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Especially, there are various predictions for the joint +spatiotemporal distribution of particle detection events on a screen, which are derived from different +formulations and interpretations of the quantum theory. Although the differences are typically small, +our studies show that these predictions can be experimentally distinguished by an unconventional +double-slit configuration, which is realizable using present-day single-atom interferometry. +I. +INTRODUCTION +In textbook quantum theory, time is a parameter in the +Schr¨odinger equation, not a self-adjoint operator, hence +there is no unique and unambiguous way to compute the +temporal probability distribution of events from the first +principles (i.e. +the Born rule) [1]. +Nonetheless, since +clocks exist and time measurements are routinely per- +formed in quantum experiments [2, 3], a complete quan- +tum theory must be able to predict the temporal statis- +tics of detection events. For example, in the famous dou- +ble slit experiment, each particle is detected at a ran- +dom time as same as at a random position on the de- +tection screen [4–8]. +Therefore, one can ask: What is +the position-time joint probability density P(x, t) on the +screen? Although this question is very old [9–12], it is +still open [13–18]. In fact, the ambiguity in the arrival +time distribution even prevents a clear prediction of cu- +mulative arrival position distribution, +� +P(x, t)dt, which +is typically measured in a non-time-resolved double-slit +experiment [19]. +Nonetheless, usual experiments are performed in the +far-field (or scattering) regime, where a semiclassical +analysis is often sufficient [13, 19]. In this analysis, it is +assumed that particles move along classical trajectories, +and the arrival time distribution is computed using the +quantum momentum distribution [8, 20, 21]. However, +because of the quantum backflow effect [22], even in free +space, the quantum mechanical time evolution of position +probability density is not consistent with the underlying +uniform motion assumption, especially in near-field inter- +ference phenomena [23]. In fact, due to recent progress in +the ultra-fast detectors technology (e.g. see [24–27]), it +will be soon possible to investigate the near-field regime, +where the semiclassical approximation breaks down and +a deeper analysis would be demanded [13, 28, 29]. +To remedy this problem, based on various interpre- +tations and formulations of quantum theory, several at- +tempts have been made to introduce a suitable arrival +∗ aliayat@physics.sharif.edu +† kazemi.j.m@gmail.com +time distribution. +On the one hand, according to the +(generalized) standard canonical interpretation, the ar- +rival distribution is considered as a generalized observ- +able, which is described by a positive-operator-valued +measure (POVM), satisfying some required symmetries +[10, 11, 30, 31]. +On the other hand, in the realistic- +trajectory-based formulations of quantum theory, such +as the Bohmian mechanics [32], Nelson stochastic me- +chanics [33], and many interacting worlds interpretation +[34], the arrival time distribution could be obtained from +particles trajectories [7, 18, 35, 36]. Moreover, in other +approaches, the arrival time distribution is computed via +phenomenological modeling of the detection process, such +as the (generalized) path integral formalism in the pres- +ence of an absorbing boundary [12, 37–39], Schr¨odinger +equation with complex potential or absorbing boundary +[40–44], and so on [45–47]. +In principle, the results of these approaches are dif- +ferent. However, in most of the experimental situations, +the differences are typically slight, and so far as we know, +in the situation where differences are significant, none of +the proposals have been backed up by experiments in a +strict manner [8, 36]. An experiment that can probe these +differences would undoubtedly enrich our understanding +of the foundations of quantum mechanics. The purpose +of the present paper is to make it evident, via numerical +simulations, that the famous two-slit experiment could be +utilized to investigate these differences if we simply use +a horizontal screen instead of a vertical one: see Fig. 1. +Using current laser cooling and magneto-optical trapping +technologies, this type of experiment can be realized by +Bose-Einstein condensates, as a controllable source of co- +herent matter waves [48–50]. Moreover, our numerical +study shows that the required space-time resolution in +particle detection is achievable using fast single-atom de- +tectors, such as the recent delay-line detectors described +in [51, 52] or the detector used in [6, 53]. +The structure of this paper is as follows: In Section +II, we study the main proposed intrinsic arrival distri- +butions. Then, in section III we compare them in the +double-slit setup with vertical and horizontal screens and +in different detection schemes. In Section IV, we study +the screen back-effect, and we summarize in section V. +arXiv:2301.02641v1 [quant-ph] 6 Jan 2023 + +2 +II. +“INTRINSIC” ARRIVAL DISTRIBUTIONS +In this section, we first review the semi-classical ap- +proximation and then scrutinize two main proposed in- +trinsic arrival time distributions [16, 36] and their asso- +ciated screen observables. In these approaches, the effect +of the detector’s presence on the wave function evolution, +before particle detection, is not considered. We discuss +this effect in section IV. +A. +Semiclassical approximation +As mentioned, in the experiments in which the detec- +tors are placed far away from the support of the initial +wave function (i.e. +far-field regime), the semiclassical +arrival time distribution is routinely used to the descrip- +tion of the particle time-of-flight [21, 54–57]. In this ap- +proximation, it is assumed that particles move classically +between the preparation and measurement. In this ap- +proach, the arrival time randomness is understood as a +result of the uncertainty of momentum, and so the arrival +time distribution is obtained from momentum distribu- +tion [13, 17, 36, 58]. +In the one-dimensional case, the +classical arrival time is given by +t = m(L − x0)/p0, +(1) +which is applicable for a freely moving particle of mass +m that at the initial t = 0 had position x0 and momen- +tum p0 arriving at a distant point L on a line. Hence, +for a particle with the momentum wave fuction ˜ψ0(p), +assuming ∆x0 ≪|L − ⟨x⟩0|, the semiclassical arrival time +distribution reads [58] +ΠSC(t|x=L) = mL +t2 | ˜ψ0(mL/t)|2. +(2) +This analysis could be generalized in three-dimensional +space. Then, the distribution of arrival time at a screen +surface S is given by [36] +ΠSC(t|x∈S) = m3 +t4 +� +S +| ˜ψ0(mx/t)|2 x · dS, +(3) +where the dS is the surface element directed outward. +The other main distribution that should be demanded +is the joint position-time probability distribution on the +screen, which is also called ”screen observable” [11]. Us- +ing the conditional probability definition, the joint prob- +ability of finding the particle in dS and in a time in- +terval [t, t+dt] could be written as P(x, t|x ∈ S)dSdt = +[Π(t|x∈S)dt] × [P(x|x∈S, t)dS] . In this regard, one can +use the fact that ψt(x) is the state of the system, con- +ditioned on the time being t in the Schr¨odinger picture. +This implies that |ψt(x)|2 refers to the position probabil- +ity density conditioned at a specific time t [14, 15, 59]. +Therefore, in the semiclassical approximation, the joint +spatiotemporal probability density reads as +PSC(x, t|x∈S) = NSCΠSC(t|x∈S) |ψt(x)|2 +(4) +in which NSC ≡1/ +� +S dS |ψt(x)|2 is the normalization con- +stant, and dS =n·dS, where n is the outward unit normal +vector at x∈S. +B. +“Standard” approach +The first attempts to investigate the arrival time prob- +lem, based on the standard rules of quantum theory, were +made at the beginning of the 1960s by Aharonov and +Bohm [60], and also Paul [61]. This approach starts with +a symmetric quantization of classical arrival time expres- +sion (1), as follows [62]: +ˆtAB = mL ˆp −1 − m +2 (ˆp −1 ˆx + ˆx ˆp −1), +(5) +where ˆx and ˆp=−i ∂/∂x are the usual position and mo- +mentum operators, respectively, and ˆtAB is called the +Aharonov-Bohm time operator. This operator satisfies +the canonical commutation relation with the free Hamil- +tonian operator, [ˆtAB, ˆp2/2m] = iℏ, which has been used +to establish the energy-time uncertainty relation [63, 64]. +However, although ˆtAB is Hermitian (or symmetric in +mathematics literature), it is not a self-adjoint operator +[65]—a fact that is in agreement with Pauli’s theorem +[1]. The origin of this non-self-adjointness can be under- +stood as a result of the singularity at p = 0 in the mo- +mentum representation, ˆtAB → (iℏm/2)(p−2 − 2p−1∂p) +[65]. Nevertheless, although the (generalized) eigenfunc- +tions of ˆtAB are not orthogonal, they constitute an over- +complete set and provide a POVM, which are used to +define the arrival-time distribution as follows [63, 65]: +ΠSTD(t|x=L)= +1 +2πℏ +� +α=± +����� +� ∞ +−∞ +dp θ(αp) +� +|p| +m +˜ψt(p)e +i +ℏ Lp +����� +2 +, +(6) +where θ(·) is Heaviside’s step function and ˜ψt(p) is the +wave function in the momentum representation which +could be obtained from the initial wave function ˜ψ0(p), as +˜ψt(p) = ˜ψ0(p) exp +� +− itp2/2mℏ +� +. The distribution ΠSTD +and its generalization in the presence of interaction po- +tential have been referred to as the ”standard arrival- +time distribution” by some authors [16, 66–69]. In fact, +Grot, Rovelli, and Tate treated the singularity of (5) by +symmetric regularization and obtained equation (6) via +the standard Born rule [64]. The generalizations of equa- +tions (5) and (6) in the presence of interaction potential +have been investigated in various works [16, 31, 70–75]. +Using these developments, it has been shown that the +non-self-adjointness of the free arrival time operator can +also be lifted by spatial confinement [71, 76], and the +above arrival time distribution could be derived from the +limit of the arrival time distribution in a confining box +as the length of the box increases to infinity [72]. Fur- +thermore, recently, the distribution (6) is derived from a +space-time-symmetric extension of non-relativistic quan- +tum mechanics [77]. + +3 +The three-dimensional generalization of (6) is derived +by Kijowski’s [10] via an axiomatic approach. The as- +sumed axioms are implied by the principle of the prob- +ability theory, the mathematical structure of standard +quantum mechanics, and the Galilei invariance [78]. +Based on these axioms, Kijowski constructed the follow- +ing arrival time distribution for a free particle that passes +through a two-dimensional plane S as +ΠSTD(t|x ∈ S) += +1 +2πℏ +� +α=± +� +R2d2p∥ +× +����� +� ∞ +−∞ +dp⊥ θ(αp.n) +� +|p⊥| +m +˜ψt(p)e +i +ℏ x.p⊥ +����� +2 +, +(7) +where p⊥ ≡(p . n)n and p∥ ≡ p − p⊥ are perpendicular +and parallel components of p relative to S respectively, +and n is the outward normal of plane S. +In fact, he +first proves the above expression for the wave functions +whose supports lie in the positive (or negative) amounts +of p⊥. Then he uniquely derives the following self-adjoint +variant of the (three-dimensional version of) Aharonov- +Bohm arrival time operator, by demanding that the time +operator be self-adjoint and leads to (7) for these special +cases via the Born rule [10, 78]: +ˆtL = sgn(ˆp⊥) +� +mLˆp−1 +⊥ − m +2 (ˆp−1 +⊥ ˆx⊥ + ˆx⊥ˆp−1 +⊥ ) +� +, +(8) +where ˆx⊥ ≡ ˆx.n and L (≡ x.n) represent the distance +between the detection surface and the origin [29]. +Fi- +nally, for an arbitrary wave function, the equation (7) +could be derived from this self-adjoint operator. More- +over, considering this time operator, besides the com- +ponents of the position operator in the detection plane, +ˆx∥ ≡ ˆx − (ˆx.n)n, Kijowski obtains the following expres- +sion as the joint position-time distribution on the detec- +tion screen via the Born rule [78]: +PSTD(x, t|x∈S) = +� +α=± +|ψα +S (x, t)|2, +(9) +in which ψ± +S (x, t) is the wave function on the basics of +joint eigenstates of the operators ˆtL and ˆx∥. Explicitly +ψ± +S (x, t) = +1 +(2πℏ)3/2 +� +d3p θ(±p.n) +� +|p⊥| +m +˜ψt(p)e +i +ℏ x.p. +(10) +Note that, the arrival time distribution (7) could be re- +produced by taking the integral of (9) over the whole of +the screen plane. The joint space-time probability distri- +bution (9), and its generalization for the particles with +arbitary spin, have been also derived by Werner in an- +other axiomatic manner [11]. Moreover, it is easy to see +that the results (7) and (9) can be obtained from a reg- +ularized version of the (three-dimensional generalization +of) Aharonov-Bohm time operator, which is the same as +the procedure used by Grot, Rovelli and Tate in one- +dimensional cases [64]. +C. +Quantum flux and Bohmian approach +Inspiring by classical intuition, another proper candi- +date for screen observables is the perpendicular compo- +nent of the quantum probability current to the screen +surface, J(x, t).n, where +J(x, t) = − ℏ +m Im [ψ∗ +t (x)∇ψt(x)] , +(11) +and n is the outward normal to the screen S. This pro- +posal is applicable for a particle in a generic external +potential and a generic screen surface, not necessarily +an infinite plane. There are several attempts to derive +this proposal in various approaches, such as Bohmian +mechanics for the scattering case in [79], decoherent his- +tories approach in [80] as an approximation, or in [81] as +an exact formula using the concept of extended probabil- +ities, and so on [45, 46, 82]. Howover, even if the wave +function contains only momentum in the same direction +as n, the J(x, t) · n could be negative due to the back- +flow effect [22]. This property is incompatible with the +standard notion of probability. Nevertheless, this prob- +lem could be treated from the Bohmian point of view: +Using Bohmian trajectories, it can be shown that the +positive and negative values of J(x, t) · n correspond to +the particles that reach the point x at S in the same di- +rection of n or the opposite direction of it, respectively +[83, 84]. In this regard, through the Bohmian mechanics +in one-dimension, Leavens demonstrates that the time +distribution of arrival to x=L from both sides could be +obtained from the absolute form of probability flux as +[35, 85] +ΠQF(t|x=L) = +|J(L, t)| +� +dt |J(L, t)|, +(12) +which is free from the aforementioned problem. +The three-dimensional justification of J(x, t) · n as an +operational formulation of the arrival time model has +been made in [82]. Also, the generalization of (12) for +arrival to the surface S is given by [7, 13, 16, 86] +ΠQF(t|x∈S) = +� +S dS|J(x, t)·n| +� +dt +� +S dS|J(x, t).n|, +(13) +with dS =n·dS the magnitude of the surface element dS +which is directed outward at x ∈ S. To illustrate (13) +and to generalize it to the case of joint arrival distri- +bution, we can use the Bohmian point of view. In this +theory, each particle has a specific trajectory, depending +on the initial position, and so the rate of passing par- +ticles through an area element dS centered at x ∈ S, in +the time interval between t and t + dt, is proportional to +ρt(x)|v(x, t)·dS|dt, where v(x, t)=J(x, t)/|ψt(x)|2 is the +Bohmian velocity of the particle. Hence, using quantum +equilibrium condition [87, 88], ρt(x) = |ψt(x)|2, and ac- +complishing normalization, the joint arrival distribution +could be represented by the absolute value of the current +density as + +4 +PQF(x, t|x∈S) = +|J(x, t)·n| +� +dt +� +S dS|J(x, t)·n|. +(14) +Now, by integrating (14) over all x ∈ S, we arrive at the +three-dimensional arrival time distribution (13) for the +screen surface S. +It should be noted that Eq. +(14) is +not necessarily followed for an ensemble of classical par- +ticles because a positive or negative current at a space- +time point, (x, t), can in general have contributions from +all the particles arriving to x at t from any direction. +Nonetheless, since the Bohmian velocity field is single- +valued, the particle trajectories cannot intersect each +other at any point of space-time and so only a single tra- +jectory contributes to the current density J(x, t) at the +particular space-time point (x, t). +Moreover, this fact +implies that when v(x, t) · n>0 we can say that the tra- +jectory and consequently the particle has passed through +the screen from the inside and vice versa for v(x, t)·n<0. +Hence, one can define the joint probability distribution +for the time of arrival to each side of S as +P± +QF(x, t|x∈S) = +J±(x, t)·n +� +dt +� +S dS J±(x, t)·n, +(15) +where J±(x, t) = ± θ(±J·n) J(x, t). In addition, note +that there may be some trajectories which cross S more +than once—and we have multi-crossing trajectories (see +the typical Bohmian trajectory in Fig. 1). The course +of the above inference to Eq. (14) was in such a manner +that multi-crossing trajectories could contribute several +times (see Fig. 2 (a)). +However, one could assume the +detection surface as a barrier that does not allow the +crossed particle to return inside (see Fig. 2 (c)). In this +case, it is suggested to use the truncated current defined +as +˜J(x, t) := +�J(x, t) +if (x, t) is a first exit through S +0 +otherwise +(16) +where (x, t) is a first exit event through the boundary +surface S, if the trajectory passing through x at time t +leaves inside S at this time, for the first time since t = 0 +[13, 79, 89]. The limiting condition in (16), imposes that +the joint probability distribution based on it should be +computed numerically using trajectories: +˜PQF(x, t|x∈S) = +˜J(x, t)·n +� +dt +� +S dS ˜J(x, t)·n +. +(17) +Of course, the detection screen is not always a barrier- +like surface (see Fig. 2 (b)), and one could assume that +there is a point-like detector that lets the multi-crossing +trajectories to contribute to the distribution and we can +use (14) in such cases. +Horizontal screen +Vertical screen +Ly +Lx +x +y +s +o +FIG. 1. +Schematic double-slit experiment setup. +The cen- +ter of two slits is considered as the coordinate origin, and +the vertical and horizontal screens are placed at x = Lx and +y = Ly, respectively. The dashed black line shows a typical +Bohmian trajectory that arrives at the horizontal screen. A +suitable single-particle detector, in addition to particle arrival +position, can record the arrival time using a proper clock. +III. +“INTRINSIC” SCREEN OBSERVABLE IN +TWO-SLIT EXPERIMENT +In this section, we study the discussed proposals in the +previous section for the double-slit experiment. We com- +pare the results of these proposals in the cases of vertical +and horizontal screens (see Fig. 1), and also in different +detection schemes. The main motivation for the study +of the horizontal screen is the non-classical particles’ mo- +tions along the y-direction, in the Bohmian perspective; +see a typical Bohmian trajectory in Fig. 1. This behav- +ior is due to changing the sign of the probability cur- +rent’s component in the y-direction. This behavior does +not occur for x-component of J and consequently for the +Bohmian motion of a particle along the x-direction. +As shown in Fig. 1, the setup contains two identical +slits at y = ±s, and screens are placed at x = Lx and +y=Ly correspond to the vertical and horizontal screens, +respectively. To avoid the mathematical complexity of +Fresnel diffraction at the sharp-edge slits, it is supposed +that the slits have soft edges that generate waves hav- +ing identical Gaussian profiles in the y-direction. So, for +each slit, we can take the wave function as an uncorre- +lated two-dimensional Gaussian wave packet, which in +each dimension has the form +ψ(i) +G (x, t) = (2πs2 +t)- 1 +4 exp +� +(x − x(i) +0 − uxt)2 +4σ0st +� +× exp +� i +ℏmux(x − x(i) +0 − uxt +2 ) +� +(i = 1, 2), +(18) +with m the particle’s mass, σ0 the initial dispersion, ux + +5 +S +S +S +(a) +(b) +(c) +First-arrival +Second-arrival +Third-arrival +FIG. 2. Different schemes of particle detection on the screen +surface S. In the Bohmian point of view, particles could have a +recursive motion on surface S and cross it more than once (e.g. +see the trajectory that plotted in Fig. 1). Assuming different +detector types, one can prob variant possible observables on +the screen. In panel (a) a conceivable particle trajectory is +depicted, which crosses S three times. In this panel, a movable +point-like detector is placed on S, which can survey the whole +screen and detect particles that arrive only from one side, +while in panel (b) a two-sided point detector is placed on +S, which can move along it and detect particles that arrive +from up and down. In addition, one can assume there is (c) +an array of side-by-side detectors covering the entire screen +surface S. The last configuration blocks the trajectory and +does not allow the crossed particle to return. In this scheme, +we only detect first-arrivals from one side. +the wave packet’s velocity, x(i) +0 +the initial position of wave +packet or in other words the location of i-th slit, and +st = σ0(1 + iℏt/(2mσ2 +0)). Therefore, when the particle +passes through the slits, we have the total wave function +as +ψ(x, y, t) = +1 +√ +2[ψ(1) +G (x, t)ψ(1) +G (y, t) + ψ(2) +G (x, t)ψ(2) +G (y, t)], +(19) +where superscripts (1) and (2) correspond to upper and +lower slits, respectively. This form of Gaussian superpo- +sition state is commonly used in the literature [7, 90–93] +and is feasible to implement by quantum technologies be- +cause such a state could be produced and controlled read- +ily [94, 95], even without using slits [49]. In this paper, +we have chosen the metastable helium atom, with mass +m = 6.64 × 10−27 kg, as the interfering particle, and the +parameters as s = 10 µm, σx = 0.04 µm, σy = 0.5 µm, +ux = 3 m/s, and uy = 0 m/s. These values are feasible +according to the performed experiments [96]. Moreover, +the meta-stable helium atom could be detected with high +efficiency because of its large internal energy [52, 97]. +A. +Vertical screen +The arrival time distribution for the vertical screen +placed at different distances from the two-slit is shown +in Fig. 3. As one can see this distribution is the same for +all methods, and their average arrival time is close to the +corresponding quantity in classical uniform motion. To +calculate the mean time of arrival to the screen, we use +the arrival time distribution of each method presented in +sec II, i.e., Eq. (3), (7) and (13), and we have +¯tS = +� ∞ +0 +dt Π(t|x∈S) t, +(20) +as the mean arrival time at the surface S. Furthermore, +we can compute the average arrival time to each point +on the screen using the joint probability distribution as +¯tx = +� ∞ +0 +dt P(x, t|x∈S) t +� ∞ +0 +dt P(x, t|x∈S) . +(21) +This observable is depicted in Fig. 4-b for a vertical screen +placed at Lx = 300 mm. Apparently, the results of the +standard and quantum flux methods are the same and +similar to one that resulted in [7] by Nelson’s mechanics. +Nevertheless, they are different from the semiclassical ap- +proximation. However, when the interference pattern is +calculated by either method, we see that their predicted +cumulative position distributions do not differ much from +the others (Fig. 4-a). This observable can be calculated +by using the joint distribution as +▲▲▲▲▲▲▲▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲▲▲▲▲▲▲ +▲ +▲ +▲ +▲ +▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ +70 +80 +90 +100 +110 +120 +130 +▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ +▲ +▲ +▲ +▲ +▲ +▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ +70 +80 +90 +100 +110 +120 +130 +▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ +▲ +▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ +70 +80 +90 +100 +110 +120 +130 +▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ +70 +80 +90 +100 +110 +120 +130 +Lx =330 mm +Lx =300 mm +Lx =270 mm +Lx =240 mm +Π(t | x = Lx) +t (ms) +0 +0.08 +0 +0.08 +0 +0.08 +0 +0.08 +Semiclassical +Quantum flux +Standard +FIG. 3. Arrival time distributions of particles that arrive at +the vertical screen of the double-slit experiment at different +screen distances. + +6 +▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ +▲ +▲ +▲▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲▲ +▲ +▲ +▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ +-4 +-2 +0 +2 +4 +▲ +▲ +▲ +▲▲▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲▲▲▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲▲▲ +▲ +▲ +▲ +-4 +-2 +0 +2 +4 +94 +96 +98 +100 +102 +104 +106 +(a) +(b) +Averaged arrival time (ms) +P(y) +y (mm) +0 +0.5 +Semiclassical +Standard +Quantum flux +FIG. 4. (a) The cumulative arrival position distribution, Eq. +(22), for the vertical screen at Lx = 300 mm, and (b) the +average arrival time at each point of the screen, Eq. (21). +P(x|x∈S)= +� ∞ +0 +dt P(x, t|x∈S) +� ∞ +0 +dt +� +S dS P(x, t|x∈S). +(22) +As mentioned, it should be noted that, |ψt(x)|2 is just the +conditional position probability density at the specific +time t, not the position-time joint probability density +and so the accumulated interference pattern, P(x|x∈S), +is not given by +� +dt|ψt(x)|2 [98]. +B. +Horizontal screen +In this section, we are going to compare the mentioned +proposals in the double-slit setup with a horizontal de- +tection screen (see Fig. 1). In this regard, in Fig. 5, the +arrival time distributions at the screen are plotted for +some horizontal screens which are located at Ly =15, 20, +25, and 30 µm. In this figure, solid-black, dashed-green, +and dash-dotted-blue curves represent the distributions +ΠST D, ΠQF and ΠSC respectively. +Also, the vertical +lines show the average time of arrival to the screen, ¯tS, +associated with these arrival time distributions. +From +this figure, one can see that, although the averages al- +most coincide, the distributions are distinct. Moreover, +as expected, when the screen’s distance from the center +of the two slits Ly decreases, the difference between dis- +tributions increases. Most of these differences occur in +the early times, which are associated with the particles +that arrive at the S in the near field. Furthermore, we +observe that the ΠSC behaves quite differently from ΠQF +and ΠST D. The distributions ΠQF and ΠST D are more +or less in agreement, however, for the screen that is lo- +cated at Ly =15 µm, a significant difference between the +standard and quantum flux distributions occurs around +t≈0.2 ms. +0.1 +0.5 +1 +5 +10 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0.1 +0.5 +1 +5 +10 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.5 +1 +5 +10 +0.0 +0.2 +0.4 +0.6 +0.8 +0.1 +0.5 +1 +5 +10 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Π(t | y = Ly) +t (ms) +Ly = 30 µm +Ly = 25 µm +Ly = 20 µm +Ly = 15 µm +Semiclassical +Standard +Quantum flux +FIG. 5. +Arrival time distributions of particles that arrive +on the horizontal screen at four different distances from the +center of two slits. The vertical lines show the average arrival +time. +To have a more comprehensive insight, we can look at +the joint spatiotemporal arrival distributions in Fig. 6. +In this figure, joint distributions, PSC, PSTD and PQF are +plotted in three panels, for the horizontal screen surface +located at Ly = 15 µm. +These density plots clearly +visualize differences between the mentioned arrival dis- +tribution proposals. In these plots, we can see separated +fringes with different shapes, which this fact imply +that the particles arrive at the screen in some detached +space-time regions. In the insets, one can see that the +shapes of these regions are different for each proposal. +In the joint density of the semiclassical approximation + +7 +(Fig.6-a), fringes are well-separated, while the standard +distribution (Fig. 6-b) exhibits more continuity in its +fringes. In addition, in the pattern of the quantum flux +proposal (Fig. 6-c) there are grooves between every two +fringes which is due to changing the sign of J(x, t) · n in +(14). In all panels of Fig.6, the duration of “temporal +no-arrival windows” between every two typical fringes +variate in the range between 0.01 and 0.2 ms which +has a spatial extension of about 0.3 to 2 mm. +These +space-time scales are utterly amenable empirically by +current technologies [53, 96], which could be used to test +these results. +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +1 +2 +3 +4 +5 +6 +0.0 +0.5 +1.0 +1.5 +2.0 +0.02 +0.04 +0.08 +0.16 +0.32 +0.64 +1.28 +3.20 +4.00 +0.00 +0.02 +0.04 +0.08 +0.16 +0.32 +0.64 +1.28 +3.20 +5.00 +0.00 +0.02 +0.04 +0.08 +0.16 +0.32 +0.64 +1.28 +3.00 +0.00 +t (ms) +t (ms) +t (ms) +x (mm) +Quantum flux +Standard +Semiclassical +(c) +(b) +(a) +FIG. 6. Density plots of joint arrival position-time distribu- +tions for particles that arrive at the horizontal screen of the +double-slit experiment. Panels (a), (b), and (c) represent PSC, +PSTD and PQF, respectively. Insets: Magnified contour plots +of the joint distributions. +The average time of arrival to each point of the screen +and cumulative position interference pattern could be cal- +culated as in the vertical screen case by Eqs. (21) and +(22). In Fig. 7(a)-(b), these two quantities are shown for +the horizontal screen which is placed at y = 15 µm. In +contrast to the vertical screen, the cumulative position +distribution of the semiclassical approximation is entirely +separate from the two other proposals. The cumulative +position distribution resulting from standard and quan- +tum flux approaches have obvious differences from each +other, as well. +As one can see in Fig. 7(b), the aver- +age arrival times are the same for all three methods at +first and begin to deviate from each other at x ≈ 5 mm; +then again, these curves converge to each other at x≈25 +mm, approximately. The maximum deviation between +the standard and quantum flux average arrival time oc- +curs at x≈19 mm, which is quite in the far-field regime— +the width of the initial wave function is ∼ O(10−3)mm +which is smaller than 19 mm. Therefore one can suggest +the average arrival time in the gray region of Fig. 7(b) as +a practical target for comparing these approaches experi- +mentally. To this end, we study arrival time distributions +at some points of this region as local arrival distributions. +The arrival time distribution conditioned at a specific +point x on the screen can be obtained as follow +Πx(t|x∈S) = +P(x, t|x∈S) +� ∞ +0 +dt P(x, t|x∈S). +(23) +Using the associated joint distribution of each proposal, +we have plotted Fig. 7(c)-(f) that show Πx(t|x ∈ S) at +the positions x=16.2, 17.4, 18.4, 19.2 mm, on the screen +placed at Ly = 15 µm. +The broken black curves in +Fig. 7 (c)-(f), resulting from the quantum flux proposal, +against the smooth curves of the other two methods could +be understood as the result of the changing the signa- +ture of the y-component of the probability current: Note +that, quantum flux distribution is given by the absolute +value of the probability current. The origin of distinc- +tions between the local average arrival times is more per- +ceptible from these local arrival distributions. In princi- +ple, these distributions could be probed using fast and +high-resolution single-atom detectors [53, 97]. In partic- +ular, the delay-line detector that is recently developed +by Keller et al. [51] seems suitable for our purpose: It +has the capability to resolve single-atom detection events +temporally with 220 ps and spatially with 177µm at rates +of several 106 events per second. +We estimate by a numerical investigation that these lo- +cal arrival distributions could be well reconstructed from +about 104 number of detection events. As an example, +in Fig. 7, the histograms associated with the probability +densities of the panel (f) are plotted in panel (g), using +104 numerical random sampling. It is easy to estimate +that the recording of 104 particle detection events can de- +termine the local average arrival time with a statistical +error of about 10−2ms, while the differences between local +average arrival times of various proposals are almost big- +ger than 10−1ms. Using cumulative position distribution, + +0.90 +0.85 +0.80 +0.75 +0.70 +0.65 +1.8 +2.0 +2.2 +2.4 +2.6 +2.8 +3.00.90 +0.85 +0.80 +0.75 +0.70 +0.65 +1.8 +2.0 +2.2 +2.4 +2.6 +2.8 +3.00.90 +0.85 +0.80 +0.75 +0.70 +0.65 +1.8 +2.0 +2.2 +2.4 +2.6 +2.8 +3.08 +5 +6 +7 +8 +5 +6 +7 +8 +5 +6 +7 +8 +5 +6 +7 +8 +5 +6 +7 +8 +x=19.2 mm +x=18.4 mm +x=17.4 mm +x=16.2 mm +Πx(t|x ∈ S) +t (ms) +∆N +(g) +(f) +(e) +(d) +(c) +0 +1.5 +0 +1.5 +0 +1 +0 +1 +0 +600 +Semiclassical +Standard +Quantum flux +0 +5 +10 +15 +20 +25 +0 +2 +4 +6 +8 +3 +4 +5 +6 +7 +8 +1.0 +1.5 +2.0 +2.5 +0 +5 +12 +14 +16 +18 +20 +22 +24 +26 +0.001 +0.003 +0.005 +Averaged arrival time (ms) +x (mm) +P(x) +(b) +(a) +0 +0.4 +FIG. 7. The space-time arrival statistics for the double-slit experiment with a horizontal screen placed at Ly =15 µm. Panel (a) +represents the average time of arrival at each point of the screen, ¯tx. Panel (b) represents the cumulative position probability den- +sity. The panels (c)-(f) show the local arrival time probability densities, Πx(t|x∈S), at the at the points x=16.2, 17.4, 18.4, 19.2 +mm on the screen, which are chosen from the gray region in panel (b). The vertical lines in these panels represent the average +arrival times. Panel (g) is Histograms associated with probability densities of panel (f), which are generated by 104 numerical +random sampling. +Fig. 7(b), one can estimate that, if the total number of +particles that arrived at the screen is about 108, we have +about 104 particles around x = 19.2 mm, in the spacial +interval (19.1, 19.3). Using recent progress in laser cool- +ing and magneto-optical trapping [97], the preparation +of a coherent ensemble of metastable helium atoms with +this number of particles is quite achievable [51]. +One might be inclined to think that the difference be- +tween the quantum flux and standard average arrival +times is just due to changing the signature of J(x, t) · n, +but in the following, we show that even without the con- +tribution of the negative part of J(x, t)·n, these proposals +are significantly distinguishable: see Fig. 8. +C. +Detection schemes +As we mentioned in section II C, according to the +Bohmian deterministic point of view, there are several +possible schemes to detect arrived particles, especially +for the horizontal screen surface which we have recursive +motions on it (see Fig. 1 and 2). One can assume that +the horizontal screen is swept with a point-like detector +that surveys all arrived particles at the surface S, which +we call spot-detection scheme. In this scheme, one option +is to use a unilateral detector to detect arrived particles +at the top or bottom of S. In this case, the positive and +negative parts of the quantum probability current have +respectively corresponded to particles that arrive at the +top or bottom of S (as shown in Fig. 2 (a)), and we must +use Eq. (15) to calculate the screen observables. Addi- +tionally, we can choose a bilateral detector (or two uni- +lateral detectors) that prob all particles that arrive from +both sides of S, along