) +Finally, we perform volume rendering via Equation 2 to produce a pixel color and supervise training based on the corresponding observation. +Figure 3. Extracting sample point appearance in the dynamic +setting from our keyframe-based representation. (1) We first +advect the sample points {xk} at time τ into the nearest keyframe +τi, using velocities {vk} outputted from the sample prediction net- +work. (2) We then query the outer products of space-time textures +in order to produce per-sample-point appearance features, which +are then converted to colors via Equation 10. We follow a similar +procedure for extracting per-sample-point opacities. +rendering in Equation 2. In this work, we use the Pl¨ucker +parameterization to represent the ray: +r = Pl¨ucker(o, ⃗ω) = (⃗ω, ⃗ω × o) . +(3) +While many designs for the sample prediction network Eφ +are possible, giving the network too much flexibility may +negatively affect view synthesis quality. For example, if +(x1, . . . , xn) are completely arbitrary points, then renderings +may not appear to be multi-view-consistent. +To address this problem, we choose to predict the pa- +rameters of a set of geometric primitives G1, . . . , Gn with +the sample prediction network, where the primitive parame- +ters can vary depending on the input ray. To get our sample +points, we then intersect the ray with each primitive: +Eφ(o, ⃗ω) = (G1, . . . , Gn) , +(4) +(x1, . . . , xn) = (inter(G1; o, ⃗ω), . . . , inter(Gn; o, ⃗ω)) . +(5) +Above, inter(Gk; o, ⃗ω) is a differentiable operation that in- +tersects the ray with the primitive Gk. In all of our exper- +iments, we use axis-aligned z-planes (for forward-facing +scenes) or concentric spherical shells centered at the origin +(for all other scenes) as our geometric primitives. +This approach is constrained in that it produces sample +points that initially lie along the ray. Further, predicting +primitives defined in a global coordinate frame makes the +sample signal smooth and easy to interpolate. For example, +if two distinct rays observe the same point in the scene, then +the sample network needs only predict one primitive for +both rays (i.e., defining a primitive that passes through the +point). In contrast, existing works such as NeuSample [12], +AdaNeRF [20], and TermiNeRF [38] predict distances or +per-segment weights that vary depending on the ray even if +these rays observe the same point in the scene. +Flexible Sampling for Challenging Appearance. +To +grant our samples additional flexibility to better represent +challenging view-dependent appearance, we also predict a +set of Tanh-activated per-sample-point offsets (e1, . . . , en), +as well as a set of scalar values (δ1, . . . , δn). We convert +these scalar values to weights with a sigmoid activation, i.e., +(γ(δ1), . . . , γ(δn)) where γ is the sigmoid operator. Specifi- +4 + +d3 +(o,w) +X3 +G1 G2(o. 3)1 +Zcally, we have: +(d1, . . . dn) = (γ(δ1)e1, . . . , γ(δn)en) +(6) +(x1, . . . xn) ← (x1 + d1, . . . , xn + dn) , +(7) +where we use (d1, . . . , dn) to denote the final displacement, +or “point-offset” added to each point. +The sample network outputs may appear to be over- +parameterized and under-constrained. However, the above +design is essential for achieving good-quality view synthesis. +In particular, initializing the scalars (δ1, . . . , δn) to negative +values, where the sigmoid is close to 0, and its gradient is +small, implicitly discourages the network from unmasking +the point offsets, while still allowing the network to use them +as necessary. +In addition to enabling real-time rendering with low sam- +ple counts, one added benefit of our sample network architec- +ture is the improved modeling of complex view-dependent +appearance. For example, distorted refractions break epipolar +geometry and appear to change the depth of the refracted con- +tent depending on the viewpoint. As illustrated in Figure 2, +our sample network, on the other hand, has the flexibility to +model sample points that warp depending on viewpoint. +Existing works like Eikonal fields [8] can be considered a +special case of this sample warping approach; they use phys- +ically derived Eikonal constraints to learn ray-conditional +warp fields. Unlike these works, our approach can handle +both reflections and refractions, and does not require evalu- +ating costly multi-step ODE solvers during rendering. See +Figure 1 and our website for additional results and compar- +isons on challenging view-dependent scenes. +3.2. Keyframe-Based Dynamic Volumes +Thus far, we have covered how to efficiently sample a 3D +scene volume, but have not yet discussed how we represent +the volume itself. In the static case, we make use of the mem- +ory efficient Tensorial Radiance Fields (TensoRF) approach +(Section 3.2.1), and in the dynamic case we extend Ten- +soRF to a keyframe-based dynamic volume representation +(Section 3.2.2). +3.2.1 +Representing 3D Volumes with TensoRF [11] +Recall that TensoRF factorizes a 3D volume as a set of +outer products between functions of one or more spatial +dimensions. Specifically, we can write the set of spherical +harmonic coefficients A (xk) capturing the appearance of a +point xk = (xk, yk, zk) as: +A (xk) = B1(f1(xk, yk) ⊙ g1(zk)) ++ B2(f2(xk, zk) ⊙ g2(yk)) +(8) ++ B3(f3(yk, zk) ⊙ g3(xk)) . +Above, fj and gj are vector-valued functions with output di- +mension M, and the operator ‘⊙’ computes an element-wise +product. In the original TensoRF work [11], the functions +fj and gj are discretized into M different 2D and and 1D +arrays, respectively. +Further, Bj denote linear transforms that map the products +of fj and gj to spherical harmonic coefficients. The color +Le(xk, ⃗ω) for point xk and direction ⃗ω is then given by +the dot product of the coefficients A (xk) and the spherical +harmonic basis functions evaluated at ray direction ⃗ω. +Similar to appearance, for density, we have: +σ(xk) = 1⊤ (h1(xk, yk) ⊙ k1(zk)) ++ 1⊤ (h2(xk, zk) ⊙ k2(yk)) +(9) ++ 1⊤ (h3(yk, zk) ⊙ k3(xk)) , +where 1 is a vector of ones, and hj and kj are vector- +valued functions with output dimension M. Given the color +Le(xk, ⃗ω) and density σ(xk) for all sample points {xk} +along a ray, we can then make use of Equation 2 to render +the final color for that ray. +3.2.2 +Representing Keyframe-Based Volumes +To handle dynamics, we adapt TensoRF to parameterize +volumetric “keyframes”, or snapshots of a dynamic volume +at a set of discrete time steps. If we denote τi as the time +step corresponding to the ith keyframe, we can write: +A (xk, τi) = B1(f1(xk, yk) ⊙ g1(zk, τi)) ++ B2(f2(xk, zk) ⊙ g2(yk, τi)) +(10) ++ B3(f3(yk, zk) ⊙ g3(xk, τi)) , +and +σ(xk, τi) = 1⊤ (h1(xk, yk) ⊙ k1(zk, τi)) ++ 1⊤ (h2(xk, zk) ⊙ k2(yk, τi)) +(11) ++ 1⊤ (h3(yk, zk) ⊙ k3(xk, τi)) , +where the only change from Section 3.2.1 is that gj and kj +now depend on time, in addition to one spatial dimension. +We note that the above factorization of the dynamic vol- +ume representing all keyframes in a video has a similar mem- +ory footprint to a static TensoRF for a single frame, assuming +that the number of keyframes is small relative to the reso- +lution of our spatial dimensions. In particular, if the spatial +resolution of our volume is (Nx, Ny, Nz) and the number of +keyframes is Nt, then we can store a single component of f1 +with an Nx × Ny array, and store a single component of g1 +with an Nz × Nt array. Because Nt ≪ Nx/y/z, the arrays +gj do not contribute significantly to the size of the model. +5 + +3.2.3 +Rendering from Keyframe-Based Volumes +In order to combine our sampling procedure (Section 3.1) +and keyframe-based volume representation (Section 3.2.2) to +complete our system for 6-DoF video, a few additional mod- +ifications are required. First, since the surfaces in a dynamic +scene move over time, the sample points {xk} should be +time dependent. We therefore augment our sample prediction +network to take the current time τ as input. +Second, the decomposition of the dynamic scene in Sec- +tion 3.2.2 creates temporal “snapshots” of the volume at +discrete keyframes τi, but we would like to sample the vol- +ume at arbitrary times τ. In order to generate the dynamic +volume at all intermediate times, we also output velocities +vk ∈ R3 from the sample prediction network, which we use +to advect the sample points into the nearest keyframe τi with +a single forward-Euler step: +xk ← xk + vk(τi − τ). +(12) +Another perspective on Equation 12 is that it defines a back- +wards warp with scene flow field vk that generates the vol- +ume at time τ. The process of warping sample points and +querying the keyframe-based dynamic volume is illustrated +in Figure 3. +After querying the keyframe-based volume with {xk}, +the equation for volume rendering is then: +C(o, ⃗ω, τ) = +N +� +k=1 +wk Le (xk, ⃗ω, τi) , +(13) +where wk = ˆT(o, xk, τi) (1 − e−σ(xk,τi)∆xk), and τi is the +time step corresponding to the closest keyframe to time τ. +This is effectively the same as Equation 2, except C, xk, wk +and Le now depend on the time τ. The sampling procedure +(Section 3.1), volume representation (Section 3.2.2), and ren- +dering scheme for keyframe-based volumes (Section 3.2.3) +comprise our 6-DoF video representation: HyperReel. +3.3. Optimization +We optimize our representation using only the training im- +ages, and apply total variation and ℓ1 sparsity regularization +to our tensor components, similar to TensoRF [11]: +L = LL2 + wL1LL1 + wTVLTV +where +(14) +LL2 = +� +o,⃗ω,τ +∥C(o, ⃗ω, τ) − CGT(o, ⃗ω, τ)∥. +(15) +The loss is summed over training rays and times, and CGT +represents the ground-truth color for a given ray and time. +We only use a subset of all training rays to make the +optimization tractable on machines with limited memory. +In all dynamic experiments, for frame numbers divisible by +4, we alternate between using all training rays and using +training rays from images downsampled by a 4× factor. For +all other instances, we downsample images by an 8× factor. +4. Experiments +Implementation Details. +We implement our method in +PyTorch [36] and run experiments on a single NVIDIA RTX +3090 GPU with 24 GB RAM. Our sample network is a 6- +layer, 256-hidden unit MLP with Leaky ReLU activations +for both static and dynamic settings. Unless otherwise speci- +fied, for forward-facing scenes, we predict 32 z-planes as our +geometric primitives with our ray-conditioned sample predic- +tion network. In all other settings, we predict the radii of 32 +spherical shells centered at the origin. We use the same space +contraction scheme for unbounded scenes as in mip-NeRF +360 [6]. For our keyframe-based volume representation, we +give the (x, y) and (z, t) textures eight components each +and four components to all other textures. For all dynamic +datasets, we use every 4th frame as a keyframe. For both +static and dynamic datasets, we use a batch size of 16,384 +rays for training, an initial learning rate of 0.02 for the param- +eters of the keyframe-based volume, and an initial learning +rate of 0.0075 for our sample prediction network. We train +all models for 1.5 hours each. For Technicolor, Google Im- +mersive, and all static scenes, we set the wTV weight in +Equation 14 to 0.05 for both appearance and density, which +is decayed by a factor of 0.1 every 30,000 iterations. On +the other hand, wL1 starts at 8·10−5 and decays to 4·10−5 +over 30,000 iterations and is only applied to the density +components. +Qualitative Results. +We show a few qualitative re- +sults and comparisons in Figure 4, and a compre- +hensive +set +of +qualitative +results +on +our +website: +hyperreel.github.io. +4.1. Comparisons on Static Scenes +DoNeRF Dataset. +The DoNeRF dataset [31] contains six +synthetic sequences with images of 800×800 pixel resolu- +tion. Here, we validate the efficacy of our sample prediction +network approach by comparing it to existing methods for +static view synthesis, including NeRF, InstantNGP, and three +sampling-network–based approaches [20, 31, 38]. +As demonstrated in Table 1, our approach outperforms all +baselines in terms of quality and improves the performance +of other sampling network schemes by a large margin. Addi- +tionally, our model is implemented in vanilla PyTorch and +renders 800×800 pixel images at 6.5 FPS on a single RTX +3090 GPU (or 29 FPS with our Tiny model). +We also compare our sampling network-based approach +to the single-sample R2L light field representation [55] +on the downsampled 400×400 resolution DoNeRF dataset +(with their provided metrics). We outperform their approach +quantitatively without using pretrained teacher networks. +Further, inference with our six-layer, 256-hidden-unit net- +work, and TensoRF volume backbone is faster than R2L’s +deep 88-layer, 256-hidden-unit MLP. +6 + +Ground truth (Technicolor [44]) +Ours +Neural 3D Video [23] +Ground truth (Neural 3D Video [23]) +Ours +NeRFPlayer [49] +Ground truth (Google Immersive LF Video [9]) +Ours +NeRFPlayer [49] +Figure 4. Qualitative comparisons of dynamic reconstruction. We show visual comparisons of our method on three datasets against two +baselines on heldout views. We pick non-keyframe time-steps for evaluation, except for the Google Immersive light field video (last row), +for which we pick the matching image to the NeRFPlayer [49] result. +LLFF Dataset. +The LLFF dataset [29] contains eight real- +world sequences with 1008×756 pixel images. In Table 1, +we compare our method to the same approaches as above on +this dataset. Our approach outperforms DoNeRF, AdaNeRF, +TermiNeRF, and InstantNGP but achieves slightly worse +quality than NeRF. This dataset is challenging for explicit +volume representations (which have more parameters and +thus can more easily overfit to the training images) due to a +combination of erroneous camera calibration and input-view +sparsity. For completeness, we also include a comparison to +R2L on the downsampled 504×378 LLFF dataset, where we +perform slightly worse in terms of quality. +4.2. Comparisons on Dynamic Scenes +Technicolor Dataset. +The Technicolor light field dataset +[44] contains videos of varied indoor environments captured +by a time-synchronized 4×4 camera rig. Each image in each +video stream is 2048×1088 pixels, and we hold out the view +in the second row and second column for evaluation. We +compare HyperReel to Neural 3D Video [23] at full image +resolution on five sequences (Birthday, Fabien, Painter, The- +ater, Trains) from this dataset, each 50 frames long. We train +Neural 3D Video on each sequence for approximately one +week on a machine with 8 NVIDIA V100 GPUs. +We show in Table 2 that the quality of HyperReel exceeds +that of Neural 3D Video [23] while also training in just 1.5 +GPU hours per sequence (rather than 1000+ GPU hours for +Neural 3D), and rendering far more quickly. +Neural 3D Video Dataset. +The Neural 3D Video dataset +[23] contains six indoor multi-view video sequences cap- +tured by 20 cameras at 2704×2028 pixel resolution. We +downsample all sequences by a factor of 2 for training and +evaluation and hold out the central view for evaluation. Met- +rics are averaged over all scenes. Additionally, due to the +challenging nature of this dataset (time synchronization er- +rors, inconsistent white balance, imperfect poses), we output +64 z-planes per ray with our sample network rather than 32. +We show in Table 2 that we outperform all baseline ap- +proaches on this dataset, including contemporary works such +as NeRFPlayer [49] and StreamRF [22]. In particular, we +outperform NeRFPlayer quantitatively while rendering ap- +proximately 40 times faster. We outperform StreamRF by a +more significant margin in terms of quality, although their +approach with a Plenoxels backbone (which uses custom +CUDA kernels for faster inference) renders faster than our +7 + +++Table 1. Static comparisons. We compare our sampling network +architecture to others on the synthetic DoNeRF dataset [31] and +real LLFF dataset [28]. FPS is normalized per megapixel; memory +in MB. +Dataset +Method +PSNR↑ +FPS↑ Memory ↓ +DoNeRF 400×400 +Single sample +R2L [55] +35.5 +— +23.7 +Ours (per-frame) +36.7 +4.0 +58.8 +DoNeRF 800×800 +Uniform sampling +NeRF [29] +30.9 +0.3 +3.8 +Instant NGP [30] +33.1 +3.8 +64.0 +Adaptive sampling +DoNeRF [31] +30.8 +2.1 +4.1 +AdaNeRF [20] +30.9 +4.7 +4.1 +TermiNeRF [38] +29.8 +2.1 +4.1 +Ours (per-frame) +35.1 +4.0 +58.8 +LLFF 504×378 +Single sample +R2L [55] +27.7 +— +23.7 +Ours (per-frame) +27.5 +4.0 +58.8 +LLFF 1008×756 +Uniform sampling +NeRF [29] +26.5 +0.3 +3.8 +Instant NGP [30] +25.6 +5.3 +64.0 +Adaptive sampling +DoNeRF [31] +22.9 +2.1 +4.1 +AdaNeRF [20] +25.7 +5.6 +4.1 +TermiNeRF [38] +23.6 +2.1 +4.1 +Ours (per-frame) +26.2 +4.0 +58.8 +Table 2. Dynamic comparisons. We compare HyperReel to exist- +ing 3D video methods on three light-field/multi-view video datasets. +Note that all FPS numbers are reported for megapixel images, and +memory is in MB per frame. †NeRFPlayer [49] and StreamRF [22] +do not provide SSIM and LPIPS scores. ‡The accompanying paper +does not provide quantitative metrics, while the concurrent NeRF- +Player does, so we provide a comparison with NeRFPlayer only. +Table 3 provides a proxy comparison of our method to Google’s, +and our website includes a qualitative comparison. +Dataset +Method +PSNR↑ +SSIM↑ +LPIPS↓ +FPS↑ Memory↓ +Technicolor [44] +Neural 3D Video [23] +31.8 +0.958 +0.140 +0.02 +0.6 +Ours +32.7 +0.906 +0.109 +4.00 +1.2 +Neural 3D Video [23] +Neural 3D Video [23] +29.6 +0.961 +0.083 +0.02 +0.1 +NeRFPlayer [49] +30.7 +—† +—† +0.06 +17.1 +StreamRF [22] +28.3 +—† +—† +10.90 +17.7 +Ours +31.1 +0.927 +0.096 +2.00 +1.2 +Google LF videos [9]‡ +NeRFPlayer [49] +27.4 +0.871 +0.295 +0.12 +17.1 +Ours +28.8 +0.874 +0.193 +4.00 +1.2 +model. Our model consumes less memory on average per +frame than both StreamRF and NeRFPlayer. +Google Immersive Dataset. +The Google Immersive +dataset [9] contains light field videos of various indoor and +outdoor environments captured by a time-synchronized 46- +fisheye camera rig. Here, we compare our approach to NeRF- +Player and select the same seven scenes as NeRFPlayer for +evaluation on this dataset (Welder, Flames, Truck, Exhibit, +Table 3. Quantitative comparisons to DeepView. In addition to +the comparison to NeRFPlayer in Table 2, we report a comparison +with DeepView [13], a variant of which is used per-frame in im- +mersive LF video [9]. We thus compare to DeepView as a proxy +for quantitative comparison. FPS normalized per megapixel. +Dataset +Method +PSNR↑ +SSIM↑ +LPIPS↓ FPS↑ +Spaces [13] +DeepView [13] +31.60 +0.965 +0.085 +>100 +Ours +35.47 +0.968 +0.080 +4.0 +Table 4. Ablations. We perform several ablations on our method, +including on the number of keyframes, the use of the sampling +network, and model size. All FPS normalized per megapixel. +Dataset +Method +PSNR↑ +SSIM↑ +LPIPS↓ FPS↑ +Technicolor +Ours (keyframe every 1 frame(s)) +32.34 +0.895 +0.117 +4.0 +Ours (keyframe every 4 frame(s)) +32.73 +0.906 +0.109 +4.0 +Ours (keyframe every 16 frame(s)) +32.07 +0.893 +0.112 +4.0 +Ours (keyframe every 50 frame(s)) +32.35 +0.896 +0.110 +4.0 +Ours (w/o sample network) +29.08 +0.815 +0.209 +1.3 +Ours (Tiny) +30.09 +0.835 +0.157 +17.5 +Ours (Small) +31.76 +0.903 +0.125 +9.1 +Face Paint 1, Face Paint 2, Cave), holding out the central +view for validation. For our approach, we split each video +into several 50-frame chunks. Our results in Table 2 outper- +form NeRFPlayer’s by a 1 dB margin in terms of quality, +while again rendering more quickly. +DeepView Dataset. +Unfortunately, Google’s Immersive +Light Field Video [9] does not provide quantitative bench- +marks for the performance of their approach in terms of +image quality. As a proxy, we compare our approach to +DeepView [13], the method upon which their representation +is built, on the static Spaces dataset in Table 3. +Our method achieves superior quality, outperforming +DeepView by a large margin. Further, HyperReel consumes +less memory per frame than the Immersive Light Field +Video’s baked layered mesh representation: 1.2 MB per +frame vs. 8.87 MB per frame (calculated from the reported +bitrate numbers [9]). Their layered mesh can render at more +than 100 FPS on commodity hardware, while our approach +renders at a little over 4 FPS. However, our approach is en- +tirely implemented in vanilla PyTorch and can be further +optimized using custom CUDA kernels or baked into a real- +time renderable representation for better performance. +4.3. Ablation Studies +Number of Keyframes. +In Table 4, we ablate our method +on the Technicolor light field dataset with different numbers +of keyframes. In general, the optimal number of keyframes +depends on the motion within a scene. Our dynamic volume +representation implicitly trades off between temporal resolu- +tion and spatial resolution, as it can use the (x, t), (y, t), and +(z, t) components to add either spatial or temporal details. +Our choice of one keyframe for every four frames strikes a +good balance between temporal resolution and spatial reso- +8 + +GT +Full model +Small +Tiny +No sampling +Figure 5. Ablations on our sampling network. We show close-up +results for various sampling networks architectures on two of the +Technicolor sequences also shown in Figure 4. +Table 5. Point offset ablation. We evaluate the performance of our +network with and without point offsets. +Scene +Point offset +PSNR↑ +SSIM↑ LPIPS↓ +DoNeRF “Forest” [31] +Without +34.86 +0.969 +0.0146 +(diffuse) +With +36.34 +0.975 +0.0122 +Shiny “Lab” [58] +Without +31.28 +0.943 +0.0416 +(highly refractive) +With +32.49 +0.959 +0.0294 +lution and achieves the best overall performance (Table 4). +Sample Network Size. +We also show the performance of +our method with different sample prediction network de- +signs in Table 4, including the performance for a Tiny model +(4-layers, 128-hidden-unit MLP with 8 predicted sample +points), and Small model (4-layers, 256-hidden-unit MLP +with 16 predicted sample points). Our Tiny model runs at +18 FPS, and our Small model runs at 9 FPS at megapixel +resolution, again without any custom CUDA code. Our Tiny +model performs reasonably well but achieves worse quality +than Neural 3D Video on the Technicolor dataset. In contrast, +our Small model achieves comparable overall performance +to Neural3D—showing that we can still achieve good quality +renderings at even higher frame rates. We show accompany- +ing qualitative results for these models in Figure 5. +With and Without Sample Prediction Network. +We +show results on the Technicolor dataset without our sam- +ple prediction network, using every frame as a keyframe, +and with 4× the number of samples (128 vs. 32). Our full +method outperforms this approach by a sizeable margin. +With and Without Point Offset. +In Table 5, we show re- +sults on two static scenes with and without point offsets +(Equation 7): one diffuse and one highly refractive scene. +Point offsets improve quality in both cases, suggesting that +they may help with better model capacity allocation in ad- +dition to view-dependence—similar to “canonical frame” +deformations used in Nerfies [34] and Neural Volumes [27]. +5. Conclusion +HyperReel is a novel representation for 6-DoF video, +which combines a ray-conditioned sampling network with a +0.1 +1 +10 +29 +30 +31 +Ours +StreamRF +N3DV +NeRFPlayer +Rendering speed [frames per second] +PSNR [dB] +Dynamic Reconstruction +0.1 +1 +10 +30 +32 +34 +Ours +NeRF +Instant NGP +DoNeRF +TermiNeRF +AdaNeRF +Rendering speed [frames per second] +PSNR [dB] +Static Reconstruction +Figure 6. Rendering speed vs. quality trade-off. We show the +speed-quality trade-off of our method and others applied to dynamic +scenes (left) on the Neural 3D Video dataset [23] and static scenes +methods (right) on the DoNeRF dataset [31]. +Our result +GT +Our result +GT +Figure 7. Limitations. Our approach can sometimes produce blurry +reconstructions due to the training ray subsampling scheme (Sec- +tion 3.3) (left) or noisy reconstructions in sparsely observed regions +due to an under-constrained sampling network (right). +keyframe-based dynamic volume representation. It achieves +a balance between high rendering quality, speed, and mem- +ory efficiency that sets it apart from existing 6-DoF video +representations. We qualitatively and quantitatively compare +our approach to prior and contemporary 6-DoF video rep- +resentations, showing that HyperReel outperforms each of +these works along multiple axes, as illustrated in Figure 6. +Limitations and Future Work. +Our sample prediction +network is supervised only by a rendering loss on the training +images. This can lead to a reduction in quality for views +outside of the convex hull of the training cameras or for scene +content that is only observed in a small number of views (see +Figure 7), where the sample network may predict erroneous +sample points. Exploring regularization methods that enable +reasonable geometry predictions even for extrapolated views +is an important future direction. +Although our keyframe-based representation is more +memory efficient than most existing 3D video formats, it can- +not be streamed like NeRFPlayer [49] or StreamRF [22]. In +practice, however, our representation is sufficiently small that +this does not pose a major issue. Additionally, our sample +network approach is compatible with any streaming-based +dynamic volume. +Currently, our approach falls short of the rendering speed +required for settings like VR (ideally 72 FPS, in stereo). As +our method is implemented in vanilla PyTorch, we expect to +gain significant speedups with more engineering effort. +9 + ++EAMTEAM+Acknowledgments +We thank Thomas Neff, Yu-Lun Liu, and Xiaoming Zhao +for valuable feedback and discussions, Zhaoyang Lv for +help running the Neural 3D Video Synthesis codebase [23], +and Liangchen Song for providing information about the +scenes from the Google Immersive Video dataset [9] used in +NeRFPlayer [49]. Matthew O’Toole acknowledges support +from NSF IIS-2008464. +References +[1] Kara-Ali Aliev, Artem Sevastopolsky, Maria Kolos, Dmitry +Ulyanov, and Victor Lempitsky. 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Additional experimental details regarding our keyframe- +based volume design. +Further, we provide a full per-scene breakdown of image +metrics for the Technicolor dataset in Table D.1. As we do +not have direct access to the outputs of other methods on the +Google Immersive ([9]) and Neural 3D Video ([62]) datasets, +we do not provide per-scene breakdowns for these datasets. +Finally, in addition to a video of our real-time demo, our +website contains: +1. Dynamic dataset results from our method on each of +Technicolor ([44]), Neural 3D Video ([23]), and Google +Immersive Video ([9]); +2. Qualitative results and comparisons on view-dependent +static scenes from the Shiny Dataset ([58]) and the +Stanford Light Field Dataset ([57]); +3. Qualitative comparison to [9]. +B. Additional Training & Evaluation Details +B.1. Training Ray-Subsampling +We provide pseudo-code for our training ray-subsampling +scheme in Algorithm 1. +B.2. LPIPS Evaluation Details +For LPIPS computation, we use the AlexNet LPIPS variant +for all of our comparisons in the main paper (as do all of the +baseline methods). +B.3. SSIM Evaluation Details +For SSIM computation, we use the structural similarity +scikit-image library function, with our images normalized +to the range of [0, 1], and the data range parameter set to 1. +We note, however, that several methods either: +1. Use their own implementation of SSIM, which are +not consistent with this standard implementation (e.g. +R2L [56]); +2. Fail to set the data range parameter appropriately, so +that it defaults to the value of 2.0 (e.g. Neural 3D Video +[23]). +In both of these cases, the SSIM function returns higher- +than-intended values (which we confirm by testing these +variants on our predicted images). While we believe that this +inconsistency makes SSIM scores somewhat less reliable, we +still report our aggregated SSIM metrics in the quantitative +result tables in the main paper. +C. Sample Prediction Network Description +C.1. Forward Facing Scenes +For forward facing scenes, we first convert all rays to nor- +malized device coordinates (NDC) [29], so that the view +frustum of a “reference” camera lives within [−1, 1]3. After +mapping a ray with origin o and direction ⃗ω to its Pl¨ucker +parameterization via +r = Pl¨ucker(o, ⃗ω) = (⃗ω, ⃗ω × o) . +(16) +we predict the parameters of a set of planes normal to +the z-axis with our embedding network. In particular, we +12 + +ALGORITHM 1: Training Ray-Subsampling Scheme +Input: Number of videos {N}, Number of frames {M} +Output: Training Rays raysGT , Ground Truth Colors CGT +// Initialize rays and colors +raysGT = {} +CGT = {} +// Iterate over all N videos +for n ∈ {1, · · · , N} do +// Iterate over all M frames in video n +for m ∈ {1, · · · , M} do +// Get frame m from video n +Cn,m = GetFrame(n, m) +// Get corresponding rays for this frame +raysn,m = GetRays(n, m) +if m is not divisible by 8 then +// Downsample rays and colors by a factor of 4 +Cn,m ← NearestNeighborDownsample(Cn,m, 4) +raysn,m ← NearestNeighborDownsample(raysn,m, 4) +if m is not divisible by 4 then +// Downsample rays and colors by an additional factor of 2 +Cn,m ← NearestNeighborDownsample(Cn,m, 2) +raysn,m ← NearestNeighborDownsample(raysn,m, 2) +end +end +// Add current rays and colors to output +CGT ← CGT + Cn,m +raysGT ← raysGT + raysn,m +end +end +predict (z1, . . . , zn), and intersect the ray with the axis- +aligned planes at these distances to produce our sample +points (x1, . . . , xn). Additionally, we initialize the values +(z1, . . . , zn) in a stratified manner, so that they uniformly +span the range of [−1, 1]. +C.2. Outward Facing Scenes +For all other (outward facing) scenes, we predict the radii +of a set of spheres centered at the origin (r1, . . . , rn), and +intersect the ray with each sphere to produce our sample +points. We initialize (r1, . . . , rn) so that they range from the +minimum distance to the maximum distance in the scene. +C.3. Differentiable Intersection +In both of the above cases, we make use of the implicit form +of each primitive (for planes normal to the z-axis, z = zk, +and for the spheres centered at the origin x2 +y2 +z2 = r2 +k) +and the parameteric equation for a ray o + tk⃗ω, to solve for +the intersection distances tk (as is done in typical ray-tracers). +The intersection distance is differentiable with respect to the +primitive parameters, so that gradients can propagate from +the color loss to the sample network. +C.4. Implicit Color Correction +In order to better handle multi-view datasets with inconsis- +tent color correction / white balancing, we also output a color +scale cscale +k +and shift cshift +k +from the sample prediction network +for each sample point xk. These are used to modulate the +color Le(xk, ⃗ω, τi) extracted from the dynamic volume via: +Le(xk, ⃗ω, τi) ← Le(xk, ⃗ω, τi) · cscale +k ++ cshift +k +. +(17) +Note that these outputs vary with low-frequency with respect +to the input ray (since we use few positional encoding fre- +quencies for the sample prediction network). Additionally, +the density from the volume remains unchanged. +D. Keyframe-Based Volume Details +We initialize our keyframe-based dynamic volume within +a 1283 grid, so that each of the spatial tensor components +have resolution 128 × 128. Our final grid size is 6403. We +upsample the volume at iterations 4000, 6000, 8000, 10000, +and 12000, interpolating the resolution linearly in log space. +13 + +Table D.1. Per-scene results from the Technicolor dataset [44]. See Section Appendix B.3 for a brief disucssion of the reliability of SSIM +metrics. +Scene +PSNR↑ +SSIM↑ +LPIPS↓ +Neural 3D Video [23] Ours Small Tiny +Neural 3D Video [23] Ours Small Tiny +Neural 3D Video [23] Ours +Small +Tiny +Birthday +29.20 +29.99 29.32 27.80 +0.952 +0.922 0.907 0.876 +0.0668 +0.0531 0.0622 0.0898 +Fabien +32.76 +34.70 33.67 32.25 +0.965 +0.895 0.882 0.860 +0.2417 +0.1864 0.1942 0.2233 +Painter +35.95 +35.91 36.09 34.61 +0.972 +0.923 0.920 0.905 +0.1464 +0.1173 0.1182 0.1311 +Theater +29.53 +33.32 32.19 30.74 +0.939 +0.895 0.880 0.845 +0.1881 +0.1154 0.1306 0.1739 +Trains +31.58 +29.74 27.51 25.02 +0.962 +0.895 0.835 0.773 +0.0670 +0.0723 0.1196 0.1660 +14 + diff --git a/O9E0T4oBgHgl3EQfTgDH/content/tmp_files/load_file.txt b/O9E0T4oBgHgl3EQfTgDH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5656231d4ddc7347f89b61e22353f7630e79a028 --- /dev/null +++ b/O9E0T4oBgHgl3EQfTgDH/content/tmp_files/load_file.txt @@ -0,0 +1,967 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf,len=966 +page_content='HyperReel: High-Fidelity 6-DoF Video with Ray-Conditioned Sampling Benjamin Attal∗ Carnegie Mellon University Jia-Bin Huang Meta & UMD Christian Richardt Reality Labs Research Michael Zollh¨ofer Reality Labs Research Johannes Kopf Meta Matthew O’Toole Carnegie Mellon University Changil Kim Meta https://hyperreel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='io Dynamic 6-DoF rendering t = 0 s t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='25 s t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='5 s t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='75 s t = 1 s t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='25 s t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='5 s Static 6-DoF rendering Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' HyperReel: A novel 6-DoF video representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' HyperReel converts synchronized multi-view video streams into a high-fidelity, memory efficient scene representation that can be rendered from novel views and time steps at interactive rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' HyperReel’s combination of high rendering quality, speed, and compactness sets it apart from existing 6-DoF video representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The upper two rows show 6-DoF (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=', varying viewpoint and viewing orientation) rendering of dynamic scenes [9, 44];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' the lower two of static scenes [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Abstract Volumetric scene representations enable photorealistic view synthesis for static scenes and form the basis of several ex- isting 6-DoF video techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' However, the volume render- ing procedures that drive these representations necessitate careful trade-offs in terms of quality, rendering speed, and memory efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In particular, existing methods fail to simultaneously achieve real-time performance, small mem- ory footprint, and high-quality rendering for challenging ∗ This work was done while Benjamin was an intern at Meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' real-world scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' To address these issues, we present Hy- perReel — a novel 6-DoF video representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The two core components of HyperReel are: (1) a ray-conditioned sample prediction network that enables high-fidelity, high frame rate rendering at high resolutions and (2) a compact and memory- efficient dynamic volume representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Our 6-DoF video pipeline achieves the best performance compared to prior and contemporary approaches in terms of visual quality with small memory requirements, while also rendering at up to 18 frames-per-second at megapixel resolution without any custom CUDA code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='02238v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='CV] 5 Jan 2023 WDGREEN WD 7mmS ALCOHOL ALSOFFALCOHOL ALSOFF mmonsongobcuALCOHOL ALSOFF allens onels!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='mmt ALCOHOL ALSOFFWD GREEN WD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='5/7mmSd 5"/7mmDi ALCOHOL ALSOFFWDGREEN WD /7mmSc 7mmD ALCOHOL ALSOFFWDGREEN zmmS ALCOHOL ALSOFFPACE FS HEHANGEDMA 工PACES HEHANGEDMAPACES QUEEN CHANGEDMAPACE S QUEEN HANGEDMAPACES QUEEN IANGEDMAPACE S QUEENF HANGEDMAPACE S QUEEN EHANGEDMA1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Introduction Six–Degrees-of-Freedom (6-DoF) videos allow for free ex- ploration of an environment by giving the users the ability to change their head position (3 degrees of freedom) and orientation (3 degrees of freedom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' As such, 6-DoF videos offer immersive experiences with many exciting applications in AR/VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The underlying methodology that drives 6-DoF video is view synthesis: the process of rendering new, unob- served views of an environment—static or dynamic—from a set of posed images or videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Volumetric scene representa- tions such as neural radiance fields [29] and instant neural graphics primitives [30] have recently made great strides toward photorealistic view synthesis for static scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' While several recent works build dynamic view synthe- sis pipelines on top of these volumetric representations [14, 23, 24, 33, 61], it remains a challenging task to cre- ate a 6-DoF video format that can achieve high quality, fast rendering, and a small memory footprint (even given many synchronized video streams from multi-view camera rigs [9, 35, 44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Existing approaches that attempt to create memory-efficient 6-DoF video can take nearly a minute to render a single megapixel image [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Works that target ren- dering speed and represent dynamic volumes directly with 3D textures require gigabytes of storage even for short video clips [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' While other volumetric methods achieve memory efficiency and speed by leveraging sparse or compressed vol- ume storage for static scenes [11, 30], only contemporary work [22, 49] addresses the extension of these approaches to dynamic scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Moreover, all of the above representations struggle to capture highly view-dependent appearance, such as reflections and refractions caused by non-planar surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In this paper, we present HyperReel, a novel 6-DoF video representation that achieves state-of-the-art quality while being memory efficient and real-time renderable at high res- olution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The first ingredient of our approach is a novel ray- conditioned sample prediction network that predicts sparse point samples for volume rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In contrast to exist- ing static view synthesis methods that use sample networks [20, 31], our design is unique in that it both (1) acceler- ates volume rendering and at the same time (2) improves rendering quality for challenging view-dependent scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Second, we introduce a memory-efficient dynamic vol- ume representation that achieves a high compression rate by exploiting the spatio-temporal redundancy of a dynamic scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Specifically, we extend Tensorial Radiance Fields [11] to compactly represent a set of volumetric keyframes, and capture intermediate frames with trainable scene flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The combination of these two techniques comprises our high-fidelity 6-DoF video representation, HyperReel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We validate the individual components of our approach and our representation as a whole with comparisons to state-of-the- art sampling network-based approaches for static scenes as well as 6-DoF video representations for dynamic scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Not only does HyperReel outperform these existing works, but it also provides high-quality renderings for scenes with chal- lenging non-Lambertian appearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Our system renders at up to 18 frames-per-second at megapixel resolution without using any custom CUDA code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In summary, the contributions of our work include the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' A novel sample prediction network for volumetric view synthesis that accelerates volume rendering and accu- rately represents complex view-dependent effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' A memory-efficient dynamic volume representation that compactly represents a dynamic scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' HyperReel, a 6-DoF video representation that achieves a desirable trade-off between speed, quality, and mem- ory, while rendering in real time at megapixel resolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Related Work Novel View Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Novel-view synthesis is the pro- cess of rendering new views of a scene given a set of input posed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Classical image-based rendering techniques use approximate scene geometry to reproject and blend source image content onto novel views [10, 37, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Recent works leverage the power of deep learning and neural fields [62] to improve image-based rendering from both structured (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=', light fields [16, 21]) and unstructured data [7, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Rather than performing image-based rendering, which re- quires storing the input images, another approach is to op- timize some 3D scene representation augmented with ap- pearance information [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Examples of such representa- tions include point clouds [1, 40], voxel grids [27, 32, 47], meshes [42, 43], or layered representations like multi-plane [13, 28, 65] or multi-sphere images [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Neural Radiance Fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' NeRFs are one such 3D scene representation for view synthesis [29] that parameterize the appearance and density of every point in 3D space with a multilayer perceptron (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' While NeRFs enable high- quality view synthesis at a small memory cost, they do not lend themselves to real-time rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' To render the color of a ray from a NeRF, one must evaluate and integrate the color and opacity of many points along a ray—necessitating, in the case of NeRF, many hundreds of MLP evaluations per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Still, due to its impressive performance for static view synthesis, many recent methods build on NeRFs in the quest for higher visual quality, more efficient training, and faster rendering speed [15, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Several works improve the quality of NeRFs by accounting for finite pixels and aper- tures [5, 60], by enabling application to unbounded scenes [6, 63, 64], or by modifying the representation to allow for better reproduction of challenging view-dependent appear- ances like distorted reflections and refractions [8, 17, 19, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 2 One can achieve significant training and inference speed im- provements by replacing the deep multilayer perceptron with a feature voxel grid in combination with a small neural net- work [11, 30, 51] or no network at all [18, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Several other works achieve both fast rendering and memory-efficient stor- age with tensor factorizations [11], learned appearance code- books, or quantized volumetric features [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Adaptive Sampling for Neural Volume Rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Other works aim to improve the speed of volumetric repre- sentations by reducing the number of volume queries re- quired to render a single ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Approaches like DoNeRF [31], TermiNeRF [38], and AdaNeRF [20] learn weights for each segment along a ray as a function of the ray itself, and use these weights for adaptive evaluation of the under- lying NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In doing so, they can achieve near-real-time rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' NeuSample [12] replaces the NeRF coarse net- work with a module that directly predicts the distance to each sample point along a ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Methods like AutoInt [26], DIVeR [59], and neural light fields [4, 25, 48] learn integrated opac- ity and color along a small set of ray segments (or just one segment), requiring only a single network evaluation per segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' A key component of our framework is a flexible sampling network, which is among one of the few schemes that both accelerates volume rendering, and also improves volume rendering quality for challenging scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 6–Degrees-of-Freedom Video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 6-DoF video is an emer- gent technology that allows users to explore new views within videos [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Systems for 6-DoF video [35] use multi- view camera rigs that capture a full 360-degree field of view and use variants of depth-based reprojection [45] for view synthesis at each frame of the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Other methods optimize time-varying multi-sphere images (MSIs) [2, 9], which can provide better visual quality but at a higher training cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 6-DoF from Monocular Captures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Due to the success of neural radiance fields for static view synthesis, many recent approaches attempt to extend volumetric scene representa- tions to dynamic scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Several such works reconstruct 6-DoF video from single-view (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' monocular) RGB se- quences [24, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' This is a highly under-constrained setting, which requires decoupling camera and object motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The natural signal priors provided by neural radiance fields help during reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' However, most methods typically rely on additional priors, such as off-the-shelf networks for pre- dicting scene flow and geometry or depth from ToF cameras [3, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Still, other approaches model the scene at different time steps as smoothly “warped” copies of some canoni- cal frame [33, 39], which works best for small temporal windows and smooth object motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 6-DoF from Multi-View Captures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Other methods, like ours, aim to produce 6-DoF video from multi-view camera rigs [9, 23, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Despite the additional constraints provided by multiple cameras, this remains a challenging task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' an ideal 6-DoF video format must simultaneously achieve high visual quality, rendering speed, and memory efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Directly extending recent volumetric methods to dynamic scenes can achieve high quality and rendering speed [56], but at the cost of substantial memory requirements, potentially giga- bytes of memory [63] for each video frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Contemporary works such as StreamRF [22] and NeRFPlayer [49] design volumetric 6-DoF video representations that mitigate storage requirements but sacrifice either rendering speed or visual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' On the other hand, our approach achieves both fast and high-quality 6-DoF video rendering while maintaining a small memory footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Method We start by considering the problem of optimizing a volu- metric representation for static view synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Volume repre- sentations like NeRF [29] model the density and appearance of a static scene at every point in the 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' More specif- ically, a function Fθ : (x, ⃗ω) → (Le(x, ⃗ω), σ(x)) maps position x and direction ⃗ω along a ray to a color Le(x, ⃗ω) and density σ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Here, the trainable parameters θ may be neural network weights, N-dimensional array entries, or a combination of both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We can then render new views of a static scene with C(o, ⃗ω) = � tf tn T(o, xt) � �� � Transmittance σ(xt) � �� � Density Le(xt, ⃗ω) � �� � Radiance dt, (1) where T (o, xt) denotes the transmittance from o to xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In practice, we can evaluate Equation 1 using numerical quadrature by taking many sample points along a given ray: C(o, ⃗ω) ≈ N � k=1 wk Le(xk, ⃗ω) , (2) where the weights wk = ˆT (o, xk) (1−e−σ(xk)∆xk) specify the contribution of each sample point’s color to the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Sample Networks for Volume Rendering Most scenes consist of solid objects whose surfaces lie on a 2D manifold within the 3D scene volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In this case, only a small set of sample points contributes to the rendered color for each ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' To accelerate volume rendering, we would like to query color and opacity only for points with non-zero wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' While most volume representations use importance sampling and pruning schemes that help reduce sample counts, they often require hundreds or even thousands of queries per ray to produce accurate renderings [11, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' As shown in Figure 2, we use a feed-forward network to predict a set of sample locations xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Specifically, we use a sample prediction network Eφ : (o, ⃗ω) → (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , xn) that maps a ray (o, ⃗ω) to the sample points xk for volume 3 𝝎 𝝎×𝒐 𝒐 (𝒐, 𝝎) 𝒓 = 𝝎, 𝝎×𝒐 Compute sample points 𝐱!