the time with several repeats of +the experiment (as shown in Fig. 2 (b)). In these circum- +stances (i.e. spot-detection scheme), there is no barrier +in front of the particles before they reach the point of +detection and we can use Eq. (14) to obtain the screen +observables as in the two previous subsections. +As we have already shown in section II C, whether the +particles arrive from the top or bottom of S, the abso- +lute value of the quantum probability current yield the +trajectories’ density and consequently give the joint dis- +tribution of the total arrival at each point of S. +This +fact is the case for the standard method, as well, how- +ever, there is a subtle difference between the two propos- +als in the spot-detection scheme. When we talk about + +9 +0 +5 +10 +15 +20 +25 +5 +20 +0 +2 +4 +6 +8 +0 +5 +10 +15 +20 +25 +0 +2 +4 +6 +8 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0 +1 +2 +3 +4 +5 +6 +0.0 +0.5 +1.0 +1.5 +2.0 +x (mm) +t (ms) +First arrivals +Second arrivals +Third arrivals +All arrivals +First arrivals +Quantum flux +FIG. 8. The space-time Bohmian arrival statistics for the double-slit experiment with a horizontal screen placed at Ly =15 µm. +The interior curves in the central figure are the averaged times of arrival obtained by different detection schemes: see Fig. 2. +The Left and top plots are marginal arrival time distributions and marginal arrival position distributions, respectively. The +scatter plot is generated using 2 × 106 Bohmian trajectories, and the black, blue, and green points of the scatter plot represent +the first, second, and third arrivals of Bohmian particles to the screen, respectively. The inset is a zoom-in of the dashed +rectangle. +the spot-detection in the Bohmian approach, it would +be considered the possibility of multi-crossing and the +distribution includes all-arrivals at S. Although, in the +standard method there is an interpretation for ψ+ +S (x, t) +and ψ− +S (x, t) in Eq. (10), which relates them to the par- +ticles arrive at S in a direction which is the same or op- +posite with the direction of outward normal of the screen +n, respectively [10, 64], nevertheless, since there are no +defined paths in this approach, it is obscure whether it +counts only the first-arrivals to each side of the screen or +includes recursive movements of particles. +Alternatively, along with the spot-detection scheme, it +could be assumed that there is a continuous flat barrier +in front of the particle’s paths as the detection surface +or screen surface that does not allow particles to cross +this surface. Depending on the screen’s length and posi- +tion, there are several possibilities for the detection pro- +cess. In each case, a specific number of particle paths +contribute to the distribution of arrival time. +In the +simplest case, the screen blocks all the trajectories that +reach the horizontal surface S, and we only detect the +first-arrivals. In such a setup, we can no longer use the +quantum flux method to represent Bohmian trajectories’ +first encounter with the surface; hence, the screen ob- +servables must be obtained by numerical analysis, due to +the definition of truncated current as in Eq. (16) and its +corresponding joint distribution, ˜PQF(x, t|x∈S), defined +in Eq. (17). By computing the Bohmian trajectories, +we can find positions and times of the first-arrivals to +the screen, and consequently calculate the arrival time +distribution which mathematically could be defined as +˜ΠQF(t|x∈S) = +� +S +˜PQF(x, t|x∈S)dS. +(24) +Also, other observable quantities such as the cumulative +spatial distribution and averaged arrival time over the +detection surface could be defined and calculated numer- +ically in a similar way—by substituting ˜PQF(x, t|x ∈ S) +in Eqs. (21) and (22). Furthermore, we can complete the +computations to find the second and third encounters to +the surface (regardless of the barrier). +In Fig. 8, we show our numerical results of Bohmian +trajectories simulation. The background scatter plot is +the position and time of arrivals of 2 × 106 trajectories. +In this plot, the second and third arrivals are shown in +blue and green, respectively. Here, it is more clear why + +10 +5 +6 +7 +8 +5 +6 +7 +8 +5 +6 +7 +8 +5 +6 +7 +8 +x=19.2 mm +x=18.4 mm +x=17.4 mm +x=16.2 mm +Π(t| x, y) +t (ms) +0 +1.5 +0 +1.5 +0 +1.2 +0 +1.5 +First arrivals +Quantum flux +All arrivals +FIG. 9. Arrival time distribution at the horizontal screen po- +sitions x = 16.2, 17.4, 18.4, 19.2 mm, and Ly = 15 µm, which +are in the gray region of Fig (8). The width of sampling in +each point is about δx = 0.25 mm, and 108 Bohmian trajec- +tories are simulated to obtain these distributions. +we interpret the grooves of the quantum flux density plot +(Fig. 6 (c)) as a result of the multi-crossing of Bohmian +trajectories. +The three middle graphs are the average +time of the first and all-arrivals, which are simulation re- +sults of 108 trajectories, and are compared by the quan- +tum flux method. As expected, the average time of all- +arrivals fits on the quantum flux curve. However, the av- +erage time of first-arrivals deviates from all-arrivals in the +area discussed in the previous section (between x = 16.2 +mm and x = 19.2 mm). +To scrutinize the deviation zone of Fig. 8 (the gray re- +gion), Fig. 9 is drawn to show the arrival time distribu- +tions of screen positions x = 16.2, 17.4, 18.4, 19.2 mm. +As one can see, at the first recursive points of quantum +flux distribution, the first-arrival distributions raise down +to zero. This implies that in the presence of a barrier- +like screen, there would be a big temporal gap between +arrived particles. These gaps could be investigated as a +result of the non-intersection property of Bohmian tra- +jectories that cause a unilateral motion of particles along +the direction of the probability current field. +IV. +SCREEN BACK-EFFECT +In principle, the presence of the detector could mod- +ify the wave function evolution, before the particle detec- +tion, which is called detector back-effect. To have a more +thorough investigation of detection statistics, we should +consider this effect. Howsoever, due to the measurement +problem and the quantum Zeno effect [9], a complete in- +vestigation of the detector effects is problematic at the +fundamental level, and it is less obvious how to model +an ideal detector. Nonetheless, some phenomenological +non-equivalent models are proposed, such as the gener- +alized Feynman path integral approach in the presence +of absorbing boundary [12, 37–39], Schr¨odinger equation +with a complex potential [44], Schr¨odinger equation with +absorbing (or complex Robin) boundary condition [40– +44], and so on. The results of these approaches are not +the same, and a detailed study of the differences is an in- +teresting topic. In this section, we provide a brief review +of the absorbing boundary rule (ABR) and path-Integral +with absorbing boundary (PAB) models, then we com- +pare them in the double-slit setup with the horizontal +screen. +A. +Absorbing Boundary Rule +Among the above-mentioned phenomenological mod- +els, the absorbing boundary condition approach has the +most compatibility with Bohmian mechanics [42]. The +application of absorbing boundary condition in arrival +time problem was first proposed by Werner [40], and re- +cently it is re-derived and generalized by Tumulka and +others using various methods [41–44]. Especially, it is re- +cently shown that in a suitable (non-obvious) limit, the +imaginary potential approach yields the distribution of +detection time and position in agreement with the ab- +sorbing boundary rule [44]. According to this rule, the +particle wave function ψ evolves according to the free +Schr¨odinger equation, while the presence of a detection +screen is modeled by imposing the following boundary +conditions on the Detection screen, x ∈ S, +n · ∇ψ = iκψ, +(25) +where κ>0 is a constant characterizing the type of detec- +tor, in which ℏκ/m represents the momentum that the +detector is most sensitive to. This boundary condition +ensures that waves with wave number κ are completely +absorbed while waves with other wave numbers are partly +absorbed and partly reflected [41, 99]. In the absorbing +boundary rule, the joint spatiotemporal distribution of +the detection event is given by quantum flux. Consider- +ing (25), this distribution reads +PABR(t, x|x∈S) = +|ψABC|2 +� +dt +� +S dS|ψABC|2 , +(26) +where ψABC represent the solution of the free Schr¨odinger +equation satisfying the aforementioned absorbing bound- +ary condition. +This distribution can be understood in +terms of Bohmian trajectories. +The Bohmian particle +equation of motion, ˙X = (ℏ/m)Im [∇ψABC/ψABC], to- +gether with the boundary condition (25), imply that tra- +jectories can cross the boundary S only outwards and so +there are no multi-crossing trajectories. If it is assumed + +11 +that the detector clicks when and where the Bohmian +particle reaches S, the probability distribution of detec- +tion events is given by (26), because the initial distribu- +tion of the Bohmian particle is |ψABC(x, 0)|2 [41]. +B. +Path-Integral with Absorbing Boundary +In several papers [12, 37–39], Marchuwka and Schuss +develop an interesting method to calculate the detec- +tion effect of absorbing surface using the Feynman path +integral method. +They postulate a separation princi- +ple for the wave function in which we could consider +the (bounded wave function) as a sum of two parts, +ψ(x, t) = ψ1(x, t) + ψ2(x, t), such that ψ1(x, t) corre- +sponds to the survival part of the wave which is orthogo- +nal to ψ2(x, t) at a time t and evolve independently [38]. +So, we can obtain the probability of survival of the parti- +cle, denoted S(t), which is the probability of the particle +not being absorbed by the time t, as +� +D d3x|ψ1(x, t)|2, +where the integral is over the domain D, outside the ab- +sorbing region. +By discretizing the path integral in a +time interval [0, t] and eliminating the trajectories that, +in each time interval [t′, t′+∆t′] for all t′ < t, are reached +to the absorbing surface S, the survival and consequently +absorbing probability would be obtained. Based on this +analysis, we could define a unidirectional probability cur- +rent into the surface as d +dt[1−S(t)], which yields a normal +component of the multidimensional probability current +density at any point on S as +J(x, t)·n= λℏ +mπ |n·∇ψ(x, t)|2 +× exp +� +− λℏ +mπ +� t +0 +dt′ +� +S +dS|n·∇ψ(x′, t′)|2 +� +, +(27) +where dS = n · dS is the magnitude of the surface ele- +ment dS, n is the unit outer normal to the absorbing +surface S, and λ is a proportionality factor with the di- +mension of length [37, 62]. Also, ψ(x, t) is the solution +of Schr¨odinger equation bounded and normalized in the +domain D. Moreover, the normal component J(x, t)·n is +supposed to be the probability density for observing the +particle at the point x on the screen at time t [12, 39]. +C. +Screen back-effect in two-slit experiment +In order to complete the investigations carried out +in section III, we are going to study the screen back- +effect in the double-slit experiment with a horizontal +screen. +In this regard, we compare the arrival distri- +butions which are resulted from the absorbing bound- +ary rule (ABR), path-Integral with absorbing boundary +(PAB), and Bohmian truncated current (BTC). +We continue with the same initial conditions as in sec- +tion III, and choose κ = 1 µm−1 for ABR. This value of +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +1 +2 +3 +4 +5 +6 +0.0 +0.5 +1.0 +1.5 +2.0 +0.02 +0.04 +0.08 +0.16 +0.32 +0.64 +1.28 +2.56 +5.12 +0.00 +0.02 +0.04 +0.08 +0.16 +0.32 +0.64 +2.50 +6.25 +12.50 +0.00 +0.02 +0.04 +0.08 +0.16 +0.32 +0.64 +1.28 +3.20 +6.00 +0.00 +t (ms) +t (ms) +t (ms) +x (mm) +Bohmian truncated current +Path-Integral with Absorbing Boundary +Absorbing Boundary Rule +(c) +(b) +(a) +FIG. 10. +Density plots of joint probability distributions of +position and time (screen observable) for the horizontal screen +placed at y = 15 µm in the double-slit experiment. +These +densities are calculated by the three methods which take the +screen effects into account. +κ leads to the maximum absorption probability—which +is almost 0.4—for the chosen initial wave function. In +addition, for a more meaningful comparison, we consider +λ = 1 µm in the PAB method, which leads to the same +absorption probability as ABR. The resulting joint ar- +rival time-position distributions of the three methods are +depicted in Fig. 10. As one can see, the distributions of +the ABR and PAB methods—i.e., panels (a) and (b) in +Fig. 10—have more compatibility with each other than +the result of the BTC method. However, there are dif- +ferences between them which are more obvious in the +zoomed areas. The joint density of the ABR is more uni- + +0.90 +0.85 +0.80 +0.75 +0.70 +0.65 +1.8 +2.0 +2.2 +2.4 +2.6 +2.8 +3.00.90 +0.85 +0.80 +0.75 +0.70 +0.65 +1.8 +2.0 +2.2 +2.4 +2.6 +2.8 +3.00.90 +0.85 +0.80 +0.75 +0.70 +0.65 +1.8 +2.0 +2.2 +2.4 +2.6 +2.8 +3.012 +▲ +▲ ▲ ▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ ▲ +▲ +▲ ▲ ▲ ▲ +▲ +▲ ▲ ▲ ▲ ▲ ▲ ▲ +▲ +▲ +▲ +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +▲ +▲ +▲ +▲ +▲ +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +▲ +▲ +▲ +▲ +▲ +0 +2 +4 +6 +8 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +▲ ▲ ▲ ▲ ▲ +▲ +▲ +▲ +▲ +▲ ▲ ▲ ▲ ▲ ▲ ▲ +▲ +▲ ▲ +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +▲ +▲ ▲ ▲ +▲ ▲ ▲ ▲ +▲ +▲ +▲ ▲ ▲ ▲ +▲ +▲ +▲ ▲ ▲ ▲ ▲ ▲ ▲ +▲ +▲ +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +2 +4 +6 +8 +▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ +▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ +▲ +▲ +▲▲ +▲ +▲ +▲▲▲▲▲▲▲▲▲▲ +▲ +▲ +▲ +▲▲▲▲▲▲▲▲▲▲ +▲ +▲ +▲ +▲ +▲ +▲▲▲▲ +▲ +▲ +▲ +▲ +▲▲▲▲ +▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲ +▲ +▲▲ +▲ +▲ +▲▲ +▲ +▲ +▲▲ +▲ +▲ +▲▲ +▲▲ +▲ +▲▲▲▲ +▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲▲▲▲▲▲▲▲▲▲ +0 +Averaged arrival time (ms) +x (mm) +P(x) +Π(t | y) +0.0 +0.8 +2.5 +0 +Absorbing Boundary Rule +Path-Integral with Absorbing Boundary +Bohmian truncated current +FIG. 11. Averaged time of arrival at each point of the screen +(central figure), cumulative interference pattern (upper fig- +ure), and distribution of time of arrival to the horizontal +screen of the double-slit experiment placed at y = 15 µm +(right-hand figure). +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ ▲ ▲ ▲ ▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +5 +6 +7 +8 +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ ▲ ▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +5 +6 +7 +8 +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +▲ +▲ +▲ +▲ +▲ +▲ ▲ +▲ ▲ ▲ +▲ +▲ +▲ +▲ +▲ +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +▲ ▲ +▲ +▲ ▲ ▲ +▲ +▲ ▲ +▲ ▲ +▲ +▲ +▲ +▲ ▲ ▲ +▲ ▲ ▲ +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +5 +6 +7 +8 +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +▲ ▲ ▲ ▲ ▲ ▲ +▲ ▲ ▲ +▲ +▲ ▲ +▲ +▲ +▲ ▲ ▲ +▲ +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +5 +6 +7 +8 +x=19.2 mm +x=18.4 mm +x=17.4 mm +x=16.2 mm +Π(t| x, y) +t (ms) +0 +1.5 +0 +1.5 +0 +1.2 +0 +1.5 +ABR +PAB +BTC +FIG. 12. Arrival time distribution at the horizontal screen po- +sitions x = 16.2, 17.4, 18.4, 19.2 mm, and Ly = 15 µm, which +are calculated for the three methods which take the screen +effects into account. +formly distributed than of the PAB method. The empty +areas between the fringes of the panel (c) of Fig. 10 are +due to the elimination of the recursive trajectories—or +in other words, are due to the elimination of second and +third arrivals in Fig. 8. +For a more detailed comparison, in Fig. 11 the spa- +tial and temporal marginal distributions are shown. In +addition, the associated local average arrival times are +compared in the central panel of this figure. The PAB +method leads to significant discrepancies in marginal dis- +tributions; The maximum difference is about 40% that +occurs around x≈0.8 mm, which seems testable clearly. +In contrast to the previous results on intrinsic distri- +butions, in which the difference between average arrival +times was significant, there is a good agreement in this +observable for the ABR and PAB methods. +However, +there is a significant difference between the average ar- +rival time in these two methods and BTC around x = 6 +mm. +In Fig. 12, the local arrival time distributions at +some points on the screen are plotted, which show simi- +lar behavior. +V. +SUMMARY AND DISCUSSION +When and where does the wave function collapse? How +one can model a detector in quantum theory? These are +the questions that we investigated in this work. We tried +to show that there is no agreed answer for these ques- +tions, even for the double-slit experiment that has in it +the heart of quantum mechanics [100]. This is a practical +encounter with the measurement problem [73]. In this +regard, we numerically investigated and compared the +main proposed answers to these questions for a double- +slit setup with a horizontal detection screen. It is shown +that these proposals lead to experimentally distinguish- +able predictions, thanks to the current single-atom de- +tection technology. +In this work, we suggest the meta-stable helium atom +as a proper coherent source of the matter wave, however, +other sources may lead to some practical improvements. +For example, using heavier condensate atoms can lead +to more clear discrepancies. Moreover, it is worth not- +ing that although the experiment with photons may have +some practical advantages, there are more complications +in its theoretical analysis. This is partially because of the +relativistic localization-causality problem [101–104]. 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Zangh`ı, Journal of Sta- +tistical Physics 116, 959 (2004). +[107] D. +D¨urr +and +S. +Teufel, +in +Multiscale Methods in Quantum Mechanics (Springer, +2004) pp. 41–58. + diff --git a/-tE0T4oBgHgl3EQfxAG2/content/tmp_files/load_file.txt b/-tE0T4oBgHgl3EQfxAG2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8359836d2128ba372ce05f321d92f97f60698f66 --- /dev/null +++ b/-tE0T4oBgHgl3EQfxAG2/content/tmp_files/load_file.txt @@ -0,0 +1,1818 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf,len=1817 +page_content='Can the double-slit experiment distinguish between quantum interpretations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Ali Ayatollah Rafsanjani,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' ∗ MohammadJavad Kazemi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' † Alireza Bahrampour,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 3 and Mehdi Golshani2 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Sharif University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Tehran,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Iran 2School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Institute for Research in Fundamental Sciences (IPM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Tehran,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Iran 3Research center for quantum engineering and photonics technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Sharif University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Tehran,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Iran Despite the astonishing successes of quantum mechanics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' due to some fundamental problems such as the measurement problem and quantum arrival time problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' the predictions of the theory are in some cases not quite clear and unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Especially, there are various predictions for the joint spatiotemporal distribution of particle detection events on a screen, which are derived from different formulations and interpretations of the quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Although the differences are typically small, our studies show that these predictions can be experimentally distinguished by an unconventional double-slit configuration, which is realizable using present-day single-atom interferometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' INTRODUCTION In textbook quantum theory, time is a parameter in the Schr¨odinger equation, not a self-adjoint operator, hence there is no unique and unambiguous way to compute the temporal probability distribution of events from the first principles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' the Born rule) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Nonetheless, since clocks exist and time measurements are routinely per- formed in quantum experiments [2, 3], a complete quan- tum theory must be able to predict the temporal statis- tics of detection events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' For example, in the famous dou- ble slit experiment, each particle is detected at a ran- dom time as same as at a random position on the de- tection screen [4–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Therefore, one can ask: What is the position-time joint probability density P(x, t) on the screen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Although this question is very old [9–12], it is still open [13–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In fact, the ambiguity in the arrival time distribution even prevents a clear prediction of cu- mulative arrival position distribution, � P(x, t)dt, which is typically measured in a non-time-resolved double-slit experiment [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Nonetheless, usual experiments are performed in the far-field (or scattering) regime, where a semiclassical analysis is often sufficient [13, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this analysis, it is assumed that particles move along classical trajectories, and the arrival time distribution is computed using the quantum momentum distribution [8, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' However, because of the quantum backflow effect [22], even in free space, the quantum mechanical time evolution of position probability density is not consistent with the underlying uniform motion assumption, especially in near-field inter- ference phenomena [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In fact, due to recent progress in the ultra-fast detectors technology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' see [24–27]), it will be soon possible to investigate the near-field regime, where the semiclassical approximation breaks down and a deeper analysis would be demanded [13, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' To remedy this problem, based on various interpre- tations and formulations of quantum theory, several at- tempts have been made to introduce a suitable arrival ∗ aliayat@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='edu † kazemi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='m@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='com time distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' On the one hand, according to the (generalized) standard canonical interpretation, the ar- rival distribution is considered as a generalized observ- able, which is described by a positive-operator-valued measure (POVM), satisfying some required symmetries [10, 11, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' On the other hand, in the realistic- trajectory-based formulations of quantum theory, such as the Bohmian mechanics [32], Nelson stochastic me- chanics [33], and many interacting worlds interpretation [34], the arrival time distribution could be obtained from particles trajectories [7, 18, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Moreover, in other approaches, the arrival time distribution is computed via phenomenological modeling of the detection process, such as the (generalized) path integral formalism in the pres- ence of an absorbing boundary [12, 37–39], Schr¨odinger equation with complex potential or absorbing boundary [40–44], and so on [45–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In principle, the results of these approaches are dif- ferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' However, in most of the experimental situations, the differences are typically slight, and so far as we know, in the situation where differences are significant, none of the proposals have been backed up by experiments in a strict manner [8, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' An experiment that can probe these differences would undoubtedly enrich our understanding of the foundations of quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The purpose of the present paper is to make it evident, via numerical simulations, that the famous two-slit experiment could be utilized to investigate these differences if we simply use a horizontal screen instead of a vertical one: see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Using current laser cooling and magneto-optical trapping technologies, this type of experiment can be realized by Bose-Einstein condensates, as a controllable source of co- herent matter waves [48–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Moreover, our numerical study shows that the required space-time resolution in particle detection is achievable using fast single-atom de- tectors, such as the recent delay-line detectors described in [51, 52] or the detector used in [6, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The structure of this paper is as follows: In Section II, we study the main proposed intrinsic arrival distri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Then, in section III we compare them in the double-slit setup with vertical and horizontal screens and in different detection schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In Section IV, we study the screen back-effect, and we summarize in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='02641v1 [quant-ph] 6 Jan 2023 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' “INTRINSIC” ARRIVAL DISTRIBUTIONS In this section, we first review the semi-classical ap- proximation and then scrutinize two main proposed in- trinsic arrival time distributions [16, 36] and their asso- ciated screen observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In these approaches, the effect of the detector’s presence on the wave function evolution, before particle detection, is not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' We discuss this effect in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Semiclassical approximation As mentioned, in the experiments in which the detec- tors are placed far away from the support of the initial wave function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' far-field regime), the semiclassical arrival time distribution is routinely used to the descrip- tion of the particle time-of-flight [21, 54–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this ap- proximation, it is assumed that particles move classically between the preparation and measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this ap- proach, the arrival time randomness is understood as a result of the uncertainty of momentum, and so the arrival time distribution is obtained from momentum distribu- tion [13, 17, 36, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In the one-dimensional case, the classical arrival time is given by t = m(L − x0)/p0, (1) which is applicable for a freely moving particle of mass m that at the initial t = 0 had position x0 and momen- tum p0 arriving at a distant point L on a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Hence, for a particle with the momentum wave fuction ˜ψ0(p), assuming ∆x0 ≪|L − ⟨x⟩0|, the semiclassical arrival time distribution reads [58] ΠSC(t|x=L) = mL t2 | ˜ψ0(mL/t)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (2) This analysis could be generalized in three-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Then, the distribution of arrival time at a screen surface S is given by [36] ΠSC(t|x∈S) = m3 t4 � S | ˜ψ0(mx/t)|2 x · dS, (3) where the dS is the surface element directed outward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The other main distribution that should be demanded is the joint position-time probability distribution on the screen, which is also called ”screen observable” [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Us- ing the conditional probability definition, the joint prob- ability of finding the particle in dS and in a time in- terval [t, t+dt] could be written as P(x, t|x ∈ S)dSdt = [Π(t|x∈S)dt] × [P(x|x∈S, t)dS] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this regard, one can use the fact that ψt(x) is the state of the system, con- ditioned on the time being t in the Schr¨odinger picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This implies that |ψt(x)|2 refers to the position probabil- ity density conditioned at a specific time t [14, 15, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Therefore, in the semiclassical approximation, the joint spatiotemporal probability density reads as PSC(x, t|x∈S) = NSCΠSC(t|x∈S) |ψt(x)|2 (4) in which NSC ≡1/ � S dS |ψt(x)|2 is the normalization con- stant, and dS =n·dS, where n is the outward unit normal vector at x∈S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' “Standard” approach The first attempts to investigate the arrival time prob- lem, based on the standard rules of quantum theory, were made at the beginning of the 1960s by Aharonov and Bohm [60], and also Paul [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This approach starts with a symmetric quantization of classical arrival time expres- sion (1), as follows [62]: ˆtAB = mL ˆp −1 − m 2 (ˆp −1 ˆx + ˆx ˆp −1), (5) where ˆx and ˆp=−i ∂/∂x are the usual position and mo- mentum operators, respectively, and ˆtAB is called the Aharonov-Bohm time operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This operator satisfies the canonical commutation relation with the free Hamil- tonian operator, [ˆtAB, ˆp2/2m] = iℏ, which has been used to establish the energy-time uncertainty relation [63, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' However, although ˆtAB is Hermitian (or symmetric in mathematics literature), it is not a self-adjoint operator [65]—a fact that is in agreement with Pauli’s theorem [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The origin of this non-self-adjointness can be under- stood as a result of the singularity at p = 0 in the mo- mentum representation, ˆtAB → (iℏm/2)(p−2 − 2p−1∂p) [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Nevertheless,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' although the (generalized) eigenfunc- tions of ˆtAB are not orthogonal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' they constitute an over- complete set and provide a POVM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' which are used to define the arrival-time distribution as follows [63,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 65]: ΠSTD(t|x=L)= 1 2πℏ � α=± ����� � ∞ −∞ dp θ(αp) � |p| m ˜ψt(p)e i ℏ Lp ����� 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (6) where θ(·) is Heaviside’s step function and ˜ψt(p) is the wave function in the momentum representation which could be obtained from the initial wave function ˜ψ0(p),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' as ˜ψt(p) = ˜ψ0(p) exp � − itp2/2mℏ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The distribution ΠSTD and its generalization in the presence of interaction po- tential have been referred to as the ”standard arrival- time distribution” by some authors [16, 66–69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In fact, Grot, Rovelli, and Tate treated the singularity of (5) by symmetric regularization and obtained equation (6) via the standard Born rule [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The generalizations of equa- tions (5) and (6) in the presence of interaction potential have been investigated in various works [16, 31, 70–75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Using these developments, it has been shown that the non-self-adjointness of the free arrival time operator can also be lifted by spatial confinement [71, 76], and the above arrival time distribution could be derived from the limit of the arrival time distribution in a confining box as the length of the box increases to infinity [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Fur- thermore, recently, the distribution (6) is derived from a space-time-symmetric extension of non-relativistic quan- tum mechanics [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 3 The three-dimensional generalization of (6) is derived by Kijowski’s [10] via an axiomatic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The as- sumed axioms are implied by the principle of the prob- ability theory, the mathematical structure of standard quantum mechanics, and the Galilei invariance [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Based on these axioms, Kijowski constructed the follow- ing arrival time distribution for a free particle that passes through a two-dimensional plane S as ΠSTD(t|x ∈ S) = 1 2πℏ � α=± � R2d2p∥ × ����� � ∞ −∞ dp⊥ θ(αp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='n) � |p⊥| m ˜ψt(p)e i ℏ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='p⊥ ����� 2 , (7) where p⊥ ≡(p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' n)n and p∥ ≡ p − p⊥ are perpendicular and parallel components of p relative to S respectively, and n is the outward normal of plane S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In fact, he first proves the above expression for the wave functions whose supports lie in the positive (or negative) amounts of p⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Then he uniquely derives the following self-adjoint variant of the (three-dimensional version of) Aharonov- Bohm arrival time operator, by demanding that the time operator be self-adjoint and leads to (7) for these special cases via the Born rule [10, 78]: ˆtL = sgn(ˆp⊥) � mLˆp−1 ⊥ − m 2 (ˆp−1 ⊥ ˆx⊥ + ˆx⊥ˆp−1 ⊥ ) � , (8) where ˆx⊥ ≡ ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='n and L (≡ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='n) represent the distance between the detection surface and the origin [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Fi- nally, for an arbitrary wave function, the equation (7) could be derived from this self-adjoint operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' More- over, considering this time operator, besides the com- ponents of the position operator in the detection plane, ˆx∥ ≡ ˆx − (ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='n)n, Kijowski obtains the following expres- sion as the joint position-time distribution on the detec- tion screen via the Born rule [78]: PSTD(x, t|x∈S) = � α=± |ψα S (x, t)|2, (9) in which ψ± S (x, t) is the wave function on the basics of joint eigenstates of the operators ˆtL and ˆx∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Explicitly ψ± S (x, t) = 1 (2πℏ)3/2 � d3p