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' by (i) intersecting ray 𝒓 with primitive 𝐺" (ii) adding displacement vector 𝐝𝐤 𝐱% = 𝑖𝑛𝑡𝑒𝑟 𝐺&;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=" 𝒐, 𝝎 + 𝐝% Input ray Plücker coordinates Plücker coordinate illustration 𝐶 𝒐, 𝝎 = 4 & 𝑤& 𝐿'(𝐱%, 𝝎) Compute volume rendering integral with numerical quadrature Rendered image {𝐝(, 𝐝), ⋯ , 𝐝𝐧} {𝐺(, 𝐺), ⋯ , 𝐺+} Displacement vectors Geometric primitives (e." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=', planes, spheres) Sample Prediction Network 𝐸$ 𝑟 𝒓 (a) Ray parameterization (b) Sample prediction network (c) Sample generation (d) Volume rendering Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Overview of HyperReel for static scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Given a set of images and camera poses, the training objective is to reconstruct the measured color associated with every ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' (a) Given an input ray originating at the camera origin o and traveling in direction ⃗ω, we first reparameterize the ray using Pl¨ucker coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' (b) A network Eφ takes this ray as input and outputs the parameters for a set of geometric primitives {Gk} (such as axis-aligned planes and spheres) and displacement vectors {dk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' (c) To generate sample points {xk} for volume rendering, we compute the intersections between the ray and the geometric primitives, and add the displacement vectors to the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Predicting geometric primitives has the advantage of making the sample signal smooth and easy to interpolate (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The displacement vectors grant additional flexibility to the sample points, enabling better capture of complex view-dependent appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' (d) Finally, we perform volume rendering via Equation 2 to produce a pixel color and supervise training based on the corresponding observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Extracting sample point appearance in the dynamic setting from our keyframe-based representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' (1) We first advect the sample points {xk} at time τ into the nearest keyframe τi, using velocities {vk} outputted from the sample prediction net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' (2) We then query the outer products of space-time textures in order to produce per-sample-point appearance features, which are then converted to colors via Equation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We follow a similar procedure for extracting per-sample-point opacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' rendering in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In this work, we use the Pl¨ucker parameterization to represent the ray: r = Pl¨ucker(o, ⃗ω) = (⃗ω, ⃗ω × o) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' (3) While many designs for the sample prediction network Eφ are possible, giving the network too much flexibility may negatively affect view synthesis quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' For example, if (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , xn) are completely arbitrary points, then renderings may not appear to be multi-view-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' To address this problem, we choose to predict the pa- rameters of a set of geometric primitives G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , Gn with the sample prediction network, where the primitive parame- ters can vary depending on the input ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' To get our sample points, we then intersect the ray with each primitive: Eφ(o, ⃗ω) = (G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , Gn) , (4) (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , xn) = (inter(G1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' o, ⃗ω), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , inter(Gn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' o, ⃗ω)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' (5) Above, inter(Gk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' o, ⃗ω) is a differentiable operation that in- tersects the ray with the primitive Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In all of our exper- iments, we use axis-aligned z-planes (for forward-facing scenes) or concentric spherical shells centered at the origin (for all other scenes) as our geometric primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' This approach is constrained in that it produces sample points that initially lie along the ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Further, predicting primitives defined in a global coordinate frame makes the sample signal smooth and easy to interpolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' For example, if two distinct rays observe the same point in the scene, then the sample network needs only predict one primitive for both rays (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=', defining a primitive that passes through the point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In contrast, existing works such as NeuSample [12], AdaNeRF [20], and TermiNeRF [38] predict distances or per-segment weights that vary depending on the ray even if these rays observe the same point in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Flexible Sampling for Challenging Appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' To grant our samples additional flexibility to better represent challenging view-dependent appearance, we also predict a set of Tanh-activated per-sample-point offsets (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , en), as well as a set of scalar values (δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , δn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We convert these scalar values to weights with a sigmoid activation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=', (γ(δ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , γ(δn)) where γ is the sigmoid operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Specifi- 4 d3 (o,w) X3 G1 G2(o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 3)1 Zcally, we have: (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' dn) = (γ(δ1)e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , γ(δn)en) (6) (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' xn) ← (x1 + d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , xn + dn) , (7) where we use (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , dn) to denote the final displacement, or “point-offset” added to each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The sample network outputs may appear to be over- parameterized and under-constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' However, the above design is essential for achieving good-quality view synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In particular, initializing the scalars (δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , δn) to negative values, where the sigmoid is close to 0, and its gradient is small, implicitly discourages the network from unmasking the point offsets, while still allowing the network to use them as necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In addition to enabling real-time rendering with low sam- ple counts, one added benefit of our sample network architec- ture is the improved modeling of complex view-dependent appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' For example, distorted refractions break epipolar geometry and appear to change the depth of the refracted con- tent depending on the viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' As illustrated in Figure 2, our sample network, on the other hand, has the flexibility to model sample points that warp depending on viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Existing works like Eikonal fields [8] can be considered a special case of this sample warping approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' they use phys- ically derived Eikonal constraints to learn ray-conditional warp fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Unlike these works, our approach can handle both reflections and refractions, and does not require evalu- ating costly multi-step ODE solvers during rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' See Figure 1 and our website for additional results and compar- isons on challenging view-dependent scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Keyframe-Based Dynamic Volumes Thus far, we have covered how to efficiently sample a 3D scene volume, but have not yet discussed how we represent the volume itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In the static case, we make use of the mem- ory efficient Tensorial Radiance Fields (TensoRF) approach (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1), and in the dynamic case we extend Ten- soRF to a keyframe-based dynamic volume representation (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 Representing 3D Volumes with TensoRF [11] Recall that TensoRF factorizes a 3D volume as a set of outer products between functions of one or more spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Specifically, we can write the set of spherical harmonic coefficients A (xk) capturing the appearance of a point xk = (xk, yk, zk) as: A (xk) = B1(f1(xk, yk) ⊙ g1(zk)) + B2(f2(xk, zk) ⊙ g2(yk)) (8) + B3(f3(yk, zk) ⊙ g3(xk)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Above, fj and gj are vector-valued functions with output di- mension M, and the operator ‘⊙’ computes an element-wise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In the original TensoRF work [11], the functions fj and gj are discretized into M different 2D and and 1D arrays, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Further, Bj denote linear transforms that map the products of fj and gj to spherical harmonic coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The color Le(xk, ⃗ω) for point xk and direction ⃗ω is then given by the dot product of the coefficients A (xk) and the spherical harmonic basis functions evaluated at ray direction ⃗ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Similar to appearance, for density, we have: σ(xk) = 1⊤ (h1(xk, yk) ⊙ k1(zk)) + 1⊤ (h2(xk, zk) ⊙ k2(yk)) (9) + 1⊤ (h3(yk, zk) ⊙ k3(xk)) , where 1 is a vector of ones, and hj and kj are vector- valued functions with output dimension M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Given the color Le(xk, ⃗ω) and density σ(xk) for all sample points {xk} along a ray, we can then make use of Equation 2 to render the final color for that ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2 Representing Keyframe-Based Volumes To handle dynamics, we adapt TensoRF to parameterize volumetric “keyframes”, or snapshots of a dynamic volume at a set of discrete time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' If we denote τi as the time step corresponding to the ith keyframe, we can write: A (xk, τi) = B1(f1(xk, yk) ⊙ g1(zk, τi)) + B2(f2(xk, zk) ⊙ g2(yk, τi)) (10) + B3(f3(yk, zk) ⊙ g3(xk, τi)) , and σ(xk, τi) = 1⊤ (h1(xk, yk) ⊙ k1(zk, τi)) + 1⊤ (h2(xk, zk) ⊙ k2(yk, τi)) (11) + 1⊤ (h3(yk, zk) ⊙ k3(xk, τi)) , where the only change from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 is that gj and kj now depend on time, in addition to one spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We note that the above factorization of the dynamic vol- ume representing all keyframes in a video has a similar mem- ory footprint to a static TensoRF for a single frame, assuming that the number of keyframes is small relative to the reso- lution of our spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In particular, if the spatial resolution of our volume is (Nx, Ny, Nz) and the number of keyframes is Nt, then we can store a single component of f1 with an Nx × Ny array, and store a single component of g1 with an Nz × Nt array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Because Nt ≪ Nx/y/z, the arrays gj do not contribute significantly to the size of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='3 Rendering from Keyframe-Based Volumes In order to combine our sampling procedure (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1) and keyframe-based volume representation (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2) to complete our system for 6-DoF video, a few additional mod- ifications are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' First, since the surfaces in a dynamic scene move over time, the sample points {xk} should be time dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We therefore augment our sample prediction network to take the current time τ as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Second, the decomposition of the dynamic scene in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2 creates temporal “snapshots” of the volume at discrete keyframes τi, but we would like to sample the vol- ume at arbitrary times τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In order to generate the dynamic volume at all intermediate times, we also output velocities vk ∈ R3 from the sample prediction network, which we use to advect the sample points into the nearest keyframe τi with a single forward-Euler step: xk ← xk + vk(τi − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' (12) Another perspective on Equation 12 is that it defines a back- wards warp with scene flow field vk that generates the vol- ume at time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The process of warping sample points and querying the keyframe-based dynamic volume is illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' After querying the keyframe-based volume with {xk}, the equation for volume rendering is then: C(o, ⃗ω, τ) = N � k=1 wk Le (xk, ⃗ω, τi) , (13) where wk = ˆT(o, xk, τi) (1 − e−σ(xk,τi)∆xk), and τi is the time step corresponding to the closest keyframe to time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' This is effectively the same as Equation 2, except C, xk, wk and Le now depend on the time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The sampling procedure (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1), volume representation (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2), and ren- dering scheme for keyframe-based volumes (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='3) comprise our 6-DoF video representation: HyperReel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Optimization We optimize our representation using only the training im- ages, and apply total variation and ℓ1 sparsity regularization to our tensor components, similar to TensoRF [11]: L = LL2 + wL1LL1 + wTVLTV where (14) LL2 = � o,⃗ω,τ ∥C(o, ⃗ω, τ) − CGT(o, ⃗ω, τ)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' (15) The loss is summed over training rays and times, and CGT represents the ground-truth color for a given ray and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We only use a subset of all training rays to make the optimization tractable on machines with limited memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In all dynamic experiments, for frame numbers divisible by 4, we alternate between using all training rays and using training rays from images downsampled by a 4× factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' For all other instances, we downsample images by an 8× factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Experiments Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We implement our method in PyTorch [36] and run experiments on a single NVIDIA RTX 3090 GPU with 24 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Our sample network is a 6- layer, 256-hidden unit MLP with Leaky ReLU activations for both static and dynamic settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Unless otherwise speci- fied, for forward-facing scenes, we predict 32 z-planes as our geometric primitives with our ray-conditioned sample predic- tion network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In all other settings, we predict the radii of 32 spherical shells centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We use the same space contraction scheme for unbounded scenes as in mip-NeRF 360 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' For our keyframe-based volume representation, we give the (x, y) and (z, t) textures eight components each and four components to all other textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' For all dynamic datasets, we use every 4th frame as a keyframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' For both static and dynamic datasets, we use a batch size of 16,384 rays for training, an initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='02 for the param- eters of the keyframe-based volume, and an initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0075 for our sample prediction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We train all models for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='5 hours each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' For Technicolor, Google Im- mersive, and all static scenes, we set the wTV weight in Equation 14 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='05 for both appearance and density, which is decayed by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 every 30,000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' On the other hand, wL1 starts at 8·10−5 and decays to 4·10−5 over 30,000 iterations and is only applied to the density components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Qualitative Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We show a few qualitative re- sults and comparisons in Figure 4, and a compre- hensive set of qualitative results on our website: hyperreel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Comparisons on Static Scenes DoNeRF Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The DoNeRF dataset [31] contains six synthetic sequences with images of 800×800 pixel resolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Here, we validate the efficacy of our sample prediction network approach by comparing it to existing methods for static view synthesis, including NeRF, InstantNGP, and three sampling-network–based approaches [20, 31, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' As demonstrated in Table 1, our approach outperforms all baselines in terms of quality and improves the performance of other sampling network schemes by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Addi- tionally, our model is implemented in vanilla PyTorch and renders 800×800 pixel images at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='5 FPS on a single RTX 3090 GPU (or 29 FPS with our Tiny model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We also compare our sampling network-based approach to the single-sample R2L light field representation [55] on the downsampled 400×400 resolution DoNeRF dataset (with their provided metrics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We outperform their approach quantitatively without using pretrained teacher networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Further, inference with our six-layer, 256-hidden-unit net- work, and TensoRF volume backbone is faster than R2L’s deep 88-layer, 256-hidden-unit MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 6 Ground truth (Technicolor [44]) Ours Neural 3D Video [23] Ground truth (Neural 3D Video [23]) Ours NeRFPlayer [49] Ground truth (Google Immersive LF Video [9]) Ours NeRFPlayer [49] Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Qualitative comparisons of dynamic reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We show visual comparisons of our method on three datasets against two baselines on heldout views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We pick non-keyframe time-steps for evaluation, except for the Google Immersive light field video (last row), for which we pick the matching image to the NeRFPlayer [49] result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' LLFF Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The LLFF dataset [29] contains eight real- world sequences with 1008×756 pixel images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In Table 1, we compare our method to the same approaches as above on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Our approach outperforms DoNeRF, AdaNeRF, TermiNeRF, and InstantNGP but achieves slightly worse quality than NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' This dataset is challenging for explicit volume representations (which have more parameters and thus can more easily overfit to the training images) due to a combination of erroneous camera calibration and input-view sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' For completeness, we also include a comparison to R2L on the downsampled 504×378 LLFF dataset, where we perform slightly worse in terms of quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Comparisons on Dynamic Scenes Technicolor Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The Technicolor light field dataset [44] contains videos of varied indoor environments captured by a time-synchronized 4×4 camera rig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Each image in each video stream is 2048×1088 pixels, and we hold out the view in the second row and second column for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We compare HyperReel to Neural 3D Video [23] at full image resolution on five sequences (Birthday, Fabien, Painter, The- ater, Trains) from this dataset, each 50 frames long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We train Neural 3D Video on each sequence for approximately one week on a machine with 8 NVIDIA V100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We show in Table 2 that the quality of HyperReel exceeds that of Neural 3D Video [23] while also training in just 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='5 GPU hours per sequence (rather than 1000+ GPU hours for Neural 3D), and rendering far more quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Neural 3D Video Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The Neural 3D Video dataset [23] contains six indoor multi-view video sequences cap- tured by 20 cameras at 2704×2028 pixel resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We downsample all sequences by a factor of 2 for training and evaluation and hold out the central view for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Met- rics are averaged over all scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Additionally, due to the challenging nature of this dataset (time synchronization er- rors, inconsistent white balance, imperfect poses), we output 64 z-planes per ray with our sample network rather than 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We show in Table 2 that we outperform all baseline ap- proaches on this dataset, including contemporary works such as NeRFPlayer [49] and StreamRF [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In particular, we outperform NeRFPlayer quantitatively while rendering ap- proximately 40 times faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We outperform StreamRF by a more significant margin in terms of quality, although their approach with a Plenoxels backbone (which uses custom CUDA kernels for faster inference) renders faster than our 7 ++Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Static comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We compare our sampling network architecture to others on the synthetic DoNeRF dataset [31] and real LLFF dataset [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' FPS is normalized per megapixel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' memory in MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Dataset Method PSNR↑ FPS↑ Memory ↓ DoNeRF 400×400 Single sample R2L [55] 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='5 — 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='7 Ours (per-frame) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='8 DoNeRF 800×800 Uniform sampling NeRF [29] 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='8 Instant NGP [30] 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0 Adaptive sampling DoNeRF [31] 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 AdaNeRF [20] 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 TermiNeRF [38] 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 Ours (per-frame) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='8 LLFF 504×378 Single sample R2L [55] 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='7 — 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='7 Ours (per-frame) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='8 LLFF 1008×756 Uniform sampling NeRF [29] 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='8 Instant NGP [30] 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='3 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0 Adaptive sampling DoNeRF [31] 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 AdaNeRF [20] 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 TermiNeRF [38] 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 Ours (per-frame) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='8 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Dynamic comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We compare HyperReel to exist- ing 3D video methods on three light-field/multi-view video datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Note that all FPS numbers are reported for megapixel images, and memory is in MB per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' †NeRFPlayer [49] and StreamRF [22] do not provide SSIM and LPIPS scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' ‡The accompanying paper does not provide quantitative metrics, while the concurrent NeRF- Player does, so we provide a comparison with NeRFPlayer only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Table 3 provides a proxy comparison of our method to Google’s, and our website includes a qualitative comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Dataset Method PSNR↑ SSIM↑ LPIPS↓ FPS↑ Memory↓ Technicolor [44] Neural 3D Video [23] 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='6 Ours 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='109 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2 Neural 3D Video [23] Neural 3D Video [23] 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='961 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 NeRFPlayer [49] 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='7 —† —† 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='06 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 StreamRF [22] 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='3 —† —† 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='90 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='7 Ours 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='927 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='096 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2 Google LF videos [9]‡ NeRFPlayer [49] 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='871 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='295 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='12 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 Ours 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='874 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='193 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Our model consumes less memory on average per frame than both StreamRF and NeRFPlayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Google Immersive Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The Google Immersive dataset [9] contains light field videos of various indoor and outdoor environments captured by a time-synchronized 46- fisheye camera rig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Here, we compare our approach to NeRF- Player and select the same seven scenes as NeRFPlayer for evaluation on this dataset (Welder, Flames, Truck, Exhibit, Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Quantitative comparisons to DeepView.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In addition to the comparison to NeRFPlayer in Table 2, we report a comparison with DeepView [13], a variant of which is used per-frame in im- mersive LF video [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We thus compare to DeepView as a proxy for quantitative comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' FPS normalized per megapixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Dataset Method PSNR↑ SSIM↑ LPIPS↓ FPS↑ Spaces [13] DeepView [13] 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='085 >100 Ours 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='968 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='080 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Ablations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We perform several ablations on our method, including on the number of keyframes, the use of the sampling network, and model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' All FPS normalized per megapixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Dataset Method PSNR↑ SSIM↑ LPIPS↓ FPS↑ Technicolor Ours (keyframe every 1 frame(s)) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='117 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0 Ours (keyframe every 4 frame(s)) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='109 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0 Ours (keyframe every 16 frame(s)) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='893 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='112 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0 Ours (keyframe every 50 frame(s)) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='110 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0 Ours (w/o sample network) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='815 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='209 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='3 Ours (Tiny) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='835 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='157 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='5 Ours (Small) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='125 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 Face Paint 1, Face Paint 2, Cave), holding out the central view for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' For our approach, we split each video into several 50-frame chunks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Our results in Table 2 outper- form NeRFPlayer’s by a 1 dB margin in terms of quality, while again rendering more quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' DeepView Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Unfortunately, Google’s Immersive Light Field Video [9] does not provide quantitative bench- marks for the performance of their approach in terms of image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' As a proxy, we compare our approach to DeepView [13], the method upon which their representation is built, on the static Spaces dataset in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Our method achieves superior quality, outperforming DeepView by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Further, HyperReel consumes less memory per frame than the Immersive Light Field Video’s baked layered mesh representation: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2 MB per frame vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='87 MB per frame (calculated from the reported bitrate numbers [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Their layered mesh can render at more than 100 FPS on commodity hardware, while our approach renders at a little over 4 FPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' However, our approach is en- tirely implemented in vanilla PyTorch and can be further optimized using custom CUDA kernels or baked into a real- time renderable representation for better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Ablation Studies Number of Keyframes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In Table 4, we ablate our method on the Technicolor light field dataset with different numbers of keyframes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In general, the optimal number of keyframes depends on the motion within a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Our dynamic volume representation implicitly trades off between temporal resolu- tion and spatial resolution, as it can use the (x, t), (y, t), and (z, t) components to add either spatial or temporal details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Our choice of one keyframe for every four frames strikes a good balance between temporal resolution and spatial reso- 8 GT Full model Small Tiny No sampling Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Ablations on our sampling network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We show close-up results for various sampling networks architectures on two of the Technicolor sequences also shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Point offset ablation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We evaluate the performance of our network with and without point offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Scene Point offset PSNR↑ SSIM↑ LPIPS↓ DoNeRF “Forest” [31] Without 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0146 (diffuse) With 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0122 Shiny “Lab” [58] Without 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='943 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0416 (highly refractive) With 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='959 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0294 lution and achieves the best overall performance (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Sample Network Size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We also show the performance of our method with different sample prediction network de- signs in Table 4, including the performance for a Tiny model (4-layers, 128-hidden-unit MLP with 8 predicted sample points), and Small model (4-layers, 256-hidden-unit MLP with 16 predicted sample points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Our Tiny model runs at 18 FPS, and our Small model runs at 9 FPS at megapixel resolution, again without any custom CUDA code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Our Tiny model performs reasonably well but achieves worse quality than Neural 3D Video on the Technicolor dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In contrast, our Small model achieves comparable overall performance to Neural3D—showing that we can still achieve good quality renderings at even higher frame rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We show accompany- ing qualitative results for these models in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' With and Without Sample Prediction Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We show results on the Technicolor dataset without our sam- ple prediction network, using every frame as a keyframe, and with 4× the number of samples (128 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Our full method outperforms this approach by a sizeable margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' With and Without Point Offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In Table 5, we show re- sults on two static scenes with and without point offsets (Equation 7): one diffuse and one highly refractive scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Point offsets improve quality in both cases, suggesting that they may help with better model capacity allocation in ad- dition to view-dependence—similar to “canonical frame” deformations used in Nerfies [34] and Neural Volumes [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Conclusion HyperReel is a novel representation for 6-DoF video, which combines a ray-conditioned sampling network with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 1 10 29 30 31 Ours StreamRF N3DV NeRFPlayer Rendering speed [frames per second] PSNR [dB] Dynamic Reconstruction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1 1 10 30 32 34 Ours NeRF Instant NGP DoNeRF TermiNeRF AdaNeRF Rendering speed [frames per second] PSNR [dB] Static Reconstruction Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Rendering speed vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' quality trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We show the speed-quality trade-off of our method and others applied to dynamic scenes (left) on the Neural 3D Video dataset [23] and static scenes methods (right) on the DoNeRF dataset [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Our result GT Our result GT Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Our approach can sometimes produce blurry reconstructions due to the training ray subsampling scheme (Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='3) (left) or noisy reconstructions in sparsely observed regions due to an under-constrained sampling network (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' keyframe-based dynamic volume representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' It achieves a balance between high rendering quality, speed, and mem- ory efficiency that sets it apart from existing 6-DoF video representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We qualitatively and quantitatively compare our approach to prior and contemporary 6-DoF video rep- resentations, showing that HyperReel outperforms each of these works along multiple axes, as illustrated in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Limitations and Future Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Our sample prediction network is supervised only by a rendering loss on the training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' This can lead to a reduction in quality for views outside of the convex hull of the training cameras or for scene content that is only observed in a small number of views (see Figure 7), where the sample network may predict erroneous sample points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Exploring regularization methods that enable reasonable geometry predictions even for extrapolated views is an important future direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Although our keyframe-based representation is more memory efficient than most existing 3D video formats, it can- not be streamed like NeRFPlayer [49] or StreamRF [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In practice, however, our representation is sufficiently small that this does not pose a major issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Additionally, our sample network approach is compatible with any streaming-based dynamic volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Currently, our approach falls short of the rendering speed required for settings like VR (ideally 72 FPS, in stereo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' As our method is implemented in vanilla PyTorch, we expect to gain significant speedups with more engineering effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 9 +EAMTEAM+Acknowledgments We thank Thomas Neff, Yu-Lun Liu, and Xiaoming Zhao for valuable feedback and discussions, Zhaoyang Lv for help running the Neural 3D Video Synthesis codebase [23], and Liangchen Song for providing information about the scenes from the Google Immersive Video dataset [9] used in NeRFPlayer [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Matthew O’Toole acknowledges support from NSF IIS-2008464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' References [1] Kara-Ali Aliev, Artem Sevastopolsky, Maria Kolos, Dmitry Ulyanov, and Victor Lempitsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Neural point-based graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In ECCV, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 2 [2] Benjamin Attal, Selena Ling, Aaron Gokaslan, Christian Richardt, and James Tompkin.' metadata={'source': 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+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=', 37(4): 65:1–12, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Appendix Overview In addition to this appendix document, we include a supple- mental website which contains a video of a prototype demo of our method running in real-time at high-resolution with- out any custom CUDA code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We also emphasize that we plan to release code and data to make our method and results as reproducible as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Within the appendix, we provide: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Additional details regarding training and evaluation for static and dynamic datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' A more complete description of the mapping from sam- ple prediction network outputs to sample points for both forward facing, and non-forward facing scenes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Additional experimental details regarding our keyframe- based volume design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Further, we provide a full per-scene breakdown of image metrics for the Technicolor dataset in Table D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' As we do not have direct access to the outputs of other methods on the Google Immersive ([9]) and Neural 3D Video ([62]) datasets, we do not provide per-scene breakdowns for these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Finally, in addition to a video of our real-time demo, our website contains: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Dynamic dataset results from our method on each of Technicolor ([44]), Neural 3D Video ([23]), and Google Immersive Video ([9]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Qualitative results and comparisons on view-dependent static scenes from the Shiny Dataset ([58]) and the Stanford Light Field Dataset ([57]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Qualitative comparison to [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Additional Training & Evaluation Details B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Training Ray-Subsampling We provide pseudo-code for our training ray-subsampling scheme in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' LPIPS Evaluation Details For LPIPS computation, we use the AlexNet LPIPS variant for all of our comparisons in the main paper (as do all of the baseline methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' SSIM Evaluation Details For SSIM computation, we use the structural similarity scikit-image library function, with our images normalized to the range of [0, 1], and the data range parameter set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We note, however, that several methods either: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Use their own implementation of SSIM, which are not consistent with this standard implementation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' R2L [56]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Fail to set the data range parameter appropriately, so that it defaults to the value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Neural 3D Video [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In both of these cases, the SSIM function returns higher- than-intended values (which we confirm by testing these variants on our predicted images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' While we believe that this inconsistency makes SSIM scores somewhat less reliable, we still report our aggregated SSIM metrics in the quantitative result tables in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Sample Prediction Network Description C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Forward Facing Scenes For forward facing scenes, we first convert all rays to nor- malized device coordinates (NDC) [29], so that the view frustum of a “reference” camera lives within [−1, 1]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' After mapping a ray with origin o and direction ⃗ω to its Pl¨ucker parameterization via r = Pl¨ucker(o, ⃗ω) = (⃗ω, ⃗ω × o) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' (16) we predict the parameters of a set of planes normal to the z-axis with our embedding network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' we 12 ALGORITHM 1: Training Ray-Subsampling Scheme Input: Number of videos {N},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Number of frames {M} Output: Training Rays raysGT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Ground Truth Colors CGT // Initialize rays and colors raysGT = {} CGT = {} // Iterate over all N videos for n ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' N} do // Iterate over all M frames in video n for m ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' M} do // Get frame m from video n Cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='m = GetFrame(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' m) // Get corresponding rays for this frame raysn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='m = GetRays(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' m) if m is not divisible by 8 then // Downsample rays and colors by a factor of 4 Cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='m ← NearestNeighborDownsample(Cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 4) raysn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='m ← NearestNeighborDownsample(raysn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 4) if m is not divisible by 4 then // Downsample rays and colors by an additional factor of 2 Cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='m ← NearestNeighborDownsample(Cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 2) raysn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='m ← NearestNeighborDownsample(raysn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 2) end end // Add current rays and colors to output CGT ← CGT + Cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='m raysGT ← raysGT + raysn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='m end end predict (z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , zn), and intersect the ray with the axis- aligned planes at these distances to produce our sample points (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Additionally, we initialize the values (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , zn) in a stratified manner, so that they uniformly span the range of [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Outward Facing Scenes For all other (outward facing) scenes, we predict the radii of a set of spheres centered at the origin (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , rn), and intersect the ray with each sphere to produce our sample points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We initialize (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' , rn) so that they range from the minimum distance to the maximum distance in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Differentiable Intersection In both of the above cases, we make use of the implicit form of each primitive (for planes normal to the z-axis, z = zk, and for the spheres centered at the origin x2 +y2 +z2 = r2 k) and the parameteric equation for a ray o + tk⃗ω, to solve for the intersection distances tk (as is done in typical ray-tracers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' The intersection distance is differentiable with respect to the primitive parameters, so that gradients can propagate from the color loss to the sample network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Implicit Color Correction In order to better handle multi-view datasets with inconsis- tent color correction / white balancing, we also output a color scale cscale k and shift cshift k from the sample prediction network for each sample point xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' These are used to modulate the color Le(xk, ⃗ω, τi) extracted from the dynamic volume via: Le(xk, ⃗ω, τi) ← Le(xk, ⃗ω, τi) · cscale k + cshift k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' (17) Note that these outputs vary with low-frequency with respect to the input ray (since we use few positional encoding fre- quencies for the sample prediction network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Additionally, the density from the volume remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Keyframe-Based Volume Details We initialize our keyframe-based dynamic volume within a 1283 grid, so that each of the spatial tensor components have resolution 128 × 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Our final grid size is 6403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' We upsample the volume at iterations 4000, 6000, 8000, 10000, and 12000, interpolating the resolution linearly in log space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' 13 Table D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Per-scene results from the Technicolor dataset [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' See Section Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='3 for a brief disucssion of the reliability of SSIM metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content=' Scene PSNR↑ SSIM↑ LPIPS↓ Neural 3D Video [23] Ours Small Tiny Neural 3D Video [23] Ours Small Tiny Neural 3D Video [23] Ours Small Tiny Birthday 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='20 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='99 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='32 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E0T4oBgHgl3EQfTgDH/content/2301.02238v1.pdf'} +page_content='952 0.' metadata={'source': 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Department of Physics, University of Manchester, M13 9PL, UK) +First Draft Date: 2021-Dec-22 + +Abstract： + The reaction of volatile matter plays an important role in the process of bringing matter from the surface of the planet to the atmosphere. +Therefore, by simulating the mixing and chemical reaction process of volatile matter in the atmosphere during volatilization and diffusion +from the planet surface, the concentration distribution of different components in the atmosphere can be studied, which is the problem to be +solved in this paper. This paper discusses the diffusion and reaction of simple components in one-dimensional scale from the diffusion +process of volatile matter and the reaction process in the atmosphere. The diffusion and reaction models of volatile matter were established, +and the basis of the model was given. + + +Key words：Concentration Distribution; Volatiles Reaction; Atmosphere; Astrochemistry + + + + +The volatilization of substances occurs on any planet with +suitable surface conditions [1]. During the volatilization of +substances on the surface of the planet, when their components +contact each other, these substances will undergo chemical +reactions [2], which can produce some substances that can +stably exist in the atmosphere [3]. + +These reactions can affect the concentrations of different +components in the atmosphere, and then cause changes in the +abundance of elements in the atmosphere over a long period of +time [4][5]. + +Therefore, in order to understand this process, it is necessary to +establish a model to analyze the chemical reaction of volatile +matter when it diffuses in the atmosphere and its impact on the +atmosphere. + +In this paper, a one-dimensional concentration distribution +model of simple components in the atmosphere will be +established from the diffusion of volatile matter and the +reaction of volatile matters. +1 Methods +1.1 Vapor and Condensation +The vapor and condensation of atmospheric compositions is +directly influenced by temperature and pressure. This section is +to build a relation between atmospheric composition and +surface elements abundance by phase change [6]. In a +thermodynamic equilibrium system, there is phase rule as +𝐹 = 𝐶 − 𝑃 + 2 +(1) +where 𝐹 is the degrees of freedom, 𝐶 at here is the number +of components and 𝑃 is the number of phases. The phases +composition can be calculated by their phase diagram. A two- +component system which contains 𝐴 and 𝐻2𝑂 was +considered as an example in this proposal. The amounts of two +condensates are fully determined by conservation of mass, the +conservation equations for them are: + +𝑋(𝐴)𝑡𝑜𝑡𝑎𝑙 = 𝑋(𝐴)𝐻2𝑂𝑖𝑐𝑒𝑀𝐻2𝑂𝑖𝑐𝑒 + 𝑋(𝐴)𝑠𝑜𝑙′𝑛𝑀𝑠𝑜𝑙′𝑛 +(2) + + +2 +𝑋(𝐻2𝑂)𝑡𝑜𝑡𝑎𝑙 = 𝑋(𝐻2𝑂)𝐻2𝑂𝑖𝑐𝑒𝑀𝐻2𝑂𝑖𝑐𝑒 + 𝑋(𝐻2𝑂)𝑠𝑜𝑙′𝑛𝑀𝑠𝑜𝑙′𝑛(3) + +The amounts of the two phases obey an above simple lever rule. +The relations between multi-phases are more complicated but +can still be built by modeling base on the lever rule. Then, the +mass of liquid and gas of one composition 𝐴 is determined +under a given temperature and pressure, the liquid can be +considered as condensed on surface, which remains a part of +abundance of elements those in 𝐴 molecule. +1.2 Chemical Reaction and Relation +The reactions take part in atmosphere always follow with +photochemical reactions, as they were exposed to much +stronger radiation than surface. To build the model from +collision, which is the main method the reactions happen at +high altitude, the rate of thermal diffusion in gas is need to be +defined to calculate the amount of substance at a height [5]. +𝐽⃗ = −𝐷∇⃗⃗⃗𝑛 +(4) +𝐽 (mol m-2 s-1) depends on the speeds of molecule and the +gradient in the concentration of the diffusing species. 𝐷 is the +diffusion coefficient. At the turbopause altitude [7], diffusion +of molecules is fast as they can be seen as move freely, the +above formulation can be written as diffusion flux relate to +gradient. +𝐽𝑍 = 𝐾𝐷𝑛 𝜕𝑥𝑖 +𝜕𝑧 +(5) +𝐾𝐷 is a function of height, 𝐷 depends on temperature and +pressure. Since the diffusion process was built, the +photochemical model is as following. A three-body collision +reaction of components A, B and C can be considered as an +example +A + B + C → AB + C +There is following formulation to express reaction rate 𝜂, 𝑛𝑖 +is the number density of species and 𝑘 is the rate constant of +the reaction: +𝜂 = 𝑑(𝑛𝐴𝐵) +𝑑𝑡 += 𝑘𝑛𝐴𝑛𝐵𝑛𝐶 +(6. 𝑎) +Obviously the rate of reaction is effected by 𝑛𝑖 , which is +related to diffusion 𝐽𝑍. The faster the diffusion is, the lower the +density is, the slower the reaction is. If cA is the concentration +of A at a height, the diffusion differential equation contains +chemical reaction can be written as: +𝑐𝐴∇𝑢 + 𝑑𝑐𝐴 +𝑑𝑡 = 𝐷𝐴∇2𝑐𝐴 + 𝑅𝐴 +(6. 𝑏) +where u is the mass average speed of fluid, RA= 𝜂𝐴 is the molar +average formation rate of A. + +One common method during atmospheric chemistry researches +is to build a chemical reaction net. Now consider an ideal net +contains 𝑛 reactants, 𝐴, 𝐵, 𝐶 ⋯. There will be 𝐶𝑛 +2 resultants +in the first level for two-body reactions. And the resultants of +the second level reactions are: +𝐶𝑛(𝑛+1) +2 +2 += 𝑛4 + 2𝑛3−𝑛2 − 2𝑛 +8 +(7) +Every two-composition reaction has an equilibrium constant 𝐾 +which is only relative to temperature. In the following +formulation, 𝐵 takes part in both two reactions, the relation +between two reactions can be built by the partial pressure of 𝐵, +𝑝(𝐵). +aA + bB → AB , b'B + cC → BC +𝐾𝐴𝐵 = +𝑝(𝐴𝐵) +𝑝𝑎(𝐴)𝑝𝑏(𝐵) , 𝐾𝐵𝐶 = +𝑝(𝐵𝐶) +𝑝𝑏′(𝐵)𝑝𝑐(𝐶) +(8) +As the 𝜂 evaluates the degree of reactions and 𝐾 shows the +relations. Then, the amount of a component 𝐴 at a moment 𝑡 +can be calculated with the reaction rate as it is on the way +building an equilibrium. The amount can be expressed by either +partial pressure or mole, as original mole plus value changed +during ∆𝑡. +𝑛𝐴 = 𝑛0 + ∑ ∫ +𝜂 +𝑡 +𝑡−∆𝑡 +𝑑𝑡 +𝑘 +(9) +𝑛𝐴 is the current mole of 𝐴 in atmosphere, 𝑛0 is the original +mole. 𝑘 is the number of reactions that 𝐴 takes part in. 𝜂 is +the rate of reaction. In each molecule A or B or C contains +element 𝑀 and N, which can be denoted as 𝑀𝑚𝑁 , the +abundance of 𝑀 at this height is 𝐴𝑎𝑡𝑚 +𝑀 +: +∑ 𝑛𝐴 × 𝑚 +𝑛𝑡𝑜𝑡𝑎𝑙 += 𝐴𝑎𝑡𝑚 +𝑀 +(10) +1.3 Volatiles Flux +There are mass exchanges between surface and atmosphere by +geological activities like volcanism and weathering both +positively [8]. The factors of volcanism were talked in previous +work [9]. With considering the light components from a general +source, vapor and condensation are mainly considered in this +article. As the diffusion coefficient 𝐽 can be seen as a function +of temperature 𝑇. The temperature is also a function of height +𝑧, then 𝐽 is a function of z. Consider plate PA and PB whose +area are both 1m2, the distance between two plates is Δ𝑧 (m). +The flux of amount of mass (mol) through the two plates during +Δ𝑡 is: +Φ𝑃𝐴 +𝑛 = ∫ +𝐽(𝑧)𝑑𝑡 , Φ𝑃𝐵 +𝑛 = ∫ +𝐽(𝑧 + Δ𝑧) +Δ𝑡 +0 +Δ𝑡 +0 +𝑑𝑧 +(11) +There is following relation between Δ𝑧 and Δ𝑡. 𝑣𝑑𝑖𝑓𝑓 is the +velocity of diffusion, which is a function of temperature. +𝑑𝑧 +𝑑𝑡 = 𝑣𝑑𝑖𝑓𝑓(𝑇) +(12) +Because the temperature is a function of height, 𝑣𝑑𝑖𝑓𝑓 can also +be seen as a function of height 𝑧 . 𝑝0 is the original partial + + +3 +pressure of component 𝐴 at 𝑧 . Then the average partial +pressure of component 𝐴 in the space formed by plates PA and +PB is the following formulation, by limit Δ𝑧 as 0, it can be seen +as a function of height z finally. +𝑝0 + +Φ𝑃𝐵 +𝑛 − Φ𝑃𝐴 +𝑛 +Δ𝑧 += 𝑝𝑀(𝑧) +(13) + +As 𝑝𝑀(0) is already known, then the partial pressure at +different heights can be calculated by the above formulation. +2 Results +2.1 Conditions Assumption +To simplify the calculation process, 3 light elements, N, H, O +were considered to verify the model. Because their components +with each other is not as much as C, which is too difficult to the +current research situation. They are also not as less as +components of Ar, which is not able to test the model. The air +composition and temperature with pressure at surface was set +as Table.1. + +Table.1 Air composition at surface, 300K, 101MPa + +Ratio +Molar mass g/mol +N2 +0.8 +28 +O2 +0.2 +32 + +The temperature of atmosphere was set as a simple piecewise +linear function of height, the pressure was set as a linear +function of height and then equal to zero. + +T= -6.490×10-3 Z + 300K (Z<=10000m) +T= 1.188×10-3 (Z-10000) + 216K (10000m 0, and for +antiferromagnets J < 0. +We use the standard Monte Carlo (MC) method to thermalize the lattice alone at T. Next, we introduce the +electrons into the lattice. We suppost that each electron spin interacts with the surrounding lattice spins inside a +sphere centered at its position, of radius D1. The electron spin also interacts with other electron spins within a sphere +of radius D2. Electrons move under the applied electric field. +In order to obtain the spin current we have to thermalize the state of the spins of conduction electrons in the lattice. +This is done by the following steps: +i) we take a conduction spin and calculate its actual energy Eold using the different interactions mentioned above, +ii) we make a trial move ⃗ℓ for the electron in a random direction between 0 and a where a is the lattice constant. +If the move ⃗ℓ takes the itinerant electron outside the sample, then we put it inside on the other sample end by virtue +of the periodic boundary condition, +iii) we then calculate the new energy Enew. If Enew < Eold, then the trial position is accepted. Otherwise, it is +accepted with the probability exp[−(Enew − Eold)/(kBT)]. +iv) we take another conduction electron and repeat the three steps above. We continue with other electrons until +all electrons are considered: this accomplishes one MC step/spin. A large number of MC steps/spin is neccessary +to arrive at a steady current state. +We next average physical quantities of interest at the temperature T under +consideration. +iv) we take another T and repeat the above four steps. We should cover the temperature region of interest. +This paper is a review of our publications over the past 15 years. In each publication, we used various sample sizes to +detect finite size effects. We increased the sample size until the results do not depend anymore on the size. The results +are considered as valid thermodynamically. The finite size effect and the finite size scaling are what the simulators +do before reaching conclusions. This problem is particularly important when one wants to study the criticality or +the order of a phase transion. In the review, we do not repeat these details which depend on the studied system. +Rather, we emphasize on the results. The reader interested on the details of the simulation methods is referred to +each original publication. +We calculate the spin resistivity ρ as : +ρ = +1 +Ns +(10) +where Ns is the number of mobile electrons passing thru a unit surface perpendicular to the direction of the applied +electric field x, per MC time unit. An application with a real material using real units is presented in subsect. III C. +For a good thermal average, we have to perform very long MC runs and we proceed as follows: for each configuration +of the lattice spins we average the spin resistivity over N1 MC steps, then we thermalize again the lattice with N2 MC +steps to get rid of the correlation between lattice spin configurations, before averaging again the resistivity for N1 MC +steps. We repeat this N1 + N2 cycle for a large number of times N3. The total MC steps of resistivity averaging is +about 4 × 105 steps per spin in our runs. We found by comparison that this ”multi-step” averaging method strongly +suppresses statistical fluctuations seen in our earlier work [20]. +It is obvious that the larger N1 and N2 yields the better statistics. We know that depending on T, the relaxation +time τL of the lattice spins varies. We have to compare τL with the relaxation time τI of conduction electron spins in +order to choose a right value of N1 in order to calculate the average of the resistivity with one lattice spin configuration +at T. We know the two limiting cases. The first case is when τL ≃ τI. In this case, we have to take N1 = 1, namely + +5 +the lattice spin configuration should change with each move of itinerant spins. The second case is when τL ≫ τI. In +this case itinerant spins can be scattered many times along its trajectory across the same lattice configuration and +for many times across the lattice. +In order to choose a right value of N1, we consider the following temperature dependence of τL in non frustrated +spin systems. The relaxation time is expressed in this case as [30–32] +τL = +A +|1 − T/TC|zν +(11) +where A is a constant, ν the correlation critical exponent, and z the dynamic exponent which depends on the spin +model and space dimension. For 3D Ising model, ν = 0.638 and z = 2.02. From this expression, we see that as T tends +to TC, τL diverges. In the critical region around TC the system encounters thus the so-called ”critical slowing down”: +the spin relaxation is extremely long due to the divergence of the spin-spin correlation. When we take into account +the temperature dependence of τL, the shape of the resistivity is strongly modified near TC where τL is very long, +in contrast to the paramagnetic phase where the relaxation time is very short due to rapid thermal fluctuations. We +should emphasize that at low T, ρ does not depend on τL since in that temperature range where the ordering of the +lattice spins is almost perfect: the spin landscape from one microscopic state to another does not change significantly, +so the motion of the itinerant electron spin does not significantly vary (see Ref. [24]). +Finally, we note that we have also used the Boltzmann’s equation combined with MC data to study the spin +resistivity [21]. However, the shape of the resistivity peak at the transition temperature does not agree well with +experiments, unlike that obtained from direct MC simulations as shown below. This proves the efficiency of MC +simulations for the calculation of the spin resistivity in magnetically ordered materials. The present review therefore +aims at promoting this method. +III. +REVIEW +A. +Spin Resistivity in Ferromagnets and Antiferromagnets +Experiments mentioned above amongst numerous other data show that the spin resistivity in ferromagnets has a +sharp peak at the transition temperature TC of the lattice. We know by the theory of phase transitions and critical +phenomena that in the region around TC, the so-called ”critical slowing-down” phenomenon happens, resulting in +extremely large τL. The peak in ρ is due to this phenomenon via Eq. (11) where τL diverges at TC. Our MC +simulations using the method described above in the case of a ferromagnet where the lattice spins are of the Ising +type show a sharp peak at TC (see Fig. 1) in agreement with experimental data. We note that the spin resistivity +for ferromagnets (as well as for antiferromagnets) increases with decreasing T at low T. This can be explained by +several causes: the freezing or crystallization of conduction electrons takes place at low T so that just a small number +of conduction spins with decreasing T is mobile, or it may be the classical counter effect of the quantum Kondo +electron-impurity scattering if one considers the few excited lattice spins at low T are independent impurities, see the +last term of Eq. (2). Note that the shape of ρ depends on the lattice type, interaction strengths encountered by the +conduction electrons, electron concentration, relaxation time, spin model, applied magnetic-field amplitude etc. In +Ref. [23], we have shown that a decrease in the interaction between conduction spins, K0, reduces the increase of ρ +as T → 0, an applied magnetic field reduces the height of the resistivity peak and the larger electron density reduces +ρ. Finally, we emphasize that ρ depends on the material intrinsically via the critical exponents ν and z, see Eq. (11). +If we wish to compare simulated spin resistivity to experimental measurements performed on a given material, +we have to use in the simulation the available experimental physical parameters of that material. An example of +quantitative comparison for semiconductor MnTe is shown in subsection III C. +Note that the magnetic field applied on the system reduces the peak height as shown in our work Ref. [23], in +agreement with experiments [2]. +Unlike ferromagnets, antiferromagnets have not been much studied. Haas [29] has shown that in contrat to ferromag- +nets where the resistivity ρ has a sharp peak at the order-disorder transition of the lattice spins, in antiferromagnets +there is no such a peak. Using MC simulations, we found that the peak does exist in an antiferromagnet but it is less +sharp compared to that of a ferromagnet, as seen in Fig. 1. We think that the alternate change of sign of the spin-spin +correlation with distance may have something to do with the absence of a sharp peak. We have tested this idea on +the effect of the cut-off distance D1 [26]: in an antiferromagnet, when we increase D1, we will include successively +up-spin shells and down-spin shells in the sphere of radius D1. Consequently, the difference between the numbers of +up and down spins in the sphere oscillates with varying D1, giving rise to the oscillatory behavior of ρ observed at +small D1, unlike in ferromagnets. + +6 +ρ +T +FIG. 1: +Resistivity ρ as a function of T, obtained by simulations using the T-dependent relaxation time, Eq. +11, for +ferromagnet (black circles) and antiferromagnet (white circles) of BCC structure, with arbitrary unit in zero magnetic field. +Other parameters: ϵ = 1, I0 = 2, K0 = 0.5, A = 1. +At this stage, we note that the presence of an itinerant spin will break the invariance between a ferromagnet and +its antiferromagnet counterpart in the local Mattis transformation (Jij → −Jij, ⃗Sj → −⃗Sj). +B. +Frustrated J1 − J2 Model on a Simple Cubic Lattice +Let us consider the simple cubic lattice with NN and NNN interactions as shown in Fig. 2. The Hamiltonian is +written as +H = −J1 +� +(i,j) +⃗Si · ⃗Sj − J2 +� +(i,m) +⃗Si · ⃗Sm +(12) +where the first sum � +(i,j) is made over the NN Ising spin pairs ⃗Si and ⃗Sj with interaction J1, and the second sum +� +(i,m) is performed over the NNN pairs with interaction J2. +We focus our attention on the region of parameters which gives rise to a frustration. For that purpose, we assume +that J1 is an antiferromagnetic interaction, namely J1 = −J < 0 (J > 0), and J2 is also antiferromagnetic. We put +J2 = −ηJ where η is a positive parameter. The ground state (GS) of this system can be obtained by minimizing the +energy, or by comparing the energies of different spin configurations. We can also numerically minimize the energy +by using the steepest descent method [33]. We obtain the GS antiferromagnetic configuration shown Fig. 3a for +|J2| < 0.25|J1|, and the GS spin configuration shown in Fig. 3b for |J2| > 0.25|J1|. This latter configuration is 3-fold +degenerate because we can choose the parallel NN spins either on x, or y or z axis. In addition, with the permutation +of black and white spins, we have the total degeneracy equal to 6. +FIG. 2: Simple cubic lattice where the NN and NNN interactions, J1 and J2, are indicated. +We note in passing that in the case of the Heisenberg model in the frustrated region (|J2| > 0.25|J1|) the phase +transition has been shown to be of first order [34]. The system is very unstable due to its large degeneracy. In the +case of the Ising spin on the SC lattice treated here, we found that the first-order character of the phase transition is +even stronger [27]. +We use J1 = −J = −1 (AF interaction) for the coupling between NN lattice spins in the simulations. The energy +is thus measured in the unit of J and the temperature is in the unit of J/kB. All distances (D1 and D2) are in the +unit of the lattice constant a. + +350 +F +AF +300 +250 +R +200 +0 +00000000000000000 +0 +150 +100 +2 +6 +8 +10 +12 +4 +T/J7 +(a) +(b) +FIG. 3: Ground state (GS) of the simple cubic lattice with Ising spins: (a) GS when |J2| < 0.25|J1|; (b) GS when |J2| > 0.25|J1|. +White (black) circles denote up (down) spins. See text for comments. +Simulations have been carried out by using the temperature-dependent relaxation time of the lattice spins given +by Eq. (11) where we have taken A = 1 and τL = 1 at T = 2TC far from TC. Such a choice leads to τL = 1 at that +temperature expected for fast thermal fluctuations in the paramagnetic phase far above TC. +Since we suppose that the interaction between conduction electron spins is attractive, a chemical potential is +required to avoid the collapse of the system, namely to avoid that all conduction spins form a cluster [cf. Eq.(7)]. The +chemical potential in thermodynamics makes the particle uniformly distributed in the space. Its strength is expressed +by D which has to be chosen in accordance with K0. Figure 4 displays the phase diagram in the space (K0, D) which +shows the collapse region. This allows us to avoid this region and choose an appropriate value of D for a given K0. +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +K +D +0 +FIG. 4: D versus K0 in the case where η = 0.26. The collapse region is in green. We have used D1 = D2 = 1, I0 = 0.5, ϵ = 1. +To see the effect of the nature of phase transition on the spin resistivity, in the following we focus on two typical +cases: η = 0.2 and η = 0.26 which belong respectively to the regions of second- and first-order transition. +A. For η = 0.2: +The spin resistivity at temperatures below TN oscillates with varying D1. By analyzing the ratio of the number of +up spins to the number of down spins in the sphere of radius D1, we found that it oscillates with varying D1: the +maxima (minima) of ρ correspond to the case of largest (smallest) numbers of parallel (antiparallel) spins in the sphere +[26, 27]. At very high temperatures where the lattice spins are disordered, the numbers of up spins and down spins +in the sphere of radius D1 should be equal. There is however a very small oscillation if the temperature is close to +TN and if D1 is small. +Figure 5 displays the resistivity versus T for D1 = 1.2. The spin resistivity shows a rounded maximum at the +transition temperature. This is in agreement with the curve experimentally observed in La0.4Ca0.6MnO3 by Lu et al. +[2] (see Fig. 6, right panel). + +8 +116 +117 +118 +119 +120 +121 +122 +123 +124 +1.4 +1.5 +1.6 +1.7 +1.8 +1.9 +2 +2.1 +2.2 +2.3 +T +ρ +FIG. 5: Spin resistivity versus T in the case η = 0.2, D1 = 1.2. We have used the lattice size Nx = Ny = 20, Nz = 6. Other +variables are I0 = K0 = 0.5, D2 = 1, D = 1, ϵ = 1. +FIG. 6: Experimental spin resistivity versus T are shown for several applied magnetic fields in the compound La0.4Ca0.6MnO3. +These data are in the figure 2 of Ref. [2]. +For D1 = 0.8 or D1 = 1, the resistivity is smaller below the transition temperature, as seen in Fig. 7. This shows +the importance of the effect of the interaction range on the spin resistivity in materials. +B. For η = 0.26: +At this value of η, the transition of the lattice spins is of first order. The dependence of the resistivity on D1 is very +similar to that of the second-phase transition, namely the resistivity at a given T oscillates as D1 varies. The physical +meaning of the oscillation has been given above. More details can be found in Refs. [26, 27]. We found that the +resistivity ρ in the frustrated regime can go downward or upward at the transition temperature depending on D1 [27], +unlike in non-frustrated ferromagnets and antiferromagnnets shown earlier. This is displayed in Fig. 8 for two values +of D1 where one observes the discontinuity of ρ at the transition temperature. The discontinuity of ρ has also been +found in other frustrated antiferromagnets such as the FCC antiferromagnet [26]. +From the results shown above for the J1 − J2 model, we come to the conclusion that the behavior of the spin + +(Ω2 cm)0 +100 +200 +300 +T(K)10 +10° +H=0 T +H=9 T +(a) y=0 +10 +120 +10 +H=0 T +MI +60 +10 +H=1 T +0 +3 +6 +H=3 T +H (T) +H=6 T +10 +(b) y=0.022 +H=0 T +1 +H=1 T +(c) y=0.05 +0 +(d) y=0.07 +1.0 +H=0 T +0.5 +120 +H=1 T +60 +0.00 +1 0.05 +T +0.0 +0 +100 +200 +3000 +100 +200 +300 +T(K)(Ω2 cm)9 +100 +105 +110 +115 +120 +125 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +2.4 +2.6 +ρ +T +FIG. 7: ρ as a function of T for η = 0.2 with D1 = 0.8 and 1 ((black circles and blue open circles, respectively). For simulations, +we used Nx = Ny = 20, Nz = 6, I0 = K0 = 0.5, D2 = 1, D = 1, and ϵ = 1. +100 +110 +120 +130 +140 +150 +160 +170 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +1.7 +1.8 +T +ρ +FIG. 8: +ρ as a function of T for η = 0.26 where D1 = 0.8 (black circles) and D1 = 1 (blue open circles). We have used +Nx = Ny = 20, Nz = 6, I0 = K0 = 0.5, D2 = 1, D = 1 and ϵ = 1. See text for comments. +resistivity is a consequence of the nature of the lattice transition. If the lattice transition is of second order, then +the resistivity of itinerant spins has a rounded peak, while if the lattice transition is of first order, the resistivity is +discontinuous at the transition temperature. +C. +The case of MnTe +The pure semiconductor MnTe has two kinds of structures: the zinc-blend structure or the hexagonal NiAs one +shown in Fig. 9 [35]. We focus on the second structure where the N´eel temperature is TN = 310 K [36], and where +many other experimental data are available. +MnTe is a semiconductor with a large gap (1.27 eV) and a carrier +concentration n = 4.3 × 1017cm−3 at room temperature [37, 38]. Without doping, MnTe is non degenerate. The +crystal is formed by ferromagnetic xy hexagonal planes antiferromagnetically stacked in the c direction. The NN +distance in the c direction is c/2 ≃ 3.36 ˚A, and the in-plane NN distance is a = 4.158 ˚A. From neutron scattering +experiments, it was found that the main exchange interactions between Mn spins in MnTe are the interaction between +NN along the c axis with the value J1/kB = −21.5 ± 0.3 K, the ferromagnetic exchange J2/kB ≈ 0.67 ± 0.05 K + +10 +FIG. 9: Structure of MnTe of NiAs type: black and white circles present respectively opposite spins. Interactions between NN, +between next NN and between third NN are indicated respectively by J1, J2, and J3. See values given in the text. +between in-plane neighboring Mn (they are next NN by distance), and the third NN antiferromagnetic interaction +J3/kB ≃ −2.87 ± 0.04 K (see Fig. 9). The spins are lying in the xy planes perpendicular to the c direction with a +small in-plane easy-axis anisotropy Da [36]. Let us emphasize that the values of the exchange integrals given above +were deduced from experimental data by fitting with a free spin-wave theory [36]. Other fittings with mean-field +theories give slightly different values: J1/kB = −16.7 K, J2/kB = 2.55 K and J3/kB = −0.28 K [37]. Note that the +Mn spin is experimentally known to be of the Heisenberg model with magnitude S = 5/2 [36]. +We write the following Hamiltonian for the lattice spins +H = −J1 +� +(i,j) +⃗Si · ⃗Sj − J2 +� +(i,m) +⃗Si · ⃗Sm − J3 +� +(i,k) +⃗Si · ⃗Sk +−Da +� +i +(Sx +i )2 +(13) +where the first sum is performed over the NN spin pairs, the second sum over the NNN pairs and the third one over +the third NN pairs. Da > 0 is an anisotropy constant which favors the in-plane x easy-axis spin configuration. +The behavior of ρ in MnTe as a function of T has been experimentally shown in several works [39–43]. We have +studied using MC simulations the spin resistivity in MnTe with the above Hamiltonian [25]. Let us summarize this +work here. +For MC simulations, we suppose the following Hamiltonian of the itinerant spins: +Hi = − +� +n +I(⃗r − ⃗Rn)⃗σ · ⃗Sn +(14) +where the sum is performed by counting all the lattice spins ⃗Sn inside the sphere of radius D1 = a centered at ⃗r. +I(⃗r − ⃗Rn) > 0 is the ferromagnetic distance-dependent interaction between the itinerant electron spin ⃗σ at ⃗r and the +Mn spin ⃗Sn at ⃗Rn . +The electron spin is supposed of the Ising type. We neglect therefore the quantum effects which may be important +at very low T but our attention is focused on the region of high-enough T where quantum effects may be neglected. +We assume the following form of I(⃗r − ⃗Rn) : +I(⃗r − ⃗Rn) = I0 exp[−α(⃗r − ⃗Rn)] +(15) +where the constants I0 and α are chosen in such a way that the interaction Hi yields an energy much smaller than +the lattice energy given by H (see the guide for the choice of different constants given below Eq. (6) and in Ref. [23]). +It is noted that the cut-off distance D1 is rather short so that only the first few neighbors are inside the sphere, the +results shown below do not therefore depend significantly on the choice of the value of α in the exponential. Finally, + +Te +Mn +C +J +ZA +3 +y +a +x11 +note that the concentration of conduction electrons in MnTe is n = 4.3 × 1017cm−3 which is five orders lower than +the concentration of its surrounding lattice spins which is ≃ 1022cm−3. This observation justifies that the interaction +between conduction electrons for MnTe can be neglected. We have assumed this in the simulations shown in the +following. +As mentioned above, the exchange interactions deduced from experimental data have slightly different values, they +depend on the theoretical Hamiltonian and the approximations used to deduce it (often the mean-field approximation +is used, see a detailed example in Ref. [44]). Note that in semiconductors, the carrier concentration varies with T +but since this concentration is very low, we do not take into account its variation. Consequently, the number of +conduction electron spins used in the simulation is important only for the statistical average. The current obtained is +proportional to the number of itinerant spins but there are no extra effects within our assumptions mentioned above. +We have calculated ρ of MnTe, using the exchange integrals slightly modified with respect to the ones given above +in order to obtain the best fit. The obtained resistivity ρ is shown in Fig. 10. Let us note that we have taken J3 +slightly larger in magnitude than the value deduced from experiments by mean-field approximation. Our value of +J3 was chosen in order to obtain TN = 310 K which is in excellent agreement with experiments. However the most +striking feature is that the simulated ρ shows a sharp maximum at TN and coincides with the experimental data +over the whole temperature range. Note that we have used A = 1 and the well-known Heisenberg critical exponents +ν = 0.707, z = 1.97 [31] for the lattice spins. It is remarkable that with the same set of param:eters, we obtain an +excellent agreement with experiments in the temperature regions below T < 140 K and above TN. We note that +we have tried earlier to use the Boltzmann’s equation [22] but the obtained result is not as good as the MC result +presented above. +From the simulated ρ, we can calculate the relaxation time of conduction spins, we obtain τI ≃ 0.1 ps. The mean +free path can be also estimated, it is equal to ¯l ≃ 20 ˚A, at the critical temperature. +FIG. 10: Comparison between the simulated spin resistivity and the experimental data of MnTe: Black circles are results +from the Monte Carlo simulation, white circles are experimental data taken from He et al.