θ(±p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='n) � |p⊥| m ˜ψt(p)e i ℏ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (10) Note that, the arrival time distribution (7) could be re- produced by taking the integral of (9) over the whole of the screen plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The joint space-time probability distri- bution (9), and its generalization for the particles with arbitary spin, have been also derived by Werner in an- other axiomatic manner [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Moreover, it is easy to see that the results (7) and (9) can be obtained from a reg- ularized version of the (three-dimensional generalization of) Aharonov-Bohm time operator, which is the same as the procedure used by Grot, Rovelli and Tate in one- dimensional cases [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Quantum flux and Bohmian approach Inspiring by classical intuition, another proper candi- date for screen observables is the perpendicular compo- nent of the quantum probability current to the screen surface, J(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='n, where J(x, t) = − ℏ m Im [ψ∗ t (x)∇ψt(x)] , (11) and n is the outward normal to the screen S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This pro- posal is applicable for a particle in a generic external potential and a generic screen surface, not necessarily an infinite plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' There are several attempts to derive this proposal in various approaches, such as Bohmian mechanics for the scattering case in [79], decoherent his- tories approach in [80] as an approximation, or in [81] as an exact formula using the concept of extended probabil- ities, and so on [45, 46, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Howover, even if the wave function contains only momentum in the same direction as n, the J(x, t) · n could be negative due to the back- flow effect [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This property is incompatible with the standard notion of probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Nevertheless, this prob- lem could be treated from the Bohmian point of view: Using Bohmian trajectories, it can be shown that the positive and negative values of J(x, t) · n correspond to the particles that reach the point x at S in the same di- rection of n or the opposite direction of it, respectively [83, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this regard, through the Bohmian mechanics in one-dimension, Leavens demonstrates that the time distribution of arrival to x=L from both sides could be obtained from the absolute form of probability flux as [35, 85] ΠQF(t|x=L) = |J(L, t)| � dt |J(L, t)|, (12) which is free from the aforementioned problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The three-dimensional justification of J(x, t) · n as an operational formulation of the arrival time model has been made in [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Also, the generalization of (12) for arrival to the surface S is given by [7, 13, 16, 86] ΠQF(t|x∈S) = � S dS|J(x, t)·n| � dt � S dS|J(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='n|, (13) with dS =n·dS the magnitude of the surface element dS which is directed outward at x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' To illustrate (13) and to generalize it to the case of joint arrival distri- bution, we can use the Bohmian point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this theory, each particle has a specific trajectory, depending on the initial position, and so the rate of passing par- ticles through an area element dS centered at x ∈ S, in the time interval between t and t + dt, is proportional to ρt(x)|v(x, t)·dS|dt, where v(x, t)=J(x, t)/|ψt(x)|2 is the Bohmian velocity of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Hence, using quantum equilibrium condition [87, 88], ρt(x) = |ψt(x)|2, and ac- complishing normalization, the joint arrival distribution could be represented by the absolute value of the current density as 4 PQF(x, t|x∈S) = |J(x, t)·n| � dt � S dS|J(x, t)·n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (14) Now, by integrating (14) over all x ∈ S, we arrive at the three-dimensional arrival time distribution (13) for the screen surface S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' It should be noted that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (14) is not necessarily followed for an ensemble of classical par- ticles because a positive or negative current at a space- time point, (x, t), can in general have contributions from all the particles arriving to x at t from any direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Nonetheless, since the Bohmian velocity field is single- valued, the particle trajectories cannot intersect each other at any point of space-time and so only a single tra- jectory contributes to the current density J(x, t) at the particular space-time point (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Moreover, this fact implies that when v(x, t) · n>0 we can say that the tra- jectory and consequently the particle has passed through the screen from the inside and vice versa for v(x, t)·n<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Hence, one can define the joint probability distribution for the time of arrival to each side of S as P± QF(x, t|x∈S) = J±(x, t)·n � dt � S dS J±(x, t)·n, (15) where J±(x, t) = ± θ(±J·n) J(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In addition, note that there may be some trajectories which cross S more than once—and we have multi-crossing trajectories (see the typical Bohmian trajectory in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The course of the above inference to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (14) was in such a manner that multi-crossing trajectories could contribute several times (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 2 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' However, one could assume the detection surface as a barrier that does not allow the crossed particle to return inside (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 2 (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this case, it is suggested to use the truncated current defined as ˜J(x, t) := �J(x, t) if (x, t) is a first exit through S 0 otherwise (16) where (x, t) is a first exit event through the boundary surface S, if the trajectory passing through x at time t leaves inside S at this time, for the first time since t = 0 [13, 79, 89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The limiting condition in (16), imposes that the joint probability distribution based on it should be computed numerically using trajectories: ˜PQF(x, t|x∈S) = ˜J(x, t)·n � dt � S dS ˜J(x, t)·n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (17) Of course, the detection screen is not always a barrier- like surface (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 2 (b)), and one could assume that there is a point-like detector that lets the multi-crossing trajectories to contribute to the distribution and we can use (14) in such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Horizontal screen Vertical screen Ly Lx x y s o FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Schematic double-slit experiment setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The cen- ter of two slits is considered as the coordinate origin, and the vertical and horizontal screens are placed at x = Lx and y = Ly, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The dashed black line shows a typical Bohmian trajectory that arrives at the horizontal screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' A suitable single-particle detector, in addition to particle arrival position, can record the arrival time using a proper clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' “INTRINSIC” SCREEN OBSERVABLE IN TWO-SLIT EXPERIMENT In this section, we study the discussed proposals in the previous section for the double-slit experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' We com- pare the results of these proposals in the cases of vertical and horizontal screens (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1), and also in different detection schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The main motivation for the study of the horizontal screen is the non-classical particles’ mo- tions along the y-direction, in the Bohmian perspective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' see a typical Bohmian trajectory in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This behav- ior is due to changing the sign of the probability cur- rent’s component in the y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This behavior does not occur for x-component of J and consequently for the Bohmian motion of a particle along the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1, the setup contains two identical slits at y = ±s, and screens are placed at x = Lx and y=Ly correspond to the vertical and horizontal screens, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' To avoid the mathematical complexity of Fresnel diffraction at the sharp-edge slits, it is supposed that the slits have soft edges that generate waves hav- ing identical Gaussian profiles in the y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' So, for each slit, we can take the wave function as an uncorre- lated two-dimensional Gaussian wave packet, which in each dimension has the form ψ(i) G (x, t) = (2πs2 t)- 1 4 exp � (x − x(i) 0 − uxt)2 4σ0st � × exp � i ℏmux(x − x(i) 0 − uxt 2 ) � (i = 1, 2), (18) with m the particle’s mass, σ0 the initial dispersion, ux 5 S S S (a) (b) (c) First-arrival Second-arrival Third-arrival FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Different schemes of particle detection on the screen surface S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In the Bohmian point of view, particles could have a recursive motion on surface S and cross it more than once (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' see the trajectory that plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Assuming different detector types, one can prob variant possible observables on the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In panel (a) a conceivable particle trajectory is depicted, which crosses S three times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this panel, a movable point-like detector is placed on S, which can survey the whole screen and detect particles that arrive only from one side, while in panel (b) a two-sided point detector is placed on S, which can move along it and detect particles that arrive from up and down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In addition, one can assume there is (c) an array of side-by-side detectors covering the entire screen surface S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The last configuration blocks the trajectory and does not allow the crossed particle to return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this scheme, we only detect first-arrivals from one side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' the wave packet’s velocity, x(i) 0 the initial position of wave packet or in other words the location of i-th slit, and st = σ0(1 + iℏt/(2mσ2 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Therefore, when the particle passes through the slits, we have the total wave function as ψ(x, y, t) = 1 √ 2[ψ(1) G (x, t)ψ(1) G (y, t) + ψ(2) G (x, t)ψ(2) G (y, t)], (19) where superscripts (1) and (2) correspond to upper and lower slits, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This form of Gaussian superpo- sition state is commonly used in the literature [7, 90–93] and is feasible to implement by quantum technologies be- cause such a state could be produced and controlled read- ily [94, 95], even without using slits [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this paper, we have chosen the metastable helium atom, with mass m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='64 × 10−27 kg, as the interfering particle, and the parameters as s = 10 µm, σx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='04 µm, σy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 µm, ux = 3 m/s, and uy = 0 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' These values are feasible according to the performed experiments [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Moreover, the meta-stable helium atom could be detected with high efficiency because of its large internal energy [52, 97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Vertical screen The arrival time distribution for the vertical screen placed at different distances from the two-slit is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' As one can see this distribution is the same for all methods, and their average arrival time is close to the corresponding quantity in classical uniform motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' To calculate the mean time of arrival to the screen, we use the arrival time distribution of each method presented in sec II, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (3), (7) and (13), and we have ¯tS = � ∞ 0 dt Π(t|x∈S) t, (20) as the mean arrival time at the surface S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Furthermore, we can compute the average arrival time to each point on the screen using the joint probability distribution as ¯tx = � ∞ 0 dt P(x, t|x∈S) t � ∞ 0 dt P(x, t|x∈S) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (21) This observable is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 4-b for a vertical screen placed at Lx = 300 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Apparently, the results of the standard and quantum flux methods are the same and similar to one that resulted in [7] by Nelson’s mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Nevertheless, they are different from the semiclassical ap- proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' However, when the interference pattern is calculated by either method, we see that their predicted cumulative position distributions do not differ much from the others (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 4-a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This observable can be calculated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='by using the joint 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='▲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='▲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='▲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='94 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='98 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='Averaged arrival time (ms) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='P(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='y (mm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 Semiclassical Standard Quantum flux FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (a) The cumulative arrival position distribution, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (22), for the vertical screen at Lx = 300 mm, and (b) the average arrival time at each point of the screen, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' P(x|x∈S)= � ∞ 0 dt P(x, t|x∈S) � ∞ 0 dt � S dS P(x, t|x∈S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (22) As mentioned, it should be noted that, |ψt(x)|2 is just the conditional position probability density at the specific time t, not the position-time joint probability density and so the accumulated interference pattern, P(x|x∈S), is not given by � dt|ψt(x)|2 [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Horizontal screen In this section, we are going to compare the mentioned proposals in the double-slit setup with a horizontal de- tection screen (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this regard, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 5, the arrival time distributions at the screen are plotted for some horizontal screens which are located at Ly =15, 20, 25, and 30 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this figure, solid-black, dashed-green, and dash-dotted-blue curves represent the distributions ΠST D, ΠQF and ΠSC respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Also, the vertical lines show the average time of arrival to the screen, ¯tS, associated with these arrival time distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' From this figure, one can see that, although the averages al- most coincide, the distributions are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Moreover, as expected, when the screen’s distance from the center of the two slits Ly decreases, the difference between dis- tributions increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Most of these differences occur in the early times, which are associated with the particles that arrive at the S in the near field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Furthermore, we observe that the ΠSC behaves quite differently from ΠQF and ΠST D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The distributions ΠQF and ΠST D are more or less in agreement, however, for the screen that is lo- cated at Ly =15 µm, a significant difference between the standard and quantum flux distributions occurs around t≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 1 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 1 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='7 Π(t | y = Ly) t (ms) Ly = 30 µm Ly = 25 µm Ly = 20 µm Ly = 15 µm Semiclassical Standard Quantum flux FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Arrival time distributions of particles that arrive on the horizontal screen at four different distances from the center of two slits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The vertical lines show the average arrival time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' To have a more comprehensive insight, we can look at the joint spatiotemporal arrival distributions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this figure, joint distributions, PSC, PSTD and PQF are plotted in three panels, for the horizontal screen surface located at Ly = 15 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' These density plots clearly visualize differences between the mentioned arrival dis- tribution proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In these plots, we can see separated fringes with different shapes, which this fact imply that the particles arrive at the screen in some detached space-time regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In the insets, one can see that the shapes of these regions are different for each proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In the joint density of the semiclassical approximation 7 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='6-a), fringes are well-separated, while the standard distribution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 6-b) exhibits more continuity in its fringes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In addition, in the pattern of the quantum flux proposal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 6-c) there are grooves between every two fringes which is due to changing the sign of J(x, t) · n in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In all panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='6, the duration of “temporal no-arrival windows” between every two typical fringes variate in the range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 ms which has a spatial extension of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='3 to 2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' These space-time scales are utterly amenable empirically by current technologies [53, 96], which could be used to test these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='00 t (ms) t (ms) t (ms) x (mm) Quantum flux Standard Semiclassical (c) (b) (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Density plots of joint arrival position-time distribu- tions for particles that arrive at the horizontal screen of the double-slit experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Panels (a), (b), and (c) represent PSC, PSTD and PQF, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Insets: Magnified contour plots of the joint distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The average time of arrival to each point of the screen and cumulative position interference pattern could be cal- culated as in the vertical screen case by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (21) and (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 7(a)-(b), these two quantities are shown for the horizontal screen which is placed at y = 15 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In contrast to the vertical screen, the cumulative position distribution of the semiclassical approximation is entirely separate from the two other proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The cumulative position distribution resulting from standard and quan- tum flux approaches have obvious differences from each other, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' As one can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 7(b), the aver- age arrival times are the same for all three methods at first and begin to deviate from each other at x ≈ 5 mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' then again, these curves converge to each other at x≈25 mm, approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The maximum deviation between the standard and quantum flux average arrival time oc- curs at x≈19 mm, which is quite in the far-field regime— the width of the initial wave function is ∼ O(10−3)mm which is smaller than 19 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Therefore one can suggest the average arrival time in the gray region of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 7(b) as a practical target for comparing these approaches experi- mentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' To this end, we study arrival time distributions at some points of this region as local arrival distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The arrival time distribution conditioned at a specific point x on the screen can be obtained as follow Πx(t|x∈S) = P(x, t|x∈S) � ∞ 0 dt P(x, t|x∈S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (23) Using the associated joint distribution of each proposal, we have plotted Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 7(c)-(f) that show Πx(t|x ∈ S) at the positions x=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm, on the screen placed at Ly = 15 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The broken black curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 7 (c)-(f), resulting from the quantum flux proposal, against the smooth curves of the other two methods could be understood as the result of the changing the signa- ture of the y-component of the probability current: Note that, quantum flux distribution is given by the absolute value of the probability current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The origin of distinc- tions between the local average arrival times is more per- ceptible from these local arrival distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In princi- ple, these distributions could be probed using fast and high-resolution single-atom detectors [53, 97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In partic- ular, the delay-line detector that is recently developed by Keller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' [51] seems suitable for our purpose: It has the capability to resolve single-atom detection events temporally with 220 ps and spatially with 177µm at rates of several 106 events per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' We estimate by a numerical investigation that these lo- cal arrival distributions could be well reconstructed from about 104 number of detection events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' As an example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 7, the histograms associated with the probability densities of the panel (f) are plotted in panel (g), using 104 numerical random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' It is easy to estimate that the recording of 104 particle detection events can de- termine the local average arrival time with a statistical error of about 10−2ms, while the differences between local average arrival times of various proposals are almost big- ger than 10−1ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Using cumulative position distribution, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='00.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='08 5 6 7 8 5 6 7 8 5 6 7 8 5 6 7 8 5 6 7 8 x=19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm x=18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 mm x=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 mm x=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm Πx(t|x ∈ S) t (ms) ∆N (g) (f) (e) (d) (c) 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 0 1 0 1 0 600 Semiclassical Standard Quantum flux 0 5 10 15 20 25 0 2 4 6 8 3 4 5 6 7 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 0 5 12 14 16 18 20 22 24 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='005 Averaged arrival time (ms) x (mm) P(x) (b) (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The space-time arrival statistics for the double-slit experiment with a horizontal screen placed at Ly =15 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Panel (a) represents the average time of arrival at each point of the screen, ¯tx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Panel (b) represents the cumulative position probability den- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The panels (c)-(f) show the local arrival time probability densities, Πx(t|x∈S), at the at the points x=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm on the screen, which are chosen from the gray region in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The vertical lines in these panels represent the average arrival times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Panel (g) is Histograms associated with probability densities of panel (f), which are generated by 104 numerical random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 7(b), one can estimate that, if the total number of particles that arrived at the screen is about 108, we have about 104 particles around x = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm, in the spacial interval (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='1, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Using recent progress in laser cool- ing and magneto-optical trapping [97], the preparation of a coherent ensemble of metastable helium atoms with this number of particles is quite achievable [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' One might be inclined to think that the difference be- tween the quantum flux and standard average arrival times is just due to changing the signature of J(x, t) · n, but in the following, we show that even without the con- tribution of the negative part of J(x, t)·n, these proposals are significantly distinguishable: see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Detection schemes As we mentioned in section II C, according to the Bohmian deterministic point of view, there are several possible schemes to detect arrived particles, especially for the horizontal screen surface which we have recursive motions on it (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' One can assume that the horizontal screen is swept with a point-like detector that surveys all arrived particles at the surface S, which we call spot-detection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this scheme, one option is to use a unilateral detector to detect arrived particles at the top or bottom of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this case, the positive and negative parts of the quantum probability current have respectively corresponded to particles that arrive at the top or bottom of S (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 2 (a)), and we must use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (15) to calculate the screen observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Addi- tionally, we can choose a bilateral detector (or two uni- lateral detectors) that prob all particles that arrive from both sides of S, along the time with several repeats of the experiment (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 2 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In these circum- stances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' spot-detection scheme), there is no barrier in front of the particles before they reach the point of detection and we can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (14) to obtain the screen observables as in the two previous subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' As we have already shown in section II C, whether the particles arrive from the top or bottom of S, the abso- lute value of the quantum probability current yield the trajectories’ density and consequently give the joint dis- tribution of the total arrival at each point of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This fact is the case for the standard method, as well, how- ever, there is a subtle difference between the two propos- als in the spot-detection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' When we talk about 9 0 5 10 15 20 25 5 20 0 2 4 6 8 0 5 10 15 20 25 0 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='30 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 x (mm) t (ms) First arrivals Second arrivals Third arrivals All arrivals First arrivals Quantum flux FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The space-time Bohmian arrival statistics for the double-slit experiment with a horizontal screen placed at Ly =15 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The interior curves in the central figure are the averaged times of arrival obtained by different detection schemes: see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The Left and top plots are marginal arrival time distributions and marginal arrival position distributions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The scatter plot is generated using 2 × 106 Bohmian trajectories, and the black, blue, and green points of the scatter plot represent the first, second, and third arrivals of Bohmian particles to the screen, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The inset is a zoom-in of the dashed rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' the spot-detection in the Bohmian approach, it would be considered the possibility of multi-crossing and the distribution includes all-arrivals at S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Although, in the standard method there is an interpretation for ψ+ S (x, t) and ψ− S (x, t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (10), which relates them to the par- ticles arrive at S in a direction which is the same or op- posite with the direction of outward normal of the screen n, respectively [10, 64], nevertheless, since there are no defined paths in this approach, it is obscure whether it counts only the first-arrivals to each side of the screen or includes recursive movements of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Alternatively, along with the spot-detection scheme, it could be assumed that there is a continuous flat barrier in front of the particle’s paths as the detection surface or screen surface that does not allow particles to cross this surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Depending on the screen’s length and posi- tion, there are several possibilities for the detection pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In each case, a specific number of particle paths contribute to the distribution of arrival time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In the simplest case, the screen blocks all the trajectories that reach the horizontal surface S, and we only detect the first-arrivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In such a setup, we can no longer use the quantum flux method to represent Bohmian trajectories’ first encounter with the surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' hence, the screen ob- servables must be obtained by numerical analysis, due to the definition of truncated current as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (16) and its corresponding joint distribution, ˜PQF(x, t|x∈S), defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' By computing the Bohmian trajectories, we can find positions and times of the first-arrivals to the screen, and consequently calculate the arrival time distribution which mathematically could be defined as ˜ΠQF(t|x∈S) = � S ˜PQF(x, t|x∈S)dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (24) Also, other observable quantities such as the cumulative spatial distribution and averaged arrival time over the detection surface could be defined and calculated numer- ically in a similar way—by substituting ˜PQF(x, t|x ∈ S) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (21) and (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Furthermore, we can complete the computations to find the second and third encounters to the surface (regardless of the barrier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 8, we show our numerical results of Bohmian trajectories simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The background scatter plot is the position and time of arrivals of 2 × 106 trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this plot, the second and third arrivals are shown in blue and green, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Here, it is more clear why 10 5 6 7 8 5 6 7 8 5 6 7 8 5 6 7 8 x=19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm x=18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 mm x=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 mm x=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm Π(t| x, y) t (ms) 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 First arrivals Quantum flux All arrivals FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Arrival time distribution at the horizontal screen po- sitions x = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm, and Ly = 15 µm, which are in the gray region of Fig (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The width of sampling in each point is about δx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='25 mm, and 108 Bohmian trajec- tories are simulated to obtain these distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' we interpret the grooves of the quantum flux density plot (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 6 (c)) as a