[43]. We used for the simulation +J1 = −21.5K, J2 = 2.55 K, J3 = −9 K, I0 = 2 K, Da = 0.12 K, D1 = a = 4.148 ˚A, ϵ = 2 × 105 V/m, L = 30a (lattice size: +L3). See text for comments. +IV. +PHASE TRANSITION AND SPIN RESISTIVITY IN THE ISING HCP LATTICE +A. +Hamiltonian and Ground State +The lattice we consider is the HCP structure illustrated in Fig. 11. The xy planes are triangular (hexagonal) and +the stacking direction is z. We suppose the following Hamiltonian + +p (2 .cm) +1.4 +1.2 +1.0 +0.8 +0.6 +0.4 +0 +100 +200 +300 +400 +T (K)12 +J1 +J2 +Z +FIG. 11: HCP lattice: the in-plane NN interaction is denoted by J1 and the inter-plane NN interaction is denoted by J2. +H = − +� +(i,j) +Ji,j ⃗Si · ⃗Sj +(16) +where Jij is the AF interaction between nearest-neighbors (NN) ⃗Si and ⃗Sj. We denote Jij = J1 if the NN are on the +xy triangular plane, and Jij = J2 if the NN are on two adjacent planes (see Fig. 11). The GS can be determined +by minimization the local energy of each spin and doing this for all spins, then repeating many times until the total +energy converges to a minimum. Normally, with a system without bond disordering, this method needs just a small +number of iterations. The GS can be checked by looking at the final snapshot: it should be periodic. This procedure +of local energy minimization is called in the literature ”the steepest-descent method”. The implementation of this +method is very simple [33] (i) we first create an initial random configuration (ii) we then calculate the local field +acting at a spin by its neighbors using (16) (iii) we align the spin under consideration along the calculated local field, +in doing so its energy is minimum (iv) we take another spin and repeat the three preceding steps until all spins are +considered: this step completes one sweep (v) we start again another sweep and we realize a large number of sweeps +until the total energy is minimm. +One can also analytically minimize the interaction energy as shown below to find the GS. Let us assume that both +interactions J1 and J2 are antiferromagnetic. For simplicity we fix J2 = −1 and vary J1. +The case of isotropic interaction, namely J1 = J2 has been studied in Ref. [45]. We summarize the result here: +for the HCP structure, each spin is common for eight tetrahedra (four in the upper half-space and four in the lower +half-space along the z axis) and a NN bond is shared by two tetrahedra. The GS spin configuration of the system is +formed by stacking neighboring tetrahedra. In the GS, one has two pairs of antiparallel spins on each tetrahedron. +Their axes form an arbitrary angle α. The GS degeneracy is therefore infinite (see Fig. 2a of Ref. [45]). Note tthat +the periodic boundary conditions will reduce a number of the GS configurations, but the degeneracy is still infinite. +One particular family of configurations of interest for both XY and Heisenberg cases is when α = 0. The GS is +then collinear with two spins up and the other two down. The stacking sequence is simple because there are three +equivalent configurations due to the fact that there are three ways to choose the parallel spin pair in the original +tetrahedron. +The case where J1 ̸= J2 has been studied in Ref. [46] for the Ising and XY cases. Let us recall some results +concerning the Ising case which allow us to understand the new results on the spin resistivity presented below. +We use the steepest descent method described above with varying J1 (J2 = −1): we find two kinds of GS spin +configuration: the first consists of xy ferromagnetic planes stacked antiferromagnetically along the z direction, while +the second one is the stacking of xy AF planes such that each tetrahedron has two up and two down spins. The +transition between the two configurations is determined as follows: one simply writes down the respective energies of +a tetrahedron and compares them + +13 +E1 = 3(−J1 + J2) +(17) +E2 = J1 + J2 +(18) +One sees that E1 < E2 when J1 > 0.5J2, i.e. +|J1| < 0.5|J2|. +Thus the first configuration is more stable when +|J1| < 0.5|J2|. +B. +Phase Transition in the case of Ising Spins on the HCP Lattice +In the following, we present the results of simulations using the Hamiltonian Eq. (16). We use the sample size +Nx × Ny × Nz with Nx = Ny = 18 and Nz = 8, namely 16 atomic planes along the z axis, and the periodic boundary +conditions in all directions. We use the first 106 MC steps per spin to reach equilibrium and we average physical +quantities with the next 106 MC steps per spin. The energy is expressed in the unit of |J2| = 1. +Let us define η = J1/J2. We have seen that the GS changes at ηc = 0.5, so we show below the properties of the +system on both sides of this value. Figure 12 displays the averaged energy per spin, the order parameter (staggered +magnetization), the specific heat and the susceptibility for η = 0.3. As seen, the transition is of second order since +there is no discontinuity of the energy and the order poarameter at the transition temperature. +-2 +-1.8 +-1.6 +-1.4 +-1.2 +-1 +-0.8 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +3 +T +E +(a) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +3 +T +M +(b) +0 +0.5 +1 +1.5 +2 +2.5 +3 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +3 +T +CV +(c) +0 +10 +20 +30 +40 +50 +60 +70 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +3 +T +χ +(d) +FIG. 12: The case of Ising spin on the antiferromagnetic HCP lattice: (a) energy per spin E, (b) order parameter M, (c) +specific heat CV and (d) susceptibility χ, versus temperature T for η = J1/J2 = 0.30. See text for comments. +For η = J1/J2 > 0.5, Fig. 13 for η = 0.85 and 1 shows that the discontinuity of E and M at the transition is very +large, a signature of a strong first-order transition in both cases. +In order to confirm the order of the phase transition, we measure the energy histogram taken during the averaging +MC time. Figure 14 shows the energy histogram taken at the transition temperature for η = 0.3 (black), 0.85 (blue) +and 1 (red). We observe here that the first case is a Gaussian distribution indicating a second-order transition, in +contrast to the last two cases which show double-peak histograms confirming a first-order transition. +Figure 15 displays the phase diagram in the space (TC, η) where zone (1) and zone (2) denote the ordering of the +first, and second kinds, respectively; (P) indicates the paramagnetic phase. Note that the transition line between (1) +and (P) is a second-order line, while that between (2) and (P) is a first-order line. +Note that in the XY case, the change of the GS takes place at ηc = 1/3. We have studied this case in details in to +Ref. [46]. +Finally, let us emphasize that all 3D frustrated systems we know so far undergo a first-order transition [28] including +the much-studied antiferromagnetic stacked triangular lattice [47–51], the FCC antiferromagnets [52], the simple cubic +fully frustrated lattices [53–56], helimagnets [57], and antiferromagnetic HCP lattice studied here (see more details in +Refs. [45, 46]). + +14 +-2 +-1.9 +-1.8 +-1.7 +-1.6 +-1.5 +-1.4 +-1.3 +-1.2 +-1.1 +-1 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +T +E +(a) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +T +M +(b) +FIG. 13: The case of Ising spin on the antiferromagnetic HCP lattice: (a) energy per spin E; (b) order parameter M versus +temperature T for η = J1/J2 = 0.85 (blue open circles) and 1 (red triangles). See text for comments. +0 +0.002 +0.004 +0.006 +0.008 +0.01 +0.012 +0.014 +-1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 +E +P(E) +FIG. 14: Energy histogram P(E) for η = 0.3 (black circles), 0.85 (blue open circles), and 1(red triangles). +See text for +comments. +C. +Spin resistivity in the HCP lattice with Ising spins +The results in this subsection are new, they are not published so far. Using the method which has been described +in subsection II, we carry out MC simulations to study the spin resistivity in the Ising case. We show in Fig. 16 the +resistivity at two temperatures, below and above the transition temperature, as a function of D1 for the GS belonging +to phase (1). We show in Fig. 17 the case of a GS belonging to phase (2) (see Fig. 15). Similar to the case of J1 − J2 +model on the simple cubic lattice considered in Ref. [26, 27], we find here an oscillation of ρ at low temperature. Note +that ρ is always smaller at low temperature than at high temperature, whatever the value of D1 is. The physical +origin of the oscillation has been discussed above. +The spin resistivity ρ for η = 0.3 and 1 is shown in Fig. 18 as a function of T, here the distances D1 and D2 +are in unit of the distance between the NN lattice spins, and I0, K0 and D which have the energy dimension are in +the unit of |J2| = 1. As in the frustrated J1 − J2 model shown above, one finds here that ρ has a broad peak in +the second-order region, in contrast to the first-order region where it undergoes a discontinuous jump at the phase +transition. Some remarks are in order: +i) At very low temperature, the resistivity increases with decreasing temperature. This behavior can be understood +by the freezing of the itinerant spins due to low T: The energy of itinerant spins is low, they occupy the low-energy +positions in the periodic lattice, it is difficult to move them out by the insufficient thermal energy. They are somewhat +frozen in almost periodic positions; namely a pseudo crystallization occurs. Note that the increase of resistivity with +decreasing T at very low T was observed in many experiments on various materials and is not limited to ferromagnets +[3, 5, 7, 40]. This increase of ρ with decreasing T in the quantum case has been explained by J. Kondo using a +third-order perturbation theory [58]: the scattering of s-electrons by d-electrons of localized magnetic impurities gives +rise to a resistivity minimum at a finite T. We have also found here this minimum of ρ at low T with the classical + +15 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +η +(1) +(2) +(P) +Tc +FIG. 15: Transition temperature TC versus η. +(1) denotes the second-order region, (2) the first-order region and (P) the +paramagnetic phase. +110 +120 +130 +140 +150 +160 +170 +180 +190 +200 +210 +220 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +D +ρ +1 +FIG. 16: ρ versus D1 for η = 0.3 at T = 1.6 < TC (black circles) and at T = 2.8 > TC (open circles) where TC ≃ 2.4. Other +parameters are Nx = Ny = 18, Nz = 8, D2 = 1, I0 = 2, K0 = 0.5, C1 = C2 = 1, A = 1, D = 0.5, ϵ = 1. +spin model. The similarity with the quantum Kondo effect can be explained by the fact that an excited localized +lattice down-spin (in a very small number at low T) can be viewed as an impurity which captures nearby conduction +up-spins. +ii) Outside this low-T region, when T increases, the thermal energy progressively unfreezes the itinerant spins. As a +consequence, ρ decreases and passes through a minimum (see discussion above). However, at higher T, the scattering +with the lattice spins is stronger, ρ increases up to the transition temperature. +iii) At the transition temperature, ρ shows a peak. The physical mechanism leading to the peak can be explained: +in a previous work [21], it was found from our simulations that the peak is due to scattering of the itinerant spins by +antiparallel-spin clusters which are numerous in the transition region. When one gets close to the transition point, +the number of clusters of down spins are the most numerous, giving rise to the peak in ρ. Note that the ”defects” +clusters (i. e. clusters of antiparallel spins) have an energy barrier to resist the passage of itinerant spins. This is also +the origin of the extremely long relaxation time in the critical region. +iv) Well above the transition temperature, in the paramagnetic phase, as temperature increases, clusters of down +and up spins will be broken more and more into independent disordered spins, namely spins with zero energy, itinerant +spins move easily on their trajectory, making a decrease of ρ with increasing T. +Note that we have also varied the radius D1 to see its effect on ρ at the transition in the present frustrated HCP +model. We found the same effect seen in other antiferromagnets we studied previously [26, 27]: at a given temperature, +an oscillation of ρ with varying D1. oscillates slightly with distance. The origin of this oscillation has been analyzed +avove in the J1 − J2 model. + +16 +100 +150 +200 +250 +300 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +D1 +ρ +FIG. 17: ρ versus D1 for η = 1 at T = 1.5 < TC (black circles) and at T = 1.9 > TC (open circles) where TC ≃ 1.7. Other +parameters are Nx = Ny = 18, Nz = 8, D2 = 1, I0 = 2, K0 = 0.5, C1 = C2 = 1, A = 1, D = 0.5, ϵ = 1. +120 +140 +160 +180 +200 +220 +240 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +T +ρ +(a) +100 +150 +200 +250 +300 +350 +400 +0.5 +1 +1.5 +2 +2.5 +3 +T +ρ +(b) +FIG. 18: Spin resistivity ρ of the Ising HCP model versus temperature T for (a) η = 0.3; (b) η = 1. Nx = Ny = 18, Nz = 8 +(namely 16 planes in the z direction), D1 = D2 = 1, ϵ = 1, I0 = 2, K0 = 0.5, C1 = C2 = 1, A = 1, D = 0.5. All distances are +in unit of the NN distance, energy constants are in unit of |J2| = 1. See text for comments. +Finally, let us look at some experimental data obtained for ferromagnets and antiferromagnets. Figure 19 shows +experiments by Du et al. performed on ε-(Mn1−xFex)3.25Ge antiferromagnets [3], experiments by McGuire et al. +performed on antiferromagnetic superconductors LaFeAsO [6], by Chandra et al. on thin Cd1−xMnxTe films [39]. +Experiments by Santos et al. on antiferromagnetic La1−xSrxMnO3 [7] are shown in Fig. 20. We see here that our +results on the shape of the spin resistivity are in agreement with these experiments. In the lack of physical data on +these experimental materials, we cannot make a quantitative comparison as we did in the MnTe case presented above. +V. +CONCLUSION +In this paper, we have reviewed some important works published on the spin resistivity in magnetically ordered +systems. We have focused on our works published over the past 15 years using mainly Monte Carlo simulations. These +works were motivated by the absence of Monte Carlo works, even at the present time except ours, in spite of the fact +that this method of simulation has proven to be very efficient when comparing its results with experimental data. +In the case of MnTe where there are sufficient experimental data, we have made a quantitative comparison between +experimental and simulated spin resistivities. The agreement between experiments and simulations is excellent. This +review therefore aims at promoting this method to study more realistic cases. +As demonstrations, we have used this method to study the spin resistivity in generic ferromagnets and antiferro- +magnets. The cases of frustrated systems have also been presented: the J1 − J2 model and the antiferromagnetic +HCP lattice. +Let us summarize the results on the two frustrated systems. +The J1 − J2 model is a simple cubic lattice with Ising spins interacting with each other via NN and NNN antiferro- +magnetic interactions, J1 and J2 respectively. The GS of this model is determined by the ratio η = J2/J1. We have +shown that the GS changes at the critical value ηc = 0.25. For the non-frustrated region in the phase space, namely + +17 +(a) +(b) +(c) +FIG. 19: Experiments on the resistivity as a function of T iperformed by (a) by Du et al on ε-(Mn1−xFex)3.25Ge antiferromagnets +[3], (b) by McGuire et al. on antiferromagnets LaFeAsO [6], and (c) by Chandra et al. on thin films of Cd1−xMnxTe [39]. +η < 0.25, the GS is simply composed periodically of two interpenetrated tetrahedra formed by the NNN sites. In the +frustrated region, namely η > 0.25, the GS can be described as composed of one line of spin up, one line of spin down, +alternately, in one crystal direction. The degeneracy is three because there is a freedom to choose one direction among +three. The total degeneracy is 6 if we count the statesw of reverse spins. The transition in the frustrated region is +theoretically of first order since the present 6-fold GS is equivalent to the q-state Potts model with q = 6. We know +that in three dimensions, the transition of the Potts model is of first order from q = 3. We have found this directly +from the simulation. In the non-frustrated region, namely η < 0.25, the transition is found to be of second order. We +have performed MC simulations to obtain ρ of the conduction spins. We found that ρ displays a broad maximum at +the second-order phase transition while it undergoes a discontinuous change at the first-order transition. +The Ising model on the antiferromagnbetic HCP lattice has been also studied in this review. We assumeed an +in-plane interaction J1 and an inter-plane interaction J2, both antiferromagnetic. We found that the GS changes at +the critical value ηc = 0.5. Below (above) which the spins in the xy planes are ferromagnetic (antiferromagnetic). +The nature of the transition changes in these two regions: it is of second order below ηc and of first order above ηc. +The spin resistivity has been simulated in both regions of η. In the second-order region, it shows a broad maximum +while in the first-order region, the resistivity ρ makes a discontinuous jump at the transition. This feature is what we +also found in other frustrated spin systems. +These findings reviewed in this paper show a close relationship between the nature of the phase transition and the +shape of the spin resistivity in real materials. We hope that this review convinces the magnetic community on the + +T +420 +T +(μ2cm) +T. +N +390 +450 +360 +(μ2cm) +X=0.20 +330 +0 +100 +200 +300 +425 +T (K) +T. +N +X=0.17 +400 +0 +100 +200 +300 +400 +T (K)(a) +4.0 +1 +3.5- +165K +(ws ) d +3.0 +6 +(%) +2.5 +4 +142K +8 T +2 +2.0. +0 +O T +100 +120 +140 +160 +180 +200 +T (K) +1.5 +0 +50 +100 +150 +200 +250 +300 +T (K)3.5 +Cd1-xMnxTe +G0000 X=0.25 +nits) +3000 X -0.60 +00000 X= 1.00 +un +3.0 +(arb. +2.5 +Q +normalized +1.5 +100 +200 +300 +(k) +T18 +FIG. 20: Resistivity versus temperature on antiferromagnetic La1−xSrxMnO3. The figures presented are taken from Fig. 7 of +[7]. +use of MC simulations for transport phenomena. +[1] Xia, J.; Siemons, W.; Koster, G.; Beasley, M. R.; Kapitulnik, A. 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Magnetic transitions in helimagnets, Phys. Rev. B 1989, 39, 397. +[58] See ”Kondo Effect - 40 Years after the Discovery” - Special Issue of the Journal of the Physical Society of Japan 2015, +74, Issue 1, with reviews from world top scientists: Jun Kondo, Philippe Nozi`eres, A. C. Hewson, Yukihiro Shimizu and +Osamu Sakai, Ian Affleck amongst others. + diff --git a/OdE0T4oBgHgl3EQf0wIq/content/tmp_files/load_file.txt b/OdE0T4oBgHgl3EQf0wIq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1596b2072902f41642af91037ac20267c97695ca --- /dev/null +++ b/OdE0T4oBgHgl3EQf0wIq/content/tmp_files/load_file.txt @@ -0,0 +1,1396 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf,len=1395 +page_content='Spin Transport in Magnetically Ordered Systems: Ferromagnets, Antiferromagnets and Frustrated Systems Danh-Tai Hoang 1 and Hung T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Diep 2∗ 1 Biological Data Science Institute, College of Science, The Australian National University, Canberra, Australia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' danhtai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='hoang@anu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='au 2 Laboratoire de Physique Th´eorique et Mod´elisation, CY Cergy Paris Universit´e, CNRS, UMR 8089 2, Avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' diep@cyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='fr (Dated: January 10, 2023) In this review, we outline the important results on the resistivity encountered by an electron in magnetically ordered materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The mechanism of the collision between the electron and the lattice spins is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Experiments on the spin resistivity in various magnetic materials as well as theoretical background are recalled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We focus on our works since 15 years using principally Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In these works, we have studied the spin resistivity in various kinds of magnetic systems ranging from ferromagnets and antiferromagnets to frustrated spin systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' It is found that the spin resistivity shows a broad peak at the transition temperature in systems with a second- order phase transition, while it undergoes a discontinuous jump at the transition temperature of a first-order transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' New results on the hexagonal-close-packed (HCP) antiferromagnet are also shown in extended details for the Ising case in both the frustrated and non-frustrated parameter regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' INTRODUCTION The resistivity encountered by the displacement of an electron driven by an applied electric field in a material is due to its collisions with the material constituents such as atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In general these collisions are caused not by direct contacts but by various potentials, magnetic and electric fields from different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' As a matter of fact, the motion of the electron is slowed down except in the superconducting regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' One can mention many simple examples such as the motion of an electron in a magnet, or in a lattice with vibrating atoms (phonons) under an applied electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The investigation of the resistivity is one of the most important tasks in condensed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Apart from the desire to understand the mechanisms lying behind the resistivity, the numerous applications using the transport prop- erties of electrons in electronic devices have motivated an increasing number of studies, experimentally, theoretically and numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The study of the resistivity has started after the discovery of the electron more than a century ago by the simple free-electron Drude theory taking into account the relaxation time τ between two successive collisions due to atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The following relation has been established σ = ne2τ m (1) where σ is the conductivity, e the electron charge, τ the electron relaxation time, m the electron mass, and n the number of electrons crossing a unit srface per unit of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Over the years, there has been a large number of more realistic theories of resistivity which take into account different interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Nevertheless, this relation is still valid if the electron mass m is replaced by its effective mass m∗ which includes effects of the interactions of the electron with its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The relaxation time τ should also be modified, it is no more a constant but it depends on the collision mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In a crystal it is known that the effective mass of the electron can be ”heavier” or ”lighter” than its mass at rest m0 because it contains the effects of various interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This strongly modifies the mobility of the electron in crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' As for τ, it has a strong effect on the temperature dependence of the resistivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' It has been established that τ depends on a power of the electron energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This power depends on the collision process such as collisions with charged impurities, neutral impurities, magnetic impurities, phonons, magnons, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Thus, in a crystal the total resistivity ρt(T) is a sum of the contributions coming from various collision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' At low temperature (T), it is given by ∗ corresponding author arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='02689v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='stat-mech] 6 Jan 2023 2 ρt(T) = ρ0 + A1T 2 + A2T 5 + A3 ln µ T (2) where A1, A2 and A3 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The first term is T-independent, the second term, proportional to T 2, stems from the scattering of the conducting electron at low T by lattice spin-waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Note that the resistivity caused by a Fermi liquid is also proportional to T 2 with another coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The T 5 term which is observed in metals comes from the diffusion of conduction electrons by atomic vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' However, the resistivity in metals show a linear-T dependence at high T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The last term expresses the contribution from the quantum Kondo effect, namely the scattering of conduction electrons by magnetic impurities at extremely low T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In this review, we focus our attention on the resistivity ρ due to the spin of the conduction electron in magnetically ordered materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' For short let us call it the ”spin resistivity” hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This spin resistivity has been widely studied both experimentally and theoretically for more than five decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The rapid development of the field is due mainly to many applications in spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We are interested in magnetic materials which show a phase transition from a magnetically ordered phase, such as ferromagnets and antiferromagnets, to the paramagnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Let us mention in the following some experiments which have been performed in magnetic materials including metals, semiconductors and superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' These experiments carried out on various materials show different shapes of the spin resistivity around the phase-transition temperature: SrRuO3 thin films [1], Ru-doped La0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4Ca0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6MnO3 [2], antiferromagnetic ϵ-(Mn1−xFex)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='25Ge [3], semiconducting Pr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='7Ca0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3MnO3 thin films [4], superconducting BaFe2As2 single crystals [5], LaFeAsO [6] and La1−xSrxMnO3 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We see in these works that depending on the nature of the compound, ρ can have a pronounced peak [8] or a change of its slope, or a curvature change at the transition temperature TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Note that in the last case, one has a maximum of the differential resistivity dρ/dT [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Theoretically, the T 2 magnetic contribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (2) has been obtained by Kasuya [11] taking into account the scattering of the electron spin by the spin waves at low T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' However, at higher T, specially in the region of the phase transition of the magnetic lattice, there has been no such clear mechanisms explaining different experimental behaviors of the spin resistivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' de Gennes and Friedel [12] have conjectured that the spin resistivity has the origin in the spin-spin correlation so it should behave as the magnetic susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' As a consequence, it should diverge at TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' However, Fisher and Langer [13], and Kataoka [14] have made the observation that the range of spin-spin correlation should not be infinite at TC due to collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This changes the shape of ρ with respect to the magnetic susceptibility near the phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Let us mention that the resistivity due to magnetic impurities has been calculated by Zarand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [15] as a function of the Anderson’s localization length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This parameter expresses in fact a kind the correlation sphere induced around each impurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Their result shows that the resistivity peak depends on this parameter, thus in agreement with the spin-spin correlation idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Note that the spin resistivity depends on the spin orientation of the environment: the electron encounters less resistance in a ferromagnet with spins parallel to its spin than in a ferromagnet with spins antiparallel to its spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Imagine a film composed of three ferromagnetic layers where the middle one is a soft ferromagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In the layer-coupling configuration ↑ − ↑ − ↑, the movement of an up spin perpendicular to the film encounters a resistance R↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' One applies now an external magnetic field in the negative direction, small enough to reverse the spins in the (middle) soft layer: one has the three-layer configuration ↑ − ↓ − ↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The up spin is found to encounter much difficulty to cross the three layers: the resistance is R↓ which is much larger than R↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This is the phenomenon Giant-Magneto-Resistance (GMR) discovered in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [16–18] which has many applications in spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The absence of Monte Carlo (MC) simulation in the literature on the spin resistivity has motivated our works since 2007: we have studied the spin current in a number of systems including ferromagnets [19–21] and antiferromagnets [22–25] by MC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The behavior of ρ as a function of T has been shown to be in general agreement with experiments and theories mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In addition to ferromagnets and antiferromagnets, we have also studied the spin resistivity in frustrated spin systems [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' These systems discovered in the early 80’s have been intensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Many unusual properties have been found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The reader is referred to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [28] for reviews on various frustrated spin systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In this paper, we will summarize the most important aspects and results of works on the spin resistivity in ferromagnets, antiferromagnets and in some frustrated magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In section II, we present our generic model and the MC method which we employ to study the spin resistivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Section III is devoted to the presentation of our main MC results since 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Comparison with some experiments is made in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In section IV, we show new results in the case of a hexagonal-close-packed (HCP) crystal where we tune a frustration parameter allowing to study both the non-frustrated case and the frustrated case in the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Conclusing remarks are given in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 3 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' MODEL AND METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Model We have investigated the spin resistivity in magnetically ordered materials by using a newly-deviced efficient MC simulation method [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The success of the method was demonstrated when we studied the semiconducting MnTe where the agreement with experiment is excellent [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This case will be reviewed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In ferromagnets, we found that the spin resistivity has a high peak at the lattice order-disorder transition tremperature TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' As said earlier, this anomaly comes from the spin-spin correlation [12–14] but remains finite at TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In antiferromagnets, one observes only a broad maximum [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In addition to ferromagnets and antiferromagnets, we have investigated the spin resistivity in the following two frustrated systems: the face-centered cubic (FCC) lattice with Ising spins [26] and the J1 − J2 simple cubic (SC) lattice [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We found that that the first-order phase transition in these frustrated systems causes a discontinuity of the spin resistivity at TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Let us recall here the model and method which have been used in our early works shown in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [23, 26]: simulations have been carried out to calculate the current of itinerant spins moving in the system under the action of an electric field ⃗ϵ applied in the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The itinerant spin σi carried by a conduction electron interacts with its surrounding lattice spins inside the sphere of radius D1 centered at its position at the time t on its trajectory across the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The Hamiltonian is supposed to be Hl = − � j Iij⃗σi · ⃗Sj (3) where the sum is carried over all lattice spins in the sphere centered at the itinerant Ising spin ⃗σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Iij denotes the distance-dependent interaction between ⃗σi and ⃗Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' To be general, we also consider the following interaction between an electron spin with neighboring conduction spins within a sphere of radius D2 Hi = − � j Kij⃗σi · ⃗σj (4) For simplicity, we suppose the distance-dependent interactions are Iij = I0 exp(−Brij) (5) Kij = K0 exp(−Crij) (6) where I0, B, K0 and C are constants to be chosen so that the energy of a conduction electron spin is much smaller than that of a lattice spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This choice is made from a physical consideration: in choosing so, we avoid the influence of itinerant spins on the ordering of the lattice spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Note that this choice is justified in the almost-free electron model where I0 ≃ K0 ≃ 0, and in semiconductors where they are larger but still weak with respect to the exchange intergrals of the lattice spins J1 and J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' A discussion in details on the choice of these parameters has been given for example in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Let us assume in the folklowing a concentration of one itinerant electron per two lattice cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This concentration is thus of the order of electron concentration in normal metals which is 1023/cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' With this concentration, it is obvious that the averaged distance between two conduction electrons is much larger than the cutoff distance D2 which is of the order of the lattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' However, due to the attractive nature of the electron-electron interaction, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (6), it is neccessary to introduce a chemical potential term to insure that itinerant spins are uniformmly dispersed in the crystal and they do not form clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This chemical potential is written as Hp = D[n(⃗r) − n0] (7) where D is a positive constant, n(⃗r) denotes the concentration of conduction spins in the sphere of cutoff radius D2 centered at the position ⃗r of the conduction spin under consideration, and n0 the averaged electron concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' I0 is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (5) which represents the magnitude of the interaction between a conduction electron and a localized lattice spin [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' K0 is the magnitude of the interaction between two conduction electrons [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (4) and (6)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' D is the magnitude of the chemical potential [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (7)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' When we simulate a real material, if we have experimental data on the resistivity and the exchange interactions, such as in the case MnTe presented in section III C, we can estimate the values of these coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Simulation Method Let us study a film of size Nx × Ny × Nz where Nz is the film thickness which is much smaller than the sizes Nx and Ny in the x and y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Usually, we use Nz = 4 − 8 and Nx = Ny = 20 − 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Itinerant spins move under the action of an electric field acting on the electron charge, applied in the direction x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The electric field energy is HE = −e⃗ϵ · ⃗ℓ (8) where e is the electron charge, ⃗ϵ the applied electrical field and ⃗ℓ the displacement vector of the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The interaction between the lattice spins is given by the Hamiltonian HL = −J � i,j ⃗Si · ⃗Sj (9) where J is the exchange interaction between nearest neighbors (NN) ⃗Si and ⃗Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' For ferromagnets J > 0, and for antiferromagnets J < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We use the standard Monte Carlo (MC) method to thermalize the lattice alone at T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Next, we introduce the electrons into the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We suppost that each electron spin interacts with the surrounding lattice spins inside a sphere centered at its position, of radius D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The electron spin also interacts with other electron spins within a sphere of radius D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Electrons move under the applied electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In order to obtain the spin current we have to thermalize the state of the spins of conduction electrons in the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This is done by the following steps: i) we take a conduction spin and calculate its actual energy Eold using the different interactions mentioned above, ii) we make a trial move ⃗ℓ for the electron in a random direction between 0 and a where a is the lattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' If the move ⃗ℓ takes the itinerant electron outside the sample, then we put it inside on the other sample end by virtue of the periodic boundary condition, iii) we then calculate the new energy Enew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' If Enew < Eold, then the trial position is accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Otherwise, it is accepted with the probability exp[−(Enew − Eold)/(kBT)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' iv) we take another conduction electron and repeat the three steps above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We continue with other electrons until all electrons are considered: this accomplishes one MC step/spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' A large number of MC steps/spin is neccessary to arrive at a steady current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We next average physical quantities of interest at the temperature T under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' iv) we take another T and repeat the above four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We should cover the temperature region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This paper is a review of our publications over the past 15 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In each publication, we used various sample sizes to detect finite size effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We increased the sample size until the results do not depend anymore on the size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The results are considered as valid thermodynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The finite size effect and the finite size scaling are what the simulators do before reaching conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This problem is particularly important when one wants to study the criticality or the order of a phase transion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In the review, we do not repeat these details which depend on the studied system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Rather, we emphasize on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The reader interested on the details of the simulation methods is referred to each original publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We calculate the spin resistivity ρ as : ρ = 1 Ns (10) where Ns is the number of mobile electrons passing thru a unit surface perpendicular to the direction of the applied electric field x, per MC time unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' An application with a real material using real units is presented in subsect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' For a good thermal average, we have to perform very long MC runs and we proceed as follows: for each configuration of the lattice spins we average the spin resistivity over N1 MC steps, then we thermalize again the lattice with N2 MC steps to get rid of the correlation between lattice spin configurations, before averaging again the resistivity for N1 MC steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We repeat this N1 + N2 cycle for a large number of times N3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The total MC steps of resistivity averaging is about 4 × 105 steps per spin in our runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We found by comparison that this ”multi-step” averaging method strongly suppresses statistical fluctuations seen in our earlier work [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' It is obvious that the larger N1 and N2 yields the better statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We know that depending on T, the relaxation time τL of the lattice spins varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We have to compare τL with the relaxation time τI of conduction electron spins in order to choose a right value of N1 in order to calculate the average of the resistivity with one lattice spin configuration at T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We know the two limiting cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The first case is when τL ≃ τI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In this case, we have to take N1 = 1, namely 5 the lattice spin configuration should change with each move of itinerant spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The second case is when τL ≫ τI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In this case itinerant spins can be scattered many times along its trajectory across the same lattice configuration and for many times across the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In order to choose a right value of N1, we consider the following temperature dependence of τL in non frustrated spin systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The relaxation time is expressed in this case as [30–32] τL = A |1 − T/TC|zν (11) where A is a constant, ν the correlation critical exponent, and z the dynamic exponent which depends on the spin model and space dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' For 3D Ising model, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='638 and z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' From this expression, we see that as T tends to TC, τL diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In the critical region around TC the system encounters thus the so-called ”critical slowing down”: the spin relaxation is extremely long due to the divergence of the spin-spin correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' When we take into account the temperature dependence of τL, the shape of the resistivity is strongly modified near TC where τL is very long, in contrast to the paramagnetic phase where the relaxation time is very short due to rapid thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We should emphasize that at low T, ρ does not depend on τL since in that temperature range where the ordering of the lattice spins is almost perfect: the spin landscape from one microscopic state to another does not change significantly, so the motion of the itinerant electron spin does not significantly vary (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Finally, we note that we have also used the Boltzmann’s equation combined with MC data to study the spin resistivity [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' However, the shape of the resistivity peak at the transition temperature does not agree well with experiments, unlike that obtained from direct MC simulations as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This proves the efficiency of MC simulations for the calculation of the spin resistivity in magnetically ordered materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The present review therefore aims at promoting this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' REVIEW A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Spin Resistivity in Ferromagnets and Antiferromagnets Experiments mentioned above amongst numerous other data show that the spin resistivity in ferromagnets has a sharp peak at the transition temperature TC of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We know by the theory of phase transitions and critical phenomena that in the region around TC, the so-called ”critical slowing-down” phenomenon happens, resulting in extremely large τL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The peak in ρ is due to this phenomenon via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (11) where τL diverges at TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Our MC simulations using the method described above in the case of a ferromagnet where the lattice spins are of the Ising type show a sharp peak at TC (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 1) in agreement with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We note that the spin resistivity for ferromagnets (as well as for antiferromagnets) increases with decreasing T at low T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This can be explained by several causes: the freezing or crystallization of conduction electrons takes place at low T so that just a small number of conduction spins with decreasing T is mobile, or it may be the classical counter effect of the quantum Kondo electron-impurity scattering if one considers the few excited lattice spins at low T are independent impurities, see the last term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Note that the shape of ρ depends on the lattice type, interaction strengths encountered by the conduction electrons, electron concentration, relaxation time, spin model, applied magnetic-field amplitude etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [23], we have shown that a decrease in the interaction between conduction spins, K0, reduces the increase of ρ as T → 0, an applied magnetic field reduces the height of the resistivity peak and the larger electron density reduces ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Finally, we emphasize that ρ depends on the material intrinsically via the critical exponents ν and z, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' If we wish to compare simulated spin resistivity to experimental measurements performed on a given material, we have to use in the simulation the available experimental physical parameters of that material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' An example of quantitative comparison for semiconductor MnTe is shown in subsection III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Note that the magnetic field applied on the system reduces the peak height as shown in our work Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [23], in agreement with experiments [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Unlike ferromagnets, antiferromagnets have not been much studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Haas [29] has shown that in contrat to ferromag- nets where the resistivity ρ has a sharp peak at the order-disorder transition of the lattice spins, in antiferromagnets there is no such a peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Using MC simulations, we found that the peak does exist in an antiferromagnet but it is less sharp compared to that of a ferromagnet, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We think that the alternate change of sign of the spin-spin correlation with distance may have something to do with the absence of a sharp peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We have tested this idea on the effect of the cut-off distance D1 [26]: in an antiferromagnet, when we increase D1, we will include successively up-spin shells and down-spin shells in the sphere of radius D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Consequently, the difference between the numbers of up and down spins in the sphere oscillates with varying D1, giving rise to the oscillatory behavior of ρ observed at small D1, unlike in ferromagnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 6 ρ T FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 1: Resistivity ρ as a function of T, obtained by simulations using the T-dependent relaxation time, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 11, for ferromagnet (black circles) and antiferromagnet (white circles) of BCC structure, with arbitrary unit in zero magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Other parameters: ϵ = 1, I0 = 2, K0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5, A = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' At this stage, we note that the presence of an itinerant spin will break the invariance between a ferromagnet and its antiferromagnet counterpart in the local Mattis transformation (Jij → −Jij, ⃗Sj → −⃗Sj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Frustrated J1 − J2 Model on a Simple Cubic Lattice Let us consider the simple