result of the multi-crossing of Bohmian trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The three middle graphs are the average time of the first and all-arrivals, which are simulation re- sults of 108 trajectories, and are compared by the quan- tum flux method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' As expected, the average time of all- arrivals fits on the quantum flux curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' However, the av- erage time of first-arrivals deviates from all-arrivals in the area discussed in the previous section (between x = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm and x = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' To scrutinize the deviation zone of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 8 (the gray re- gion), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 9 is drawn to show the arrival time distribu- tions of screen positions x = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' As one can see, at the first recursive points of quantum flux distribution, the first-arrival distributions raise down to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This implies that in the presence of a barrier- like screen, there would be a big temporal gap between arrived particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' These gaps could be investigated as a result of the non-intersection property of Bohmian tra- jectories that cause a unilateral motion of particles along the direction of the probability current field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' SCREEN BACK-EFFECT In principle, the presence of the detector could mod- ify the wave function evolution, before the particle detec- tion, which is called detector back-effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' To have a more thorough investigation of detection statistics, we should consider this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Howsoever, due to the measurement problem and the quantum Zeno effect [9], a complete in- vestigation of the detector effects is problematic at the fundamental level, and it is less obvious how to model an ideal detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Nonetheless, some phenomenological non-equivalent models are proposed, such as the gener- alized Feynman path integral approach in the presence of absorbing boundary [12, 37–39], Schr¨odinger equation with a complex potential [44], Schr¨odinger equation with absorbing (or complex Robin) boundary condition [40– 44], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The results of these approaches are not the same, and a detailed study of the differences is an in- teresting topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this section, we provide a brief review of the absorbing boundary rule (ABR) and path-Integral with absorbing boundary (PAB) models, then we com- pare them in the double-slit setup with the horizontal screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Absorbing Boundary Rule Among the above-mentioned phenomenological mod- els, the absorbing boundary condition approach has the most compatibility with Bohmian mechanics [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The application of absorbing boundary condition in arrival time problem was first proposed by Werner [40], and re- cently it is re-derived and generalized by Tumulka and others using various methods [41–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Especially, it is re- cently shown that in a suitable (non-obvious) limit, the imaginary potential approach yields the distribution of detection time and position in agreement with the ab- sorbing boundary rule [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' According to this rule, the particle wave function ψ evolves according to the free Schr¨odinger equation, while the presence of a detection screen is modeled by imposing the following boundary conditions on the Detection screen, x ∈ S, n · ∇ψ = iκψ, (25) where κ>0 is a constant characterizing the type of detec- tor, in which ℏκ/m represents the momentum that the detector is most sensitive to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This boundary condition ensures that waves with wave number κ are completely absorbed while waves with other wave numbers are partly absorbed and partly reflected [41, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In the absorbing boundary rule, the joint spatiotemporal distribution of the detection event is given by quantum flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Consider- ing (25), this distribution reads PABR(t, x|x∈S) = |ψABC|2 � dt � S dS|ψABC|2 , (26) where ψABC represent the solution of the free Schr¨odinger equation satisfying the aforementioned absorbing bound- ary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This distribution can be understood in terms of Bohmian trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The Bohmian particle equation of motion, ˙X = (ℏ/m)Im [∇ψABC/ψABC], to- gether with the boundary condition (25), imply that tra- jectories can cross the boundary S only outwards and so there are no multi-crossing trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' If it is assumed 11 that the detector clicks when and where the Bohmian particle reaches S, the probability distribution of detec- tion events is given by (26), because the initial distribu- tion of the Bohmian particle is |ψABC(x, 0)|2 [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Path-Integral with Absorbing Boundary In several papers [12, 37–39], Marchuwka and Schuss develop an interesting method to calculate the detec- tion effect of absorbing surface using the Feynman path integral method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' They postulate a separation princi- ple for the wave function in which we could consider the (bounded wave function) as a sum of two parts, ψ(x, t) = ψ1(x, t) + ψ2(x, t), such that ψ1(x, t) corre- sponds to the survival part of the wave which is orthogo- nal to ψ2(x, t) at a time t and evolve independently [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' So, we can obtain the probability of survival of the parti- cle, denoted S(t), which is the probability of the particle not being absorbed by the time t, as � D d3x|ψ1(x, t)|2, where the integral is over the domain D, outside the ab- sorbing region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' By discretizing the path integral in a time interval [0, t] and eliminating the trajectories that, in each time interval [t′, t′+∆t′] for all t′ < t, are reached to the absorbing surface S, the survival and consequently absorbing probability would be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Based on this analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' we could define a unidirectional probability cur- rent into the surface as d dt[1−S(t)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' which yields a normal component of the multidimensional probability current density at any point on S as J(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' t)·n= λℏ mπ |n·∇ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' t)|2 × exp � − λℏ mπ � t 0 dt′ � S dS|n·∇ψ(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' t′)|2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (27) where dS = n · dS is the magnitude of the surface ele- ment dS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' n is the unit outer normal to the absorbing surface S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' and λ is a proportionality factor with the di- mension of length [37,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Also, ψ(x, t) is the solution of Schr¨odinger equation bounded and normalized in the domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Moreover, the normal component J(x, t)·n is supposed to be the probability density for observing the particle at the point x on the screen at time t [12, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Screen back-effect in two-slit experiment In order to complete the investigations carried out in section III, we are going to study the screen back- effect in the double-slit experiment with a horizontal screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this regard, we compare the arrival distri- butions which are resulted from the absorbing bound- ary rule (ABR), path-Integral with absorbing boundary (PAB), and Bohmian truncated current (BTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' We continue with the same initial conditions as in sec- tion III, and choose κ = 1 µm−1 for ABR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Density plots of joint probability distributions of position and time (screen observable) for the horizontal screen placed at y = 15 µm in the double-slit experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' These densities are calculated by the three methods which take the screen effects into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' κ leads to the maximum absorption probability—which is almost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4—for the chosen initial wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In addition, for a more meaningful comparison, we consider λ = 1 µm in the PAB method, which leads to the same absorption probability as ABR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The resulting joint ar- rival time-position distributions of the three methods are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' As one can see, the distributions of the ABR and PAB methods—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=', panels (a) and (b) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 10—have more compatibility with each other than the result of the BTC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' However, there are dif- ferences between them which are more obvious in the zoomed areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The joint density of the ABR is more uni- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='85 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm x=18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 mm x=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 mm x=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm Π(t| x, y) t (ms) 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 ABR PAB BTC FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Arrival time distribution at the horizontal screen po- sitions x = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm, and Ly = 15 µm, which are calculated for the three methods which take the screen effects into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' formly distributed than of the PAB method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The empty areas between the fringes of the panel (c) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 10 are due to the elimination of the recursive trajectories—or in other words, are due to the elimination of second and third arrivals in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' For a more detailed comparison, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 11 the spa- tial and temporal marginal distributions are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In addition, the associated local average arrival times are compared in the central panel of this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The PAB method leads to significant discrepancies in marginal dis- tributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The maximum difference is about 40% that occurs around x≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='8 mm, which seems testable clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In contrast to the previous results on intrinsic distri- butions, in which the difference between average arrival times was significant, there is a good agreement in this observable for the ABR and PAB methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' However, there is a significant difference between the average ar- rival time in these two methods and BTC around x = 6 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 12, the local arrival time distributions at some points on the screen are plotted, which show simi- lar behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' SUMMARY AND DISCUSSION When and where does the wave function collapse?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' How one can model a detector in quantum theory?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' These are the questions that we investigated in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' We tried to show that there is no agreed answer for these ques- tions, even for the double-slit experiment that has in it the heart of quantum mechanics [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This is a practical encounter with the measurement problem [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this regard, we numerically investigated and compared the main proposed answers to these questions for a double- slit setup with a horizontal detection screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' It is shown that these proposals lead to experimentally distinguish- able predictions, thanks to the current single-atom de- tection technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this work, we suggest the meta-stable helium atom as a proper coherent source of the matter wave, however, other sources may lead to some practical improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' For example, using heavier condensate atoms can lead to more clear discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Moreover, it is worth not- ing that although the experiment with photons may have some practical advantages, there are more complications in its theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This is partially because of the relativistic localization-causality problem [101–104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The theoretical investigation of a proposed experiment for photons would be an interesting extension of the present work, which has been left for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' ACKNOWLEDGMENTS We sincerely thank Mohammad Hossein Barati for carefully reviewing the manuscript, and Sheldon Gold- stein for his helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 13 [1] As shown by Pauli [105], if the Hamiltonian spectrum is discrete or has a lower bound, then there is no self-adjoint time operator canonically conjugate to the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Zimmermann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Mishra, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Doran, D.' 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VOL. 14, NO. 8, AUGUST 2021 +1 +The state-of-the-art 3D anisotropic intracranial +hemorrhage segmentation on non-contrast head CT: +The INSTANCE challenge +Xiangyu Li, Gongning Luo, Kuanquan Wang, Hongyu Wang, Shuo Li, Jun Liu, Xinjie Liang, Jie Jiang, +Zhenghao Song, Chunyue Zheng, Haokai Chi, Mingwang Xu, Yingte He, Xinghua Ma, Jingwen Guo, Yifan Liu, +Chuanpu Li, Zeli Chen, Md Mahfuzur Rahman Siddiquee, Andriy Myronenko, Antoine P. Sanner, Anirban +Mukhopadhyay, Ahmed E. Othman, Xingyu Zhao, Weiping Liu, Jinhuang Zhang, Xiangyuan Ma, Qinghui Liu, +Bradley J MacIntosh, Wei Liang, Moona Mazher, Abdul Qayyum, Valeriia Abramova, Xavier Llad´o +Abstract—Automatic intracranial hemorrhage segmentation in +3D non-contrast head CT (NCCT) scans is significant in clinical +practice. Existing hemorrhage segmentation methods usually +ignores the anisotropic nature of the NCCT, and are evaluated +on different in-house datasets with distinct metrics, making it +highly challenging to improve segmentation performance and +perform objective comparisons among different methods. The +2022 intracranial hemorrhage segmentation on non-contrast head +CT (INSTANCE 2022) was a grand challenge held in conjunc- +tion with the 2022 International Conference on Medical Image +Computing and Computer Assisted Intervention (MICCAI). It is +intended to resolve the above-mentioned problems and promote +the development of both intracranial hemorrhage segmentation +and anisotropic data processing. The INSTANCE released a +training set of 100 cases with ground-truth and a validation set +with 30 cases without ground-truth labels that were available to +the participants. A held-out testing set with 70 cases is utilized +for the final evaluation and ranking. The methods from different +participants are ranked based on four metrics, including Dice +Similarity Coefficient (DSC), Hausdorff Distance (HD), Relative +Volume Difference (RVD) and Normalized Surface Dice (NSD). +A total of 13 teams submitted distinct solutions to resolve +the challenges, making several baseline models, pre-processing +strategies and anisotropic data processing techniques available to +future researchers. The winner method achieved an average DSC +of 0.6925, demonstrating a significant growth over our proposed +baseline method. To the best of our knowledge, the proposed +INSTANCE challenge releases the first intracranial hemorrhage +segmentation benchmark, and is also the first challenge that +intended to resolve the anisotropic problem in 3D medical image +segmentation, which provides new alternatives in these research +fields. +Index Terms—Intracranial hemorrhage Segmentation Chal- +lenge Anisotropic data +I. INTRODUCTION +I +NTRACRANIAL hemorrhage (ICH) is a severe brain dis- +ease and a main cause of stroke [1], [2]. It has a high +mortality rate of 40% within one month [3], [4]. Furthermore, +ICH even causes significant disability in survivor patients, +with only 20% of patients expected to be capable of living +independently in half year [5]. Therefore, early and accurate +diagnosis of the ICH is important for saving patients’ lives +and improve their prognosis in clinical practice [1], [6]. +Non-contract head computerized tomography (NCCT) is the +primary imaging modality to diagnosing ICH for its widely +availability in most emergency rooms and high sensitivity for +detecting ICH. Moreover, NCCT enables accurate monitoring +of hemorrhage progression, and effectively quantify hematoma +volumes in ICH [1], [4], [7], making it a gold standard +examination for the diagnosis of ICH. +Hematoma volume estimation is significant for the prog- +nosis and treatment decisions for ICH patients. In recent +clinical trials, the hematoma volume has been utilized as +a reliable indicator to determine the optimal candidates for +intervention [8]–[10]. Thus, volume quantification of ICH has +become an essential procedure for outcome predictions and +ICH therapy. The hematoma volume can be estimated by +semiautomated methods with the aid of radiologists, which +is time-consuming [11] and suffers from inter-rater variability +[12]. The ABC/2 method [13] is an effective technique to +estimate hematoma volume in clinical practice since it is +simple to implement. However, the estimation accuracy of the +ABC/2 method dramatically decreases with irregular or large +scale hemorrhages [8], [14]. The ICH segmentation methods, +enabling accurate and rapid hematoma volume quantification, +have become the leading criterion in ICH diagnosis. +However, there exists plenty of challenges to segment ICH +for automatic methods. For example, the hemorrhage struc- +tures vary considerably across patients in terms of shape, size, +and localization, preventing the use of valuable location and +shape priors that are significant elements in the segmentation +of many other anatomical structures. The blurred boundaries +for the ICH region further improve the difficulty of the +segmentation task [15]. +Because of the clinical significance and the intrinsic chal- +lenges, the task of automatic intracranial hemorrhage segmen- +tation has attracted extensive attention in the past few years. +Recently, deep learning–based ICH segmentation models that +segment ICH regions and quantify hematoma volume have +been performed to effectively diagnose ICH and have achieved +competitive results [6], [16]–[20]. However, all those above- +mentioned ICH segmentation methods ignore the anisotropic +nature of the NCCT volume by simply performing 2D or 3D +convolutional networks, and they were evaluated on different +in-house hemorrhage segmentation datasets with distinct met- +rics, making it highly challenging to improve segmentation +performance and perform objective comparisons among these +arXiv:2301.03281v1 [eess.IV] 9 Jan 2023 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +2 +methods. Consequently, it remains hard to determine which +kinds of segmentation techniques may be valuable to follow +in clinical practice and research; what exactly the performance +is of the state-of-the-art methods. +To resolve the above-mentioned challenges on fair com- +parisons of different methods, we organized the INtracranial +hemorrhage SegmenTAtioN ChallengE on non-contrast head +CT (INSTANCE) in conjunction with the 2022 international +conference on Medical Image Computing and Computer As- +sisted Interventions (MICCAI) in Singapore. To this end, +we collected and released an ICH segmentation dataset of +200 3D volumes with refined labeling from several experi- +enced radiologists, and encouraged the participants to develop +novel algorithms to effectively segment hematoma region with +anisotropic NCCT volumes. Moreover, we evaluate different +benchmark ICH segmentation methods with the same metrics, +including Dice Similarity Coefficient (DSC), Hausdorff dis- +tance (HD), relative volume difference (RVD) and normalized +surface dice (NSD). Each of these benchmark methods was +implemented by different challenge participants on a subset of +the ICH dataset, and tested on a isolated testing dataset against +the manually delineated groundtruth labels. To the best of our +knowledge, INSTANCE is the first public intracranial hemor- +rhage segmentation challenge, and also the first challenge that +intended to deal with the anisotropic problem in 3D biomedical +image segmentation. It is served as a solid benchmark for ICH +segmentation tasks, and would also promote the development +of intracranial hemorrhage segmentation and anisotropic data +processing. +II. PRIOR WORKS +A. Related intracranial hemorrhage segmentation methods +A large numbers of methods have been proposed to automat- +ically segment intracranial hemorrhage in CT scans. Among +them, deep learning techniques are widely adopted for its +excellent performance in medical image segmentation tasks +[15], [21]. Ironside et al. utilized U-Net [22] to automati- +cally segment ICH and estimate the hematoma volume. They +achieved comparable accuracy and greater efficiency compared +to manual and semi-automated segmentation techniques [8]. +To address the issue of insufficient annotation data for ICH +segmentation tasks, Kuo et al. proposed a patch-based FCN +network and segmented ICH in an active learning manner [23]. +Chang et al. proposed an ROI-based framework that is opti- +mized specifically for ICH detection and segmentation tasks by +projecting 3D features to 2D networks in the feature pyramid +network [18]. Kwon et al. proposed a Siamese U-Net method +to segment ICH by leveraging the dissimilarity between +learned features of healthy templates and input images [20]. +Kyung et al. proposed a supervised multi-task aiding represen- +tation transfer learning network for ICH, which was divided +into upstream and downstream. In the upstream, effective +representation learning was performed by multi-task learning +(classification, segmentation, reconstruction) and differences +in the specific head of the consistency loss mitigation target are +added. For downstream, feature extractor trained upstream is +combined with 3D operator (classifier or divider) to implement +specific tasks [16]. Wu et al. proposed a combination of an +attention-based convolutional neural network and a variational +Gaussian process for multiple instance learning method for +predicting intracranial hemorrhage slices [24]. Toikkanen et +al. proposed a residual segmentation method based on gener- +ative adversarial network, which generates the image without +bleeding in the original section through the model, and then +calculates the difference between the generated image and the +original image, so as to obtain the segmented image [17]. +Abramova et al. introduced the squeeze-excitation block into +3D U-Net to solve the problem of segment hemorrhagic stroke +lesions. Moreover, a restrictive patch sampling is proposed to +alleviate the class imbalance problem and also to deal with +the issue of intra-ventricular hemorrhage [25]. Kuang et al. +designed new self-attention blocks and contextual attention +blocks that take full advantage of both in-chip and inter- +chip information. In addition, multilevel training strategies are +proposed to reduce the influence of inter-class imbalance [26]. +Wang et al. propose a Masked Multi-Task Network method +to detect brain CT volumes with intracranial hemorrhage and +distinguish hemorrhage type by leveraging different types of +intracranial hemorrhage at different locations [27]. Guo et al. +propose a full convolutional neural network for simultaneous +classification and segmentation of ICH, and the ConvLSTM +module was used to address this issue of the loss of spatial +information [28]. Kadam et al. propose architectures combined +Xception and LSTM/GRU for classification of Intracranial +Hemorrhage. It is also found through experiments that Xcep- +tion GRU model has better performance on most of the metrics +as compared to the Xception and Xception LSTM models [29]. +Despite the excelent results reported in the above pa- +pers, it is still challenging to identify the best performing +method among them because of the varied testing datasets +and evaluation metrics. The proposed INSTANCE challenge +provides a standardized procedure to systematically evaluate +and compare different SOTA methods on the same testing +dataset and consistent evaluation metrics, enabling objective +and fair comparison among different techniques. +B. Medical Image Segmentation Challenges +Recently years have witnessed the growing popularity for +biomedical image analysis challenges, especially for medical +image segmentation challenges. To name a few, there were 25, +20, and 40 accepted challenges at the International Conference +on Medical Image Computing and Computer-Assisted Inter- +vention (MICCAI) 2020, 2021, and 2022, respectively. From +2020 to 2022, the number of challenges nearly doubled, and +the segmentation-related challenges occupied 38% of all the +challenges1. Similarly, in the largest biomedical image chal- +lenge platform ’Grand Challenge2’, 149 out of 315 (47.3%) +challenges are designed for segmentation tasks. There are lots +of successful challenges in medical image segmentation, for +example, the Brain Tumor Segmentation (BraTS) challenge +[30] provide a solid benchmark for multimodal brain tumor +1https://www.biomedical-challenges.org/ +2https://grand-challenge.org/ + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +3 +segmentation task, numerous methods on brain tumor seg- +mentation and multi-modal learning have been validated on +this benchmark, significantly improving the development of +those research fields. The Head and neck tumor segmentation +challenge (Hecktor) [31] organized a novel challenge for head +and neck tumor segmentation on PET/CT modalities, which +claimed to be the pioneer work on this field. The abdomen ct +organ segmentation [32] first consider the inference time, and +GPU memory consumption as extra evaluation metrics instead +of simply focusing on the segmentation accuracy, providing a +novel benchmark with more comprehensive evaluation metrics. +The Kidney Tumor Segmentation (KiTS) ( [33] ) challenge +allow participants to compare their methods on kidney and +kidney tumor segmentation tasks. +Those above challenges have made great progress in pro- +moting the development of specific medical field. However, to +the best of our knowledge, there are no challenges intended +to resolve the ICH segmentation with anisotropic 3D volumes. +Hence, the INSTANCE is the first released grand challenge for +the ICH segmentation and also the first challenge that intended +to deal with the anisotropic problem in 3D medical image +segmentation. We believe that the ICH data and algorithms +provided in this benchmark would be helpful to promote +the development of both ICH diagnosis and anisotropic data +processing. +III. THE ORGANIZATION OF THE INSTANCE CHALLENGE +The proposed INSTANCE challenge was organized in 2022 +and was in conjunction with the 25rd MICCAI conference +as a satellite event. It was deployed on the Grand Challenge +platform. The official webpage of the INSTANCE challenge +is https://instance.grand-challenge.org/. Meanwhile, we also +construct a Github repository +3 which provides plenty of +resources related to the challenge, for example, the agree- +ment files for accessing the dataset, the docker rules and +submission examples, and also the baseline models. For the +challenge schedule, the registration is open to the public on +March 28, 2022. The training and validation dataset were +released on April 6 and July 15, respectively. The dead- +line of the open validation phase and the testing phase is +on August 7 and August 14, respectively. In the validation +phase, the participants uploaded their segmentation results to +the Grand challenge website, and the platform automatically +calculated the evaluation metrics by comparing them with +the ground-truth labels we provided, and then displayed the +calculated metrics on the validation leaderboard 4 In the testing +phase, the participants are required to submit one successful +docker image that contains their algorithms, and we ran the +docker images from different participants on the closed testing +dataset. The dataset of the INSTANCE challenge are currently +available to the public on Grand Challenge platform after +signing an agreement file and the post-challenge leaderboard +submission is open for researches in this community. The +following sections summarizes the detailed implementation of +the INSTANCE challenge. +3https://github.com/PerceptionComputingLab/INSTANCE2022 +4https://instance.grand-challenge.org/evaluation/challenge/leaderboard/ +A. Dataset +We obtained the approval from Peking university, shougang +hospital to perform a retrospective analysis of the patients that +were diagnosed as intracranial hemorrhage between 2017 and +2019. We then collected 200 non-contrast head CT volumes +of those patients to construct challenge dataset. For these +200 cases, they were diagnosed as different kinds of ICHs, +including intraparenchymal hemorrhage (IPH), intraventricular +hemorrhage (IPH), subarachnoid hemorrhage (SAH), subdural +hemorrhage (SDH), and epidural hemorrhage (EDH), an exam- +ple for each kind of ICH is illustrated in Fig. 1. We then split +the 200 cases into training, validation and testing, with 100, 30, +and 70 cases respectively. The CT scans and the labels of the +training set are available to the participant for model training, +while only the CT scans are provided for them to tune their +algorithms on the Grand Challenge platform. Finally, in the +testing phase, we provide a closed test set for fair comparison +between different methods. +For each of the subject in INSTANCE dataset, we first +converted the traditional Digital Imaging and Communications +in Medicine (DICOM) files to the Neuroimaging Informatics +Technology Initiative (NIfTI) format. In this way, each subject +only has one single NIfTI file instead of a bunch of DICOM +files, making it easier to process in a image segmentation +program. The volume sizes ranges from 512 × 512 × 20 to +512 × 512 × 70, and the pixel spacing of a CT volume is +0.42mm × 0.42mm × 5mm, hence the volume is anisotropic +with inter-slice resolution much smaller than the within-slice +resolution. The window width and the window center is 90HU +and 40HU, respectively. We kept the original Hu value in +the NIfTI volume since the participants can conduct different +windowing strategies. +For the data annotation, we gathered several experienced +radiologists and some postgraduate students majored in med- +ical imaging to perform hemorrhage region annotation in +the NCCT scans. To improve the efficiency of the annota- +tion process, we adopted a coarse-to-fine annotation strategy. +Specifically, the ICH lesions were first manually delineated in +the NCCT volumes with a popular annotation software in med- +ical imaging, Seg3D5 [34]. Then the experienced radiologists +checked the coarse annotations and manually refined them. +Finally, all the radiologists double-check the annotations from +other annotators, and discuss to achieve the final annotations +with majority voting strategy. +B. Evaluation Measures and Ranking Method +The INSTANCE challenge adopted four accuracy-related +evaluation metrics: Dice Similarity Coefficient (DSC), Haus- +dorff Distance (HD), Relative Volume Difference (RVD) and +Surface Dice (NSD) [35]. We utilized DSC and HD since +they are widely used in different medical image segmentation +challenges. They are complementary metrics for evaluating +segmentation performance. DSC was utilized to measure the +region overlapping error between ground truth and segmen- +tation results, while HD is used to evaluate the coincidence +5https://www.sci.utah.edu/cibc-software/seg3d.html + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +4 +Fig. 1: Different kinds of intracranial hemorrhages, including intraparenchymal hemorrhage (IPH), intraventricular hemorrhage +(IPH), subarachnoid hemorrhage (SAH), subdural hemorrhage (SDH), and epidural hemorrhage (EDH). The varied shapes and +positions for different kinds of hemorrhages promote the difficulties of the segmentation task. +TABLE I: The Correspondence between the Team names and +the aliases. +Team +Alias +vegetable +T1 +nvauto +T2 +mec-lab +T3 +ibot +T4 +stubmers +T5 +crainet +T6 +superembrace +T7 +scan +T8 +dolphins +T9 +nic-vicorob +T10 +2i mtl +T11 +avich +T12 +visal +T13 +between segmented surface and target surface. We used the +RVD since the purpose for the ICH segmentation is to quantify +the hematoma volume, making the volume differences be- +tween the predictions and the labels significant for the results +analysis. Moreover, we further added the NSD metric as a +complement evaluation for the HD metric because the HD +would become infinite when the prediction is a normal head +CT scan without hemorrhages. The NSD also measures the +discrepancy between the target and predicted boundaries. +We intended to rank different algorithms based on the +above-mentioned four metrics. Motivated from the former +challenges [31], [36], we utilized a “aggregate-then-rank” +scheme for ranking, including the following three steps: (1) +Calculate the average DSC, HD, RVD and NSD metrics for +all cases in the testing dataset. (2) Rank all the participant +teams on these