cubic lattice with NN and NNN interactions as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The Hamiltonian is written as H = −J1 � (i,j) ⃗Si · ⃗Sj − J2 � (i,m) ⃗Si · ⃗Sm (12) where the first sum � (i,j) is made over the NN Ising spin pairs ⃗Si and ⃗Sj with interaction J1, and the second sum � (i,m) is performed over the NNN pairs with interaction J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We focus our attention on the region of parameters which gives rise to a frustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' For that purpose, we assume that J1 is an antiferromagnetic interaction, namely J1 = −J < 0 (J > 0), and J2 is also antiferromagnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We put J2 = −ηJ where η is a positive parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The ground state (GS) of this system can be obtained by minimizing the energy, or by comparing the energies of different spin configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We can also numerically minimize the energy by using the steepest descent method [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We obtain the GS antiferromagnetic configuration shown Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 3a for |J2| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='25|J1|, and the GS spin configuration shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 3b for |J2| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='25|J1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This latter configuration is 3-fold degenerate because we can choose the parallel NN spins either on x, or y or z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In addition, with the permutation of black and white spins, we have the total degeneracy equal to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 2: Simple cubic lattice where the NN and NNN interactions, J1 and J2, are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We note in passing that in the case of the Heisenberg model in the frustrated region (|J2| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='25|J1|) the phase transition has been shown to be of first order [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The system is very unstable due to its large degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In the case of the Ising spin on the SC lattice treated here, we found that the first-order character of the phase transition is even stronger [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We use J1 = −J = −1 (AF interaction) for the coupling between NN lattice spins in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The energy is thus measured in the unit of J and the temperature is in the unit of J/kB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' All distances (D1 and D2) are in the unit of the lattice constant a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 350 F AF 300 250 R 200 0 00000000000000000 0 150 100 2 6 8 10 12 4 T/J7 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 3: Ground state (GS) of the simple cubic lattice with Ising spins: (a) GS when |J2| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='25|J1|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (b) GS when |J2| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='25|J1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' White (black) circles denote up (down) spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' See text for comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Simulations have been carried out by using the temperature-dependent relaxation time of the lattice spins given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (11) where we have taken A = 1 and τL = 1 at T = 2TC far from TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Such a choice leads to τL = 1 at that temperature expected for fast thermal fluctuations in the paramagnetic phase far above TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Since we suppose that the interaction between conduction electron spins is attractive, a chemical potential is required to avoid the collapse of the system, namely to avoid that all conduction spins form a cluster [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='(7)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The chemical potential in thermodynamics makes the particle uniformly distributed in the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Its strength is expressed by D which has to be chosen in accordance with K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Figure 4 displays the phase diagram in the space (K0, D) which shows the collapse region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This allows us to avoid this region and choose an appropriate value of D for a given K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='9 1 K D 0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 4: D versus K0 in the case where η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The collapse region is in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We have used D1 = D2 = 1, I0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5, ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' To see the effect of the nature of phase transition on the spin resistivity, in the following we focus on two typical cases: η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='26 which belong respectively to the regions of second- and first-order transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' For η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2: The spin resistivity at temperatures below TN oscillates with varying D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' By analyzing the ratio of the number of up spins to the number of down spins in the sphere of radius D1, we found that it oscillates with varying D1: the maxima (minima) of ρ correspond to the case of largest (smallest) numbers of parallel (antiparallel) spins in the sphere [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' At very high temperatures where the lattice spins are disordered, the numbers of up spins and down spins in the sphere of radius D1 should be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' There is however a very small oscillation if the temperature is close to TN and if D1 is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Figure 5 displays the resistivity versus T for D1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The spin resistivity shows a rounded maximum at the transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This is in agreement with the curve experimentally observed in La0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4Ca0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6MnO3 by Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [2] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 6, right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 8 116 117 118 119 120 121 122 123 124 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='9 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3 T ρ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 5: Spin resistivity versus T in the case η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2, D1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We have used the lattice size Nx = Ny = 20, Nz = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Other variables are I0 = K0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5, D2 = 1, D = 1, ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 6: Experimental spin resistivity versus T are shown for several applied magnetic fields in the compound La0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4Ca0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6MnO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' These data are in the figure 2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' For D1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 or D1 = 1, the resistivity is smaller below the transition temperature, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This shows the importance of the effect of the interaction range on the spin resistivity in materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' For η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='26: At this value of η, the transition of the lattice spins is of first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The dependence of the resistivity on D1 is very similar to that of the second-phase transition, namely the resistivity at a given T oscillates as D1 varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The physical meaning of the oscillation has been given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' More details can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We found that the resistivity ρ in the frustrated regime can go downward or upward at the transition temperature depending on D1 [27], unlike in non-frustrated ferromagnets and antiferromagnnets shown earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 8 for two values of D1 where one observes the discontinuity of ρ at the transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The discontinuity of ρ has also been found in other frustrated antiferromagnets such as the FCC antiferromagnet [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' From the results shown above for the J1 − J2 model, we come to the conclusion that the behavior of the spin (Ω2 cm)0 100 200 300 T(K)10 10° H=0 T H=9 T (a) y=0 10 120 10 H=0 T MI 60 10 H=1 T 0 3 6 H=3 T H (T) H=6 T 10 (b) y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='022 H=0 T 1 H=1 T (c) y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='05 0 (d) y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='0 H=0 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 120 H=1 T 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='00 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='05 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='0 0 100 200 3000 100 200 300 T(K)(Ω2 cm)9 100 105 110 115 120 125 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 ρ T FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 7: ρ as a function of T for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 with D1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 and 1 ((black circles and blue open circles, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' For simulations, we used Nx = Ny = 20, Nz = 6, I0 = K0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5, D2 = 1, D = 1, and ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 100 110 120 130 140 150 160 170 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 T ρ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 8: ρ as a function of T for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='26 where D1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 (black circles) and D1 = 1 (blue open circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We have used Nx = Ny = 20, Nz = 6, I0 = K0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5, D2 = 1, D = 1 and ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' See text for comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' resistivity is a consequence of the nature of the lattice transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' If the lattice transition is of second order, then the resistivity of itinerant spins has a rounded peak, while if the lattice transition is of first order, the resistivity is discontinuous at the transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The case of MnTe The pure semiconductor MnTe has two kinds of structures: the zinc-blend structure or the hexagonal NiAs one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 9 [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We focus on the second structure where the N´eel temperature is TN = 310 K [36], and where many other experimental data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' MnTe is a semiconductor with a large gap (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='27 eV) and a carrier concentration n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3 × 1017cm−3 at room temperature [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Without doping, MnTe is non degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The crystal is formed by ferromagnetic xy hexagonal planes antiferromagnetically stacked in the c direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The NN distance in the c direction is c/2 ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='36 ˚A, and the in-plane NN distance is a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='158 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' From neutron scattering experiments, it was found that the main exchange interactions between Mn spins in MnTe are the interaction between NN along the c axis with the value J1/kB = −21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3 K, the ferromagnetic exchange J2/kB ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='05 K 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 9: Structure of MnTe of NiAs type: black and white circles present respectively opposite spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Interactions between NN, between next NN and between third NN are indicated respectively by J1, J2, and J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' See values given in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' between in-plane neighboring Mn (they are next NN by distance), and the third NN antiferromagnetic interaction J3/kB ≃ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='04 K (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The spins are lying in the xy planes perpendicular to the c direction with a small in-plane easy-axis anisotropy Da [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Let us emphasize that the values of the exchange integrals given above were deduced from experimental data by fitting with a free spin-wave theory [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Other fittings with mean-field theories give slightly different values: J1/kB = −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='7 K, J2/kB = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='55 K and J3/kB = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='28 K [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Note that the Mn spin is experimentally known to be of the Heisenberg model with magnitude S = 5/2 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We write the following Hamiltonian for the lattice spins H = −J1 � (i,j) ⃗Si · ⃗Sj − J2 � (i,m) ⃗Si · ⃗Sm − J3 � (i,k) ⃗Si · ⃗Sk −Da � i (Sx i )2 (13) where the first sum is performed over the NN spin pairs, the second sum over the NNN pairs and the third one over the third NN pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Da > 0 is an anisotropy constant which favors the in-plane x easy-axis spin configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The behavior of ρ in MnTe as a function of T has been experimentally shown in several works [39–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We have studied using MC simulations the spin resistivity in MnTe with the above Hamiltonian [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Let us summarize this work here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' For MC simulations, we suppose the following Hamiltonian of the itinerant spins: Hi = − � n I(⃗r − ⃗Rn)⃗σ · ⃗Sn (14) where the sum is performed by counting all the lattice spins ⃗Sn inside the sphere of radius D1 = a centered at ⃗r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' I(⃗r − ⃗Rn) > 0 is the ferromagnetic distance-dependent interaction between the itinerant electron spin ⃗σ at ⃗r and the Mn spin ⃗Sn at ⃗Rn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The electron spin is supposed of the Ising type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We neglect therefore the quantum effects which may be important at very low T but our attention is focused on the region of high-enough T where quantum effects may be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We assume the following form of I(⃗r − ⃗Rn) : I(⃗r − ⃗Rn) = I0 exp[−α(⃗r − ⃗Rn)] (15) where the constants I0 and α are chosen in such a way that the interaction Hi yields an energy much smaller than the lattice energy given by H (see the guide for the choice of different constants given below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (6) and in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' It is noted that the cut-off distance D1 is rather short so that only the first few neighbors are inside the sphere, the results shown below do not therefore depend significantly on the choice of the value of α in the exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Finally, Te Mn C J ZA 3 y a x11 note that the concentration of conduction electrons in MnTe is n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3 × 1017cm−3 which is five orders lower than the concentration of its surrounding lattice spins which is ≃ 1022cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This observation justifies that the interaction between conduction electrons for MnTe can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We have assumed this in the simulations shown in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' As mentioned above, the exchange interactions deduced from experimental data have slightly different values, they depend on the theoretical Hamiltonian and the approximations used to deduce it (often the mean-field approximation is used, see a detailed example in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Note that in semiconductors, the carrier concentration varies with T but since this concentration is very low, we do not take into account its variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Consequently, the number of conduction electron spins used in the simulation is important only for the statistical average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The current obtained is proportional to the number of itinerant spins but there are no extra effects within our assumptions mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We have calculated ρ of MnTe, using the exchange integrals slightly modified with respect to the ones given above in order to obtain the best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The obtained resistivity ρ is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Let us note that we have taken J3 slightly larger in magnitude than the value deduced from experiments by mean-field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Our value of J3 was chosen in order to obtain TN = 310 K which is in excellent agreement with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' However the most striking feature is that the simulated ρ shows a sharp maximum at TN and coincides with the experimental data over the whole temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Note that we have used A = 1 and the well-known Heisenberg critical exponents ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='707, z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='97 [31] for the lattice spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' It is remarkable that with the same set of param:eters, we obtain an excellent agreement with experiments in the temperature regions below T < 140 K and above TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We note that we have tried earlier to use the Boltzmann’s equation [22] but the obtained result is not as good as the MC result presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' From the simulated ρ, we can calculate the relaxation time of conduction spins, we obtain τI ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='1 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The mean free path can be also estimated, it is equal to ¯l ≃ 20 ˚A, at the critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 10: Comparison between the simulated spin resistivity and the experimental data of MnTe: Black circles are results from the Monte Carlo simulation, white circles are experimental data taken from He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='[43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We used for the simulation J1 = −21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5K, J2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='55 K, J3 = −9 K, I0 = 2 K, Da = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='12 K, D1 = a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='148 ˚A, ϵ = 2 × 105 V/m, L = 30a (lattice size: L3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' See text for comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' PHASE TRANSITION AND SPIN RESISTIVITY IN THE ISING HCP LATTICE A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Hamiltonian and Ground State The lattice we consider is the HCP structure illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The xy planes are triangular (hexagonal) and the stacking direction is z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We suppose the following Hamiltonian p (2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='cm) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 0 100 200 300 400 T (K)12 J1 J2 Z FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 11: HCP lattice: the in-plane NN interaction is denoted by J1 and the inter-plane NN interaction is denoted by J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' H = − � (i,j) Ji,j ⃗Si · ⃗Sj (16) where Jij is the AF interaction between nearest-neighbors (NN) ⃗Si and ⃗Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We denote Jij = J1 if the NN are on the xy triangular plane, and Jij = J2 if the NN are on two adjacent planes (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The GS can be determined by minimization the local energy of each spin and doing this for all spins, then repeating many times until the total energy converges to a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Normally, with a system without bond disordering, this method needs just a small number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The GS can be checked by looking at the final snapshot: it should be periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This procedure of local energy minimization is called in the literature ”the steepest-descent method”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The implementation of this method is very simple [33] (i) we first create an initial random configuration (ii) we then calculate the local field acting at a spin by its neighbors using (16) (iii) we align the spin under consideration along the calculated local field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' in doing so its energy is minimum (iv) we take another spin and repeat the three preceding steps until all spins are considered: this step completes one sweep (v) we start again another sweep and we realize a large number of sweeps until the total energy is minimm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' One can also analytically minimize the interaction energy as shown below to find the GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Let us assume that both interactions J1 and J2 are antiferromagnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' For simplicity we fix J2 = −1 and vary J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The case of isotropic interaction, namely J1 = J2 has been studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We summarize the result here: for the HCP structure, each spin is common for eight tetrahedra (four in the upper half-space and four in the lower half-space along the z axis) and a NN bond is shared by two tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The GS spin configuration of the system is formed by stacking neighboring tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In the GS, one has two pairs of antiparallel spins on each tetrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Their axes form an arbitrary angle α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The GS degeneracy is therefore infinite (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 2a of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Note tthat the periodic boundary conditions will reduce a number of the GS configurations, but the degeneracy is still infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' One particular family of configurations of interest for both XY and Heisenberg cases is when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The GS is then collinear with two spins up and the other two down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The stacking sequence is simple because there are three equivalent configurations due to the fact that there are three ways to choose the parallel spin pair in the original tetrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The case where J1 ̸= J2 has been studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [46] for the Ising and XY cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Let us recall some results concerning the Ising case which allow us to understand the new results on the spin resistivity presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We use the steepest descent method described above with varying J1 (J2 = −1): we find two kinds of GS spin configuration: the first consists of xy ferromagnetic planes stacked antiferromagnetically along the z direction, while the second one is the stacking of xy AF planes such that each tetrahedron has two up and two down spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The transition between the two configurations is determined as follows: one simply writes down the respective energies of a tetrahedron and compares them 13 E1 = 3(−J1 + J2) (17) E2 = J1 + J2 (18) One sees that E1 < E2 when J1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5J2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' |J1| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5|J2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Thus the first configuration is more stable when |J1| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5|J2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Phase Transition in the case of Ising Spins on the HCP Lattice In the following, we present the results of simulations using the Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We use the sample size Nx × Ny × Nz with Nx = Ny = 18 and Nz = 8, namely 16 atomic planes along the z axis, and the periodic boundary conditions in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We use the first 106 MC steps per spin to reach equilibrium and we average physical quantities with the next 106 MC steps per spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The energy is expressed in the unit of |J2| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Let us define η = J1/J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We have seen that the GS changes at ηc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5, so we show below the properties of the system on both sides of this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Figure 12 displays the averaged energy per spin, the order parameter (staggered magnetization), the specific heat and the susceptibility for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' As seen, the transition is of second order since there is no discontinuity of the energy and the order poarameter at the transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 3 T E (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 3 T M (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 3 T CV (c) 0 10 20 30 40 50 60 70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 3 T χ (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 12: The case of Ising spin on the antiferromagnetic HCP lattice: (a) energy per spin E, (b) order parameter M, (c) specific heat CV and (d) susceptibility χ, versus temperature T for η = J1/J2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' See text for comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' For η = J1/J2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 13 for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='85 and 1 shows that the discontinuity of E and M at the transition is very large, a signature of a strong first-order transition in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In order to confirm the order of the phase transition, we measure the energy histogram taken during the averaging MC time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Figure 14 shows the energy histogram taken at the transition temperature for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3 (black), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='85 (blue) and 1 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We observe here that the first case is a Gaussian distribution indicating a second-order transition, in contrast to the last two cases which show double-peak histograms confirming a first-order transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Figure 15 displays the phase diagram in the space (TC, η) where zone (1) and zone (2) denote the ordering of the first, and second kinds, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (P) indicates the paramagnetic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Note that the transition line between (1) and (P) is a second-order line, while that between (2) and (P) is a first-order line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Note that in the XY case, the change of the GS takes place at ηc = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We have studied this case in details in to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Finally, let us emphasize that all 3D frustrated systems we know so far undergo a first-order transition [28] including the much-studied antiferromagnetic stacked triangular lattice [47–51], the FCC antiferromagnets [52], the simple cubic fully frustrated lattices [53–56], helimagnets [57], and antiferromagnetic HCP lattice studied here (see more details in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [45, 46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 14 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='1 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 T E (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='9 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 T M (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 13: The case of Ising spin on the antiferromagnetic HCP lattice: (a) energy per spin E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (b) order parameter M versus temperature T for η = J1/J2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='85 (blue open circles) and 1 (red triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' See text for comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='014 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='9 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='7 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='1 E P(E) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 14: Energy histogram P(E) for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3 (black circles), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='85 (blue open circles), and 1(red triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' See text for comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Spin resistivity in the HCP lattice with Ising spins The results in this subsection are new, they are not published so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Using the method which has been described in subsection II, we carry out MC simulations to study the spin resistivity in the Ising case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 16 the resistivity at two temperatures, below and above the transition temperature, as a function of D1 for the GS belonging to phase (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 17 the case of a GS belonging to phase (2) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Similar to the case of J1 − J2 model on the simple cubic lattice considered in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' [26, 27], we find here an oscillation of ρ at low temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Note that ρ is always smaller at low temperature than at high temperature, whatever the value of D1 is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The physical origin of the oscillation has been discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The spin resistivity ρ for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3 and 1 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 18 as a function of T, here the distances D1 and D2 are in unit of the distance between the NN lattice spins, and I0, K0 and D which have the energy dimension are in the unit of |J2| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' As in the frustrated J1 − J2 model shown above, one finds here that ρ has a broad peak in the second-order region, in contrast to the first-order region where it undergoes a discontinuous jump at the phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Some remarks are in order: i) At very low temperature, the resistivity increases with decreasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This behavior can be understood by the freezing of the itinerant spins due to low T: The energy of itinerant spins is low, they occupy the low-energy positions in the periodic lattice, it is difficult to move them out by the insufficient thermal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' They are somewhat frozen in almost periodic positions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' namely a pseudo crystallization occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Note that the increase of resistivity with decreasing T at very low T was observed in many experiments on various materials and is not limited to ferromagnets [3, 5, 7, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This increase of ρ with decreasing T in the quantum case has been explained by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Kondo using a third-order perturbation theory [58]: the scattering of s-electrons by d-electrons of localized magnetic impurities gives rise to a resistivity minimum at a finite T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We have also found here this minimum of ρ at low T with the classical 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='9 1 η (1) (2) (P) Tc FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 15: Transition temperature TC versus η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (1) denotes the second-order region, (2) the first-order region and (P) the paramagnetic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 110 120 130 140 150 160 170 180 190 200 210 220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 2 D ρ 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 16: ρ versus D1 for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3 at T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 < TC (black circles) and at T = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 > TC (open circles) where TC ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Other parameters are Nx = Ny = 18, Nz = 8, D2 = 1, I0 = 2, K0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5, C1 = C2 = 1, A = 1, D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5, ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' spin model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The similarity with the quantum Kondo effect can be explained by the fact that an excited localized lattice down-spin (in a very small number at low T) can be viewed as an impurity which captures nearby conduction up-spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' ii) Outside this low-T region, when T increases, the thermal energy progressively unfreezes the itinerant spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' As a consequence, ρ decreases and passes through a minimum (see discussion above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' However, at higher T, the scattering with the lattice spins is stronger, ρ increases up to the transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' iii) At the transition temperature, ρ shows a peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The physical mechanism leading to the peak can be explained: in a previous work [21], it was found from our simulations that the peak is due to scattering of the itinerant spins by antiparallel-spin clusters which are numerous in the transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' When one gets close to the transition point, the number of clusters of down spins are the most numerous, giving rise to the peak in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Note that the ”defects” clusters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' clusters of antiparallel spins) have an energy barrier to resist the passage of itinerant spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This is also the origin of the extremely long relaxation time in the critical region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' iv) Well above the transition temperature, in the paramagnetic phase, as temperature increases, clusters of down and up spins will be broken more and more into independent disordered spins, namely spins with zero energy, itinerant spins move easily on their trajectory, making a decrease of ρ with increasing T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Note that we have also varied the radius D1 to see its effect on ρ at the transition in the present frustrated HCP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We found the same effect seen in other antiferromagnets we studied previously [26, 27]: at a given temperature, an oscillation of ρ with varying D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' oscillates slightly with distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The origin of this oscillation has been analyzed avove in the J1 − J2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 16 100 150 200 250 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='8 2 D1 ρ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 17: ρ versus D1 for η = 1 at T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 < TC (black circles) and at T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='9 > TC (open circles) where TC ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Other parameters are Nx = Ny = 18, Nz = 8, D2 = 1, I0 = 2, K0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5, C1 = C2 = 1, A = 1, D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5, ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 120 140 160 180 200 220 240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 4 T ρ (a) 100 150 200 250 300 350 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 3 T ρ (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 18: Spin resistivity ρ of the Ising HCP model versus temperature T for (a) η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' (b) η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Nx = Ny = 18, Nz = 8 (namely 16 planes in the z direction), D1 = D2 = 1, ϵ = 1, I0 = 2, K0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5, C1 = C2 = 1, A = 1, D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' All distances are in unit of the NN distance, energy constants are in unit of |J2| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' See text for comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Finally, let us look at some experimental data obtained for ferromagnets and antiferromagnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Figure 19 shows experiments by Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' performed on ε-(Mn1−xFex)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='25Ge antiferromagnets [3], experiments by McGuire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' performed on antiferromagnetic superconductors LaFeAsO [6], by Chandra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' on thin Cd1−xMnxTe films [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Experiments by Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' on antiferromagnetic La1−xSrxMnO3 [7] are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We see here that our results on the shape of the spin resistivity are in agreement with these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In the lack of physical data on these experimental materials, we cannot make a quantitative comparison as we did in the MnTe case presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' CONCLUSION In this paper, we have reviewed some important works published on the spin resistivity in magnetically ordered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We have focused on our works published over the past 15 years using mainly Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' These works were motivated by the absence of Monte Carlo works, even at the present time except ours, in spite of the fact that this method of simulation has proven to be very efficient when comparing its results with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In the case of MnTe where there are sufficient experimental data, we have made a quantitative comparison between experimental and simulated spin resistivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The agreement between experiments and simulations is excellent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This review therefore aims at promoting this method to study more realistic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' As demonstrations, we have used this method to study the spin resistivity in generic ferromagnets and antiferro- magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The cases of frustrated systems have also been presented: the J1 − J2 model and the antiferromagnetic HCP lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Let us summarize the results on the two frustrated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The J1 − J2 model is a simple cubic lattice with Ising spins interacting with each other via NN and NNN antiferro- magnetic interactions, J1 and J2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The GS of this model is determined by the ratio η = J2/J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We have shown that the GS changes at the critical value ηc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' For the non-frustrated region in the phase space, namely 17 (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 19: Experiments on the resistivity as a function of T iperformed by (a) by Du et al on ε-(Mn1−xFex)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='25Ge antiferromagnets [3], (b) by McGuire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' on antiferromagnets LaFeAsO [6], and (c) by Chandra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' on thin films of Cd1−xMnxTe [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='25, the GS is simply composed periodically of two interpenetrated tetrahedra formed by the NNN sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In the frustrated region, namely η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='25, the GS can be described as composed of one line of spin up, one line of spin down, alternately, in one crystal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The degeneracy is three because there is a freedom to choose one direction among three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The total degeneracy is 6 if we count the statesw of reverse spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The transition in the frustrated region is theoretically of first order since the present 6-fold GS is equivalent to the q-state Potts model with q = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We know that in three dimensions, the transition of the Potts model is of first order from q = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We have found this directly from the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In the non-frustrated region, namely η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='25, the transition is found to be of second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We have performed MC simulations to obtain ρ of the conduction spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We found that ρ displays a broad maximum at the second-order phase transition while it undergoes a discontinuous change at the first-order transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The Ising model on the antiferromagnbetic HCP lattice has been also studied in this review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We assumeed an in-plane interaction J1 and an inter-plane interaction J2, both antiferromagnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We found that the GS changes at the critical value ηc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' Below (above) which the spins in the xy planes are ferromagnetic (antiferromagnetic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The nature of the transition changes in these two regions: it is of second order below ηc and of first order above ηc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The spin resistivity has been simulated in both regions of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' In the second-order region, it shows a broad maximum while in the first-order region, the resistivity ρ makes a discontinuous jump at the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' This feature is what we also found in other frustrated spin systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' These findings reviewed in this paper show a close relationship between the nature of the phase transition and the shape of the spin resistivity in real materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' We hope that this review convinces the magnetic community on the T 420 T (μ2cm) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' N 390 450 360 (μ2cm) X=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='20 330 0 100 200 300 425 T (K) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' N X=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='17 400 0 100 200 300 400 T (K)(a) 4.' metadata={'source': 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+page_content='5 0 50 100 150 200 250 300 T (K)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 Cd1-xMnxTe G0000 X=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='25 nits) 3000 X -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='60 00000 X= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='00 un 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='0 (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 Q normalized 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content='5 100 200 300 (k) T18 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 20: Resistivity versus temperature on antiferromagnetic La1−xSrxMnO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' The figures presented are taken from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' 7 of [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQf0wIq/content/2301.02689v1.pdf'} +page_content=' use of MC simulations for transport phenomena.' 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Aydemir∗ +(Dated: January 11, 2023) +1 +arXiv:2301.04052v1 [econ.GN] 10 Jan 2023 + +Abstract +The optimal age that a retiree claims social security retirement benefits is in general a complicated +function of many factors. However, if the beneficiary’s finances and health are not the constraining +factors, it is possible to formally derive mathematical models that maximize a well-defined measure +of his total benefits. A model that takes into account various factors such as the increase in the +benefits for delayed claims and the penalties for early retirement, the advantages of investing some +of the benefits in the financial markets, and the effects of cost-of-living adjustments shows that not +waiting until age 70 is almost always the better option. The optimal claiming age that maximizes +the total benefits, however, depends on the expected market returns and the rate of cost-of-living +adjustments, with the higher market rates in general pushing the optimal age lower. The models +presented here can be easily tailored to address the particular circumstances and goals of any +individual. +Keywords: Social Security; optimal timing +I. +Introduction +Social Security retirement benefits are an essential part of the “Social Safety Net” in +the US and provide much needed income to many retirees. Thus, at what age they are +claimed is usually not an option but dictated by financial necessity; a beneficiary files for +benefits as soon as or soon after he is qualified. However, there are undoubtably many +cases where the beneficiary can afford to wait for an optimal age for claiming his benefits. +Optimization criteria can vary greatly, of course, depending on the circumstances and goals +of the individual. In general, maximizing some measure of the total lifetime benefits can be +an appropriate goal when health issues and financial stress are not important concerns. +A number of authors over the years have studied the question of the “optimal age for +claiming Social Security benefits” from various points of view, with differing methodologies. +Here we give only a very brief summary and refer the reader to the extensive references in +the works cited here. Coile et al.[1] establish the benefits of delayed claims using financial +simulations and find that their results are in general consistent with the behavior observed +in various cohort groups. Friedman and Phillips [2] take what they refer to as a “sequen- +tial approach” in which a decision to claim or postpone is made each year. Perhaps not +∗ aydemir@utexas.edu +2 + +surprisingly, they find that a particular decision that is appropriate for a given year may +not be the correct one for the following year. Alleva[3] uses a range of discount rates to +calculate the present value of a future benefits stream to find an optimal age for claiming +benefits. As expected, higher discount rates lead to a correspondingly lower optimal age. Ali +et al.[4] do a similar optimization of the discounted present value of the future benefits but +make use of numerical simulations to take into account various factors such as interest rate +and life expectancy risks. A model based on income and wealth-level changes that includes +asset risks and mortality by Diamond et al.[5] seems to suggest that delaying until age 70 is +usually the better option unless market returns of over 7% can be guaranteed, which differs +from our conclusions. However, we are unable to pinpoint the source of this difference due +to an apparent lack of transparency in their methodology. +The problem of determining an optimal age for claiming benefits, stripped of concerns +about longevity, market risks and general economic fluctuations, essentially boils down to +an interesting optimization exercise in a multi-dimensional parameter space. In our view, +with some simplifying assumptions, this problem can be attacked essentially analytically, +yielding expressions immediately useful for most individuals. +They can also be used in +future stochastic simulations or other studies to explore the effects of various risk factors. +The opportunity for optimization arises from the way the benefits stream is structured. +Briefly, the benefits are reduced if claimed before the full retirement age (FRA) at 66 or +67[6]; similarly they increase for delayed retirement, until age 70[7]. However, there are +opportunity costs associated with delayed claims, since the benefits claimed early can be +invested in the financial markets. These factors all combine to make the task of determining +the optimal timing of Social Security retirement benefits a challenging problem, even without +longevity or market risk concerns. +Section II is a review of the usual break even analysis, the simplest calculation that a +potential beneficiary can do to determine the age he would break even if he were to delay +his claim. This Section also examines how the cost-of-living adjustments (COLAs) would +modify these calculations. Section III defines a gain function, a measure of the relative merits +of claiming early versus late (at age 70) and studies how it varies with various parameters +such as the claiming age, an assumed average market return rate and COLAs. Section IV +discusses two possible optimization paths for the claiming age. Finally Sect. V presents a +summary and conclusions of this study. +3 + +II. +A break-even analysis +Here we introduce some of the parameters and terminology, all in the context of a +commonly-performed break-even analysis. +Assume that S0 is the yearly benefit amount +that would be available at age 70, i.e., the maximum benefit allowed under the Social Secu- +rity rules for a given prospective retiree. Since the benefits can be claimed as early as age +62, we will always be comparing two cases: +• The early scenario: The beneficiary claims his benefits starting K number of years +before age 70, where 1 ≤ K ≤ 8, or +• The late scenario: He waits until age 70 when his benefits reach the maximum +amount S0. +Presently the social security benefits increase by 8% for every year they are delayed +beyond the full retirement age (FRA); the increases stop at age 70[7]. +We will use the +parameter p to denote this rate of increase, which, unless otherwise noted, will have the +value p = 0.08 throughout this work. The benefits are reduced, but at a smaller rate, if +claimed before the full-retirement age[6]. In order to simplify the calculations we will assume +symmetry about FRA and use p for the rate of reduction also. This simplification will make +our results somewhat more pessimistic, from the point of view of an early claimer, for claims +made before FRA. +With this assumption, if the benefits are claimed K years before age 70, the annual +amount is reduced to +SK = +S0 +(1 + p)K , +(1) +where p = 0.08. As stated earlier, the comparison is always with the “late scenario” where +the beneficiary claims at age 70 when his annual benefit amount is S0. +Since S0 > SK, the total earnings with the late scenario, TL, will eventually exceed those +from the early scenario, TE. After n number of years past age 70, the total benefits for the +early and late scenarios are +TE(K, n, p) = (K + n)SK = (K + n)S0 +(1 + p)K , and +(2) +TL(n) = nS0. +(3) +Note that we find it convenient to count the years before and after age 70 with different +variables, K and n, respectively. Thus, K = 2 will refer to age 70 − K = 68. Similarly +n = 10 will refer to age 70 + n = 80. +4 + +Let n = n1(K, p) be the solution of the equation TE = TL, the break-even point. We can +easily show that +n1(K, p) = +K +(1 + p)K − 1, +K ≥ 1. +(4) +A beneficiary who opts for the early scenario (1 ≤ K ≤ 8) starts falling behind at age +70 + n1; beyond that point, his cumulative benefits (TE in Eq. 2) will be less than TL that +he would have enjoyed if he had waited until age 70. In other words, for given values of the +parameters K, p, we have +TE(K, n, p) < TL(n) for n > n1. +(5) +Equivalently, viewed from the point of view of a late claimer, the break-even point comes at +age 70 + n1, after which he gets further ahead every year. (In this work, we will view the +results mostly from an early claimer’s point of view.) +For some numerical examples, we have (with p = 0.08): +K = 1 : n1 = 1 +p = 12.5 years, +K = 8 : n1 = 9.4 years. +(6) +Thus, someone who claims benefits at age 69 (K = 1) will start falling behind at age +70 + 12.5 ≃ 83. A more impatient beneficiary who starts at age 62 (K = 8) will be behind +by the time he reaches his 80th birthday. Under these conditions, and excluding all other +factors (market gains, poor health, spousal benefits, etc.) it is better to wait until age 70 +to claim your benefits, assuming that you will live beyond age 83. Note that this analysis +is independent of S0, the benefit amount at age 70, and depends only on p, the annual rate +of increase in the benefits before age 70. Further results for 1 ≤ K ≤ 8 are shown in the +column labeled n1(q = 0) of Table I. +Next we examine how these results are modified by the cost-of-living adjustments (CO- +LAs). +A. +Effects of the cost-of-living adjustments +The benefits increase in general every year due to cost-of-living adjustments implemented +to compensate for inflation. The data for these annual increases is available for the years +1975-2022 at the Social Security Administration website[8]. +Future COLA’s are of course unknown. For the purposes of these calculations we choose +to use an average COLA q based on N years of historical data; here the average q is defined +by +(1 + q)N = +N +� +i=1 +(1 + qi), +(7) +5 + +where qi is the COLA for the ith year. More explicitly, q can be put in the form +q = exp +� +1 +N +N +� +i=1 +ln(1 + qi) +� +− 1. +(8) +Using the full set of data for 1975-2022 in Eq. 8 results in an average COLA of q = 0.03745 +or 3.7%. If we exclude the high-inflation period of the late 1970’s and average only over the +last 40 years (1983-2022), we find q = 0.02508 or 2.5%. +Since the cumulative effect of these adjustments can be substantial over time, here we +examine how they modify our earlier results, in particular Eqs. 1-4. We again assume S0 is +the annual benefit amount at age 70. As in the “early scenario” above, if the beneficiary +claims benefits K years earlier, SK of Eq. 1 has to be discounted by K number of future +COLA increases; thus, the starting year benefits will now be +SC +K = +SK +(1 + q)K = +S0 +(1 + p)K(1 + q)K , +(9) +where we use the superscript C to denote COLA-adjusted values. Then, n years after age +70, the total benefits received will be +T C +E = SC +K +K+n−1 +� +i=0 +(1 + q)i = +S0 +(1 + p)K(1 + q)K +(1 + q)K+n − 1 +q +. +(10) +Since limq→0[(1 + q)K+n − 1]/q = (K + n), T C +E reduces to TE of Eq. 2 for q → 0 as expected. +In the “late scenario” where the benefits start at age 70, the total after n years will now be +T C +L = S0 +n−1 +� +i=0 +(1 + q)i = S0 +(1 + q)n − 1 +q +. +(11) +Again, T C +L reduces to TL of Eq. 3 in the limit q → 0. In order to find when an early claimer +starts to fall behind, for given K, p, q, we need to solve the equation +T C +E (K, n, p, q) = T C +L (n, q) +(12) +for n. We can easily show that the solution is (for q > 0) n = n1, where +n1 = a1(K, p, q) +a2(q) +, +(13) +a1 = ln +� (1 + p)K(1 + q)K − 1 +(1 + q)K[(1 + p)K − 1] +� +, +a2 = ln (1 + q). +Note that Eq. 13 is valid in the limit q → 0 but is not defined at q = 0. Recall that +n1 is the break-even point: For given K, p, q, and early claimer starts falling behind at age +6 + +K +n1(q = 0) +n1(q = 0.025) +n1(q = 0.037) +1 +12.5 +10.78 +10.15 +2 +12.02 +10.30 +9.67 +3 +11.55 +9.84 +9.21 +4 +11.10 +9.39 +8.77 +5 +10.65 +8.95 +8.34 +6 +10.22 +8.54 +7.93 +7 +9.81 +8.13 +7.53 +8 +9.40 +7.74 +7.15 +TABLE I: The break-even point n1(q) for three different values of the average COLA +parameter q is shown: q = 0 (no COLA), q = 2.5% (average for the years 1983-2022), and +q = 3.7% (average for the years 1975-2022). p = 0.08 is assumed. Recall that n1 measures +the number of years after age 70. +70 + n1, i.e., T C +E (n) < T C +L (n) for n > n1. Table I shows that n1 decreases as the COLA +rate q increases. For example, if the beneficiary claims at age 66 (K=4), he starts falling +behind at age 81 (n1 = 11.1) with no COLA (q = 0). If the average COLA is q = 3.7%, +he falls behind by age 79 (n1 = 8.77), two years earlier. This general decrease in n1 with +COLA is mostly due to the decrease in the initial benefits: SC +K = SK/(1 + q)K < SK (All +scenarios assume a fixed yearly benefit S0 at age 70). In summary, taking into account the +cost-of-living adjustments makes early claims less attractive. Below we investigate another +scenario where claiming early may lead to a more positive outcome. +III. +Taking advantage of possible market gains +In the previous section, we assumed that, if the beneficiary claimed his Social Security +benefits early (before age 70), he would be using them to meet his financial needs. The +analysis in this section is geared towards those beneficiaries who can afford to wait until age +70 but may nevertheless want to claim earlier, with the intention of investing the payments +they receive in the financial markets. Clearly there is no unique way to proceed in this +scenario. Investments can be made in myriad different ways. In order to obtain some clear, +quantitative results, we assume a rather simple scheme that can be improved with more +realistic details at a later date if necessary. +7 + +A. +“Early claim with market gains” scenario–no COLAs +We consider a rather straightforward scheme and, for the time being, we ignore the cost- +of-living adjustments. The assumptions we make are as follows: As before, the beneficiary +files for benefits K years before age 70, where 1 ≤ K ≤ 8. But now, at the end of each year, +until he is 70, he invests the benefits received that year in a financial instrument with an +average annual return rate of r. Starting at 70, he stops diverting his benefits to the financial +markets but allows the already-accumulated sum to grow indefinitely at the assumed annual +rate of r. +Because of the exponential growth of the funds in the market (earnings compounded +annually), it is possible for this scenario to generate more total income (integrated over +some relevant period) than the late scenario of the previous section. Here we determine a +range of values for the parameters K, r for which such a positive outcome is possible. +The average rate of return r, averaged over some number of years N (for our purposes, +N ≃ 30 years) is defined similarly to the COLA parameter q of Eqs. 7, 8: +r = exp +� +1 +N +N +� +i=1 +ln(1 + ri) +� +− 1, +(14) +where ri is the rate for the ith year. In order to explore the possible advantages of this +scenario, we start with a calculation of the total sum, TM, that the beneficiary accumulates +in the market before he reaches age 70, i.e., the total value of his social security investments +in the market when he turns 70. Recalling Eq. 1 and using the assumptions outlined above, +this sum is given by +TM = SK +K−1 +� +i=0 +(1 + r)i = +S0 +(1 + p)K +(1 + r)K − 1 +r +, +K ≥ 1. +(15) +By assumption, this sum will be left in the market to grow; therefore, n years after age 70, +the total benefits for the early claim with market gains scenario will be +T M +E = TM(1 + r)n + nSK. +(16) +The first term on the right-hand side represents the growing amount in the markets. The +second term is the total received (and not invested) after age 70. T M +E can be written explicitly +as +T M +E = +S0 +(1 + p)K +� +(1 + r)n � +(1 + r)K − 1 +� +r ++ n +� +. +(17) +This equation replaces Eq. 2 of the previous section. Note that in the limit r → 0 (thus +removing the market gains), Eq. 17 reduces to Eq. 2 as expected, since +lim +r→0 +(1 + r)n � +(1 + r)K − 1 +� +r += K. +(18) +8 + +At this point it will be useful to define a relative gain function g that will measure the +early claimer’s gains (or losses) as a function of time: +g(K, n, p, r) ≡ T M +E (K, n, p, r) − TL(n) +TL(n) +, +(19) +where TL(n) is given by Eq. 3. For fixed K, p, r, a positive gain function implies that the +beneficiary is ahead (n years after 70) with respect to the “late scenario” where he waits +until age 70 to start collecting his benefits. +Using Eqs. 3, 17, the gain function can be written explicitly in the form +g(K, n, p, r) = +1 +(1 + p)K +��(1 + r)K − 1 +r +� �(1 + r)n +n +� ++ 1 +� +− 1. +(20) +For fixed K, p, r we have +lim +n→0 g(K, n, p, r) = ∞, +lim +n→∞ g(K, n, p, r) = ∞. +(21) +Thus, g(K, n, p, r) is a non-monotonic, convex function of n with a minimum at some n = n∗. +The point where the gain function attains its minimum value (again for fixed K, p, r) can +be calculated easily. Treating n as a continuous time variable and letting +∂g +∂n +���� +n=n∗ += 0 +(22) +leads to +n∗(r) = +1 +ln (1 + r), +(23) +which is independent of the parameters K, p. Making use of n∗, we find the minimum gain +gmin(K, p, r) = g[K, n∗(r), p, r]. +(24) +In the following analysis, the return rate r = r∗ for which gmin = 0 will play an important +role; it can be calculated by solving the equation +g[K, n∗(r∗), p, r∗] = 0 +(25) +for r∗ with given K, p. In general this step requires a numerical solution, but the case K = 1 +can be treated analytically because of simplifications in the function g(K, n, p, r). For fixed +p and K = 1, Eqs. 23 and 25 lead to +n∗ = e +p, +r∗ = ep/e − 1 +(K = 1), +(26) +where e is the base of the natural logarithm. +For K > 1 the analytic results are not +very illuminating, and we use Mathematica[9] for numerical solutions. Values of n∗, r∗ for +9 + +1 ≤ K ≤ 8 without the COLA modifications are tabulated in the columns labeled n∗(q = 0) +and r∗(q = 0), respectively, of Table II. +An important property of the gain function that will be useful below is +r1 > r2 ⇒ g(K, n, p, r1) > g(K, n, p, r2), +(27) +which follows directly from Eq. 20. It is also intuitively obvious, since higher market yields +will naturally lead to higher relative gains. +r=r*-0.005 +r=r* +r=r*+0.005 +20 +40 +60 +80 +0 +0.025 +0.050 +0.075 +0.100 +n +g +FIG. 1: The gain function g(K, n, p, r), with no COLAs, for various values of r, and +K = 1, p = 0.08. For r = r∗ = 0.02987, g(K, n, p, r) ≥ 0; the minimum (zero) occurs at +n = n∗ = 33.98 (see the text). The horizontal axis shows the number of years after age 70. +As an example, behavior of g(K, n, p, r) is shown in Fig. 1 for various values of r and +K = 1, p = 0.08. As seen in the figure, for given K, p, r, the equation g(K, n, p, r) = 0 can +have zero, one, or two solutions for n: +• No solutions (the solid green curve); here g(K, n, p, r) > 0 for all n, implying that the +early-claimer is always ahead for this value of r, the average market return rate. +• Only one solution, at n = n∗ for r = r∗ (the dotted red curve); here the beneficiary +is always ahead except at n = n∗, where his position temporarily becomes neutral +(g = 0). The parameters n∗, r∗ will play a critical role in our analysis, since r > r∗ +implies g(K, n, p, r) > g(K, n, p, r∗) ≥ g(K, n∗, p, r∗) = 0 for all n. +• Two distinct solutions, n = n1, n2 (the dashed blue curve). For this value of r, the +beneficiary starts falling behind at n = n1 (age 70+n1) since g < 0 for n1 < n < n2. +The gain function is positive again for n > n2; however, 70 + n2 is too large a number +to be relevant for most cases. For the parameters used in Fig. 1, we have n1 = 20.87 +and n2 = 70.32, again found numerically with Mathematica[9]. +10 + +K +n∗(q = 0) +r∗(q = 0) +n∗(q = 0.025) +r∗(q = 0.025) +n∗(q = 0.037) +r∗(q = 0.037) +1 +33.98 +0.02987 +34.58 +0.04394 +35.69 +0.05128 +2 +33.17 +0.03061 +33.53 +0.04483 +34.45 +0.05221 +3 +32.39 +0.03135 +32.54 +0.04573 +33.30 +0.05314 +4 +31.65 +0.03210 +31.62 +0.04662 +32.24 +0.05406 +5 +30.93 +0.03286 +30.75 +0.04751 +31.26 +0.05498 +6 +30.23 +0.03363 +29.93 +0.04840 +30.35 +0.05589 +7 +29.57 +0.03440 +29.16 +0.04928 +29.52 +0.05679 +8 +28.93 +0.03517 +28.44 +0.05016 +28.74 +0.05767 +TABLE II: The critical parameters n∗ and r∗ for the “early claim with market gains” +scenario for various values of K, and p = 0.08. The columns with q = 0 (no COLAs) are +solutions of Eqs. 23, 25. The remaining columns with q > 0 (with COLAs) are solutions of +Eqs. 36, 37. +In order to emphasize the meaning of these important parameters, we will go through +an example. Making use of Table II, assume that a beneficiary claims his benefits at age +62 (K = 70 − 62 = 8) and invests his Social Security income in some market instrument +(as detailed previously) with an average annual return rate of r∗ ≃ 3.5%. +With these +assumptions, his cumulative benefits will continue to exceed those from the “late scenario,” +T M +E +> TL, until he is 70 + n∗ ≃ 99 years old. At that time his position becomes neutral +(T M +E = TL) briefly (the break-even point from the point of view of the “late scenario”), but +he gets ahead again beyond that age. And a final crucial point is that, for any given K, if +r > r∗ the beneficiary remains always ahead: T M +E > TL for all n. +Next we examine how these results are modified when the cost-of-living adjustments are +included in the analysis. +B. +“Early claim with market gains” scenario–with COLAs +Making use of some of our earlier results, here we look at the effects of cost-of-living +adjustments. With COLA’s, the starting year benefit SC +K (see Eq. 9) will grow both due to +market gains and COLAs; thus, the total sum the beneficiary accumulates in the market +before age 70 will now be +T C +M = SC +K +K−1 +� +i=0 +(1 + q)i(1 + r)i = +S0 +(1 + p)K(1 + q)K +(1 + q)K(1 + r)K − 1 +(1 + q)(1 + r) − 1 , +(28) +11 + +which replaces TM of Eq. 15. (Recall that the COLA-modified terms have a superscript C). +By assumption, T C +M is left in the market after reaching 70 and continues to grow at the rate +r. After 70, however, the new COLA-modified benefits are not invested but presumably +used for other purposes. Thus, T M +E of Eq. 17 representing the accumulated benefits at age +70 + n now becomes +T MC +E += T C +M(1 + r)n + SK +n−1 +� +i=0 +(1 + q)i, or +T MC +E += +S0(1 + r)n +(1 + p)K(1 + q)K +(1 + q)K(1 + r)K − 1 +(1 + q)(1 + r) − 1 ++ +S0 +(1 + p)K +(1 + q)n − 1 +q +. +(29) +The total benefits at age 70 + n with the “late scenario” and including COLAs is still given +by Eq. 11: +T C +L = S0 +(1 + q)n − 1 +q +. +(30) +At this point we can do a couple of consistency checks: Since limq→0[(1 + q)n − 1]/q = n, +T MC +E +and T C +L reduce to T M +E and TL of Eqs. 17, 3, respectively, when the COLAs are ignored +(q → 0). Similarly, when the COLAs are retained but the market gains are ignored (r → 0), +T MC +E +reduces to T C +E of Eq. 10 as expected. +As we saw earlier in Sect. II A on the effects of COLAs without the market gains, we +intuitively expect that the benefits of claiming early will be reduced with the COLAs, even +when the market gains are taken into account. In particular, we expect that the ratio of the +average market return to the average COLA, r/q, will play an important role in determining +how useful an early claim will be. In order to understand these issues better, we again look +at the gain function and its time derivative. The new gain function with the COLAs can be +written as +gc(K, n, p, q, r) ≡ T MC +E +− T C +L +T C +L +, +or +(31) +gc(K, n, p, q, r) = +1 +(1 + p)K +�� +(1 + q)K(1 + r)K − 1 +(1 + q)K[(1 + q)(1 + r) − 1] +� � q(1 + r)n +(1 + q)n − 1 +� ++ 1 +� +− 1, +which reduces to g(K, n, p, r) of Eq. 20 in the limit q → 0, as expected. Letting +A(K, p, q, r) ≡ +q +(1 + p)K +(1 + q)K(1 + r)K − 1 +(1 + q)K[(1 + q)(1 + r) − 1], +(32) +we can write the time derivative ∂gc/∂n in the form +∂gc +∂n = A(1 + r)n +�(1 + q)n ln[(1 + r)/(1 + q)] − ln(1 + r) +[(1 + q)n − 1]2 +� +. +(33) +By simple inspection, we can draw some general conclusions from Eqs. 31-33: +12 + +• The coefficient A(K, p, q, r) > 0 for r, q > 0. Thus, +∂gc +∂n < 0 for 0 < r ≤ q. +(34) +In other words, when the market return rate is less than the average COLA, the +relative gain will be a monotonically decreasing function of time (for fixed K, p, q, r). +In fact, for strict inequality, r < q, we have +lim +n→∞ gc = +1 +(1 + p)K − 1 < 0. +(35) +Under these circumstances, the equation gc = 0 will have only one solution (n = n1). +An early-claiming beneficiary will fall behind at age 70 + n1 and will never recover +(gc(K, n, p, q, r) ≤ 0 for n ≥ n1). +• For r > q > 0, limn→∞ gc = +∞, and the gain function displays a behavior similar to +what we observed earlier when we considered the market gains without the COLAs. +It will have a minimum at n = n∗, where n∗ is a solution of the equation ∂gc/∂n = 0. +We can easily show that +n∗(q, r) = +1 +ln (1 + q) ln +� +ln (1 + r) +ln [(1 + r)/(1 + q)] +� +. +(36) +Note that limr→q n∗ = ∞, consistent with the discussion above. Now the equation +gc = 0 may again have zero, one or two solutions, depending on the parameters +K, p, q, r. As in Sect. III A, the parameter r∗ will be a solution, for fixed K, p, q, of the +equation +gc[K, n∗(q, r∗), p, q, r∗] = 0. +(37) +In order to re-emphasize the significance of the parameters n∗, r∗, we summarize here +some of their important properties: +gmin(K, p, q, r) = gc[K, n∗(q, r), p, q, r], +(38) +gmin(K, p, q, r∗) = gc[K, n∗(q, r∗), p, q, r∗] = 0. +(39) +In other words, the gain function has a minimum at n = n∗, and that minimum is +zero for r = r∗. A consequence of Eqs. 27, 38, 39, for fixed K, p, q, is +gc(K, n, p, q, r > r∗) > 0 for all n. +(40) +Thus, an early-claiming beneficiary will always be ahead if he can guarantee an average +market return rate of r > r∗, even with the COLAs. However, we will find that, with +everything else fixed, r∗ will be higher with the COLAs. +13 + +These different types of behavior for the gain function gc are illustrated in Fig. 2 for +K = +1, p = 0.08 and q = 0.025, the average COLA for the years 1983 − 2022. +The +dashed blue curve has r = 0.02 < q; hence the negative slope and a zero at n1 = 13.6. The +early claimer falls behind at 70 + n1 ≃ 84 and never recovers. The dotted red curve is for +r = r∗ = 0.04394 > q. It has a minimum (zero) at n∗ = 34.58 : gc(n∗, r∗) = 0. The solid +green curve has r = 0.05 > r∗ > q; hence gc > 0 for all n. +r=0.02 +r=r*=0.04394 +r=0.05 +20 +40 +60 +80 +-0.05 +0 +0.05 +0.10 +0.15 +n +g +FIG. 2: The gain function gc(n) (with COLAs) for three different values of r, and +K = 1, p = 0.08, q = 0.025. For these parameters we have r∗ = 0.04394, n∗ = 34.58 +(Table II). The dashed blue curve: r = 0.02 < q. The dotted red curve: r = r∗ > q; it has +a minimum at n = n∗ where gc(n∗, r∗) = 0. The solid green curve: r = 0.05 > r∗ > q. +In Table II, columns with q > 0 are solutions of the coupled Eqs. 36 and 37. Since they +are both complicated functions of the parameters, they are solved iteratively for {n⋆, r}, +with the converged solution yielding {n∗(q, r∗), r∗ = r}. Note that the solutions of Eqs. 36, +37 (q > 0, with COLA) will agree with those of Eqs. 23, 25 (q = 0, no COLA) in the limit +q → 0. We find that we are able to reproduce the q = 0 results in the Table (obtained using +Eqs. 23, 25) from Eqs. 36, 37 with 0 < q < 10−5. However, q = 0 is a singular limit for the +equations with COLA, since they are undefined at q = 0. +Some general comments regarding the {n∗, r∗} values in Table II may be helpful. Recall +that the gain function (Eq. 31) has a minimum at n = n∗(q, r). The minimum value is zero +for r = r∗, i.e., gc(n∗, r∗) = 0, where we assume the other parameters, K, p, q are held fixed. +Recalling Eqs. 38-40, for r > r∗ we have +gc(n, r) ≥ gc(n∗, r) > gc(n∗, r∗) = 0. +(41) +Thus, a beneficiary who files early will have a positive gain throughout his life, if he can +maintain an average market return rate of r > r∗. The values of n∗ in Table II are ap- +proximately in the range 28 − 36; thus, the minimum of the gain function occurs when the +14 + +beneficiary is in his late 90’s (70+28) at the earliest (for r ≥ r∗). In order to maintain posi- +tive gains during this time, he has to ensure an average market return rate of approximately +3 − 3.5% (depending on how early he claims), if the COLAs are not taken into account +(q = 0 columns in the table). With COLAs, the required rate of return is approximately +4.4 − 5.0% for q = 2.5%, the average COLA for the years 1983 − 2022. The required rate +may be as high as 5.8% if q = 3.7%, the average for the years 1975 − 2022 (the entire data +set at [8]), is used. +IV. +Optimization +In the previous section we were concerned with finding parameter regimes where an early +claimer can have positive gains with respect to the “late scenario,” possibly for the rest of +his life, when both the cost-of-living adjustments and possible market gains are taken into +account. It is possible to go a step further and find an optimal time for claiming benefits as a +function of the remaining parameters in the problem. Mathematically, the problem reduces +to finding a maximum for the gain function gc = gc(K, n, p, q, r) in a 5-dimensional space. +Clearly the problem is quite unwieldy in this form; by choosing p = 0.08, as we have done +throughout this work, and using q = 0.025 for the average COLA parameter, the dimension +can be reduced to a more manageable three. Then it is possible to find an optimal time for +claiming benefits, or a Kopt(n, r), as a function of an assumed average market return rate, +r, and a time, n, in the future. Below we examine a couple of different optimization paths. +A. +Maximizing the minimum gain +Here instead of choosing a specific time in the future, we let n = n∗ (see Eq. 36) where the +gain function has its minimum, i.e., we seek a Kopt that maximizes gc +min. Recall that for the +range of parameters q, r we have been considering, n∗ ≃ 28 − 36 (Table II), corresponding +to an age range of 98-106. +The mathematical problem can now be stated as follows: +• Find K = Kopt(n∗, r) that maximizes gc(K, n∗, p, q, r) for p = 0.08, q = 0.025 and an +arbitrary r > q (Recall that gc is monotonically decreasing for r ≤ q, a regime we try +to avoid). +15 + +We start by slightly rewriting gc of Eq. 31: +gc(K, n, p, q, r) = B(n, q, r) +� +[(1 + r)/(1 + p)]K − +1 +[(1 + p)(1 + q)]K +� ++ +1 +(1 + p)K − 1, +where B(n, q, r) ≡ +q(1 + r)n +[(1 + q)(1 + r) − 1][(1 + q)n − 1]. +(42) +Then treating K as a continuous variable and setting ∂gc/∂K = 0 leads to +B +� +(1 + r)K ln +�1 + r +1 + p +� ++ ln[(1 + p)(1 + q)] +(1 + q)K +� +− ln(1 + p) = 0. +(43) +Although Eq. 43 can be solved for K = Kopt analytically in certain limits (e.g., r = p), in +general it requires a numerical solution. Figure 3 shows Kopt(n∗, r) (from Eq. 43), n∗(q, r) +(from Eq. 36) and gc +min = gc(Kopt, n∗, q, p, r) as functions of r. The limits for the average +Kopt +0.045 +0.050 +0.055 +0.060 +0 +2 +4 +6 +8 +r +Kopt +(a) +n* +22 +24 +26 +28 +30 +32 +34 +n* +(b) +0.045 +0.050 +0.055 +0.060 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +r +gmin +FIG. 3: (a) Kopt that maximizes the minimum of the relative gain, gc +min, as a function of +the assumed average market return rate r. Recall that the optimal age is given by +70 − Kopt. Also shown is the parameter n∗(q, r) where the minimum occurs. Again recall +that n measures years after age 70; thus, gc +min occurs at age 70 + n∗. (b) gc +min as a function +of r. +market return rate, 0.0440 ≤ r ≤ 0.0598, are chosen to keep Kopt within the allowed arrange +of 0 < Kopt ≤ 8 (the optimal age has to be within 62 − 70). A number of observations +follows from Fig. 3: +• Kopt increases approximately linearly with the market return rate (Fig. 3(a)); thus, +the higher the market rate, the earlier is the optimal claiming age, 70 − Kopt, which is +intuitively obvious. +• The optimized minimum gain, gc +min, increases faster than linearly with r (Fig. 3(b)). +At r = 6%, the optimal claiming age is 62 (Kopt = 7.99). The beneficiary’s minimum +gain with respect to the late scenario is 14%, which occurs at age 70 + n∗ = 92.4. At +any other age, his relative gain is higher since gc(Kopt, n, p, q, r) > gmin for n ̸= n∗. +16 + +• The sum Kopt +n∗, representing the time span from the claiming age to when the gain +function attains its minimum, is relatively constant: 30.4 ≤ Kopt + n∗ ≤ 35.0, which +has two implications: +– When calculating the average COLA rate q (Eq. 8) or the expected average +market return rate r (Eq. 14), the time span to consider is approximately 30-35 +years from the start of the benefits. This is a fairly long period that should be +enough to smooth out the market or economic fluctuations. Thus r = 5 − 6% +and q = 2.5% are not unreasonable assumptions. +– Regardless of where the optimal claiming age happens to be, the optimized mini- +mum gain is approximately 30-35 years in the future from that point. And neither +ill-health nor extreme longevity will leave the beneficiary worse off than what is +implied by gc +min. Therefore, finding a claiming age that maximizes the minimum +of the relative gain is a reasonable strategy. +B. +Optimizing for a particular age +At this point a complication that we should consider is the following: Optimizing gc +min +does not imply that the gain function itself has been optimized for all ages. In other words, +we may still have gc(Kopt, n, p, q, r) < gc(K, n, p, q, r) for some values of K ̸= Kopt, n ̸= n∗. +In fact, the equation +gc(Kopt, n, p, q, r) = gc(K, n, p, q, r), +(44) +with fixed p, q, r tends to have two distinct solutions, n = n1, n2, for K > Kopt. Thus the +gain function for K > Kopt will be higher for n < n1 and n > n2. This point is illustrated +in Fig. 4 where we plot gc as a function of n for various values of K, and p = 0.08, q = +0.025, r = 0.0525. The minima for all three curves are at n = n∗(q, r) = 26.7, and clearly the +minimum for K = Kopt = 4.69 curve is higher than those of the K = 2, 7 curves. However, +the K = 7 curve (dotted red) crosses above the K = Kopt curve (solid green) at n1 = 17.28 +and n2 = 39.13. In other words, although gc +min is optimized with K = Kopt = 4.69, for the +years corresponding to n < n1 (earlier than age 87), or n > n2 (later than age 109), the +beneficiary may be better off with K = 7, i.e., claiming at age 63 instead of 70 − Kopt ≃ 65. +In order to address this particular complication, the gain function itself (not its mini- +mum) can be optimized for any given age using the analytic expressions given above. The +mathematical problem we need to solve now is the following: +• Find solutions of Eq. 43 for K = Kopt(n, r), which maximizes gc(K, n, p, q, r) with +given p, q, r, and for various values of n while ensuring that 0 < Kopt ≤ 8. +17 + +K=2 +K=Kopt=4.69 +K=7 +20 +25 +30 +35 +40 +0.04 +0.06 +0.08 +0.10 +n +g +FIG. 4: The relative gain function gc(K, n, p, q, r) for three different values of K, with +p = 0.08, q = 0.025 and r = 0.0525. The solid green curve is for K = Kopt, optimized for +n = n∗. The dashed blue curve for K = 2 < Kopt always remains below the optimized +curve; however, the dotted red curve for K = 7 > Kopt crosses above the optimal curve at +n = n1 = 17.28 and n = n2 = 39.13. +Kopt +10 +20 +30 +40 +50 +0 +2 +4 +6 +8 +n +Kopt +(a) r=0.045 +gc +0.00 +0.05 +0.10 +0.15 +0.20 +gc +Kopt +15 +20 +25 +30 +7.0 +7.2 +7.4 +7.6 +7.8 +8.0 +n +Kopt +(b) r=0.0575 +gc +0.11 +0.12 +0.13 +0.14 +0.15 +gc +FIG. 5: Kopt and the optimized value of the gain function gc as a function of n, the +number of years after age 70. Recall that the optimal claiming age is given by 70 − Kopt, +and gc measures the gain with respect to the “late scenario” where the beneficiary waits +until age 70 to claim. (a) Average market return rate r = 0.045. (b) r = 0.0575. Note that +in (b) although the two curves nearly coincide, the scales are different. +A couple of examples are shown in Fig. 5, again for p = 0.08, q = 0.025. Recall that +gc +min(K, p, q, r∗) = gc(K, n∗, p, q, r∗) = 0. For the first example, in order to ensure gc +min > 0, +we choose r = 0.045, which is just above the minimum r∗ for q = 0.025 in Table II. The +results are shown in Fig. 5(a). For any given n, the figure shows Kopt and the resulting +optimized gain. For example, for n = 10 (age 80), we have Kopt = 7.29, gc = 0.175, a again +of 17.5% over the late scenario. For n = 20 (age 90), we get Kopt = 2.70 and gc = 0.018, +a gain of only 1.8% over the late scenario. The minimum gain is at n = n∗ = 33.34 (age +103), where it is a mere 0.3%. In Fig. 5(b) where we assume a slightly higher return rate +of 5.75%, the gains are markedly higher. For 15 < n < 35 (ages 85-105), the optimized gc +18 + +varies between a minimum of 10.8% at n = n∗ = 23.6 and 15.2% at the end points of the +range. Similarly, Kopt varies little: 6.94 ≤ Kopt ≤ 8 for this value of r. Thus, with a rather +modest market return of 5.8%, a beneficiary who claims at age 62-63 would indeed be well +ahead for the rest of his life. +V. +Summary and conclusions +The Social Security benefits provide a financial lifeline to most retirees, and a retiree +typically claims his benefits as soon as he reaches his full-retirement age (FRA), or as early +as age 62 if he is willing to accept some penalties. If a retiree is not financially constrained, +however, there is an incentive for waiting until age 70 because of the built-in yearly increases +to benefits (presently 8%) between his FRA and age 70. Ironically, precisely in those cases +where filing at FRA or earlier is not a financial necessity, it may be beneficial to file early +without waiting until age 70. This work focused on the decision of whether to do so and +when the optimal time might be. +Starting with a simple “break-even” analysis that retirees typically do to find out when +they would break even if they delayed retirement beyond FRA, we developed a series of +successively more comprehensive analytic models that compare early-claiming scenarios with +the “late scenario” where the beneficiary waits until age 70. First improvement on the break- +even analysis was the inclusion of inflation through the cost-of-living adjustments (COLAs) +in Sect. II. From the point of view of an early-claimer (a view adopted throughout this +work) COLAs make early claims less attractive. Conversely, from the point of view of a late +claimer, they bring the break-even point closer (Table I). Results of filing early and investing +the benefits received in the financial markets were examined in Sect. III. As summarized in +Table II, if the COLAs are ignored, even a modest market return of 3.5% would be sufficient +for a beneficiary who files at age 62 to be ahead of the late scenario all his life. If the COLAs +are taken into account, that return rate would have to be 5% if the average COLA is 2.5%, +and 5.8% if the COLA rate is at its historical average of 3.7%. Since the timespan relevant +to our discussion is approximately three decades from the time of claiming benefits, these +market rates, averaged over 30-35 years, are not unreasonable. +If the beneficiary wants to be not just ahead of the late scenario but to optimize his +gains, then he has a number of options, of which we examined only two. First, we found +an optimal filing age that maximizes his minimum gain, gc +min, as a function of the assumed +market return rate (Fig. 3). For example, if the average market rate r = 5.8%, we find +Kopt = 6.9, implying an optimal age of approximately 63 (70 − Kopt) for claiming; the +minimum gain of 11% occurs at n∗ = 23.6, corresponding to an age of over 93 (70 + n∗). +19 + +Both before and after that age, the gain is higher (on either side of the minimum). +The second option we examined was finding an optimal claiming age that maximizes +the relative gain at a specific point in time, for two different market rates (Fig. 5). We +found that the claiming age is rather insensitive to the age for which the gain is maximized +(Kopt ≃ 7 − 8) if r = 5.8%, although there is considerable variation in both Kopt and the +maximized gain for r = 4.5%. +The short conclusion from these analyses is that, under the conditions explained at the +beginning of Sect. III, there is no financial incentive for waiting until age 70 for claiming +Social Security benefits. A beneficiary would be almost always ahead if he were to claim +quite early (age 62-63), if the market return rate on his investments over the next 30-35 +years is approximately 5 − 6%. +Real-life scenarios are usually more complex than the simplified mathematical models +discussed in this work. We made some assumptions in order to make analytical progress, +and we did not consider important factors such as taxes and spousal benefits. It would +also be beneficial to examine optimization that takes into account a probabilistic survival +function. We hope to address these issues in a future publication. It is important to note +that, while we provided examples in our figures and tables for a variety of parameters, the +analytic expressions derived here can be useful more generally. An individual can customize +the choice of parameters to his own circumstances and use these models to make a timing +decision that aligns with his goals better. +Acknowledgments +The author gratefully acknowledges useful comments by Gary Hallock that helped im- +prove the manuscript. This research did not receive any specific grant from funding agencies +in the public, commercial, or not-for-profit sectors. +[1] C. Coile, P. Diamond, J. Gruber, and A. Jousten, Journal of Public Economics 84, 357 (2002). +[2] J. Friedman and H. Phillips, Financial Services Review 17, 155 (2008). +[3] B. J. Alleva, Social Security Bulletin 76, 1 (2016). +[4] Y. Ali, M. Fang, P. A. A. Sota, S. Taylor, and X. Wang, Risks 7, 124 (2019). +[5] S. Diamond, S. Boyd, D. Greenberg, M. Kochenderfer, and A. Ang, https://doi.org/10. +48550/arXiv.2106.00125 (2021). +20 + +[6] https://www.ssa.gov/benefits/retirement/planner/agereduction.html. +[7] https://www.ssa.gov/benefits/retirement/planner/delayret.html. +[8] https://www.ssa.gov/oact/cola/colaseries.html. +[9] Wolfram Research, Inc., Mathematica, Version 13.1, Champaign, IL, 2022, URL https:// +www.wolfram.com/mathematica. +21 + diff --git a/P9E2T4oBgHgl3EQfrwhD/content/tmp_files/load_file.txt b/P9E2T4oBgHgl3EQfrwhD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..728ec41b707e73dd5d152b52a0c6f151f0755ff0 --- /dev/null +++ b/P9E2T4oBgHgl3EQfrwhD/content/tmp_files/load_file.txt @@ -0,0 +1,681 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf,len=680 +page_content='Optimal social security timing A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Aydemir∗ (Dated: January 11, 2023) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='04052v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='GN] 10 Jan 2023 Abstract The optimal age that a retiree claims social security retirement benefits is in general a complicated function of many factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' However, if the beneficiary’s finances and health are not the constraining factors, it is possible to formally derive mathematical models that maximize a well-defined measure of his total benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' A model that takes into account various factors such as the increase in the benefits for delayed claims and the penalties for early retirement, the advantages of investing some of the benefits in the financial markets, and the effects of cost-of-living adjustments shows that not waiting until age 70 is almost always the better option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The optimal claiming age that maximizes the total benefits, however, depends on the expected market returns and the rate of cost-of-living adjustments, with the higher market rates in general pushing the optimal age lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The models presented here can be easily tailored to address the particular circumstances and goals of any individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Keywords: Social Security;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' optimal timing I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Introduction Social Security retirement benefits are an essential part of the “Social Safety Net” in the US and provide much needed income to many retirees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Thus, at what age they are claimed is usually not an option but dictated by financial necessity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' a beneficiary files for benefits as soon as or soon after he is qualified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' However, there are undoubtably many cases where the beneficiary can afford to wait for an