four metrics, hence each team would get four +ranks. (3)Based on the rankings generated from (2), we then +averaged these rankings and achieved the final ranking for each +team. Moreover, for some extreme cases, e.g., the HD metric +is infinite because the algorithm mistakenly treated some hard +ICH cased as normal head scans. In this case, we treat all +‘inf’ teams the same rank on HD which are inferior to others. +Because we believe effectively diagnosis hard samples is also +important in our challenge. +IV. RESULTS +A. Participation and submissions +The INSTANCE 2022 received over 500 applications on +grand-challenge platform and 70 teams were approved to be +able to access the challenge dataset. The reason why we +refused the other applications was that they didn’t submit the +signed agreement files that we provided in the participation +rules. In the validation phase, 30 teams uploaded their results +with over 350 valid submissions on the grand challenge +website. The final validation leaderboard is available on Grand +challenge website. In the testing phase, 13 teams successfully +submitted the Docker containers and the short papers. +B. Algorithm summary +We adopted the SLEX-NET [6] as the baseline model in +the proposed INSTANCE challenge. It is noted that the dataset +utilized in the SLEX-NET is different from INSTANCE 2022. +Therefore, we re-trained the algorithm of baseline model +on the INSTANCE 2022 dataset, with other training details +consistent with the settings in the original paper. +For the participants’ models, we find out that all the partic- +ipants chose U-Net-related architectures, including attention +U-Net [37], U-Net [22], 3D U-Net [38], nnU-Net [39], etc. +Among them, nnUNet is still the most popular model, 7 out +of 13 teams adopted it as their backbone network. Moreover, +we also summarized other key factors in the methods by those +participants, including data augmentation, loss functions, pre- +processing, post-processing, and etc. The detailed summaries +are illustrated in Table II. It shows that all teams used data +augmentation, and 10 out of 13 teams conducted ensemble +learning to improve their performance. In addition, four teams +utilized the 2D implementation, seven teams adopted the +3D implementation, and two teams combined 2D/3D imple- +mentations. For the pre-processing and post-processing, all +teams conducted different kinds of pre-processings, including +normalization, windowing, skull-stripping, and etc, while only +one team applied post-processing. To improve the learning +of deep models, each team utilized different losses, such as +Dice loss, cross-entropy loss, focal loss, and etc. Detailed +descriptions of their methods can be found in the Appendix A. +More importantly, we also released their submitted papers on + +IPH +IVH +SAH +SDH +EDHJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +5 +TABLE II: Summary of the algorithms in terms of key factors in the methods by those participants: backbone network, +2D/3D, stages, pre-processing, data augmentation, loss functions, ensembles, post-processing. Abbreviation: Normalization +(N), Windowing(W), Skull stripping(SS), Cropping (CP), Cross-Entropy(CE), Tversky (TV), Contour loss (CT) +Team +Backbone +2D/3D +Preprocess +Stage +Augmentation +Loss +Ensemble +Postprocess +Patch-based +T1 +nnU-Net +2D/3D +N +1 +✓ +Dice+WCE +✓ +� +✓ +T2 +ResUNet +2D +N +2 +✓ +Dice+CE +✓ +� +� +T3 +nnU-Net +3D +SS +1 +✓ +Dice+CE+CT +✓ +� +� +T4 +nnU-Net +3D +N+W+CP +1 +✓ +Dice+CE +� +� +✓ +T5 +nnU-Net +3D +N+SS +1 +✓ +Dice+CE +✓ +� +✓ +T6 +nnU-Net +3D +N+W +2 +✓ +Dice+Focal +✓ +� +✓ +T7 +nnU-Net +3D +N+W +1 +✓ +Dice+CE +✓ +� +✓ +T8 +U-Net +3D +W +1 +✓ +CE +✓ +� +✓ +T9 +nnU-Net +2D/3D +N +2 +✓ +CE +✓ +� +✓ +T10 +U-Net +3D +N+SS +1 +✓ +Dice+CE +✓ +✓ +✓ +T11 +Attention U-Net +2D +N +2 +✓ +Dice+CE+TV +✓ +� +� +T12 +U-Net +2D +N +1 +✓ +Dice +� +� +� +T13 +U-Net3+ +2D +SS +1 +✓ +Dice+CE +� +� +� +TABLE III: Summary of the INSTANCE 2022 validation phase. The average DSC, RVD, NSD and HD are reported for the +baseline models and the submitted algorithms from each participant. The unit of HD is [mm]. Bold values represent the best +scores for each metric. +Team +DSC(%)↑ +NSD(%)↑ +RVD↓ +HD↓ +T1 +79.12±23.00 +50.26±19.91 +0.21±0.20 +29.02±26.34 +T2 +78.21±18.45 +55.28±12.67 +0.20±0.18 +32.30±30.04 +T3 +71.60±30.10 +50.60±21.30 +0.29±0.30 +inf +T4 +73.55±26.74 +51.57±18.10 +0.24±0.24 +27.16±32.41 +T5 +73.39±27.38 +51.93±18.99 +0.25±0.27 +inf +T6 +79.53 ±17.18 +56.81±12.47 +0.20±0.18 +21.56±25.02 +T7 +71.12±29.38 +50.19±20.56 +0.27±0.30 +inf +T8 +72.34±28.52 +48.93±19.57 +0.58±1.65 +35.37±29.53 +T9 +69.96±30.26 +48.75±19.66 +0.26±0.27 +inf +T10 +69.28±28.39 +46.34±19.54 +0.36±0.44 +36.23±2.01 +T11 +52.87±29.66 +27.36±14.38 +2.16±4.86 +149.77±44.52 +T12 +64.76±31.42 +40.26±19.93 +0.52±0.76 +57.13±22.53 +T13 +67.16±33.19 +45.58±22.35 +0.27±0.29 +38.88±39.56 +Baseline [6] +64.08±27.48 +46.21±20.12 +0.514±1.14 +277.63±163.00 +the official challenge website 6 for comprehensive introduction +of their methods. +C. Evaluation results and Analysis +1) Segmentation performance: The segmentation perfor- +mance of the baseline model and other participants’ algorithms +for validation and testing set are illustrated in Table. III +and Table. IV respectively. In Table. IV, we reported the +average DSC, RVD, NSD and HD in the table, respectively. +Our baseline model, SLEX-Net [6] obtained a DSC score of +52.83%. Most of other teams improved the baseline model +in all four metrics. The average DSC score, RVD, NSD for +the participants lies in [40.22%,72.06%], [0.21, 1.55], and +[25.11%, 53.59%], respectively. The best results on DSC, +RVD, and NSD metrics achieved only 72.06%, 0.21, 53.59%, +respectively. The overall performances are much lower than +many other segmentation tasks, proving the great challenge of +intracranial hemorrhage segmentation task. More importantly, +most of the teams obtained ’infinite’ for the averaged HD +because their method mistakenly diagnosed some difficult ICH +6https://instance.grand-challenge.org/results/ +cases with tiny hemorrhages as normal subjects. The infinite +results made it challenging to effectively rank the HD metric +for different methods. In our challenge, we treat all ‘inf’ teams +the same rank on HD which are inferior to others. Because we +believe effectively diagnosis hard samples is also important +in this task. Moreover, Fig. 2(a)-(d) demonstrate the results +distribution across all the subjects in the testing dataset with +box plots. It can be inferred that the standard deviations of the +results distribution for top ranking teams are smaller that that +of lower ranking ones, and also fewer outliers exists for them +as well. +2) Hematoma Volume Analysis: In this section, we ana- +lyzed the relationship between hematoma volume size and +the segmentation performances for different algorithms. The +volume sizes of ICH are calculated by multiplying the +voxel numbers of ICH and the pixel spacing in x,y,z di- +mensions, which is consistent with the method in [6], [8]. +Fig. 3 highlights the correlation between volume size and +the DSC scores with a scatter plot. It demonstrates that +hemorrhages with small volume sizes are difficult to seg- +ment, while large hematoma ICHs are relatively easier to +achieve better segmentation results. Fig. 4 shows the segmen- + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +6 +TABLE IV: Summary of the INSTANCE 2022 testing phase. The average DSC, RVD, NSD and HD are reported for the +baseline models and the submitted algorithms from each participant. The unit of HD is [mm]. The ranking is only provided +for teams that successfully submitted the docker image and the technical paper descriptions in the testing phase. Bold values +represent the best scores for each metric. +Team +DSC(%)↑ +NSD(%)↑ +RVD↓ +HD↓ +Ranking +T1 +69.25±19.14 +53.59±15.65 +0.21±0.20 +35.27±28.47 +1 +T2 +72.06±21.07 +53.43±16.45 +0.26±0.25 +inf +2 +T3 +69.00±24.68 +51.25±18.94 +0.31±0.28 +inf +3 +T4 +68.94±25.06 +50.36±19.35 +0.32±0.29 +inf +4 +T5 +67.97±25.07 +49.46±18.73 +0.32±0.28 +inf +5 +T6 +67.39±26.91 +48.40±20.84 +0.32±0.31 +inf +6 +T7 +66.84±24.75 +48.09±18.68 +0.33±0.27 +43.90±33.78 +7 +T8 +65.28±27.98 +47.49±21.70 +0.37±0.31 +inf +8 +T9 +64.97±26.78 +46.86±19.58 +0.34±0.31 +inf +9 +T10 +62.14±27.70 +42.01±19.88 +0.33±0.30 +inf +10 +T11 +61.95±25.91 +42.80±18.75 +0.40±0.78 +55.36±26.53 +11 +T12 +57.04±28.20 +36.73±19.17 +0.43±0.50 +60.81±25.22 +12 +T13 +40.22±32.35 +25.11±21.54 +1.55±4.67 +68.36±41.79 +13 +Baseline [6] +52.83±28.92 +38.42±21.04 +0.725±2.06 +309.06±287.31 +(a) Dice Coefficient +(b) Normalized Surface Dice +(c) Relative Volume Difference +(d) Hausdorff Distance +Fig. 2: Box plots of the experimental results on different evaluation metrics for all the submitted teams. The dots denote the +individual scores of the 70 cases in the testing set. + +0.8 +DiceCoefficient(% +0.6 +0.4 +0.2 +0.00.8 +SurfaceDice(%) +0.6 +0.4 +0.2 +0.0 +T6T7T8T9Relative VolumeDifference +1.00 +0.75 +0.50 +0.25 +0.00 +T1T2 T3 T4 T5 T6 T7 T8 T9 T10T11T12T13150 +HausdorffDistance +100 +50 +T1 T2 T3 T4 T5 T6 T7 T8 T9 T10T11JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +7 +Fig. 3: The relationship between different Dice coefficients +and the hematoma volume sizes demonstrates that the cases +with smaller hematoma volumes are hard cases. +Fig. +4: +The +team-wise +’Volumn-DSC’ +relationship +fig- +ure shows that the DSC scores improve with the in- +crease of volume sizes for different algorithms from the +participants. It is generated by separating the 70 test- +ing cases with four different volume size groups: in- +cluding [0, 4213], [4213, 7235], [7235, 19640], [19640, inf], re- +spectively, and the average DSC score was calculated based +on the results in each group. +Fig. 5: The bar chart on Dice Coefficient for different kinds of +intracranial hemorrhages shows that SAH is the most difficult +class to segment. +tation performance for all the methods with four hematoma +volume size groups. It is generated by separating the 70 +testing cases with four different volume size groups: in- +cluding [0, 4213], [4213, 7235], [7235, 19640], [19640, inf], re- +spectively, and the average DSC score was calculated based on +the results in each group. Fig. 4 further proves that the DSC +scores improve with the increase of volume sizes for different +algorithms from the participants. +3) Hemorrhage Sub-type Analysis: Different sub-types of +the intracranial hemorrhages are located at distinct positions +of the brain, and patients can suffer from combinations of +several kinds of hemorrhages. Certain types of hemorrhages +usually present various different characteristics, leading to +varied difficulties for distinguishing from normal brain tissues. +Fig. 5 illustrates the average DSC value for different kinds +of hemorrhages. It demonstrates that the SAH achieved the +worst results in all metrics compared to other four kinds of +ICHs. Hence, how to effectively segment SAH might be the +most urgent problem needed to be solved to improve the ICH +segmentation. +D. Challenge Ranking Analysis +Similar to the significance analysis in many biomedical +image segmentation challenges [31], [32], we utilized the +significance map to demonstrate the pairwise significant su- +periority between different algorithms, as is illustrated in Fig. +6. Specifically, we choose to perform significant test with one- +sided Wilcoxon signed rank test at 5% significance level. In +Fig. 6 (a-d), most of the yellow blocks are above the diagonal +and the blue blocks are under the diagonal, indicating that most +of the teams with smaller rank are significantly superior to +those with larger ranks. Moreover, it also shows that different +metrics have distinct ability to distinguish the good and bad + +1.0 +0.8 +0.6 +Dice +0.4 +0.2 +0.0 +0 +30000 +00009 +00006 +120000 150000 180000 +Volumein[mm"T1 +0.8 +T2 +T3 +T5 +T6 +T7 +T8 +Average test DSC +0.6 +T9 +T10 +T11 +T12 +T13 +0.4 +0.2 +0.0 +[0,4213] +[4213,7235] +[7235,19640] +[19640,inf] +Volume in [mm3]0.88 +DICE +RVD +0.8 +NSD +0.73 +0.68 +0.67 +0.62 +0.63 +0.6 +0.52 +0.54 +0.41 +0.43 +0.4 +0.39 +0.36 +0.32 +0.29 +0.2 +0.09 +0.0 +SDH +EDH +SAH +IPH +IVHJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +8 +(a) Significance map for Dice Coefficient +(b) Significance map for Normalized Surface Dice +(c) Significance map for Relative Volume Difference +(d) Significance map for Hausdorff Distance +Fig. 6: The significant superiority maps for ranking robustness analysis of different evaluation metrics. In each of the four +maps, yellow blocks means that the evaluation metric for teams on the x-axis are significantly superior to those from the teams +on the y-axis, which blue blocks means no significant superiority. The pairwise significant test with one-sided Wilcoxon signed +rank test at 5% significance level is adopted in our experiment. +performances among different algorithms. For example, the +DSC, NSD and HD of T7 are significantly superior to that of +T12, however, there exists no significant superiority on RVD +metric. +V. DISCUSSIONS +A. 2D/3D architecture Choice +The algorithm summary in section IV-B shows that the +participants chose different algorithm implementations for 2D +or 3D methods. We noticed that the winner method adopted +the 2D/3D combination method, and most of the 3D methods +outperformed the 2D implementations, yet we cannot draw +definite conclusions on which kinds of methods are superior +to another since there are many other factors contributing to +the final results. However, we believe that directly utilizing 2D +networks would lose significant context information among +slices, which has been proved in numerous medical image +segmentation tasks [6], [40]–[42]. Therefore, how to effec- +tive incorporate inter-slice contextual information would be +a fundamental problem for improving ICH segmentation. To +this end, many participants utilized 3D UNet implementation, +however, this might not be the optimal solution considering +that the CT volumes in this challenge are anisotropic (pixel +spacing: 0.42mm×0.42mm×5mm) [43], thus more effective +techniques for exploiting inter-slice context for anisotropic +volumes are needed. +B. Bottlenecks for ICH segmentation +The hematoma volume analysis in section IV-C2 demon- +strates the inferior segmentation performance for hemorrhages + +T13 +T12 +TII +T10 +T9 +T8 +T7 +T6 +T5 +T4 +T3 +T2 +T1 +T1 +T13T13 +T12 +T11 +T10 +T9 +T8 +T7 +T6 +T5 +T4 +T3 +T2 +T1 +T1 +T2T3T4T5T6T7T8T9T10T11T12T13T13 +T12 +T11 +T10 +T9 +T8 +T7 +T6 +T5 +T4 +T3 +T2 +T1 +T1 +T2T3T4T5 +T6T7T8T9T10 T11T12 T13T13 +T12 +T11 +T10 +T9 +T8 +T7 +T6 +T5 +T4 +T3 +T2 +T1 +T1 +T2JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +9 +with small volume sizes. The degradation of the segmentation +indicates that the hemorrhage cases with small volume sizes +are hard to segment. Fig. 3 shows that all the methods pro- +posed by the participants have trouble dealing with very small +hemorrhages. The majority of the cases that achieve a DSC +score lower than 0.3 are those subjects with hemorrhage vol- +ume smaller than 15000m3, and the overall DSC performances +for all the subjects significantly deteriorate with substantial +low DSC scores. Therefore, one important bottleneck for ICH +segmentation is the small hemorrhage lesion segmentation, and +effectively resolving this problem would certainty improve the +overall segmentation performance and achieve better ranking +in the challenge. Besides, the hemorrhage sub-type analysis in +section IV-C3 shows that the subarachnoid hemorrhage (SAH) +achieved the worst results in all metrics, with average DSC +score for only 0.41. Thus, another bottleneck for ICH seg- +mentation is how to deal with the subarachnoid hemorrhage. +In conclusion, the future directions for the researches of ICH +segmentation may be concentrated on the above-mentioned +two bottlenecks. The researches of the hemorrhage diagnosis +would be greatly improved by resolving these extremely hard +cases. +C. Evaluation Metrics Analysis +We highly suggest the use of DSC, NSD and the RVD as +the evaluation metrics for the ICH segmentation benchmark. +According to the descriptions in section III-B, and section +IV-C1. The HD and NSD are similar metrics that are used +to evaluate the discrepancy between the target and predicted +boundaries. However, we came across multiple extreme cases +with average HD metrics equal to infinite when the predicted +methods mistakenly diagnosed those hard cases with small +hemorrhage lesions as normal head scans. The infinite values +make it challenging to effectively rank different algorithms +on that metric. However, the NSD metric has the same +upper bound as DSC (100%), and there will be no such +circumstances occur. Therefore, Hausdorff distance might not +be a good metric choice for the INSTANCE challenge, and we +consider abandoning it in the future INSTANCE challenges. +D. Limitations and Future work +Although this year’s INSTANCE challenge has achieved +great success with numerous participants around the world, it +still suffers from lots of limitations. They are mainly consist +of three aspects: +1) Data collection and annotation: +Even though the +INSTANCE2022 challenge has provided a relatively large +dataset, they are mainly collected from a single institution +with the same CT scanner. Although it could work in our +challenge, it would definitely restrict the generalization of the +model developed by different participants. In addition, for the +data annotation, we only delineate the hemorrhage regions as +foreground without considering the ICH sub-types, which are +actually important information in clinical diagnosis and can +also guide the segmentation of ICH. +2) Task designs: In this years’ INSTANCE challenge, we +only consider the hemorrhage segmentation task. However, it +is also significant to perform ICH classification and hematoma +volume quantification, which are highly clinical-related. The +design of multiple tasks would simultaneously make the chal- +lenge more comprehensive and provide more diverse research +directions for the participants. In conclude, we will enhance +the single-task challenge to a multi-task one in the future +challenges. +3) Source code Availability: In this years’ INSTANCE +challenge, we highly recommended the participants to make +their implementations to the public, and didn’t make it a +mandatory option. As a result, we only find out one team make +their code available. We didn’t demand them to share the code +because we don’t expect it to be an obstacle for participating +in this challenge. However, we notice that the code is too +significant to be ignored for promoting the development in this +research field. Therefore, we consider making it mandatory for +top participants to make their code public available for future +INSTANCE challenges. +4) Future works for INSTANCE: We are currently working +to promote the INSTANCE 2022 Challenge in many different +aspects. Detailed improving directions are as follows: +• More multi-institutional data. We will collect more ICH +data from different CT scanner and different hospitals to +improve the generalization of methods that are trained +based on the INSTANCE benchmark. +• More annotations and comprehensive task designs. +We will annotate the different ICH sub-types of each CT +scans and also calculate the hematoma volume of each +cases to provide more clinical-related datasets. Mean- +while, based on the above-mentioned extra annotations, +we further expand the single-task challenge to a multi- +task one, which simultaneously performs hemorrhage +segmentation, classification and volume quantification +tasks. +• Mandatory options for open-source code. To pro- +mote the advancement of the intracranial hemorrhage +diagnosis, the top participants in the future INSTANCE +challenge are required to share their code to the public. +VI. CONCLUSION +The INSTANCE challenge provides a novel benchmark +for objectively measuring different intracranial hemorrhage +segmentation methods in non-contrast head CT scans. A total +of 13 teams successfully submitted their methods, and the +winner solution achieved a DSC score of 0.6925 on the +testing set, dramatically improving our baseline network. We +have made the training set, the methodology descriptions and +evaluation code public available on the challenge website, +we hope this would promote the development in the ICH +segmentation field. The challenge is now remains open for +post-challenge submissions via Grand Challenge platform for +benchmarking further algorithm exploitation. In the future, +we will collect more multi-institutional data to improve the +generalization of methods that are trained on the benchmark, +and also perform more clinical-relevant annotations on ICH + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +10 +Fig. 7: Segmentation results for different ICH sub-types in terms of DSC and NSD scores. The blue color denotes the ICH +lesion. +sub-type and hematoma volumes and expand the single-task +challenge to a multi-task one. +ACKNOWLEDGMENTS +We sincerely appreciate all the members in INSTANCE2022 +organization team for their hard work. Without your con- +tinuous devotion to this challenge, it would not be that +successful. This work was supported by the National Nat- +ural Science Foundation of China under Grant 62001144, +62272135 and 62001141, and by Science and Technology +Innovation Committee of Shenzhen Municipality under Grant +RCBS20210609103820029 and JCYJ20210324131800002. +APPENDIX +In (Li and Chen, 2022), Li and Chen used a combination +of nnU-Net and uncertainty estimation ensemble strategy. +Their experiments showed that even though the 2D nnU- +Net could not achieve the overall dice accuracy of 3D nnU- +Net, it performed better results than 3D nnU-Net when the +intracranial hemorrhage had very small area or blurred bound- +aries. Therefore, they use both 2D and 3D nnU-Net to predict +the final result. Furthermore, in order to further alleviate the +segmentation issue of small area intracranial hemorrhage and +maintain stability during training, they utilized the weighted +cross-entropy loss to replace simple cross-entropy loss in +the nnU-Net. Due to the unbalanced intracranial hemorrhage +types and intracranial hemorrhage areas, the models trained +in different folds might predict completely different results. +Simply average the predicted results from the models provide +no additional benefit for these cases. To this end, they propose +a simple but efficient uncertainty estimation ensemble strategy. + +DSC:89.9 NSD:65.4 +DSC:90.2 NSD:60.8 +DSC:90.4 NSD:60 +Case 1 +DSC:89.9 NSD:64.6 +DSC:93.3 NSD:69.9 +DSC:91.4 NSD:65.3 +Case 2 +DSC:55.3 NSD:46.3 +DSC:47.9 NSD:43.8 +DSC:48.4 NSD:39.8 +N +Case 3 +DSC:65.1 NSD:45.9 +DSC:72.2 NSD:4S.6 +DSC:67.8 NSD:43.2 +Case 4 +DSC:67.7 NSD:48.9 +DSC:43.5 NSD:37.0 +DSC:29.1 NSD:19.7 +Case 5 +(a)Image +(b)Ground Truth +(c)T1 +(d)T3 +(e)T9JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +11 +For those cases with high uncertainty values, they use the +voting method to get the final result. Use nnU-Net’s own data +augmentation methods. +In (Siddiquee et al., 2022 [44]), Siddiquee et al. used the +2D version of encoder-decoder backbone based on with an +asymmetrically larger encoder to extract image features and +a smaller decoder to reconstruct the segmentation mask7. For +the encoder part, they used 5 stages of down-sampling and +2D ResNet blocks that each block’s output is followed by an +additive identity skip connection. Furthermore, they used batch +normalization and ReLU. For the decoder part, the decoder +structure is similar to the encoder one, but with a single +block per each spatial level. Each decoder level begins with +upsizing with transposed convolution. In the preprocessing, +they applied random rotation and random zoom on each axis +with a probability of 0.4 and random contrast adjustment +and random Gaussian noise with a probability of 0.2. The +random coarse shuffle and random flips were applied on each +axis with a probability of 0.5. In the training, they randomly +split the entire dataset into 5-folds and trained a model for +each. Moreover, they used L2 norm regularization on the +convolution kernel parameters with a weight of 1e−5. The +DiceCE loss is used for training. +In (Sanner and Mukhopadhyay, 2022), Sanner and +Mukhopadhyay used nnU-Net for the segmentation and pro- +pose an evaluation of contour-based losses. Specifically, they +integrated both the Hausdorff-distance loss as proposed by +[45] and the contour loss proposed by [46]. While the former +estimates the Hausdorff distance, the latter extracts the contour +of both the prediction and the ground truth and minimizes the +mean square error between them. In practice, Dice loss and +CE loss were used as loss function and the Hausdorff-distance +loss or the contour loss was used depending on the experiment. +Furthermore, rather than using the standard z-normalization +of nnU-Net for input images, they chose to clip the intensity +values to [0 - 100]. A five-fold cross-validation was used to +train five models and all models were ensembled to make the +final prediction. The ”insane DA” scheme was used for data +augmentation. +In (Zhao et al., 2022), Zhao et al. used two stage 3D +cascade U-Net network for ICH segmentation. For the stage 1, +the basic module of the encoder and decoder is Conv-Instance +Norm-LeakyReLU [47]. The operation of downsampling in +the encoder is achieved by max pooling. The upsampling +operation in the decoder is achieved by using the transpose +convolution of 2 × 2 × 2. For the stage 2, a 3D U-Net was +cascaded to the model, whose input is the output of probability +map of the first stage. The 5-fold cross-validation was used for +the training. In the preprocessing, the HU of CT images were +clipped according to three different windows and levels, and +corresponding range of HU were [0, 80], [-20, 180] and [-150, +230]. The intensity of the voxel above the range were assigned +the value of upper limit in range, and the intensity below the +range is assigned the value of lower limit in range. Then the +three images with different HU range clip were served as three +channels and treated as one image. +7Implementation: https://monai.io/apps/auto3dseg +In (Zhang and Ma, 2022), Zhang and Ma used the +standard nnU-Net.First, a 3D U-Net processes downsampled +data, the resulting segmentation maps are upsampled to the +original resolution.Then, these segmentations are concatenated +as one-hot encodings to the full resolution data and refined +by a second 3D U-Net. The preprocessing includes crop- +ping,resampling and normalization. Meanwhile, random rota- +tion, random scaling, random elastic transformation, gamma +correction, and mirror were used to augment the data. The 3D +nnU-Net was trained with an weighted combination of Dice +loss and cross-entropy loss. The results on the test set were +obtained as an ensemble of five models. +In (Liu et al., 2022 [48]), Liu et al. used an ensemble +model that combined viola-Unet and nnU-Net networks8. +For the viola-Unet, it relies on voxels in feature space that +intersect along orthogonal levels to provide an attention U-Net, +which is an asymmetric encoder-decoder architecture with 7- +depth layers. Overall, the Viola module is composed of three +key blocks, i.e., the adaptive average pooling (AdaAvgPool) +module that squeezes the input feature volume into three latent +representation spaces along each axis of the input feature +patch. The customized dense dilated convolutions merging +(DDCM) networks fuses cross-channel and non-local contex- +tual information on each orthogonal direction with adaptive +kernel sizes, dilated ratios and strides. The Viola unit con- +structs the voxels intersecting along orthogonal level attention +volume based on fused and reshaped cross-channel-direction +latent representation spaces. They trained all networks with +randomly sampled patches of fixed size as input and applied +a combination loss function of the dice loss and Focal loss +for all their experiments. In preprocessing, CT image and +ground truth labels were reoriented into ”RAS” format, then +resized to a standard spacing of 1×1×5 mm3 using trilinear +interpolation for the image and nearest-neighbor interpolation +for the label. Each CT image was windowed into three image +intensity ranges, and re-scaled to the range [0, 1] by min-max +normalization and then stacked as 3-channel volumes to serve +as inputs with the (C, H, W, D) shape, and then the 3-channel +3D volume was normalized on only non-zero values with +calculated mean and std on each channel separately. The data +augmentations include random crop, random zoom, Gaussian +noise, Gaussian smooth, rotation, random shift, random scale, +flips, random contrast. Furthermore, they manually select +the best prediction on each validation example from each +submission as the pseudo-label and put them into our training +set to fine-tune our models repeatedly in practice. +In (Liang, 2022), Liang proposed a nnUNet-based method +for 3-dimensional intracranial hemorrhage segmentation. In +the preprocessing, the authors first deal with the data in method +windowing and decide to choose a width of 59 and a center of +96 for the image windowing by experiment. After windowing, +in order to arrange the information of image, the author used +a threshold to ensure the gray value of the image in a certain +standard interval, unified data input. Then, downsampling the +X and y axes, normalize the spacing of the slice axis to +Slice down scale. A sampling includes maximum sampling, +8Implementation: https://github.com/samleoqh/Viola-Unet + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +12 +average sampling, summation area sampling, and random area +sampling. Finally, nnU-Net does the rest of the preprocessing. +In the training, the author uses CE loss + DICE loss as +loss function. Furthermore, to deal with category imbalance, +oversampling was used, with 66.7% of the samples coming +from random locations in the selected training sample’s, while +33.3% of the patches were guaranteed to contain one of the +foreground classes present in the selected training sample +(randomly selected). The number of foreground patches was +rounded to force a minimum value of 1 (resulting in one +random patch and one foreground patch with a batch size of +2). Use nnU-Net’s own data augmentation methods. +In (Geiger et al., 2022), Geiger et al. used classic U- +Net architecture and the network was conducted with the jax +version of the e3nn library which enables the creation of neural +networks equivariant to translations, rotations, and mirroring. +Specially, the convolution kernels in the original architecture +are replaced by a 3D e3nn voxel convolution of diameter +5 mm. Furthermore, they used three 2x2x2 downsampling +operations which halve the resolution in the encoding path +and three corresponding trilinear upsampling operations on the +decoding path. A Gaussian error linear unit activation function +and instance normalization was used after each convolution. +For the preprocessing, each CT volume was windowed to three +different Hounsfield unit value ranges, scaled, and added to a +separate channel which served as the model input. To increase +the variety in the data, a random diffeomorphic deformation +was performed on each training sample. The loss function em- +ployed was cross-entropy loss. Eight models were trained, each +on 80 randomly sampled subsets from the training dataset. The +final prediction was performed by applying each of the eight +models to patches of size 144x144x13 with padding discarding +22x22x2 pixels on each side, a sliding window with an overlap +of 26 pixels and Gaussian weighing, and then averaging the +model outputs. For the final prediction, they take the ensemble +average of the eight models. +In (Qayyum et al., 2022), Qayyum et al. developed a +coarse and fine segmentation model for intracranial hemor- +rhage segmentations. They trained two different models for +intracranial hemorrhage segmentations. In the first model, they +trained 2DDensNet for coarse segmentation and cascaded the +coarse segmentation masks output in the fine segmentation +model along with input training samples. The proposed model +is implemented made by a dense encoder followed by a non- +dense decoder. The dense encoder consists of 5 dense blocks, +each consisting of 6 dense layers followed by a transition +layer. Each dense layer consists of 2 convolutional layers +with batch normalization and ReLU activation functions. The +model is trained using 5-fold cross-validation. To compute the +final prediction, 2D images are stacked to make a 3D seg- +mentation mask. The predicted segmentation mask is further +cascaded in a fine segmentation model. In the fine stage, they +used the nnU-Net model with fivefold cross-validation. The +binary cross-entropy function was used as loss function. Hor- +izontalFlip (p=0.5), VerticalFlip (p=0.5), and RandomGamma +(p=0.8) were used to augment the dataset for training the +proposed model. In addition, the dataset is normalized between +0 and 1 using the max and min intensity normalization method. +The training shape of each volume is fixed (256x256x16) and +resample the prediction mask to the original shape for each +validation volume using the linear interpolation method. +In (Abramova et al., 2022), Abramova et al. used an +approach based on a 3D U-Net architecture which incorporates +squeeze-and-excitation blocks that similarly to their previous +work [25]. For the preprocessing, coil removal and skull +stripping were used, and a symmetric image was created +for each case by flipping the original non-contrast CT and +registering it to the initial one using the FLIRT algorithm from +the FSL toolbox. For the normalization of input images, they +performed percentile based range adjustment and used 0.5 and +99.5 percentiles of brain-related voxels for clipping together +with image-based calculated mean and standard deviation +normalization. For the issue of class imbalance, they used +a balanced sampling patch extraction technique, where we +extracted an equal number of patches representing both classes +from each image. Specifically, to avoid extracting a lot of +patches from image background, they restricted the area to +extract the negative patches within the brain mask and set a +target number of patches to extract from each image in the +training set. Half of them are uniformly extracted from the +brain tissue area and represent negative class, while the other +half is extracted from the lesion voxels. They augment the +proposed dataset by choosing difficult cases and adding them +into the training set again, meanwhile performing flipping and +rotation, ensuring that more difficult patches are generated for +training. The Dice loss and cross-entropy loss was used as +loss function. To prevent overfitting, they used early stopping +technique when approaching the minimal loss on validation +set. The five-fold cross-validation strategy was used for the +training. For the validation and testing stages, an ensemble +with all the 5 models obtained in the cross-validation exper- +iment was used to generate the final prediction masks. The +probability masks obtained from the 5 models were averaged +and thresholded to obtain the final binary mask for each case. +Considering the results on the validation set, postprocessing +was added to their pipeline to reduce the number of false +positives. Specifically, as sizes of lesions vastly vary in the +provided images, they remove all the lesions with the volume +less than 10% of the biggest one in the post-processed image. +In (Montagnon and Letourneau-Guillon, 2022), Mon- +tagnon and Letourneau-Guillon used an ensemble approach +including the Attention U-Net and SegResNet (with or without +variational autoencoder) architectures combined with different +loss functions. Specifically, they trained U-Net and SegResNet +separately to use different loss functions including combina- +tions of Dice with either Cross-Entropy loss or Focal loss, +Tversky loss and Generalised Dice loss. Then leveraging all +predictions, an ensemble voting approach allowed prediction +of a final volume. Finally, to further remove potential false pos- +itive predictions, predicted clusters were filtered by preserving +ones with a volume larger than 36 pixels, an elevation above or +equal to 3 slices and a mean density within [40; 80] HU range. +In the preprocessing, in order to assess hemorrhage properties, +they used DBSCAN, a density-based clustering algorithm, in +order to extract connected pixels corresponding to hemorrhagic +areas in each exam. Then they clipped images in the range [- + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +13 +10; 140] HU. Taking into account the intracranial hemorrhage +subtype distribution in the training dataset, they using Euler +transforms consisting of rotations of either - π +2 or - π +2 around +z-axis and translations ranging from -30 to 30 pixels, 10 pixels +stepwise for subarachnoid and subdural hemorrhage subtypes +images. Considering the limited size of the dataset, they used +random orthogonal rotations and cropping for images in the +training phase. In order to limit class imbalance issues, models +were trained only on images containing at least one pixel of +positive class. All models were trained using original images +size (i.e. 512 × 512), clipped within [-10;140] and divided +by the range of considered densities, which is 150 in their +configuration. +In (Roca et al., 2022), Roca et al. used a simple 2D Unet- +like model and trained it with a binary cross-entropy loss. +Especially, the model input is a layer that performs the clipping +operation between [0, 256] and a normalization between [- +0.5, 3.5] directly inside the model. In the preprocessing, +they clipping the HU intensities in the soft tissue range of +interest. For the data augmentation, they performed rotations +and mirroring in the axial plane, plus some amount of intensity +shift. Due to the data stratification was based on the presence +of a segmentation on a given slice (positive cohort) vs. absence +of segmentation (negative cohort), they used during training a +balanced 50% / 50% of each cohort per mini-batch. +In (Sindhura et al., 2022), Sindhura et al. proposed a deep +learning framework which involves clinical knowledge and +used U-Net3+ network for the segmentation. Specifically, they +proposed a new data augmentation approach that leverages +from the clinical knowledge that the two hemispheres of +the human brain exhibit approximate symmetry. Due to the +brain is approximately divided into two equal hemispheres by +the midsagittal plane (MSP). So they use the MSP flipped +versions of the CT scans as extra data. To extract MSP, +they first apply the sobel edge detection method followed by +thresholding to obtain the outline of the skull. An initial plane +of reference is chosen to be the exact middle slice in the +sagittal direction. A similarity metric is computed between the +two hemispheres that are divided with the plane of reference. +The reference plane is rotated by an angle of ±0.5◦. The +plane which yields maximum similarity is the required MSP. +Furthermore, to improve the robustness of the model, the +usual data augmentations such as shear, rotation, zoom, flip, +elastic transform, noise etc are being used. In view of there +exists a very high class imbalance between the hematoma +and non-hematoma pixels. So only the slices which contain +hemorrhages are used in the training process and all slices +of each scan are tested in the testing phase. In addition, to +differentiate between the hemorrhage region and skull bone, +which share similar intensities, they have performed skull +stripping on each scan for both the training and testing process. +The sum of focal loss and Dice similarity loss is used as the +loss function in the training process. +REFERENCES +[1] J. A. Caceres and J. N. 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Bjørnerud, “Voxels intersecting along orthogonal levels attention u- +net (viola-unet) to segment intracerebral haemorrhage using computed +tomography head scans,” arXiv preprint arXiv:2208.06313, 2022. + diff --git a/1NE1T4oBgHgl3EQflATa/content/tmp_files/load_file.txt b/1NE1T4oBgHgl3EQflATa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c1184be284cf6d37bce0a78b1a2d32c84cba22b --- /dev/null +++ b/1NE1T4oBgHgl3EQflATa/content/tmp_files/load_file.txt @@ -0,0 +1,1544 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf,len=1543 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 8, AUGUST 2021 1 The state-of-the-art 3D anisotropic intracranial hemorrhage segmentation on non-contrast head CT: The INSTANCE challenge Xiangyu Li, Gongning Luo, Kuanquan Wang, Hongyu Wang, Shuo Li, Jun Liu, Xinjie Liang, Jie Jiang, Zhenghao Song, Chunyue Zheng, Haokai Chi, Mingwang Xu, Yingte He, Xinghua Ma, Jingwen Guo, Yifan Liu, Chuanpu Li, Zeli Chen, Md Mahfuzur Rahman Siddiquee, Andriy Myronenko, Antoine P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Sanner, Anirban Mukhopadhyay, Ahmed E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Othman, Xingyu Zhao, Weiping Liu, Jinhuang Zhang, Xiangyuan Ma, Qinghui Liu, Bradley J MacIntosh, Wei Liang, Moona Mazher, Abdul Qayyum, Valeriia Abramova, Xavier Llad´o Abstract—Automatic intracranial hemorrhage segmentation in 3D non-contrast head CT (NCCT) scans is significant in clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Existing hemorrhage segmentation methods usually ignores the anisotropic nature of the NCCT, and are evaluated on different in-house datasets with distinct metrics, making it highly challenging to improve segmentation performance and perform objective comparisons among different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The 2022 intracranial hemorrhage segmentation on non-contrast head CT (INSTANCE 2022) was a grand challenge held in conjunc- tion with the 2022 International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' It is intended to resolve the above-mentioned problems and promote the development of both intracranial hemorrhage segmentation and anisotropic data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The INSTANCE released a training set of 100 cases with ground-truth and a validation set with 30 cases without ground-truth labels that were available to the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' A held-out testing set with 70 cases is utilized for the final evaluation and ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The methods from different participants are ranked based on four metrics, including Dice Similarity Coefficient (DSC), Hausdorff Distance (HD), Relative Volume Difference (RVD) and Normalized Surface Dice (NSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' A total of 13 teams submitted distinct solutions to resolve the challenges, making several baseline models, pre-processing strategies and anisotropic data processing techniques available to future researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The winner method achieved an average DSC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='6925, demonstrating a significant growth over our proposed baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' To the best of our knowledge, the proposed INSTANCE challenge releases the first intracranial hemorrhage segmentation benchmark, and is also the first challenge that intended to resolve the anisotropic problem in 3D medical image segmentation, which provides new alternatives in these research fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Index Terms—Intracranial hemorrhage Segmentation Chal- lenge Anisotropic data I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' INTRODUCTION I NTRACRANIAL hemorrhage (ICH) is a severe brain dis- ease and a main cause of stroke [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' It has a high mortality rate of 40% within one month [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Furthermore, ICH even causes significant disability in survivor patients, with only 20% of patients expected to be capable of living independently in half year [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Therefore, early and accurate diagnosis of the ICH is important for saving patients’ lives and improve their prognosis in clinical practice [1], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Non-contract head computerized tomography (NCCT) is the primary imaging modality to diagnosing ICH for its widely availability in most emergency rooms and high sensitivity for detecting ICH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Moreover, NCCT enables accurate monitoring of hemorrhage progression, and effectively quantify hematoma volumes in ICH [1], [4], [7], making it a gold standard examination for the diagnosis of ICH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Hematoma volume estimation is significant for the prog- nosis and treatment decisions for ICH patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In recent clinical trials, the hematoma volume has been utilized as a reliable indicator to determine the optimal candidates for intervention [8]–[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Thus, volume quantification of ICH has become an essential procedure for outcome predictions and ICH therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The hematoma volume can be estimated by semiautomated methods with the aid of radiologists, which is time-consuming [11] and suffers from inter-rater variability [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The ABC/2 method [13] is an effective technique to estimate hematoma volume in clinical practice since it is simple to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' However, the estimation accuracy of the ABC/2 method dramatically decreases with irregular or large scale hemorrhages [8], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The ICH segmentation methods, enabling accurate and rapid hematoma volume quantification, have become the leading criterion in ICH diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' However, there exists plenty of challenges to segment ICH for automatic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For example, the hemorrhage struc- tures vary considerably across patients in terms of shape, size, and localization, preventing the use of valuable location and shape priors that are significant elements in the segmentation of many other anatomical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The blurred boundaries for the ICH region further improve the difficulty of the segmentation task [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Because of the clinical significance and the intrinsic chal- lenges, the task of automatic intracranial hemorrhage segmen- tation has attracted extensive attention in the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Recently, deep learning–based ICH segmentation models that segment ICH regions and quantify hematoma volume have been performed to effectively diagnose ICH and have achieved competitive results [6], [16]–[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' However, all those above- mentioned ICH segmentation methods ignore the anisotropic nature of the NCCT volume by simply performing 2D or 3D convolutional networks, and they were evaluated on different in-house hemorrhage segmentation datasets with distinct met- rics, making it highly challenging to improve segmentation performance and perform objective comparisons among these arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='03281v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='IV] 9 Jan 2023 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 8, AUGUST 2021 2 methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Consequently, it remains hard to determine which kinds of segmentation techniques may be valuable to follow in clinical practice and research;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' what exactly the performance is of the state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' To resolve the above-mentioned challenges on fair com- parisons of different methods, we organized the INtracranial hemorrhage SegmenTAtioN ChallengE on non-contrast head CT (INSTANCE) in conjunction with the 2022 international conference on Medical Image Computing and Computer As- sisted Interventions (MICCAI) in Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' To this end, we collected and released an ICH segmentation dataset of 200 3D volumes with refined labeling from several experi- enced radiologists, and encouraged the participants to develop novel algorithms to effectively segment hematoma region with anisotropic NCCT volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Moreover, we evaluate different benchmark ICH segmentation methods with the same metrics, including Dice Similarity Coefficient (DSC), Hausdorff dis- tance (HD), relative volume difference (RVD) and normalized surface dice (NSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Each of these benchmark methods was implemented by different challenge participants on a subset of the ICH dataset, and tested on a isolated testing dataset against the manually delineated groundtruth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' To the best of our knowledge, INSTANCE is the first public intracranial hemor- rhage segmentation challenge, and also the first challenge that intended to deal with the anisotropic problem in 3D biomedical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' It is served as a solid benchmark for ICH segmentation tasks, and would also promote the development of intracranial hemorrhage segmentation and anisotropic data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' PRIOR WORKS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Related intracranial hemorrhage segmentation methods A large numbers of methods have been proposed to automat- ically segment intracranial hemorrhage in CT scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Among them, deep learning techniques are widely adopted for its excellent performance in medical image segmentation tasks [15], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Ironside et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' utilized U-Net [22] to automati- cally segment ICH and estimate the hematoma volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' They achieved comparable accuracy and greater efficiency compared to manual and semi-automated segmentation techniques [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' To address the issue of insufficient annotation data for ICH segmentation tasks, Kuo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' proposed a patch-based FCN network and segmented ICH in an active learning manner [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' proposed an ROI-based framework that is opti- mized specifically for ICH detection and segmentation tasks by projecting 3D features to 2D networks in the feature pyramid network [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' proposed a Siamese U-Net method to segment ICH by leveraging the dissimilarity between learned features of healthy templates and input images [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Kyung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' proposed a supervised multi-task aiding represen- tation transfer learning network for ICH, which was divided into upstream and downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In the upstream, effective representation learning was performed by multi-task learning (classification, segmentation, reconstruction) and differences in the specific head of the consistency loss mitigation target are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For downstream, feature extractor trained upstream is combined with 3D operator (classifier or divider) to implement specific tasks [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' proposed a combination of an attention-based convolutional neural network and a variational Gaussian process for multiple instance learning method for predicting intracranial hemorrhage slices [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Toikkanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' proposed a residual segmentation method based on gener- ative adversarial network, which generates the image without bleeding in the original section through the model, and then calculates the difference between the generated image and the original image, so as to obtain the segmented image [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Abramova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' introduced the squeeze-excitation block into 3D U-Net to solve the problem of segment hemorrhagic stroke lesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Moreover, a restrictive patch sampling is proposed to alleviate the class imbalance problem and also to deal with the issue of intra-ventricular hemorrhage [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Kuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' designed new self-attention blocks and contextual attention blocks that take full advantage of both in-chip and inter- chip information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In addition, multilevel training strategies are proposed to reduce the influence of inter-class imbalance [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' propose a Masked Multi-Task Network method to detect brain CT volumes with intracranial hemorrhage and distinguish hemorrhage type by leveraging different types of intracranial hemorrhage at different locations [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' propose a full convolutional neural network for simultaneous classification and segmentation of ICH, and the ConvLSTM module was used to address this issue of the loss of spatial information [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Kadam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' propose architectures combined Xception and LSTM/GRU for classification of Intracranial Hemorrhage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' It is also found through experiments that Xcep- tion GRU model has better performance on most of the metrics as compared to the Xception and Xception LSTM models [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Despite the excelent results reported in the above pa- pers, it is still challenging to identify the best performing method among them because of the varied testing datasets and evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The proposed INSTANCE challenge provides a standardized procedure to systematically evaluate and compare different SOTA methods on the same testing dataset and consistent evaluation metrics, enabling objective and fair comparison among different techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Medical Image Segmentation Challenges Recently years have witnessed the growing popularity for biomedical image analysis challenges, especially for medical image segmentation challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' To name a few, there were 25, 20, and 40 accepted challenges at the International Conference on Medical Image Computing and Computer-Assisted Inter- vention (MICCAI) 2020, 2021, and 2022, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' From 2020 to 2022, the number of challenges nearly doubled, and the segmentation-related challenges occupied 38% of all the challenges1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Similarly, in the largest biomedical image chal- lenge platform ’Grand Challenge2’, 149 out of 315 (47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='3%) challenges are designed for segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' There are lots of successful challenges in medical image segmentation, for example, the Brain Tumor Segmentation (BraTS) challenge [30] provide a solid benchmark for multimodal brain tumor 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='biomedical-challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='org/ 2https://grand-challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='org/ JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 8, AUGUST 2021 3 segmentation task, numerous methods on brain tumor seg- mentation and multi-modal learning have been validated on this benchmark, significantly improving the development of those research fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The Head and neck tumor segmentation challenge (Hecktor) [31] organized a novel challenge for head and neck tumor segmentation on PET/CT modalities, which claimed to be the pioneer work on this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The abdomen ct organ segmentation [32] first consider the inference time, and GPU memory consumption as extra evaluation metrics instead of simply focusing on the segmentation accuracy, providing a novel benchmark with more comprehensive evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The Kidney Tumor Segmentation (KiTS) ( [33] ) challenge allow participants to compare their methods on kidney and kidney tumor segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Those above challenges have made great progress in pro- moting the development of specific medical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' However, to the best of our knowledge, there are no challenges intended to resolve the ICH segmentation with anisotropic 3D volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Hence, the INSTANCE is the first released grand challenge for the ICH segmentation and also the first challenge that intended to deal with the anisotropic problem in 3D medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' We believe that the ICH data and algorithms provided in this benchmark would be helpful to promote the development of both ICH diagnosis and anisotropic data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' THE ORGANIZATION OF THE INSTANCE CHALLENGE The proposed INSTANCE challenge was organized in 2022 and was in conjunction with the 25rd MICCAI conference as a satellite event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' It was deployed on the Grand Challenge platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The official webpage of the INSTANCE challenge is https://instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='grand-challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Meanwhile, we also construct a Github repository 3 which provides plenty of resources related to the challenge, for example, the agree- ment files for accessing the dataset, the docker rules and submission examples, and also the baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For the challenge schedule, the registration is open to the public on March 28, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The training and validation dataset were released on April 6 and July 15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The dead- line of the open validation phase and the testing phase is on August 7 and August 14, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In the validation phase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' the participants uploaded their segmentation results to the Grand challenge website,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' and the platform automatically calculated the evaluation metrics by comparing them with the ground-truth labels we provided,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' and then displayed the calculated metrics on the validation leaderboard 4 In the testing phase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' the participants are required to submit one successful docker image that contains their algorithms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' and we ran the docker images from different participants on the closed testing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The dataset of the INSTANCE challenge are currently available to the public on Grand Challenge platform after signing an agreement file and the post-challenge leaderboard submission is open for researches in this community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The following sections summarizes the detailed implementation of the INSTANCE challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='com/PerceptionComputingLab/INSTANCE2022 4https://instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='grand-challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='org/evaluation/challenge/leaderboard/ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Dataset We obtained the approval from Peking university, shougang hospital to perform a retrospective analysis of the patients that were diagnosed as intracranial hemorrhage between 2017 and 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' We then collected 200 non-contrast head CT volumes of those patients to construct challenge dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For these 200 cases, they were diagnosed as different kinds of ICHs, including intraparenchymal hemorrhage (IPH), intraventricular hemorrhage (IPH), subarachnoid hemorrhage (SAH), subdural hemorrhage (SDH), and epidural hemorrhage (EDH), an exam- ple for each kind of ICH is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' We then split the 200 cases into training, validation and testing, with 100, 30, and 70 cases respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The CT scans and the labels of the training set are available to the participant for model training, while only the CT scans are provided for them to tune their algorithms on the Grand Challenge platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Finally, in the testing phase, we provide a closed test set for fair comparison between different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For each of the subject in INSTANCE dataset, we first converted the traditional Digital Imaging and Communications in Medicine (DICOM) files to the Neuroimaging Informatics Technology Initiative (NIfTI) format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In this way, each subject only has one single NIfTI file instead of a bunch of DICOM files, making it easier to process in a image segmentation program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The volume sizes ranges from 512 × 512 × 20 to 512 × 512 × 70, and the pixel spacing of a CT volume is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='42mm × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='42mm × 5mm, hence the volume is anisotropic with inter-slice resolution much smaller than the within-slice resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The window width and the window center is 90HU and 40HU, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' We kept the original Hu value in the NIfTI volume since the participants can conduct different windowing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For the data annotation, we gathered several experienced radiologists and some postgraduate students majored in med- ical imaging to perform hemorrhage region annotation in the NCCT scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' To improve the efficiency of the annota- tion process, we adopted a coarse-to-fine annotation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Specifically, the ICH lesions were first manually delineated in the NCCT volumes with a popular annotation software in med- ical imaging, Seg3D5 [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Then the experienced radiologists checked the coarse annotations and manually refined them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Finally, all the radiologists double-check the annotations from other annotators, and discuss to achieve the final annotations with majority voting strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Evaluation Measures and Ranking Method The INSTANCE challenge adopted four accuracy-related evaluation metrics: Dice Similarity Coefficient (DSC), Haus- dorff Distance (HD), Relative Volume Difference (RVD) and Surface Dice (NSD) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' We utilized DSC and HD since they are widely used in different medical image segmentation challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' They are complementary metrics for evaluating segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' DSC was utilized to measure the region overlapping error between ground truth and segmen- tation results, while HD is used to evaluate the coincidence 5https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='utah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='edu/cibc-software/seg3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='html JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 8, AUGUST 2021 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 1: Different kinds of intracranial hemorrhages, including intraparenchymal hemorrhage (IPH), intraventricular hemorrhage (IPH), subarachnoid hemorrhage (SAH), subdural hemorrhage (SDH), and epidural hemorrhage (EDH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The varied shapes and positions for different kinds of hemorrhages promote the difficulties of the segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' TABLE I: The Correspondence between the Team names and the aliases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Team Alias vegetable T1 nvauto T2 mec-lab T3 ibot T4 stubmers T5 crainet T6 superembrace T7 scan T8 dolphins T9 nic-vicorob T10 2i mtl T11 avich T12 visal T13 between segmented surface and target surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' We used the RVD since the purpose for the ICH segmentation is to quantify the hematoma volume, making the volume differences be- tween the predictions and the labels significant for the results analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Moreover, we further added the NSD metric as a complement evaluation for the HD metric because the HD would become infinite when the prediction is a normal head CT scan without hemorrhages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The NSD also measures the discrepancy between the target and predicted boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' We intended to rank different algorithms based on the above-mentioned four metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Motivated from the former challenges [31], [36], we utilized a “aggregate-then-rank” scheme for ranking, including the following three steps: (1) Calculate the average DSC, HD, RVD and NSD metrics for all cases in the testing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' (2) Rank all the participant teams on these four metrics, hence each team would get four ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' (3)Based on the rankings generated from (2), we then averaged these rankings and achieved the final ranking for each team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Moreover, for some extreme cases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=', the HD metric is infinite because the algorithm mistakenly treated some hard ICH cased as normal head scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In this case, we treat all ‘inf’ teams the same rank on HD which are inferior to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Because we believe effectively diagnosis hard samples is also important in our challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Participation and submissions The INSTANCE 2022 received over 500 applications on grand-challenge platform and 70 teams were approved to be able to access the challenge dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The reason why we refused the other applications was that they didn’t submit the signed agreement files that we provided in the participation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In the validation phase, 30 teams uploaded their results with over 350 valid submissions on the grand challenge website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The final validation leaderboard is available on Grand challenge website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In the testing phase, 13 teams successfully submitted the Docker containers and the short papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Algorithm summary We adopted the SLEX-NET [6] as the baseline model in the proposed INSTANCE challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' It is noted that the dataset utilized in the SLEX-NET is different from INSTANCE 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Therefore, we re-trained the algorithm of baseline model on the INSTANCE 2022 dataset, with other training details consistent with the settings in the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For the participants’ models, we find out that all the partic- ipants chose U-Net-related architectures, including attention U-Net [37], U-Net [22], 3D U-Net [38], nnU-Net [39], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Among them, nnUNet is still the most popular model, 7 out of 13 teams adopted it as their backbone network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Moreover, we also summarized other key factors in the methods by those participants, including data augmentation, loss functions, pre- processing, post-processing, and etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The detailed summaries are illustrated in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' It shows that all teams used data augmentation, and 10 out of 13 teams conducted ensemble learning to improve their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In addition, four teams utilized the 2D implementation, seven teams adopted the 3D implementation, and two teams combined 2D/3D imple- mentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For the pre-processing and post-processing, all teams conducted different kinds of pre-processings, including normalization, windowing, skull-stripping, and etc, while only one team applied post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' To improve the learning of deep models, each team utilized different losses, such as Dice loss, cross-entropy loss, focal loss, and etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Detailed descriptions of their methods can be found in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' More importantly, we also released their submitted papers on IPH IVH SAH SDH EDHJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 8, AUGUST 2021 5 TABLE II: Summary of the algorithms in terms of key factors in the methods by those participants: backbone network, 2D/3D, stages, pre-processing, data augmentation, loss functions, ensembles, post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Abbreviation: Normalization (N),' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='TABLE III: Summary of the INSTANCE 2022 validation phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The average DSC, RVD, NSD and HD are reported for the baseline models and the submitted algorithms from each participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The unit of HD is [mm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Bold values represent the best scores for each metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Team DSC(%)↑ NSD(%)↑ RVD↓ HD↓ T1 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='12±23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='00 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='26±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='21±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='20 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='02±26.