optimal age for claiming his benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Optimization criteria can vary greatly, of course, depending on the circumstances and goals of the individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In general, maximizing some measure of the total lifetime benefits can be an appropriate goal when health issues and financial stress are not important concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' A number of authors over the years have studied the question of the “optimal age for claiming Social Security benefits” from various points of view, with differing methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Here we give only a very brief summary and refer the reader to the extensive references in the works cited here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Coile et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' [1] establish the benefits of delayed claims using financial simulations and find that their results are in general consistent with the behavior observed in various cohort groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Friedman and Phillips [2] take what they refer to as a “sequen- tial approach” in which a decision to claim or postpone is made each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Perhaps not ∗ aydemir@utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='edu 2 surprisingly, they find that a particular decision that is appropriate for a given year may not be the correct one for the following year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Alleva[3] uses a range of discount rates to calculate the present value of a future benefits stream to find an optimal age for claiming benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' As expected, higher discount rates lead to a correspondingly lower optimal age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Ali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' [4] do a similar optimization of the discounted present value of the future benefits but make use of numerical simulations to take into account various factors such as interest rate and life expectancy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' A model based on income and wealth-level changes that includes asset risks and mortality by Diamond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' [5] seems to suggest that delaying until age 70 is usually the better option unless market returns of over 7% can be guaranteed, which differs from our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' However, we are unable to pinpoint the source of this difference due to an apparent lack of transparency in their methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The problem of determining an optimal age for claiming benefits, stripped of concerns about longevity, market risks and general economic fluctuations, essentially boils down to an interesting optimization exercise in a multi-dimensional parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In our view, with some simplifying assumptions, this problem can be attacked essentially analytically, yielding expressions immediately useful for most individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' They can also be used in future stochastic simulations or other studies to explore the effects of various risk factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The opportunity for optimization arises from the way the benefits stream is structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Briefly, the benefits are reduced if claimed before the full retirement age (FRA) at 66 or 67[6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' similarly they increase for delayed retirement, until age 70[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' However, there are opportunity costs associated with delayed claims, since the benefits claimed early can be invested in the financial markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' These factors all combine to make the task of determining the optimal timing of Social Security retirement benefits a challenging problem, even without longevity or market risk concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Section II is a review of the usual break even analysis, the simplest calculation that a potential beneficiary can do to determine the age he would break even if he were to delay his claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' This Section also examines how the cost-of-living adjustments (COLAs) would modify these calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Section III defines a gain function, a measure of the relative merits of claiming early versus late (at age 70) and studies how it varies with various parameters such as the claiming age, an assumed average market return rate and COLAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Section IV discusses two possible optimization paths for the claiming age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Finally Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' V presents a summary and conclusions of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 3 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' A break-even analysis Here we introduce some of the parameters and terminology, all in the context of a commonly-performed break-even analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Assume that S0 is the yearly benefit amount that would be available at age 70, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=', the maximum benefit allowed under the Social Secu- rity rules for a given prospective retiree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Since the benefits can be claimed as early as age 62, we will always be comparing two cases: The early scenario: The beneficiary claims his benefits starting K number of years before age 70, where 1 ≤ K ≤ 8, or The late scenario: He waits until age 70 when his benefits reach the maximum amount S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Presently the social security benefits increase by 8% for every year they are delayed beyond the full retirement age (FRA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' the increases stop at age 70[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' We will use the parameter p to denote this rate of increase, which, unless otherwise noted, will have the value p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='08 throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The benefits are reduced, but at a smaller rate, if claimed before the full-retirement age[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In order to simplify the calculations we will assume symmetry about FRA and use p for the rate of reduction also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' This simplification will make our results somewhat more pessimistic, from the point of view of an early claimer, for claims made before FRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' With this assumption, if the benefits are claimed K years before age 70, the annual amount is reduced to SK = S0 (1 + p)K , (1) where p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' As stated earlier, the comparison is always with the “late scenario” where the beneficiary claims at age 70 when his annual benefit amount is S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Since S0 > SK, the total earnings with the late scenario, TL, will eventually exceed those from the early scenario, TE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' After n number of years past age 70, the total benefits for the early and late scenarios are TE(K, n, p) = (K + n)SK = (K + n)S0 (1 + p)K , and (2) TL(n) = nS0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (3) Note that we find it convenient to count the years before and after age 70 with different variables, K and n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Thus, K = 2 will refer to age 70 − K = 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Similarly n = 10 will refer to age 70 + n = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 4 Let n = n1(K, p) be the solution of the equation TE = TL, the break-even point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' We can easily show that n1(K, p) = K (1 + p)K − 1, K ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (4) A beneficiary who opts for the early scenario (1 ≤ K ≤ 8) starts falling behind at age 70 + n1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' beyond that point, his cumulative benefits (TE in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 2) will be less than TL that he would have enjoyed if he had waited until age 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In other words, for given values of the parameters K, p, we have TE(K, n, p) < TL(n) for n > n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (5) Equivalently, viewed from the point of view of a late claimer, the break-even point comes at age 70 + n1, after which he gets further ahead every year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (In this work, we will view the results mostly from an early claimer’s point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=') For some numerical examples, we have (with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='08): K = 1 : n1 = 1 p = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='5 years, K = 8 : n1 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='4 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (6) Thus, someone who claims benefits at age 69 (K = 1) will start falling behind at age 70 + 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='5 ≃ 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' A more impatient beneficiary who starts at age 62 (K = 8) will be behind by the time he reaches his 80th birthday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Under these conditions, and excluding all other factors (market gains, poor health, spousal benefits, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=') it is better to wait until age 70 to claim your benefits, assuming that you will live beyond age 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Note that this analysis is independent of S0, the benefit amount at age 70, and depends only on p, the annual rate of increase in the benefits before age 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Further results for 1 ≤ K ≤ 8 are shown in the column labeled n1(q = 0) of Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Next we examine how these results are modified by the cost-of-living adjustments (CO- LAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Effects of the cost-of-living adjustments The benefits increase in general every year due to cost-of-living adjustments implemented to compensate for inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The data for these annual increases is available for the years 1975-2022 at the Social Security Administration website[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Future COLA’s are of course unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' For the purposes of these calculations we choose to use an average COLA q based on N years of historical data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' here the average q is defined by (1 + q)N = N � i=1 (1 + qi), (7) 5 where qi is the COLA for the ith year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' More explicitly, q can be put in the form q = exp � 1 N N � i=1 ln(1 + qi) � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (8) Using the full set of data for 1975-2022 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 8 results in an average COLA of q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='03745 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' If we exclude the high-inflation period of the late 1970’s and average only over the last 40 years (1983-2022), we find q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='02508 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Since the cumulative effect of these adjustments can be substantial over time, here we examine how they modify our earlier results, in particular Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' We again assume S0 is the annual benefit amount at age 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' As in the “early scenario” above, if the beneficiary claims benefits K years earlier, SK of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 1 has to be discounted by K number of future COLA increases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' thus, the starting year benefits will now be SC K = SK (1 + q)K = S0 (1 + p)K(1 + q)K , (9) where we use the superscript C to denote COLA-adjusted values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Then, n years after age 70, the total benefits received will be T C E = SC K K+n−1 � i=0 (1 + q)i = S0 (1 + p)K(1 + q)K (1 + q)K+n − 1 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (10) Since limq→0[(1 + q)K+n − 1]/q = (K + n), T C E reduces to TE of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 2 for q → 0 as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In the “late scenario” where the benefits start at age 70, the total after n years will now be T C L = S0 n−1 � i=0 (1 + q)i = S0 (1 + q)n − 1 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (11) Again, T C L reduces to TL of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 3 in the limit q → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In order to find when an early claimer starts to fall behind, for given K, p, q, we need to solve the equation T C E (K, n, p, q) = T C L (n, q) (12) for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' We can easily show that the solution is (for q > 0) n = n1, where n1 = a1(K, p, q) a2(q) , (13) a1 = ln � (1 + p)K(1 + q)K − 1 (1 + q)K[(1 + p)K − 1] � , a2 = ln (1 + q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 13 is valid in the limit q → 0 but is not defined at q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Recall that n1 is the break-even point: For given K, p, q, and early claimer starts falling behind at age 6 K n1(q = 0) n1(q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='025) n1(q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='037) 1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='78 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='15 2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='02 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='30 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='67 3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='55 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='84 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='21 4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='39 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='77 5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='65 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='95 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='34 6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='22 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='54 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='93 7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='81 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='13 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='53 8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='40 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='74 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='15 TABLE I: The break-even point n1(q) for three different values of the average COLA parameter q is shown: q = 0 (no COLA), q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='5% (average for the years 1983-2022), and q = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='7% (average for the years 1975-2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='08 is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Recall that n1 measures the number of years after age 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 70 + n1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=', T C E (n) < T C L (n) for n > n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Table I shows that n1 decreases as the COLA rate q increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' For example, if the beneficiary claims at age 66 (K=4), he starts falling behind at age 81 (n1 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='1) with no COLA (q = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' If the average COLA is q = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='7%, he falls behind by age 79 (n1 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='77), two years earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' This general decrease in n1 with COLA is mostly due to the decrease in the initial benefits: SC K = SK/(1 + q)K < SK (All scenarios assume a fixed yearly benefit S0 at age 70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In summary, taking into account the cost-of-living adjustments makes early claims less attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Below we investigate another scenario where claiming early may lead to a more positive outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Taking advantage of possible market gains In the previous section, we assumed that, if the beneficiary claimed his Social Security benefits early (before age 70), he would be using them to meet his financial needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The analysis in this section is geared towards those beneficiaries who can afford to wait until age 70 but may nevertheless want to claim earlier, with the intention of investing the payments they receive in the financial markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Clearly there is no unique way to proceed in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Investments can be made in myriad different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In order to obtain some clear, quantitative results, we assume a rather simple scheme that can be improved with more realistic details at a later date if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 7 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' “Early claim with market gains” scenario–no COLAs We consider a rather straightforward scheme and, for the time being, we ignore the cost- of-living adjustments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The assumptions we make are as follows: As before, the beneficiary files for benefits K years before age 70, where 1 ≤ K ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' But now, at the end of each year, until he is 70, he invests the benefits received that year in a financial instrument with an average annual return rate of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Starting at 70, he stops diverting his benefits to the financial markets but allows the already-accumulated sum to grow indefinitely at the assumed annual rate of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Because of the exponential growth of the funds in the market (earnings compounded annually), it is possible for this scenario to generate more total income (integrated over some relevant period) than the late scenario of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Here we determine a range of values for the parameters K, r for which such a positive outcome is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The average rate of return r, averaged over some number of years N (for our purposes, N ≃ 30 years) is defined similarly to the COLA parameter q of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 7, 8: r = exp � 1 N N � i=1 ln(1 + ri) � − 1, (14) where ri is the rate for the ith year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In order to explore the possible advantages of this scenario, we start with a calculation of the total sum, TM, that the beneficiary accumulates in the market before he reaches age 70, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=', the total value of his social security investments in the market when he turns 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Recalling Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 1 and using the assumptions outlined above, this sum is given by TM = SK K−1 � i=0 (1 + r)i = S0 (1 + p)K (1 + r)K − 1 r , K ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (15) By assumption, this sum will be left in the market to grow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' therefore, n years after age 70, the total benefits for the early claim with market gains scenario will be T M E = TM(1 + r)n + nSK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (16) The first term on the right-hand side represents the growing amount in the markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The second term is the total received (and not invested) after age 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' T M E can be written explicitly as T M E = S0 (1 + p)K � (1 + r)n � (1 + r)K − 1 � r + n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (17) This equation replaces Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 2 of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Note that in the limit r → 0 (thus removing the market gains), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 17 reduces to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 2 as expected, since lim r→0 (1 + r)n � (1 + r)K − 1 � r = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (18) 8 At this point it will be useful to define a relative gain function g that will measure the early claimer’s gains (or losses) as a function of time: g(K, n, p, r) ≡ T M E (K, n, p, r) − TL(n) TL(n) , (19) where TL(n) is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' For fixed K, p, r, a positive gain function implies that the beneficiary is ahead (n years after 70) with respect to the “late scenario” where he waits until age 70 to start collecting his benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 3, 17, the gain function can be written explicitly in the form g(K, n, p, r) = 1 (1 + p)K ��(1 + r)K − 1 r � �(1 + r)n n � + 1 � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (20) For fixed K, p, r we have lim n→0 g(K, n, p, r) = ∞, lim n→∞ g(K, n, p, r) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (21) Thus, g(K, n, p, r) is a non-monotonic, convex function of n with a minimum at some n = n∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The point where the gain function attains its minimum value (again for fixed K, p, r) can be calculated easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Treating n as a continuous time variable and letting ∂g ∂n ���� n=n∗ = 0 (22) leads to n∗(r) = 1 ln (1 + r), (23) which is independent of the parameters K, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Making use of n∗, we find the minimum gain gmin(K, p, r) = g[K, n∗(r), p, r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (24) In the following analysis, the return rate r = r∗ for which gmin = 0 will play an important role;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' it can be calculated by solving the equation g[K, n∗(r∗), p, r∗] = 0 (25) for r∗ with given K, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In general this step requires a numerical solution, but the case K = 1 can be treated analytically because of simplifications in the function g(K, n, p, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' For fixed p and K = 1, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 23 and 25 lead to n∗ = e p, r∗ = ep/e − 1 (K = 1), (26) where e is the base of the natural logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' For K > 1 the analytic results are not very illuminating, and we use Mathematica[9] for numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Values of n∗, r∗ for 9 1 ≤ K ≤ 8 without the COLA modifications are tabulated in the columns labeled n∗(q = 0) and r∗(q = 0), respectively, of Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' An important property of the gain function that will be useful below is r1 > r2 ⇒ g(K, n, p, r1) > g(K, n, p, r2), (27) which follows directly from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' It is also intuitively obvious, since higher market yields will naturally lead to higher relative gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' r=r*-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='005 r=r* r=r*+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='005 20 40 60 80 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='100 n g FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 1: The gain function g(K, n, p, r), with no COLAs, for various values of r, and K = 1, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' For r = r∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='02987, g(K, n, p, r) ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' the minimum (zero) occurs at n = n∗ = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='98 (see the text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The horizontal axis shows the number of years after age 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' As an example, behavior of g(K, n, p, r) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 1 for various values of r and K = 1, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' As seen in the figure, for given K, p, r, the equation g(K, n, p, r) = 0 can have zero, one, or two solutions for n: No solutions (the solid green curve);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' here g(K, n, p, r) > 0 for all n, implying that the early-claimer is always ahead for this value of r, the average market return rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Only one solution, at n = n∗ for r = r∗ (the dotted red curve);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' here the beneficiary is always ahead except at n = n∗, where his position temporarily becomes neutral (g = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The parameters n∗, r∗ will play a critical role in our analysis, since r > r∗ implies g(K, n, p, r) > g(K, n, p, r∗) ≥ g(K, n∗, p, r∗) = 0 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Two distinct solutions, n = n1, n2 (the dashed blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' For this value of r, the beneficiary starts falling behind at n = n1 (age 70+n1) since g < 0 for n1 < n < n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The gain function is positive again for n > n2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' however, 70 + n2 is too large a number to be relevant for most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' For the parameters used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 1, we have n1 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='87 and n2 = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='32, again found numerically with Mathematica[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 10 K n∗(q = 0) r∗(q = 0) n∗(q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='025) r∗(q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='025) n∗(q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='037) r∗(q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='037) 1 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='98 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='03517 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='05016 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='05767 TABLE II: The critical parameters n∗ and r∗ for the “early claim with market gains” scenario for various values of K, and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The columns with q = 0 (no COLAs) are solutions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 23, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The remaining columns with q > 0 (with COLAs) are solutions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 36, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In order to emphasize the meaning of these important parameters, we will go through an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Making use of Table II, assume that a beneficiary claims his benefits at age 62 (K = 70 − 62 = 8) and invests his Social Security income in some market instrument (as detailed previously) with an average annual return rate of r∗ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' With these assumptions, his cumulative benefits will continue to exceed those from the “late scenario,” T M E > TL, until he is 70 + n∗ ≃ 99 years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' At that time his position becomes neutral (T M E = TL) briefly (the break-even point from the point of view of the “late scenario”), but he gets ahead again beyond that age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' And a final crucial point is that, for any given K, if r > r∗ the beneficiary remains always ahead: T M E > TL for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Next we examine how these results are modified when the cost-of-living adjustments are included in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' “Early claim with market gains” scenario–with COLAs Making use of some of our earlier results, here we look at the effects of cost-of-living adjustments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' With COLA’s, the starting year benefit SC K (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 9) will grow both due to market gains and COLAs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' thus, the total sum the beneficiary accumulates in the market before age 70 will now be T C M = SC K K−1 � i=0 (1 + q)i(1 + r)i = S0 (1 + p)K(1 + q)K (1 + q)K(1 + r)K − 1 (1 + q)(1 + r) − 1 , (28) 11 which replaces TM of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (Recall that the COLA-modified terms have a superscript C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' By assumption, T C M is left in the market after reaching 70 and continues to grow at the rate r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' After 70, however, the new COLA-modified benefits are not invested but presumably used for other purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Thus, T M E of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 17 representing the accumulated benefits at age 70 + n now becomes T MC E = T C M(1 + r)n + SK n−1 � i=0 (1 + q)i, or T MC E = S0(1 + r)n (1 + p)K(1 + q)K (1 + q)K(1 + r)K − 1 (1 + q)(1 + r) − 1 + S0 (1 + p)K (1 + q)n − 1 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (29) The total benefits at age 70 + n with the “late scenario” and including COLAs is still given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 11: T C L = S0 (1 + q)n − 1 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (30) At this point we can do a couple of consistency checks: Since limq→0[(1 + q)n − 1]/q = n, T MC E and T C L reduce to T M E and TL of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 17, 3, respectively, when the COLAs are ignored (q → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Similarly, when the COLAs are retained but the market gains are ignored (r → 0), T MC E reduces to T C E of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 10 as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' As we saw earlier in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' II A on the effects of COLAs without the market gains, we intuitively expect that the benefits of claiming early will be reduced with the COLAs, even when the market gains are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In particular, we expect that the ratio of the average market return to the average COLA, r/q, will play an important role in determining how useful an early claim will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In order to understand these issues better, we again look at the gain function and its time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The new gain function with the COLAs can be written as gc(K, n, p, q, r) ≡ T MC E − T C L T C L , or (31) gc(K, n, p, q, r) = 1 (1 + p)K �� (1 + q)K(1 + r)K − 1 (1 + q)K[(1 + q)(1 + r) − 1] � � q(1 + r)n (1 + q)n − 1 � + 1 � − 1, which reduces to g(K, n, p, r) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 20 in the limit q → 0, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Letting A(K, p, q, r) ≡ q (1 + p)K (1 + q)K(1 + r)K − 1 (1 + q)K[(1 + q)(1 + r) − 1], (32) we can write the time derivative ∂gc/∂n in the form ∂gc ∂n = A(1 + r)n �(1 + q)n ln[(1 + r)/(1 + q)] − ln(1 + r) [(1 + q)n − 1]2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (33) By simple inspection, we can draw some general conclusions from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 31-33: 12 The coefficient A(K, p, q, r) > 0 for r, q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Thus, ∂gc ∂n < 0 for 0 < r ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (34) In other words, when the market return rate is less than the average COLA, the relative gain will be a monotonically decreasing function of time (for fixed K, p, q, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In fact, for strict inequality, r < q, we have lim n→∞ gc = 1 (1 + p)K − 1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (35) Under these circumstances, the equation gc = 0 will have only one solution (n = n1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' An early-claiming beneficiary will fall behind at age 70 + n1 and will never recover (gc(K, n, p, q, r) ≤ 0 for n ≥ n1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' For r > q > 0, limn→∞ gc = +∞, and the gain function displays a behavior similar to what we observed earlier when we considered the market gains without the COLAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' It will have a minimum at n = n∗, where n∗ is a solution of the equation ∂gc/∂n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' We can easily show that n∗(q, r) = 1 ln (1 + q) ln � ln (1 + r) ln [(1 + r)/(1 + q)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (36) Note that limr→q n∗ = ∞, consistent with the discussion above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Now the equation gc = 0 may again have zero, one or two solutions, depending on the parameters K, p, q, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' As in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' III A, the parameter r∗ will be a solution, for fixed K, p, q, of the equation gc[K, n∗(q, r∗), p, q, r∗] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (37) In order to re-emphasize the significance of the parameters n∗, r∗, we summarize here some of their important properties: gmin(K, p, q, r) = gc[K, n∗(q, r), p, q, r], (38) gmin(K, p, q, r∗) = gc[K, n∗(q, r∗), p, q, r∗] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (39) In other words, the gain function has a minimum at n = n∗, and that minimum is zero for r = r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' A consequence of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 27, 38, 39, for fixed K, p, q, is gc(K, n, p, q, r > r∗) > 0 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (40) Thus, an early-claiming beneficiary will always be ahead if he can guarantee an average market return rate of r > r∗, even with the COLAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' However, we will find that, with everything else fixed, r∗ will be higher with the COLAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 13 These different types of behavior for the gain function gc are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 2 for K = 1, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='08 and q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='025, the average COLA for the years 1983 − 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The dashed blue curve has r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='02 < q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' hence the negative slope and a zero at n1 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The early claimer falls behind at 70 + n1 ≃ 84 and never recovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The dotted red curve is for r = r∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='04394 > q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' It has a minimum (zero) at n∗ = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='58 : gc(n∗, r∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The solid green curve has r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='05 > r∗ > q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' hence gc > 0 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='02 r=r*=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='04394 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='05 20 40 60 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='15 n g FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 2: The gain function gc(n) (with COLAs) for three different values of r, and K = 1, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='08, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' For these parameters we have r∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='04394, n∗ = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='58 (Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The dashed blue curve: r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='02 < q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The dotted red curve: r = r∗ > q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' it has a minimum at n = n∗ where gc(n∗, r∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The solid green curve: r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='05 > r∗ > q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In Table II, columns with q > 0 are solutions of the coupled Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 36 and 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Since they are both complicated functions of the parameters, they are solved iteratively for {n⋆, r}, with the converged solution yielding {n∗(q, r∗), r∗ = r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Note that the solutions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 36, 37 (q > 0, with COLA) will agree with those of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 23, 25 (q = 0, no COLA) in the limit q → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' We find that we are able to reproduce the q = 0 results in the Table (obtained using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 23, 25) from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 36, 37 with 0 < q < 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' However, q = 0 is a singular limit for the equations with COLA, since they are undefined at q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Some general comments regarding the {n∗, r∗} values in Table II may be helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Recall that the gain function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 31) has a minimum at n = n∗(q, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The minimum value is zero for r = r∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=', gc(n∗, r∗) = 0, where we assume the other parameters, K, p, q are held fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Recalling Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 38-40, for r > r∗ we have gc(n, r) ≥ gc(n∗, r) > gc(n∗, r∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (41) Thus, a beneficiary who files early will have a positive gain throughout his life, if he can maintain an average market return rate of r > r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The values of n∗ in Table II are ap- proximately in the range 28 − 36;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' thus, the minimum of the gain function occurs when the 14 beneficiary is in his late 90’s (70+28) at the earliest (for r ≥ r∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In order to maintain posi- tive gains during this time, he has to ensure an average market return rate of approximately 3 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='5% (depending on how early he claims), if the COLAs are not taken into account (q = 0 columns in the table).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' With COLAs, the required rate of return is approximately 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='4 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='0% for q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='5%, the average COLA for the years 1983 − 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The required rate may be as high as 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='8% if q = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='7%, the average for the years 1975 − 2022 (the entire data set at [8]), is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Optimization In the previous section we were concerned with finding parameter regimes where an early claimer can have positive gains with respect to the “late scenario,” possibly for the rest of his life, when both the cost-of-living adjustments and possible market gains are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' It is possible to go a step further and find an optimal time for claiming benefits as a function of the remaining parameters in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Mathematically, the problem reduces to finding a maximum for the gain function gc = gc(K, n, p, q, r) in a 5-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Clearly the problem is quite unwieldy in this form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' by choosing p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='08, as we have done throughout this work, and using q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='025 for the average COLA parameter, the dimension can be reduced to a more manageable three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Then it is possible to find an optimal time for claiming benefits, or a Kopt(n, r), as a function of an assumed average market return rate, r, and a time, n, in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Below we examine a couple of different optimization paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Maximizing the minimum gain Here instead of choosing a specific time in the future, we let n = n∗ (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 36) where the gain function has its minimum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=', we seek a Kopt that maximizes gc min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Recall that for the range of parameters q, r we have been considering, n∗ ≃ 28 − 36 (Table II), corresponding to an age range of 98-106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The mathematical problem can now be stated as follows: Find K = Kopt(n∗, r) that maximizes gc(K, n∗, p, q, r) for p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='08, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='025 and an arbitrary r > q (Recall that gc is monotonically decreasing for r ≤ q, a regime we try to avoid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 15 We start by slightly rewriting gc of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 31: gc(K, n, p, q, r) = B(n, q, r) � [(1 + r)/(1 + p)]K − 1 [(1 + p)(1 + q)]K � + 1 (1 + p)K − 1, where B(n, q, r) ≡ q(1 + r)n [(1 + q)(1 + r) − 1][(1 + q)n − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (42) Then treating K as a continuous variable and setting ∂gc/∂K = 0 leads to B � (1 + r)K ln �1 + r 1 + p � + ln[(1 + p)(1 + q)] (1 + q)K � − ln(1 + p) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (43) Although Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 43 can be solved for K = Kopt analytically in certain limits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=', r = p), in general it requires a numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Figure 3 shows Kopt(n∗, r) (from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 43), n∗(q, r) (from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 36) and gc min = gc(Kopt, n∗, q, p, r) as functions of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The limits for the average Kopt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='060 0 2 4 6 8 r Kopt (a) n* 22 24 26 28 30 32 34 n* (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='14 r gmin FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 3: (a) Kopt that maximizes the minimum of the relative gain, gc min, as a function of the assumed average market return rate r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Recall that the optimal age is given by 70 − Kopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Also shown is the parameter n∗(q, r) where the minimum occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Again recall that n measures years after age 70;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' thus, gc min occurs at age 70 + n∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (b) gc min as a function of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' market return rate, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='0440 ≤ r ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='0598, are chosen to keep Kopt within the allowed arrange of 0 < Kopt ≤ 8 (the optimal age has to be within 62 − 70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' A number of observations follows from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 3: Kopt increases approximately linearly with the market return rate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 3(a));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' thus, the higher the market rate, the earlier is the optimal claiming age, 70 − Kopt, which is intuitively obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The optimized minimum gain, gc min, increases faster than linearly with r (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' At r = 6%, the optimal claiming age is 62 (Kopt = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The beneficiary’s minimum gain with respect to the late scenario is 14%, which occurs at age 70 + n∗ = 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' At any other age, his relative gain is higher since gc(Kopt, n, p, q, r) > gmin for n ̸= n∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 16 The sum Kopt +n∗, representing the time span from the claiming age to when the gain function attains its minimum, is relatively constant: 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='4 ≤ Kopt + n∗ ≤ 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='0, which has two implications: – When calculating the average COLA rate q (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 8) or the expected average market return rate r (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 14), the time span to consider is approximately 30-35 years from the start of the benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' This is a fairly long period that should be enough to smooth out the market or economic fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Thus r = 5 − 6% and q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='5% are not unreasonable assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' – Regardless of where the optimal claiming age happens to be, the optimized mini- mum gain is approximately 30-35 years in the future from that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' And neither ill-health nor extreme longevity will leave the beneficiary worse off than what is implied by gc min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Therefore, finding a claiming age that maximizes the minimum of the relative gain is a reasonable strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Optimizing for a particular age At this point a complication that we should consider is the following: Optimizing gc min does not imply that the gain function itself has been optimized for all ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In other words, we may still have gc(Kopt, n, p, q, r) < gc(K, n, p, q, r) for some values of K ̸= Kopt, n ̸= n∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In fact, the equation gc(Kopt, n, p, q, r) = gc(K, n, p, q, r), (44) with fixed p, q, r tends to have two distinct solutions, n = n1, n2, for K > Kopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Thus the gain function for K > Kopt will be higher for n < n1 and n > n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' This point is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 4 where we plot gc as a function of n for various values of K, and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='08, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='025, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='0525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The minima for all three curves are at n = n∗(q, r) = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='7, and clearly the minimum for K = Kopt = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='69 curve is higher than those of the K = 2, 7 curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' However, the K = 7 curve (dotted red) crosses above the K = Kopt curve (solid green) at n1 = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='28 and n2 = 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In other words, although gc min is optimized with K = Kopt = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='69, for the years corresponding to n < n1 (earlier than age 87), or n > n2 (later than age 109), the beneficiary may be better off with K = 7, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=', claiming at age 63 instead of 70 − Kopt ≃ 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In order to address this particular complication, the gain function itself (not its mini- mum) can be optimized for any given age using the analytic expressions given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The mathematical problem we need to solve now is the following: Find solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 43 for K = Kopt(n, r), which maximizes gc(K, n, p, q, r) with given p, q, r, and for various values of n while ensuring that 0 < Kopt ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 17 K=2 K=Kopt=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='69 K=7 20 25 30 35 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='10 n g FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 4: The relative gain function gc(K, n, p, q, r) for three different values of K, with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='08, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='025 and r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='0525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The solid green curve is for K = Kopt, optimized for n = n∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The dashed blue curve for K = 2 < Kopt always remains below the optimized curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' however, the dotted red curve for K = 7 > Kopt crosses above the optimal curve at n = n1 = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='28 and n = n2 = 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Kopt 10 20 30 40 50 0 2 4 6 8 n Kopt (a) r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='045 gc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='20 gc Kopt 15 20 25 30 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='0 n Kopt (b) r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='0575 gc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='15 gc FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 5: Kopt and the optimized value of the gain function gc as a function of n, the number of years after age 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Recall that the optimal claiming age is given by 70 − Kopt, and gc measures the gain with respect to the “late scenario” where the beneficiary waits until age 70 to claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (a) Average market return rate r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' (b) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='0575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Note that in (b) although the two curves nearly coincide, the scales are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' A couple of examples are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 5, again for p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='08, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Recall that gc min(K, p, q, r∗) = gc(K, n∗, p, q, r∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' For the first example, in order to ensure gc min > 0, we choose r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='045, which is just above the minimum r∗ for q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='025 in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' For any given n, the figure shows Kopt and the resulting optimized gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' For example, for n = 10 (age 80), we have Kopt = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='29, gc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='175, a again of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='5% over the late scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' For n = 20 (age 90), we get Kopt = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='70 and gc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='018, a gain of only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='8% over the late scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The minimum gain is at n = n∗ = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='34 (age 103), where it is a mere 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 5(b) where we assume a slightly higher return rate of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='75%, the gains are markedly higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' For 15 < n < 35 (ages 85-105), the optimized gc 18 varies between a minimum of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='8% at n = n∗ = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='6 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='2% at the end points of the range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Similarly, Kopt varies little: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='94 ≤ Kopt ≤ 8 for this value of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Thus, with a rather modest market return of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='8%, a beneficiary who claims at age 62-63 would indeed be well ahead for the rest of his life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Summary and conclusions The Social Security benefits provide a financial lifeline to most retirees, and a retiree typically claims his benefits as soon as he reaches his full-retirement age (FRA), or as early as age 62 if he is willing to accept some penalties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' If a retiree is not financially constrained, however, there is an incentive for waiting until age 70 because of the built-in yearly increases to benefits (presently 8%) between his FRA and age 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Ironically, precisely in those cases where filing at FRA or earlier is not a financial necessity, it may be beneficial to file early without waiting until age 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' This work focused on the decision of whether to do so and when the optimal time might be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Starting with a simple “break-even” analysis that retirees typically do to find out when they would break even if they delayed retirement beyond FRA, we developed a series of successively more comprehensive analytic models that compare early-claiming scenarios with the “late scenario” where the beneficiary waits until age 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' First improvement on the break- even analysis was the inclusion of inflation through the cost-of-living adjustments (COLAs) in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' From the point of view of an early-claimer (a view adopted throughout this work) COLAs make early claims less attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Conversely, from the point of view of a late claimer, they bring the break-even point closer (Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Results of filing early and investing the benefits received in the financial markets were examined in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' As summarized in Table II, if the COLAs are ignored, even a modest market return of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='5% would be sufficient for a beneficiary who files at age 62 to be ahead of the late scenario all his life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' If the COLAs are taken into account, that return rate would have to be 5% if the average COLA is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='5%, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='8% if the COLA rate is at its historical average of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Since the timespan relevant to our discussion is approximately three decades from the time of claiming benefits, these market rates, averaged over 30-35 years, are not unreasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' If the beneficiary wants to be not just ahead of the late scenario but to optimize his gains, then he has a number of options, of which we examined only two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' First, we found an optimal filing age that maximizes his minimum gain, gc min, as a function of the assumed market return rate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' For example, if the average market rate r = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='8%, we find Kopt = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='9, implying an optimal age of approximately 63 (70 − Kopt) for claiming;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' the minimum gain of 11% occurs at n∗ = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='6, corresponding to an age of over 93 (70 + n∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 19 Both before and after that age, the gain is higher (on either side of the minimum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The second option we examined was finding an optimal claiming age that maximizes the relative gain at a specific point in time, for two different market rates (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' We found that the claiming age is rather insensitive to the age for which the gain is maximized (Kopt ≃ 7 − 8) if r = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='8%, although there is considerable variation in both Kopt and the maximized gain for r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' The short conclusion from these analyses is that, under the conditions explained at the beginning of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' III, there is no financial incentive for waiting until age 70 for claiming Social Security benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' A beneficiary would be almost always ahead if he were to claim quite early (age 62-63), if the market return rate on his investments over the next 30-35 years is approximately 5 − 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Real-life scenarios are usually more complex than the simplified mathematical models discussed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' We made some assumptions in order to make analytical progress, and we did not consider important factors such as taxes and spousal benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' It would also be beneficial to examine optimization that takes into account a probabilistic survival function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' We hope to address these issues in a future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' It is important to note that, while we provided examples in our figures and tables for a variety of parameters, the analytic expressions derived here can be useful more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' An individual can customize the choice of parameters to his own circumstances and use these models to make a timing decision that aligns with his goals better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Acknowledgments The author gratefully acknowledges useful comments by Gary Hallock that helped im- prove the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' Coile, P.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='00125 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 20 [6] https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='ssa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='gov/benefits/retirement/planner/agereduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' [7] https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='ssa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='gov/benefits/retirement/planner/delayret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' [8] https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='ssa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='gov/oact/cola/colaseries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' [9] Wolfram Research, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=', Mathematica, Version 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='1, Champaign, IL, 2022, URL https:// www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='wolfram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content='com/mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} +page_content=' 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E2T4oBgHgl3EQfrwhD/content/2301.04052v1.pdf'} diff --git a/PS0xJYoBy6fLz0q3Z1Pk/content/tmp_files/2301.07503v1.pdf.txt b/PS0xJYoBy6fLz0q3Z1Pk/content/tmp_files/2301.07503v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..38b8bf6fd8c9e8b2caf9cdd6536c923403810ca9 --- /dev/null +++ b/PS0xJYoBy6fLz0q3Z1Pk/content/tmp_files/2301.07503v1.pdf.txt @@ -0,0 +1,1474 @@ +Model-free machine learning of conservation laws from data +Shivam Arora‡, Alex Bihlo‡, Rüdiger Brecht♭ and Pavel Holba† +‡Department of Mathematics and Statistics, Memorial University of Newfoundland, +St. John’s (NL) A1C 5S7, Canada +♭Department of Mathematics, University of Hamburg +Hamburg, Germany +†Mathematical Institute, Silesian University in Opava +Opava, Czech Republic +E-mails: sarora17@mun.ca, abihlo@mun.ca, ruediger.brecht@uni-hamburg.de, pavel.holba@math.slu.cz +January 19, 2023 +We present a machine learning based method for learning first integrals of systems of ordi- +nary differential equations from given trajectory data. The method is model-free in that it +does not require explicit knowledge of the underlying system of differential equations that +generated the trajectories. As a by-product, once the first integrals have been learned, also +the system of differential equations will be known. We illustrate our method by considering +several classical problems from the mathematical sciences. +1 +Introduction +Conservation laws play an important role in the mathematical sciences. +They provide cru- +cial physical constraints and significant information about structural symmetries to real-world +systems [15], and are used extensively in numerical methods for differential equations, see +e.g. [10, 17, 21]. +While both the analytical and numerical investigation of conservation laws have reached a +certain maturity in the past several decades, recently also machine learning has been proposed +as a means to find conservation laws, see e.g. [5, 7, 8]. A main appeal of machine learning is that +it allows one to work with raw data, making it particularly relevant for the solution of inverse +problems. Indeed, machine learning has been used extensively to identify differential equations +from data, see [6, 18] for some examples. +The current methods to find conservation laws using machine learning are limited in the sense +that either they focus on learning a single conservation law or require the prior knowledge of +the mathematical model governing the system. The present paper can be seen as a contribution +to the inverse problem on conservation laws in that our method can be use to identify multiple +conservation laws directly from data. +More specifically, in this paper we propose a model-free method based on machine learning to +learn conservation laws from data. We refer to this method as model-free as it does not require +1 +arXiv:2301.07503v1 [cs.LG] 12 Jan 2023 + +the knowledge of an underlying system of differential equations, but rather can work with given +trajectory data only. As a by-product, once the conservation laws inherent in the given data +have been successfully identified, our method will have also learned the system of differential +equations generating the given data, and it will be possible to simulate new trajectories using +the learned conservation laws. This approach has the following advantages over training the +network to directly learn the trajectory: +• The conservation laws provide insights into the physics and underlying symmetries of the +system +• The model is trained to learn the underlying dynamics of the system rather than a pure +non-linear approximation which is prone to over-fitting and missing the underlying physics, +see [4]. +In contrast to the inverse problem on symmetries of differential equations [15], the inverse +problem on conservation laws, i.e. the problem of finding which differential equations admit +prescribed conservation laws, has received comparatively little attention [16]. +In particular, +these ideas can be further investigated to streamline differential equations discovery directly +through data. +The further organization of this paper is as follows. In the subsequent Section 2 we give an +overview of recent work on conservation laws relevant for the present consideration. Section 3 +introduces our method for learning conservation laws from data and, as a by-product, identi- +fying the associated differential equations underlying the given data. Section 4 provides the +numerical results for identifying conservation laws for several important physical systems. The +final section 5 provides a short summary and discusses potential future directions of research in +this field. +2 +Related work +Conservation laws have been studied extensively analytically in the past, with such celebrated +results as Noether’s theorem relating symmetries of Lagrangian systems to conservation laws +via a constructive formula [15]. More generally, conservation laws of differential equations can +be found constructively using conservation law characteristics (or multipliers), which can be +computed by solving a suitable linear system of determining partial differential equations [3, 15]. +This relation between conservation laws and conservation law characteristics is at the heart of +the multiplier method for finding conservative discretization schemes to systems of differential +equations [21]. +Machine learning for problems related to conservation laws is a relatively recent field. Here, +a key result is the development of Hamiltonian neural networks [4], where a neural network is +used to learn/model the Hamiltonian function of a Hamiltonian system. The loss function for +this problem are the Hamiltonian equations of motion, which are approximated using trajectory +data. Owing to the form of the problem, Hamiltonian neural networks numerically conserve the +learned Hamiltonian. Learning conservation laws from trajectory data was the subject of [5, 8], +where a new variance decreasing loss and methods of dimensionality reduction have been used, +respectively. In [7] the authors use the defining equations of the considered systems of differential +equations to learn their conservation laws. +Our method has flavours of the works [5, 7, 8] in that we do use trajectory data for training +but end up learning the conservation laws inherent in this data by identifying the differential +equation responsible for this data. +2 + +3 +Method +In this section we detail our method for identifying conservation laws of physical systems from +trajectory data. We work exclusively with data stemming from systems of ordinary differential +equations and will thus first review the associated terminology of conservation laws for such +systems. +3.1 +First integrals of ordinary differential equations +We consider the system of ordinary differential equation (ODEs) +˙x = f(x) +x(t0) = x0, +(1) +where t ∈ T ⊂ R, and x(t) = (x1(t), . . . , xd(t)) ∈ X ⊂ Rd. If f ∈ Cp−1(T ×X → Rd) is a p-times +continuously differentiable function over the domain T × X then system (1) admits a unique +solution x × Cp(T → X) in a neighbourhood of (t0, x0) ∈ T × X. +Conservation laws of systems of ordinary differential equations are typically referred to as +constants of motion or first integrals. In the following, we will refer to them as first integrals. +Mathematically, a first integral is a function I ∈ C1(T × X → R) such that +DtI(t, x)| ˙x−f(x)=0 = 0 +(2) +holds for any t ∈ T and any Cp solution of x of system (1), where Dt is the total derivative with +respect to t. That is, I(t0, x0) = I(t, x), and I is constant on t along solutions of system (1). +Note that in the following we will allow for system (1) to also include arbitrary parameters +γ ∈ Rs, i.e. we will consider models of the form ˙x = f(x; γ). Also, while first integrals may +depend on the time t, such as for the damped harmonic oscillator [7], in the following we will +consider only the case of time-independent first integrals, i.e. I = I(x). +If system (1) admits n first integrals Ii with 0 < i ⩽ n, whose gradients are linearly indepen- +dent, then one can write system (1) in the form +˙x = ϵ(x)∇I1∇I2 · · · ∇In, +or, component-wise, using Einstein’s summation rule, +˙xi = ϵij1j2...jn∂j1I1∂j2I2 · · · ∂jnIn, +(3) +with each jk = 1, . . . , d, k ∈ {1, . . . , n}, and where ϵ is a totally anti-symmetric (n + 1)-tensor. +See [10] for further discussions on the skew-gradient form (3). Two prominent special cases arise +from the general skew-gradient form (3). +If n = 1 then this form reduces to +˙xi = ϵij∂jI +which for the special case of ϵij being the two-dimensional Levi-Civita symbol and I being the +Hamiltonian is the general form of a canonical Hamiltonian system [15]. +If n = 2 then the skew-gradient from (3) reads +˙xi = ϵijk∂jI1∂kI2 +(4) +which was first suggested by Nambu as a generalization to Hamiltonian dynamics [12]. Two +prominent examples for Nambu systems are the Euler equation for rigid rotations [12] and the +conservative Lorenz-1963 model [13]. We will consider the latter in Section 4. +3 + +3.2 +Discrete gradient methods +The skew-gradient form (3) is at the heart of the discrete gradient method [10, 14, 17], which +is a general purpose method for constructing conservative discretization schemes for systems of +ordinary differential equations. +The discrete gradient method requires consistent choices for discretizing the skew-gradient +tensor ϵ(x) and the gradients ∇Ii, i = 1, . . . , n. Moreover, the gradients have to be chosen in +such a manner as to guarantee that the resulting discretization indeed numerically preserves Ii. +Consider the case where n = 1, i.e. a single first integral exists. Then it was show in [17] that +for a one-step method for system (1) of the form +x′ − x +∆t += ˜f(x, x′, ∆t) +(5) +the following is a discrete gradient of I, +¯i(x, x′) · (x′ − x) = I(x′) − I(x), +¯i(x, x) = ∇I(x), +(6) +and that if the discretization (5) is chosen as +x′ − x +∆t += ˜ϵ(x, x′, ∆t)¯i(x, x′), +(7) +then (7) indeed numerically preserves I as long as ˜ϵ(x, x′, ∆t) is a consistent approximation to +the anti-symmetric matrix ϵ(x), i.e. as long as lim∆t→0 ˜ϵ(x, x′, ∆t) = ϵ(x). +Analogously, for general n, the one-step method +x′ +i − xi +∆t += ˜ϵij1j2...jn¯i1 +j1¯i2 +j2 · · ·¯in +jn, +(8) +will preserve the first integrals I1, . . . , In numerically provided that ˜ϵ is consistent with the +anti-symmetric (n + 1)-tensor ϵ and ¯ik are discrete gradients of Ik, k = 1, . . . , n. +Various formulas for discrete gradients exist that satisfy the consistency condition (6), see [10]. +In the following, we use the simple expression +¯i(x, x′) = +� +� +� +� +� +� +� +� +� +I(x′ +1,x2,...,xd)−I(x1,x2,...,xd) +x′ +1−x1 +I(x′ +1,x′ +2,...,xd)−I(x′ +1,x2,...,xd) +x′ +2−x2 +... +I(x′ +1,x2,...,x′ +d)−I(x′ +1,x′ +2,...,x′ +d−1,xd) +x′ +d−xd +, +� +� +� +� +� +� +� +� +� +, +(9) +which does satisfy (6). +We should like to note that there is a self-consistency among the skew-gradient forms. In +particular, if system (1) possesses n first integrals then it can be written both as +˙xi = ϵij1j2...jn∂j1I1∂j2I2 · · · ∂jnIn, +(10a) +and as +˙xi = ˜ϵij1j2...jn−1∂j1I1∂j2I2 · · · ∂jn−1In−1, +(10b) +where the n-tensor ˜ϵ is totally anti-symmetric and defined as. +˜ϵij1j2...jn−1 = ϵij1j2...jn∂jnIn. +(10c) +4 + +As a particular example for the self-consistency condition (10), consider the case when n = 2. +System (4) can then be written as +˙xi = ϵijk∂jI1∂kI2 = ˜ϵij∂jI1 = ¯ϵik∂kI2, +(11) +where +˜ϵij = ϵijk∂kI2, +¯ϵik = ϵijk∂jI1. +In the following, we will make extensive use of the discrete gradient method. +3.3 +Learning first integrals using discrete gradient methods +We assume that we have collected trajectory data of the form (x0, x1, . . . , xN) for a physical +system stemming from an unknown system of ordinary differential equations, where superscripts +denote the time step. For the sake of simplicity, we assume that this trajectory data is sampled +at a uniform time step ∆t, although this is not necessary for our method as long as the (variable) +sampling time step is known. +Assuming this system has a single first integral, then it can be written as +˙x = ϵ(x)∇I(x), +for some unknown anti-symmetric matrix ϵ(x) and first integral I(x). A discrete gradient for- +mulation for this equation is given by (7). Since the trajectory data is given, the task is to +identify the unknown anti-symmetric matrix and first integral. +In this work we use a standard feedforward neural network with weights θ for this identi- +fication task. Specifically, our neural network accepts as input pairs of consecutive trajectory +time steps (xk, xk+1), k = 0, . . . , N − 1 and outputs the associated anti-symmetric matrix +ϵθ(xk, xk+1, ∆t) and first integral Iθ(xk) by minimizing the loss function +L1 +θ = 1 +N +N−1 +� +k=0 +� +xk+1 − xk +∆t +− ϵθ(xk, xk+1, ∆t)¯iθ(xk, xk+1) +�2 +, +(12) +where ¯iθ(xk, xk+1) denotes the evaluation of the discrete gradient (9) using the neural network +model Iθ for the first integral I. +Remark 1. If the unknown system has no first integral then the loss function (12) will not +go to zero. We will present some numerical evidence for a system that does not admit a first +integral in Section 4. +Assuming that the unknown system of differential equations has more than one first integral, +there are several computationally feasible ways for determining them. +1. If the number of first integrals was known to be n, then one could try to directly minimize +the loss analogous to (12) derived from using Eq. (8). A main downside of this approach, +besides requiring to know n, is that the anti-symmetric (n+1)-tensor ˜ϵ is increasingly +difficult to learn for larger n or d. +2. Rather than learning all first integrals and the anti-symmetric (n+1)-tensor ˜ϵ at once, one +can use the self-consistency among skew-gradient forms (10). One begins assuming that +a single first integral exists, minimizing the loss (12) and then subsequently trains addi- +tional neural networks using loss functions based on the self-consistency of anti-symmetric +tensors (10c). In particular, once the first anti-symmetric matrix ϵ and first integral I1 +5 + +has been found by training the first neural network with weights θ1, one defines a second +neural network with weights θ2 to minimize +L2 +θ = 1 +N +N−1 +� +l=0 +� +i,j +� +(ϵθ1)ij(xl, xl+1, ∆t) − (˜ϵθ2)ijk(xl, xl+1, ∆t)¯iθ2(xl, xl+1, ∆t) +�2 , +to find the anti-symmetric 3-tensor ˜ϵ and the second first integral I2. We can then proceed +with higher degree tensors and further first integrals in an analogous fashion using (10c). +An advantage of this method is that the number of first integrals does not have to be +known beforehand. Rather, the optimization procedure is carried out step-by-step until +it eventually fails, marking the upper limit of first integrals found. A downside of this +approach is that again large anti-symmetric tensors have to be defined, which we in practice +found to slow down learning. +3. The last possibility lies in using (11). We proceed as in the second method by training +the neural network with weights θ1 to learn the anti-symmetric matrix and first integral +using the loss (12). We refer to this anti-symmetric matrix as ˜ϵθ1 here. If more than one +first integral exists, then we can use the equivalency illustrated in (11). It is then required +to penalise the second neural network with weights θ2 to not recover the already learned +anti-symmetric matrix ˜ϵθ1 and first integral I1 +θ1 again. To do so, we follow the ideas of [7] +and define the loss function of the second neural network as +L2 +θ = 1 +N +N−1 +� +k=0 +� � +xk+1 − xk +∆t +− ¯ϵθ2(xk, xk+1, ∆t)¯iθ2(xk, xk+1) +�2 ++ +α +� +¯iθ1(xk, xk+1)¯iθ2(xk, xk+1) +�2 � +, +where the second term, the gradient penalty with penalization constant α ∈ R, should +prevent the second neural network from learning I1 again. If more than two first integrals +exist, then the above loss will simply be extended with further gradient penalties. +Remark 2. A main issue in the case of multiple first integrals is that any function of first +integrals is again a first integral. This can make it potentially difficult to identify the most +elementary, or the most canonical, form of a set of first integrals. Also, and more importantly, it +becomes potentially challenging to identify the minimal set of first integrals that are functionally +independent. We will address this issue in Section 3.5. +3.4 +Neural network architecture and training +We use a standard feedforward neural network in this work and report the number of layers and +units used for each example below. We use the exponential linear unit as activation function +except for the output layer where a linear activation function is used. +The input to the network is a pair of d-dimensional data point (xk, xk+1) on the sampled +trajectory. +If the system under consideration admits some model parameters γ, and if the +trajectory data is sampled for different parameter values, then the neural network will also +have to accept these model parameters as input. We illustrate this below for the case of the +harmonic oscillator, where γ = (m, k), the mass and spring constant of the oscillator, for the +conservative Lorenz model, where γ = (σ, ρ), which are related to the Prandtl and Rayleigh +numbers, respectively [9], and for the point-vortex equations where γ = (Γ1, . . . , Γp) are the +vortex strengths of the p point vortices. +The output of the network depends on which of the above three variations of our method +is chosen. If all first integrals are learned at once (assuming the total number of first integrals +6 + +is known), then the network will output an n-dimensional vector of first integrals and an anti- +symmetric (n+1)-tensor ϵ. +Similarly, if we aim to learn the first integrals in a sequential manner, then the output of the +neural network for the first first integral will be a single scalar quantity and an appropriately +shaped anti-symmetric matrix. For the second first integral the neural network will then have to +output another scalar quantity and either an anti-symmetric 3-tensor (using the second method) +or another anti-symmetric matrix (using the third method). Additional first integrals are then +learned analogously with additional neural networks. +All models have been trained in TensorFlow 2.11 using a single NVIDIA RTX 8000 GPU. +3.5 +Evaluation metrics +In Remark 2 we mentioned that the representation of a set of first integrals is not unique as +any function of first integrals is again a first integral. Additionally, owing to the skew-gradient +form (3) any multiplication of each first integral can be offset by an associate re-scaling of the +skew-symmetric tensor ϵ. Thus, it may be challenging to find the equations for the predicted +first integral in its simplest form. +Here, instead we evaluate how well the neural network identifies first integrals by +• computing how well the first integrals are conserved (see Eq. (2)). We compute (I(x0) − +I(xk))/I(x0), k = 0, ..., N for a trajectory (x0, ..., xn). +• using the predicted right hand side to simulate a new trajectory (x0 +θ, ..., xN +θ ). Then we +compute |xk − xk +θ|. +To identify the correct number of first integrals we can also follow the strategy proposed +in [7]. Having computed two first k > 1 integrals we form the matrix +A = +� +� +� +� +� +� +I1(x1) +I2(x1) +· · · +Ik(x1) +I1(x2) +I2(x2) +· · · +Ik(x2) +... +... +... +... +I1(xN) +I2(xN) +· · · +Ik(xN) +� +� +� +� +� +� +where N ≫ k is the number of points alone one trajectory. We then compute the singular +value decomposition of the matrix A and determine the rank of this matrix by the number of +non-vanishing singular values. As argued in [7], if N ≫ k then it is exponentially unlikely that +this method will under-sample the true manifold dimensionality, and thus this method will yield +the correct rank, and hence the correct number of functionally independent first integrals. This +will be verified explicitly in Section 4. +4 +Application to models from the mathematical sciences +In this section we illustrate our method with several examples from the mathematical sciences. +All trajectory computations were done using the discrete gradient method (7) with the learned +anti-symmetric tensor and first integrals being used to construct the right hand side. By default +we use the third method described in Section 3 for all examples shown in this section. We +illustrate the other two methods for the conservative Lorenz model as well, with the associated +results being presented in Appendix A. +7 + +4.1 +Harmonic oscillator +The harmonic oscillator is a Hamiltonian system +˙q = p +m +˙p = −kq +(13) +where m is the mass of the oscillator and k is the associated spring constant. The Hamiltonian +(which is a conserved quantity) is +I1 = H = 1 +2 +� +q2 +m + kp2 +� +. +(14) +Let x = (q, p) and J = +� +1 +0 +0 +−1 +� +be the canonical Poisson tensor. We can then rewrite (13) in +the form +˙x = J∇H, +(15) +and the goal of the neural network is to identify both J and H. +The initial conditions x0 = (q0, p0) are sampled randomly using a two-dimensional uniform +distribution over the domain [−1, 1]×[−1, 1]. In addition, we randomly sample associated masses +and spring constants each from the uniform distribution over [0.1, 1]. The associated harmonic +oscillators are then integrated by solving (13) using the symplectic Euler method [2] from t = 0 +to t = 1 using ∆t = 0.01, yielding a total of 1000 points per each trajectory. A total of 100 such +harmonic oscillators are sampled and then integrated. +We note here in particular that the symplectic Euler method used for integrating (13) is, in +contrast to the midpoint method, not conservative, i.e. it does not preserve the Hamiltonian +exactly but only approximately. We believe this is a reasonable choice for an numerical integra- +tor as trajectories sampled from real-world phenomena will never exhibit exactly conservative +behaviour due to noise in the data acquisition process. As such, we should like to test here the +robustness of our algorithm for data that has only approximately conservative properties. +We use a neural network with 4 layers and 40 units each, which is trained for 5000 epochs +using the Adam optimizer with a learning rate of 10−3. +We evaluate how well the neural network learned the first integral H for a randomly sampled +harmonic oscillator with m = 0.6 and k = 0.8. We note in particular that this is a harmonic os- +cillator that the neural network has never encountered during training, and hence illustrates how +well the neural network has learned to understand the overall physics of this class of harmonic +oscillators. +Using the matrix A identified that there exists one conserved quantity. In the following we +further analyse the learned quantities, see Fig. 1. In Fig. 1a we see that the shape of the second +learned quantity does not look like the true Hamiltonian function for the harmonic oscillator. +Then, when evaluating how well the learned quantities are conserved, see Fig. 1b and Fig. 1c, +we see that I1 +θ is conserved in the order of O(10−2) and I2 +θ in the order of O(1). When using +the learned quantities to simulate a trajectory we see that only the simulation using I1 +θ is close +to the true trajectory, see Fig. 1d and Fig. 1e. +Remark 3. We have also carried out experiments with a fully conservative numerical integrator +and found the results to be qualitatively and quantitatively the same as with the non-conservative +symplectic Euler method shown here, underlying the robustness of the proposed method. +8 + +(a) The shape of the learned quantities. +(b) Conservation of I1 +θ. +(c) Conservation of I2 +θ. +(d) Using I1 +θ to simulate a trajectory. +(e) Using I2 +θ to simulate a trajectory +Figure 1: Evaluation of the learned quantities for the harmonic oscillator. +4.2 +Conservative Lorenz–1963 system +As our next example, we consider the conservative Lorenz–1963 system given by +˙x = σy +˙y = x(ρ − z) +˙z = xy +(16) +9 + +True Hamiltonian +LearnedHamiltonian1 +Learned Hamiltonian 2 +0.7 +0.6 +0.1 +0.0 +0.5 +0.2 H +0.4 H +0.5 +EO +E'O- +0.2 +1.0 +0.1 +0.4 +1.5 +1.0 +1.0 +1.0 +0.5 +0.5 +0.5 +1.0 +0.0 +1.0 +0.5 +0.0 +1.0 +0.5 +0.0 +0.5 +0.0 +0.5 +p +0'0 +0.5 +p +0.0 +0.5 +p +q +0.5 +1.0 +q +0.5 +1.0 +q +0.5 +1.0 +1.0 +1.0 +O'T-0.025 +0.020 +0.015 +0.010 +0.005 +0.000 +0.005 +0.010 +20 +40 +60 +80 +100 +steps15.0 +12.5 +10.0 +lative +7.5 +Rel +5.0 +2.5 +0.0 +2.5 +0 +20 +40 +60 +80 +100 +steps0.2 +0.1 +Q +0.0 +0.1 +0.2 +0.4 +0.3 +0.2 +0.1 +0'0 +0.1 +0.2 +E'0 +0.4 +qle6 +nn +1.50 +true +1.25 +1.00 +d +0.75 +0.50 +0.25 +0.00 +. +0.0 +0.5 +10 +1'5 +2.0 +2.5 +q +le6and it has the two first integrals +I1 = z − x2 +2σ +(17) +I2 = y2 +2 + z2 +2 − ρz, +(18) +see [13] for further discussions. It was shown in [13] that one can rewrite system (16) as +˙x = ϵ∇H1∇H2 +(19) +where x = (x, y, z) and ϵ is the constant totally anti-symmetric 3 × 3 × 3 Levi-Civita tensor. +The initial conditions x0 = (x0, y0, z0) are sampled randomly using a three-dimensional uni- +form distribution over the domain [−2, 2] × [−2, 2] × [0, 1]. The two model parameters σ and ρ +are sampled from uniform distributions over the intervals [0.1, 4] and [0.1, 2], respectively. The +associated conservative Lorenz models are numerically integrated from t = 0 to t = 2 with a +time step of ∆t = 0.01 providing 2000 data points for each trajectory using the trapezoidal +method. 100 of such trajectories are generated. +We note here that the trapezoidal method is again not a conservative method. In contrast to +the symplectic Euler method used for integrating the harmonic oscillator, which only oscillates +around the true value of the first integral, the trapezoidal method for the Lorenz model actually +shows a systematic drift in numerical conservation, depicted in Figure 2, and thus presents an +even more challenging dataset to work with. +Figure 2: Conservation of the conserved quantities of the Lorenz system using the trapezoidal +scheme. +We trained neural networks with 6 hidden layers with 40 units each, that were trained for +1000 epochs using the Adam optimizer with a learning rate of 10−3. The results of the trained +neural networks are depicted in Fig. 3. +Figure 3 shows that the neural networks have indeed learned the correct shape of the first +integrals, with these first integrals being approximately conserved along trajectories of the Lorenz +model. +4.3 +Three-species Lotka–Volterra equations +We consider the three-species Lotka–Volterra system +˙x = x(y − z) +˙y = y(z − x) +˙z = z(x − y), +(20) +10 + +le-5 +00'0 +0.25 +relative change +0.50 +0.75 +1.00 +1.25 +1.50 +P +1.75 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +t(a) True shape of the two first integrals I1 and I2. +(b) Learned shape of the two first integrals I1 +θ and I2 +θ. +(c) Conservation of I1 +θ and I2 +θ along trajectories. +(d) Using I1 +θ and I2 +θ to simulate a trajectory. +Figure 3: Evaluation of the learned quantities for the Lorenz system. +which has two conserved quantities +I1 = x + y + z, +I2 = xyz, +see figure 4 and refer [20] for further details. +The initial conditions x0 = (x0, y0, z0) for this problem are sampled from a three-dimensional +uniform distribution over the domain [0.1, 2] × [0.1, 2] × [0.1, 2] accounting for the fact that +each variable, the number of that respective species, should be positive. We then integrate the +associated model from t = 0 to t = 20 with a time step of ∆t = 0.05 leading to a trajectory with +400 points. Integration is done with the numerical scheme proposed in [21], which conserves +both integrals up to machine precision. 100 of such trajectories are generated. +The associated neural network uses 6 hidden layers with 40 units per each layer. +We evaluate how well the neural network conserves the learned conservation law, see Fig. 5. +11 + +4.5 +4.0 +3.5 +2.0 +15 +1.0 +a +2 +3 +1 +2 +3 +4 +1 +5 +00.10 +0.08 +0.06 +(0)T +0.04 +0.02 +0.00 +-0.02 +0.04 +250 +500 +750 +1000 +1250 +1500 +1750 +20000.6 +0.4 +I_2-I_2(0) +0.2 +0.0 +0.2 +-0.4 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +tnn1 +true +2.00 +1.75 +1.50 +1.25 +1.00 +0.75 +0.50 +0.25 +1.0 +0.5 +-2 +0.0 +-1 +0 +1 +0.5 +2 +E +1.0nn2 +tue +2.00 +1.75 +1.50 +1.25 +1.00 +0.75 +0.50 +1.0 +0.5 +-3 +0.0 +-1 +0 +0.5 +2 +3 +1.04.5 +4.0 +3.5 +2.0 +1.5 +1.0 +5 +4 +0 +3 +1 +2 +7 +1 +4 +04.0 +3.5 +3525 +15 +10 +0.5 +4 +3 +1 +2 +2 +3 +4 +1 +n4.5 +4.0 +3.5 +3.0 +2.5 +2.0 +1.5 +4 +D +3 +1 +2 +2 +3 +4 +1 +5 +0(a) Shape of I1 +(b) Shape of I2 +Figure 4: The shape of the two first integrals. +(a) The shape of the two learned quantities I1 +θ and I2 +θ. +(b) Conservation of I1 +θ and I2 +θ. +(c) Using I1 +θ and I2 +θ to simulate a trajectory. +Figure 5: Evaluation of the learned quantities for the Lotka–Volterra. +4.4 +Point vortex dynamics +The equations for the p-point vortex problem in the plane are given by +˙xi = − 1 +2π +p +� +j=1,j̸=i +Γj +yij +r2 +ij +, +˙yi = 1 +2π +p +� +j=1,j̸=i +Γj +xij +r2 +ij +, +(21) +12 + +89 +6°0 +0.2 +0.1 +0.0 +0.8 +0.0 +0.6 +0.2 +0.4 +0.4 +0.6 +0.2 +0.8 +0.06'0 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.989 +0.9 +0.1 +0.0 +0.8 +0.0 +0.6 +0.2 +0.4 +0.4 +0.6 +0.2 +0.8 +0.0 0.9 +0.8 +0.7 +0.1 +0.0 +0.8 +0.0 +0.6 +0.2 +0.4 +0.4 +0.6 +0.2 +0.8 +0.00.020 +0.015 +0.005 +0.000 +50 +100 +150 +200 +250 +300 +350 +400 +t0.004 +0.002 +Relative error +0.000 +0.002 +0.004 +50 +100 +150 +200 +250 +300 +350 +400nn1 +true +2.0 +1.8 +16 +14 +12 +1.0 +0.8 +0.6 +0.500.751.00,251.50175 2.00 +0.50nn2 +tue +2.0 +1.8 +1.6 +1.0 +8°0 +0.6 +0.50o0.751.001.251.501.752.00 +0.50where i = 1, . . . , p, and (xi, yi) denotes the position of the ith vortex with vortex strength Γi, +and xij = xi − xj, yij = yi − yj and rij = +� +x2 +ij + y2 +ij. This system admits the follow four first +integrals, +I1 = +p +� +i=1 +Γixi, +I2 = +p +� +i=1 +Γiyi, +I3 = +p +� +i=1 +Γi(x2 +i + y2 +i ), +I4 = − 1 +2π +� +1⩽i