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='16±32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='41 T5 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='39±27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='38 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='93±18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='25±0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='23±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='01 T11 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='87±29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='66 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='36±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='16±4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='21±20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='514±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='14 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='63±163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='00 the official challenge website 6 for comprehensive introduction of their methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Evaluation results and Analysis 1) Segmentation performance: The segmentation perfor- mance of the baseline model and other participants’ algorithms for validation and testing set are illustrated in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' III and Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' IV respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' IV, we reported the average DSC, RVD, NSD and HD in the table, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Our baseline model, SLEX-Net [6] obtained a DSC score of 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='83%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Most of other teams improved the baseline model in all four metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The average DSC score, RVD, NSD for the participants lies in [40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='22%,72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='06%], [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='21, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='55], and [25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='11%, 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='59%], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The best results on DSC, RVD, and NSD metrics achieved only 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='06%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='21, 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='59%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The overall performances are much lower than many other segmentation tasks, proving the great challenge of intracranial hemorrhage segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' More importantly, most of the teams obtained ’infinite’ for the averaged HD because their method mistakenly diagnosed some difficult ICH 6https://instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='grand-challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='org/results/ cases with tiny hemorrhages as normal subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The infinite results made it challenging to effectively rank the HD metric for different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In our challenge, we treat all ‘inf’ teams the same rank on HD which are inferior to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Because we believe effectively diagnosis hard samples is also important in this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Moreover, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 2(a)-(d) demonstrate the results distribution across all the subjects in the testing dataset with box plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' It can be inferred that the standard deviations of the results distribution for top ranking teams are smaller that that of lower ranking ones, and also fewer outliers exists for them as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 2) Hematoma Volume Analysis: In this section, we ana- lyzed the relationship between hematoma volume size and the segmentation performances for different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The volume sizes of ICH are calculated by multiplying the voxel numbers of ICH and the pixel spacing in x,y,z di- mensions, which is consistent with the method in [6], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 3 highlights the correlation between volume size and the DSC scores with a scatter plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' It demonstrates that hemorrhages with small volume sizes are difficult to seg- ment, while large hematoma ICHs are relatively easier to achieve better segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 4 shows the segmen- JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 8, AUGUST 2021 6 TABLE IV: Summary of the INSTANCE 2022 testing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The average DSC, RVD, NSD and HD are reported for the baseline models and the submitted algorithms from each participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The unit of HD is [mm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The ranking is only provided for teams that successfully submitted the docker image and the technical paper descriptions in the testing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Bold values represent the best scores for each metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Team DSC(%)↑ NSD(%)↑ RVD↓ HD↓ Ranking T1 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='25±19.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='725±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='06 309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='06±287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='31 (a) Dice Coefficient (b) Normalized Surface Dice (c) Relative Volume Difference (d) Hausdorff Distance Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 2: Box plots of the experimental results on different evaluation metrics for all the submitted teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The dots denote the individual scores of the 70 cases in the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='8 DiceCoefficient(% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='00.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='00 T1T2 T3 T4 T5 T6 T7 T8 T9 T10T11T12T13150 HausdorffDistance 100 50 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10T11JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 8, AUGUST 2021 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 3: The relationship between different Dice coefficients and the hematoma volume sizes demonstrates that the cases with smaller hematoma volumes are hard cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 4: The team-wise ’Volumn-DSC’ relationship fig- ure shows that the DSC scores improve with the in- crease of volume sizes for different algorithms from the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' It is generated by separating the 70 test- ing cases with four different volume size groups: in- cluding [0, 4213], [4213, 7235], [7235, 19640], [19640, inf], re- spectively, and the average DSC score was calculated based on the results in each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 5: The bar chart on Dice Coefficient for different kinds of intracranial hemorrhages shows that SAH is the most difficult class to segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' tation performance for all the methods with four hematoma volume size groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' It is generated by separating the 70 testing cases with four different volume size groups: in- cluding [0, 4213], [4213, 7235], [7235, 19640], [19640, inf], re- spectively, and the average DSC score was calculated based on the results in each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 4 further proves that the DSC scores improve with the increase of volume sizes for different algorithms from the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 3) Hemorrhage Sub-type Analysis: Different sub-types of the intracranial hemorrhages are located at distinct positions of the brain, and patients can suffer from combinations of several kinds of hemorrhages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Certain types of hemorrhages usually present various different characteristics, leading to varied difficulties for distinguishing from normal brain tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 5 illustrates the average DSC value for different kinds of hemorrhages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' It demonstrates that the SAH achieved the worst results in all metrics compared to other four kinds of ICHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Hence, how to effectively segment SAH might be the most urgent problem needed to be solved to improve the ICH segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Challenge Ranking Analysis Similar to the significance analysis in many biomedical image segmentation challenges [31], [32], we utilized the significance map to demonstrate the pairwise significant su- periority between different algorithms, as is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Specifically, we choose to perform significant test with one- sided Wilcoxon signed rank test at 5% significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 6 (a-d), most of the yellow blocks are above the diagonal and the blue blocks are under the diagonal, indicating that most of the teams with smaller rank are significantly superior to those with larger ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Moreover, it also shows that different metrics have distinct ability to distinguish the good and bad 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='6 Dice 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='0 0 30000 00009 00006 120000 150000 180000 Volumein[mm"T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='8 T2 T3 T5 T6 T7 T8 Average test DSC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='6 T9 T10 T11 T12 T13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='0 [0,4213] [4213,7235] [7235,19640] [19640,inf] Volume in [mm3]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='88 DICE RVD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='8 NSD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='0 SDH EDH SAH IPH IVHJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 8, AUGUST 2021 8 (a) Significance map for Dice Coefficient (b) Significance map for Normalized Surface Dice (c) Significance map for Relative Volume Difference (d) Significance map for Hausdorff Distance Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 6: The significant superiority maps for ranking robustness analysis of different evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In each of the four maps, yellow blocks means that the evaluation metric for teams on the x-axis are significantly superior to those from the teams on the y-axis, which blue blocks means no significant superiority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The pairwise significant test with one-sided Wilcoxon signed rank test at 5% significance level is adopted in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' performances among different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For example, the DSC, NSD and HD of T7 are significantly superior to that of T12, however, there exists no significant superiority on RVD metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' DISCUSSIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 2D/3D architecture Choice The algorithm summary in section IV-B shows that the participants chose different algorithm implementations for 2D or 3D methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' We noticed that the winner method adopted the 2D/3D combination method, and most of the 3D methods outperformed the 2D implementations, yet we cannot draw definite conclusions on which kinds of methods are superior to another since there are many other factors contributing to the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' However, we believe that directly utilizing 2D networks would lose significant context information among slices, which has been proved in numerous medical image segmentation tasks [6], [40]–[42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Therefore, how to effec- tive incorporate inter-slice contextual information would be a fundamental problem for improving ICH segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' To this end, many participants utilized 3D UNet implementation, however, this might not be the optimal solution considering that the CT volumes in this challenge are anisotropic (pixel spacing: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='42mm×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='42mm×5mm) [43], thus more effective techniques for exploiting inter-slice context for anisotropic volumes are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Bottlenecks for ICH segmentation The hematoma volume analysis in section IV-C2 demon- strates the inferior segmentation performance for hemorrhages T13 T12 TII T10 T9 T8 T7 T6 T5 T4 T3 T2 T1 T1 T13T13 T12 T11 T10 T9 T8 T7 T6 T5 T4 T3 T2 T1 T1 T2T3T4T5T6T7T8T9T10T11T12T13T13 T12 T11 T10 T9 T8 T7 T6 T5 T4 T3 T2 T1 T1 T2T3T4T5 T6T7T8T9T10 T11T12 T13T13 T12 T11 T10 T9 T8 T7 T6 T5 T4 T3 T2 T1 T1 T2JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 8, AUGUST 2021 9 with small volume sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The degradation of the segmentation indicates that the hemorrhage cases with small volume sizes are hard to segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 3 shows that all the methods pro- posed by the participants have trouble dealing with very small hemorrhages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The majority of the cases that achieve a DSC score lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='3 are those subjects with hemorrhage vol- ume smaller than 15000m3, and the overall DSC performances for all the subjects significantly deteriorate with substantial low DSC scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Therefore, one important bottleneck for ICH segmentation is the small hemorrhage lesion segmentation, and effectively resolving this problem would certainty improve the overall segmentation performance and achieve better ranking in the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Besides, the hemorrhage sub-type analysis in section IV-C3 shows that the subarachnoid hemorrhage (SAH) achieved the worst results in all metrics, with average DSC score for only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Thus, another bottleneck for ICH seg- mentation is how to deal with the subarachnoid hemorrhage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In conclusion, the future directions for the researches of ICH segmentation may be concentrated on the above-mentioned two bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The researches of the hemorrhage diagnosis would be greatly improved by resolving these extremely hard cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Evaluation Metrics Analysis We highly suggest the use of DSC, NSD and the RVD as the evaluation metrics for the ICH segmentation benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' According to the descriptions in section III-B, and section IV-C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The HD and NSD are similar metrics that are used to evaluate the discrepancy between the target and predicted boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' However, we came across multiple extreme cases with average HD metrics equal to infinite when the predicted methods mistakenly diagnosed those hard cases with small hemorrhage lesions as normal head scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The infinite values make it challenging to effectively rank different algorithms on that metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' However, the NSD metric has the same upper bound as DSC (100%), and there will be no such circumstances occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Therefore, Hausdorff distance might not be a good metric choice for the INSTANCE challenge, and we consider abandoning it in the future INSTANCE challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Limitations and Future work Although this year’s INSTANCE challenge has achieved great success with numerous participants around the world, it still suffers from lots of limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' They are mainly consist of three aspects: 1) Data collection and annotation: Even though the INSTANCE2022 challenge has provided a relatively large dataset, they are mainly collected from a single institution with the same CT scanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Although it could work in our challenge, it would definitely restrict the generalization of the model developed by different participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In addition, for the data annotation, we only delineate the hemorrhage regions as foreground without considering the ICH sub-types, which are actually important information in clinical diagnosis and can also guide the segmentation of ICH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 2) Task designs: In this years’ INSTANCE challenge, we only consider the hemorrhage segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' However, it is also significant to perform ICH classification and hematoma volume quantification, which are highly clinical-related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The design of multiple tasks would simultaneously make the chal- lenge more comprehensive and provide more diverse research directions for the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In conclude, we will enhance the single-task challenge to a multi-task one in the future challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 3) Source code Availability: In this years’ INSTANCE challenge, we highly recommended the participants to make their implementations to the public, and didn’t make it a mandatory option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' As a result, we only find out one team make their code available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' We didn’t demand them to share the code because we don’t expect it to be an obstacle for participating in this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' However, we notice that the code is too significant to be ignored for promoting the development in this research field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Therefore, we consider making it mandatory for top participants to make their code public available for future INSTANCE challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 4) Future works for INSTANCE: We are currently working to promote the INSTANCE 2022 Challenge in many different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Detailed improving directions are as follows: More multi-institutional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' We will collect more ICH data from different CT scanner and different hospitals to improve the generalization of methods that are trained based on the INSTANCE benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' More annotations and comprehensive task designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' We will annotate the different ICH sub-types of each CT scans and also calculate the hematoma volume of each cases to provide more clinical-related datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Mean- while, based on the above-mentioned extra annotations, we further expand the single-task challenge to a multi- task one, which simultaneously performs hemorrhage segmentation, classification and volume quantification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Mandatory options for open-source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' To pro- mote the advancement of the intracranial hemorrhage diagnosis, the top participants in the future INSTANCE challenge are required to share their code to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' CONCLUSION The INSTANCE challenge provides a novel benchmark for objectively measuring different intracranial hemorrhage segmentation methods in non-contrast head CT scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' A total of 13 teams successfully submitted their methods, and the winner solution achieved a DSC score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='6925 on the testing set, dramatically improving our baseline network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' We have made the training set, the methodology descriptions and evaluation code public available on the challenge website, we hope this would promote the development in the ICH segmentation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The challenge is now remains open for post-challenge submissions via Grand Challenge platform for benchmarking further algorithm exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In the future, we will collect more multi-institutional data to improve the generalization of methods that are trained on the benchmark, and also perform more clinical-relevant annotations on ICH JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 8, AUGUST 2021 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 7: Segmentation results for different ICH sub-types in terms of DSC and NSD scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The blue color denotes the ICH lesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' sub-type and hematoma volumes and expand the single-task challenge to a multi-task one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' ACKNOWLEDGMENTS We sincerely appreciate all the members in INSTANCE2022 organization team for their hard work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Without your con- tinuous devotion to this challenge, it would not be that successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' This work was supported by the National Nat- ural Science Foundation of China under Grant 62001144, 62272135 and 62001141, and by Science and Technology Innovation Committee of Shenzhen Municipality under Grant RCBS20210609103820029 and JCYJ20210324131800002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' APPENDIX In (Li and Chen, 2022), Li and Chen used a combination of nnU-Net and uncertainty estimation ensemble strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Their experiments showed that even though the 2D nnU- Net could not achieve the overall dice accuracy of 3D nnU- Net, it performed better results than 3D nnU-Net when the intracranial hemorrhage had very small area or blurred bound- aries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Therefore, they use both 2D and 3D nnU-Net to predict the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Furthermore, in order to further alleviate the segmentation issue of small area intracranial hemorrhage and maintain stability during training, they utilized the weighted cross-entropy loss to replace simple cross-entropy loss in the nnU-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Due to the unbalanced intracranial hemorrhage types and intracranial hemorrhage areas, the models trained in different folds might predict completely different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Simply average the predicted results from the models provide no additional benefit for these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' To this end, they propose a simple but efficient uncertainty estimation ensemble strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' DSC:89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='9 NSD:65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='4 DSC:90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='2 NSD:60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='8 DSC:90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='4 NSD:60 Case 1 DSC:89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='9 NSD:64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='6 DSC:93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='3 NSD:69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='9 DSC:91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='4 NSD:65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='3 Case 2 DSC:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='3 NSD:46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='3 DSC:47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='9 NSD:43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='8 DSC:48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='4 NSD:39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='8 N Case 3 DSC:65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='1 NSD:45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='9 DSC:72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='2 NSD:4S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='6 DSC:67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='8 NSD:43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='2 Case 4 DSC:67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='7 NSD:48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='9 DSC:43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='5 NSD:37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='0 DSC:29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='1 NSD:19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='7 Case 5 (a)Image (b)Ground Truth (c)T1 (d)T3 (e)T9JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 8, AUGUST 2021 11 For those cases with high uncertainty values, they use the voting method to get the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Use nnU-Net’s own data augmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In (Siddiquee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=', 2022 [44]), Siddiquee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' used the 2D version of encoder-decoder backbone based on with an asymmetrically larger encoder to extract image features and a smaller decoder to reconstruct the segmentation mask7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For the encoder part, they used 5 stages of down-sampling and 2D ResNet blocks that each block’s output is followed by an additive identity skip connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Furthermore, they used batch normalization and ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For the decoder part, the decoder structure is similar to the encoder one, but with a single block per each spatial level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Each decoder level begins with upsizing with transposed convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In the preprocessing, they applied random rotation and random zoom on each axis with a probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='4 and random contrast adjustment and random Gaussian noise with a probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The random coarse shuffle and random flips were applied on each axis with a probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In the training, they randomly split the entire dataset into 5-folds and trained a model for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Moreover, they used L2 norm regularization on the convolution kernel parameters with a weight of 1e−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The DiceCE loss is used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In (Sanner and Mukhopadhyay, 2022), Sanner and Mukhopadhyay used nnU-Net for the segmentation and pro- pose an evaluation of contour-based losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Specifically, they integrated both the Hausdorff-distance loss as proposed by [45] and the contour loss proposed by [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' While the former estimates the Hausdorff distance, the latter extracts the contour of both the prediction and the ground truth and minimizes the mean square error between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In practice, Dice loss and CE loss were used as loss function and the Hausdorff-distance loss or the contour loss was used depending on the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Furthermore, rather than using the standard z-normalization of nnU-Net for input images, they chose to clip the intensity values to [0 - 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' A five-fold cross-validation was used to train five models and all models were ensembled to make the final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The ”insane DA” scheme was used for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=', 2022), Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' used two stage 3D cascade U-Net network for ICH segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For the stage 1, the basic module of the encoder and decoder is Conv-Instance Norm-LeakyReLU [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The operation of downsampling in the encoder is achieved by max pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The upsampling operation in the decoder is achieved by using the transpose convolution of 2 × 2 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For the stage 2, a 3D U-Net was cascaded to the model, whose input is the output of probability map of the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The 5-fold cross-validation was used for the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In the preprocessing, the HU of CT images were clipped according to three different windows and levels, and corresponding range of HU were [0, 80], [-20, 180] and [-150, 230].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The intensity of the voxel above the range were assigned the value of upper limit in range, and the intensity below the range is assigned the value of lower limit in range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Then the three images with different HU range clip were served as three channels and treated as one image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 7Implementation: https://monai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='io/apps/auto3dseg In (Zhang and Ma, 2022), Zhang and Ma used the standard nnU-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='First, a 3D U-Net processes downsampled data, the resulting segmentation maps are upsampled to the original resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='Then, these segmentations are concatenated as one-hot encodings to the full resolution data and refined by a second 3D U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The preprocessing includes crop- ping,resampling and normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Meanwhile, random rota- tion, random scaling, random elastic transformation, gamma correction, and mirror were used to augment the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The 3D nnU-Net was trained with an weighted combination of Dice loss and cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The results on the test set were obtained as an ensemble of five models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=', 2022 [48]), Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' used an ensemble model that combined viola-Unet and nnU-Net networks8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For the viola-Unet, it relies on voxels in feature space that intersect along orthogonal levels to provide an attention U-Net, which is an asymmetric encoder-decoder architecture with 7- depth layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Overall, the Viola module is composed of three key blocks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=', the adaptive average pooling (AdaAvgPool) module that squeezes the input feature volume into three latent representation spaces along each axis of the input feature patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The customized dense dilated convolutions merging (DDCM) networks fuses cross-channel and non-local contex- tual information on each orthogonal direction with adaptive kernel sizes, dilated ratios and strides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The Viola unit con- structs the voxels intersecting along orthogonal level attention volume based on fused and reshaped cross-channel-direction latent representation spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' They trained all networks with randomly sampled patches of fixed size as input and applied a combination loss function of the dice loss and Focal loss for all their experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In preprocessing, CT image and ground truth labels were reoriented into ”RAS” format, then resized to a standard spacing of 1×1×5 mm3 using trilinear interpolation for the image and nearest-neighbor interpolation for the label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Each CT image was windowed into three image intensity ranges, and re-scaled to the range [0, 1] by min-max normalization and then stacked as 3-channel volumes to serve as inputs with the (C, H, W, D) shape, and then the 3-channel 3D volume was normalized on only non-zero values with calculated mean and std on each channel separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The data augmentations include random crop, random zoom, Gaussian noise, Gaussian smooth, rotation, random shift, random scale, flips, random contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Furthermore, they manually select the best prediction on each validation example from each submission as the pseudo-label and put them into our training set to fine-tune our models repeatedly in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In (Liang, 2022), Liang proposed a nnUNet-based method for 3-dimensional intracranial hemorrhage segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In the preprocessing, the authors first deal with the data in method windowing and decide to choose a width of 59 and a center of 96 for the image windowing by experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' After windowing, in order to arrange the information of image, the author used a threshold to ensure the gray value of the image in a certain standard interval, unified data input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Then, downsampling the X and y axes, normalize the spacing of the slice axis to Slice down scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' A sampling includes maximum sampling, 8Implementation: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='com/samleoqh/Viola-Unet JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 8, AUGUST 2021 12 average sampling, summation area sampling, and random area sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Finally, nnU-Net does the rest of the preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In the training, the author uses CE loss + DICE loss as loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Furthermore, to deal with category imbalance, oversampling was used, with 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='7% of the samples coming from random locations in the selected training sample’s, while 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='3% of the patches were guaranteed to contain one of the foreground classes present in the selected training sample (randomly selected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The number of foreground patches was rounded to force a minimum value of 1 (resulting in one random patch and one foreground patch with a batch size of 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Use nnU-Net’s own data augmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In (Geiger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=', 2022), Geiger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' used classic U- Net architecture and the network was conducted with the jax version of the e3nn library which enables the creation of neural networks equivariant to translations, rotations, and mirroring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Specially, the convolution kernels in the original architecture are replaced by a 3D e3nn voxel convolution of diameter 5 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Furthermore, they used three 2x2x2 downsampling operations which halve the resolution in the encoding path and three corresponding trilinear upsampling operations on the decoding path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' A Gaussian error linear unit activation function and instance normalization was used after each convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For the preprocessing, each CT volume was windowed to three different Hounsfield unit value ranges, scaled, and added to a separate channel which served as the model input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' To increase the variety in the data, a random diffeomorphic deformation was performed on each training sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The loss function em- ployed was cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Eight models were trained, each on 80 randomly sampled subsets from the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The final prediction was performed by applying each of the eight models to patches of size 144x144x13 with padding discarding 22x22x2 pixels on each side, a sliding window with an overlap of 26 pixels and Gaussian weighing, and then averaging the model outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For the final prediction, they take the ensemble average of the eight models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In (Qayyum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=', 2022), Qayyum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' developed a coarse and fine segmentation model for intracranial hemor- rhage segmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' They trained two different models for intracranial hemorrhage segmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In the first model, they trained 2DDensNet for coarse segmentation and cascaded the coarse segmentation masks output in the fine segmentation model along with input training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The proposed model is implemented made by a dense encoder followed by a non- dense decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The dense encoder consists of 5 dense blocks, each consisting of 6 dense layers followed by a transition layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Each dense layer consists of 2 convolutional layers with batch normalization and ReLU activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The model is trained using 5-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' To compute the final prediction, 2D images are stacked to make a 3D seg- mentation mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The predicted segmentation mask is further cascaded in a fine segmentation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In the fine stage, they used the nnU-Net model with fivefold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The binary cross-entropy function was used as loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Hor- izontalFlip (p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='5), VerticalFlip (p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='5), and RandomGamma (p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='8) were used to augment the dataset for training the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In addition, the dataset is normalized between 0 and 1 using the max and min intensity normalization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The training shape of each volume is fixed (256x256x16) and resample the prediction mask to the original shape for each validation volume using the linear interpolation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In (Abramova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=', 2022), Abramova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' used an approach based on a 3D U-Net architecture which incorporates squeeze-and-excitation blocks that similarly to their previous work [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For the preprocessing, coil removal and skull stripping were used, and a symmetric image was created for each case by flipping the original non-contrast CT and registering it to the initial one using the FLIRT algorithm from the FSL toolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For the normalization of input images, they performed percentile based range adjustment and used 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='5 and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='5 percentiles of brain-related voxels for clipping together with image-based calculated mean and standard deviation normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For the issue of class imbalance, they used a balanced sampling patch extraction technique, where we extracted an equal number of patches representing both classes from each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Specifically, to avoid extracting a lot of patches from image background, they restricted the area to extract the negative patches within the brain mask and set a target number of patches to extract from each image in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Half of them are uniformly extracted from the brain tissue area and represent negative class, while the other half is extracted from the lesion voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' They augment the proposed dataset by choosing difficult cases and adding them into the training set again, meanwhile performing flipping and rotation, ensuring that more difficult patches are generated for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The Dice loss and cross-entropy loss was used as loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' To prevent overfitting, they used early stopping technique when approaching the minimal loss on validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The five-fold cross-validation strategy was used for the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For the validation and testing stages, an ensemble with all the 5 models obtained in the cross-validation exper- iment was used to generate the final prediction masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The probability masks obtained from the 5 models were averaged and thresholded to obtain the final binary mask for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Considering the results on the validation set, postprocessing was added to their pipeline to reduce the number of false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Specifically, as sizes of lesions vastly vary in the provided images, they remove all the lesions with the volume less than 10% of the biggest one in the post-processed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In (Montagnon and Letourneau-Guillon, 2022), Mon- tagnon and Letourneau-Guillon used an ensemble approach including the Attention U-Net and SegResNet (with or without variational autoencoder) architectures combined with different loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Specifically, they trained U-Net and SegResNet separately to use different loss functions including combina- tions of Dice with either Cross-Entropy loss or Focal loss, Tversky loss and Generalised Dice loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Then leveraging all predictions, an ensemble voting approach allowed prediction of a final volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Finally, to further remove potential false pos- itive predictions, predicted clusters were filtered by preserving ones with a volume larger than 36 pixels, an elevation above or equal to 3 slices and a mean density within [40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 80] HU range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In the preprocessing, in order to assess hemorrhage properties, they used DBSCAN, a density-based clustering algorithm, in order to extract connected pixels corresponding to hemorrhagic areas in each exam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Then they clipped images in the range [- JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 8, AUGUST 2021 13 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 140] HU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Taking into account the intracranial hemorrhage subtype distribution in the training dataset, they using Euler transforms consisting of rotations of either - π 2 or - π 2 around z-axis and translations ranging from -30 to 30 pixels, 10 pixels stepwise for subarachnoid and subdural hemorrhage subtypes images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Considering the limited size of the dataset, they used random orthogonal rotations and cropping for images in the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In order to limit class imbalance issues, models were trained only on images containing at least one pixel of positive class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' All models were trained using original images size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' 512 × 512), clipped within [-10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='140] and divided by the range of considered densities, which is 150 in their configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In (Roca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=', 2022), Roca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' used a simple 2D Unet- like model and trained it with a binary cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Especially, the model input is a layer that performs the clipping operation between [0, 256] and a normalization between [- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='5] directly inside the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In the preprocessing, they clipping the HU intensities in the soft tissue range of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' For the data augmentation, they performed rotations and mirroring in the axial plane, plus some amount of intensity shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Due to the data stratification was based on the presence of a segmentation on a given slice (positive cohort) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' absence of segmentation (negative cohort), they used during training a balanced 50% / 50% of each cohort per mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In (Sindhura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=', 2022), Sindhura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' proposed a deep learning framework which involves clinical knowledge and used U-Net3+ network for the segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Specifically, they proposed a new data augmentation approach that leverages from the clinical knowledge that the two hemispheres of the human brain exhibit approximate symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Due to the brain is approximately divided into two equal hemispheres by the midsagittal plane (MSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' So they use the MSP flipped versions of the CT scans as extra data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' To extract MSP, they first apply the sobel edge detection method followed by thresholding to obtain the outline of the skull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' An initial plane of reference is chosen to be the exact middle slice in the sagittal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' A similarity metric is computed between the two hemispheres that are divided with the plane of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The reference plane is rotated by an angle of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content='5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' The plane which yields maximum similarity is the required MSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' Furthermore, to improve the robustness of the model, the usual data augmentations such as shear, rotation, zoom, flip, elastic transform, noise etc are being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' In view of there exists a very high class imbalance between the hematoma and non-hematoma pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQflATa/content/2301.03281v1.pdf'} +page_content=' So only the slices which contain hemorrhages are used in the training process and all slices of each scan are tested in the testing phase.' metadata={'source': 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https://git-lfs.github.com/spec/v1 +oid sha256:e47c776720682598c3c3aee3f063a4d6cc7a787e76881d3aba0032439c3a18ba +size 146430 diff --git a/2dE1T4oBgHgl3EQflgSY/content/tmp_files/2301.03286v1.pdf.txt b/2dE1T4oBgHgl3EQflgSY/content/tmp_files/2301.03286v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f33ccd79ab624301726136b1ed814bd9b00fe99f --- /dev/null +++ b/2dE1T4oBgHgl3EQflgSY/content/tmp_files/2301.03286v1.pdf.txt @@ -0,0 +1,3015 @@ +arXiv:2301.03286v1 [eess.SP] 9 Jan 2023 +1 +A Dual-Function Radar-Communication System +Empowered by Beyond Diagonal Reconfigurable +Intelligent Surface +Bowen Wang, Student Member, IEEE, Hongyu Li, Student Member, IEEE, +Ziyang Cheng, Member, IEEE, Shanpu Shen, Member, IEEE, +and Bruno Clerckx, Fellow, IEEE +Abstract—This work focuses on the use of reconfigurable +intelligent surface (RIS) in dual-function radar-communication +(DFRC) systems to improve communication capacity and sensing +precision, and enhance coverage for both functions. In contrast +to most of the existing RIS aided DFRC works where the RIS +is modeled as a diagonal phase shift matrix and can only reflect +signals to half space, we propose a novel beyond diagonal RIS +(BD-RIS) aided DFRC system. Specifically, the proposed BD-RIS +supports the hybrid reflecting and transmitting mode, and is com- +patible with flexible single/group/fully-connected architectures, +enabling the system to realize full-space coverage. To achieve the +expected benefits, we jointly optimize the transmit waveform, the +BD-RIS coefficients, and sensing receive filters, by maximizing +the minimum signal-to-clutter-plus-noise ratio for fair target +detection, subject to the constraints of the communication quality +of service, different BD-RIS architectures and power budget. +To solve the non-convex and non-smooth max-min problem, a +general solution based on the alternating direction method of +multipliers is provided for all considered BD-RIS architectures. +Numerical simulations validate the efficacy of the proposed +algorithm and show the superiority of the BD-RIS aided DFRC +system in terms of both communication and sensing compared +to conventional RIS aided DFRC. +Index Terms—Beyond diagonal reconfigurable intelligent sur- +faces, dual-function radar-communication, full-space coverage, +max-min optimization. +I. INTRODUCTION +In recent years, spectrum resources are becoming increas- +ingly limited and valuable due to the exponential growth of +services in wireless communications. Meanwhile, radar sys- +tems are competing for the same scarce sources, which moti- +vates the emergence of the dual-function radar-communication +(DFRC) technology to achieve spectrum sharing between +communication and radar. In DFRC systems, communication +and radar functionalities are integrated on a common platform, +which brings the benefit of enhanced spectrum efficiency while +(Corresponding author: Ziyang Cheng, Shanpu Shen). +B. Wang and Z. Cheng are with the School of Information & Communica- +tion Engineering, University of Electronic Science and Technology of China, +Chengdu, China. (email: B W Wang@163.com, zycheng@uestc.edu.cn). +H. Li is with the Department of Electrical & Electronic Engineering, Impe- +rial College London, London SW7 2AZ, U.K. (email: c.li21@imperial.ac.uk). +S. Shen is with the Department of Electronic and Computer Engineering, +The Hong Kong University of Science and Technology, Clear Water Bay, +Kowloon, Hong Kong (email: sshenaa@connect.ust.hk). +B. Clerckx is with the Department of Electrical & Electronic Engineering, +Imperial College London, London, SW7 2AZ, U.K. and with Silicon Austria +Labs (SAL), Graz A-8010, Austria (email: b.clerckx@imperial.ac.uk). +reducing power consumption and hardware costs. Therefore, +DFRC is envisioned to play an important role in emerging +environment-aware applications [1], such as vehicular net- +works, environmental monitoring, and smart houses. +Due to the benefits of DFRC, plenty of technical efforts have +been devoted to designing DFRC systems. The design method- +ology can be roughly divided into three categories: radar- +centric design [2]–[4], communication-centric design [5]–[7], +and joint waveform design [8], [9]. Radar-centric approaches +utilize the radar waveform as the information carrier, where +the communication symbols are embedded in conventional +radar signals, such as linear frequency modulation [2] and +frequency hopping [4]. On the other hand, communication- +centric approaches realize the radar sensing tasks by modifying +existing communication protocols [5] and waveforms [6], [7]. +In contrast to the first two categories [2]–[7], the DFRC +waveforms can be jointly designed to provide more design +freedoms so as to enhance both functionalities [8], [9]. Despite +the above works [2]–[9] achieve satisfactory sensing and +communication performance, one limitation is that they rely on +the line-of-sight (LoS) links between the base station (BS) and +communication users/sensing targets, which however yields +the following two issues in practice: 1) The LoS link toward +sensing targets or communication users can be easily blocked +by obstacles. 2) The LoS channels may suffer from severe +path loss especially for high frequencies. +To overcome these issues, a promising technology named +reconfigurable intelligent surface (RIS) [10]–[13] can be lever- +aged. Specifically, RIS consists of numerous passive reconfig- +urable scattering elements with low hardware cost and power +consumption [10]–[13]. By properly placing and adjusting +the RIS, it can establish virtual non-LOS (NLoS) links to +“bypass” obstacles, and therefore compensate for the path +loss and enhance system performance. Due to its advantages, +RIS has been investigated for communications [14]–[16] and +sensing [17]–[20] fields. Furthermore, RIS has been explored +in various DFRC systems [21]–[27] to enhance both the +communication and sensing performance, which are classified +into the following two categories. The first category assumes +LoS links exist from BS to users and targets. In this category, +the RIS is used to compensate for the propagation loss and +to improve the performance [21]–[23]. The second category +focuses on the scenario where either communication users or +sensing targets are blocked by barriers. In this category, RIS + +2 +is utilized to establish a NLoS link to bypass the barriers and +thus enable DFRC [24]–[27]. +The limitation of the aforementioned works [21]–[27] is that +they assume the RIS can only reflect signals towards the same +side as the BS. In this case, both communication users and +sensing targets should be located at the same side of RIS, +i.e., within the same half-space, which limits the coverage +and beam control flexibility of the RIS enabled DFRC sys- +tem. To address this limitation, a novel hybrid transmissive +and reflective RIS, namely simultaneously transmitting and +reflecting RIS (STAR-RIS) [28] or intelligent omni-surface +[29], is proposed to support signal reflection and transmission +and thus extend the coverage. The integration of STAR- +RIS and DFRC is first studied in [30], where the system is +designed by minimizing the Cram´er-Rao bound (CRB) for +radar target estimation subject to communication constraints. +Then, a STAR-RIS is deployed at the vehicle to improve both +sensing and communication performance [31]. Nevertheless, +the achievable performance of STAR-RIS aided DFRC in [30], +[31] is limited by the simple architecture of STAR-RIS without +fully exploiting the architecture of RIS. +To enhance the performance of RIS, a novel branch, namely +beyond diagonal RIS (BD-RIS) [32]–[35], is proposed by +exploring different architectures/modes of RIS. BD-RIS with +group/fully-connected architectures under the reflective mode +is first proposed in [32], which provides more controllable +scattering matrices than conventional RIS. Then, the hybrid +reflective and transmissive BD-RIS is proposed in [33] to +achieve full-space coverage. It is proved that STAR-RIS is +essentially a particular instance of two-port group-connected +reconfigurable impedance network when each two antenna +ports are connected to each other, namely cell-wise single- +connected (CW-SC) architecture in [33]. More general cell- +wise group/fully-connected (CW-GC/FC) architectures are +also proposed based on the flexible connections among more +antenna ports, which achieves better performance than STAR- +RIS. Furthermore, a multi-sector BD-RIS is proposed in [35], +which not only achieves full-space coverage but also provides +higher performance gain than hybrid BD-RIS. +Due to the benefits of BD-RIS, in this paper, we propose +to adopt BD-RIS in DFRC systems to achieve full-space cov- +erage and better performance. To the best of our knowledge, +adopting BD-RIS in DFRC has not been investigated in the +literature. In addition, in contrast to [30], [31] which ignore +the signal-dependent clutters, we consider a more general and +practical multi-target detection scenario with the presence of +multiple clutters. The main contributions of this work are +summarized as follows: +Proposing BD-RIS aided DFRC. We propose a BD-RIS +aided DFRC system, which consists of a BD-RIS enabling +the full-space coverage, multiple users, and multiple sensing +targets corrupted by multiple clutters. The BD-RIS divides +the space into two sides and establishes virtual NLoS links +for communication and sensing, where the dual-function BS +(DFBS) performs communication tasks in one half space and +sensing tasks in another side. To avoid multi-step path loss, +we implement the radar sensing receiver on the BD-RIS for +multi-target detection. +Formulating Max-min fairness problem. We formulate +the optimization problem to jointly design the transmit wave- +form at the DFBS, the reflective and transmissive beamforming +at the BD-RIS, and matched filters at the radar sensing +receiver, to maximize the minimum radar output signal-to- +clutter-plus-noise ratio (SCNR), subject to the communication +quality of service (QoS) requirement for downlink communi- +cations, the transmit power constraint at the DFBS, and the +BD-RIS constraints with different architectures. +Developing joint design framework. The joint design of +BD-RIS aided DFRC is challenging due to the complicated +and non-smooth objective, and newly introduced non-convex +constraints of BD-RIS. To overcome these difficulties, we +propose to decouple the BD-RIS constraints by the alternat- +ing direction method of multipliers (ADMM) framework so +that the resulting sub-problems are reformulated into easily +handled forms and iteratively solved until convergence. +Providing insights and numerical validation. We provide +simulation results to illustrate the performance improvement +achieved by BD-RIS. It is shown that benefiting from the +high flexibility of BD-RIS, and the joint design of transmit +waveform, BD-RIS, and the matched filters, the CW-GC/FC +BD-RISs can achieve higher radar SCNR than CW-SC (STAR- +RIS) ones under the same communication requirement. It +is also shown the BD-RIS can substantially improve the +performance and coverage compared to the conventional RIS, +which shows the high flexibility of BD-RIS in manipulating +the incident signal for enhancing the DFRC system. +Organization: Section II presents the system model of +the proposed BD-RIS aided DFRC. Section III formulates +the max-min fairness problem and provides a joint design +algorithm. Section IV evaluates the performance of the pro- +posed algorithm and compares different BD-RIS architectures. +Section V concludes this work. +Notation: Scalars, vectors and matrices are denoted by stan- +dard lowercase letter a, lower case boldface letter a and upper +case boldface letter A, respectively. Cn and Cm×n denote +the n-dimensional complex-valued vector space and m × n +complex-valued matrix space, respectively. (·)T , (·)H, and +(·)−1 denote the transpose, conjugate-transpose operations, +and inversion, respectively. ℜ{·} and ℑ{·} denote the real and +imaginary part of a complex number, respectively. ∥ · ∥F and +| · | denote the Frobenius norm and magnitude, respectively. +Diag(·) denotes a diagonal matrix. BlkDiag(·) denotes a block +matrix such that the main-diagonal blocks are matrices and all +off-diagonal blocks are zero matrices. IL indicates an L × L +identity matrix.  denotes imaginary unit. ∠(·) represent the +phase values of a matrix. Tr(·) denotes the summation of +diagonal elements of a matrix. ⌊·⌋ is the round-down operation. +II. SYSTEM MODEL +As depicted in Fig. 1, we consider a DFRC system, where +an NT-antenna DFBS simultaneously sends communication +symbols to U single-antenna users and detects K targets in +the presence of Q strong clutters with the assistance of an +NS-cell BD-RIS. The BD-RIS adopts the hybrid transmissive +and reflective mode, which divides the whole space into two + +3 +Target +Target +Clutter +Clutter +Transmissive Area +for Radar +Cell 1 +BD-RIS +Target 1 +Target K +Clutter 1 +Clutter Q +Reflective Area for +Communication +Transmissive Area +for Radar +User +User +DFBS +Reflective Area for +Communication +RIS elements +Sensor elements +User 1 +User NU +DFBS +NT +Fig. 1. Illustration of a BD-RIS aided DFRC system. +half areas, i.e., the transmissive and reflective areas. The DFBS +provides communication services at the reflective area while +performing radar sensing at the transmissive area aided by BD- +RIS. The radar sensing receiver with NR antennas is installed +adjacent to the BD-RIS to collect target echos and conduct +target detection tasks. In the following subsections, we will +review the modeling of BD-RIS with different architectures, +and establish the communication and radar models. +A. BD-RIS Architecture Model +According to [33], the hybrid reflective and transmissive +mode is essentially based on the group-connected reconfig- +urable impedance network. Specifically, each two antenna +ports are connected to each other, constructing one cell as +illustrated in Fig. 1. Within each cell, two antennas with uni- +directional radiation pattern are back to back placed such that +each antenna covers half space. Mathematically, the BD-RIS +with hybrid reflective and transmissive mode is characterized +by two matrices, i.e., ΦR ∈ CNS×NS and ΦT ∈ CNS×NS. +Depending on the inter-cell connection strategies, the BD-RIS +can be categorized into the following three architectures. +1) CW-SC BD-RIS Architecture: As shown in Fig. 2(a), +we provide a simple example of CW-SC BD-RIS with 2 cells, +from which we can observe that different RIS cells are not +connected to each other. Therefore, matrices ΦT, ΦR are all +restricted to be diagonal, i.e., ΦT = Diag(φT,1, . . . , φT,NS) and +ΦR = Diag(φR,1, . . . , φR,NS), and satisfy +|φT,i|2 + |φR,i|2 = 1, ∀i = 1, · · · , NS, +(1) +which conforms to the STAR-RIS constraints, indicating that +the STAR-RIS is a special case of BD-RIS with CW-SC +architecture [28], [29]. +2) CW-FC BD-RIS Architecture: Fig. 2(b) depicts an exam- +ple of CW-FC BD-RIS with 2 cells. In contrast to CW-SC case, +all cells of the CW-FC BD-RIS are connected to each other +through reconfigurable impedance components. Accordingly, +ΦT, ΦR are all full matrices satisfying +ΦH +T ΦT + ΦH +R ΦR = INS. +(2) +3) CW-GC BD-RIS Architecture: As a balance between the +above two extreme cases, CW-GC divides all cells into several +Cell� +User� +User� +BD�RIS +DFBS +Target� +Target� +Clutter� +Clutter� +Reflective�Area�for� +Communication +Transmissive�Area� +for�Radar +Antenna�3 +Antenna�1 +Antenna�4 +Z3,4 +Z3 +Z1,3 +Z1 +Z2,4 +Z1,2 +Z1,4 +Z2,3 +Z2 +Z4 +Antenna�2 +2�Cell�CW�FC�BD�RIS +(b) +Cell�1 +Cell�2 +Antenna�4 +Z2,4 +Z2 +Z4 +Antenna�2 +2�Cell�CW�SC�BD�RIS� +(a) +Cell�2 +Antenna�5 +Antenna�1 +Antenna�6 +Z5,6 +Z5 +Z1,5 +Z1 +Z2,6 +Z1,2 +Z1,6 +Z2,5 +Z2 +Z6 +Antenna�2 +4�Cell�CW�GC�BD�RIS +Antenna�7 +Antenna�3 +Antenna�8 +Z7,8 +Z7 +Z3,7 +Z3 +Z4,8 +Z3,4 +Z3,8 +Z4,7 +Z4 +Z8 +Antenna�4 +Group�1 +Group�2 +(c) +Cell�1 +Cell�2 +Cell�3 +Cell�4 +Antenna�3 +Antenna�1 +Z3 +Z1,3 +Z1 +Cell�1 +Fig. 2. Examples of (a) CW-SC BD-RIS, (b) CW-FC BD-RIS, and (c) CW- +GC BD-RIS. +groups and cells in each group adopt the the fully-connected +architecture. Depending on the group division strategies, there +are plenty of CW-GS architectures. For simplicity, here we +consider the case where NS cells of the BD-RIS are uniformly +divided into G groups and each group has the same size M = +NS/G. For ease of understanding, an example of a 4-cell BD- +RIS with CW-GC architecture having 2 groups is illustrated +in Fig. 2(c). Hence, the model for CW-GC BD-RIS can be +expressed as +ΦT = BlkDiag(ΦT,1, . . . , ΦT,G), +ΦR = BlkDiag(ΦR,1, . . . , ΦR,G), +ΦH +T,gΦT,g + ΦH +R,gΦR,g = IM, ∀g = 1, · · · , G. +(3) +where ΦT,g ∈ CM×M and ΦR,g ∈ CM×M. +Remark 1. The CW-GC architecture of BD-RIS is a general +case, which becomes the CW-SC architecture (STAR-RIS) with +a simple circuit when G = NS, and the CW-FC architecture +achieving the best performance as G = 1. This means that +CW-SC and CW-FC architectures are special cases of CW-GC +architecture and the beam control flexibility/ability of CW-GC +BD-RIS can be improved by decreasing G, but at the expense +of increasing circuit complexity. +B. Communication Model +In this paper, we consider a standard multiuser multiple +input single output (MISO) downlink scenario, where the +DFBS provides communication services to the reflective area +aided by the BD-RIS. We assume the direct links between the +DFBS and downlink users are blocked and the channel state +information (CSI) is available at the DFBS. The data symbol +vector sl = [sl [1] , · · · , sl [U]]T ∈ CU contains the overall +U data symbols in the l-th time slot, which are assumed to + +4 +be drawn from a standard M order phase-shift keying (M- +PSK) modulation constellation. Furthermore, the data symbol +vector sl is mapped to the transmit waveform w [l] ∈ CNT at +the DFBS. Accordingly, the received signal of the u-th user +at symbol time t is +yu (t) = e2πfct +L +� +l=1 +hH +u ΦRGw [l] rect (t − l∆t) + nc,u (t) , +(4) +where fc is the carrier frequency, L is the number of time slots +during one transmission duration, G ∈ CNS×NT and hu ∈ +CNS stand for the channel coefficients of the communication +links DFBS→BD-RIS and BD-RIS→u-th user, ∆t stands for +symbol duration, rect (t) is the rectangle window function that +takes the value 1 for t ∈ [0, ∆t] and 0 otherwise, and nc,u (t) +is the additive white Gaussian noise (AWGN). +By down converting the signal into baseband and sampling +received signal yu (t) at the rate fs = 1/∆t within the symbol +duration, the discrete baseband signal at the l-th time slot is +yu [l] = hH +u ΦRGw [l] + nc,u [l] , +(5) +where nc,u [l] is the AWGN with zero mean and variance σ2 +C,u. +In this work, we adopt the recently emerged symbol level +beamforming (SLB) technology for communication in DFRC. +Specifically, SLB technology utilizes the constructive inter- +ference (CI), which is defined as the multi-user interference +(MUI) that pushes the received symbols away from the detec- +tion thresholds of the modulation constellation, to enhance the +communication QoS while reducing BER [36], [37]. Here we +briefly review the concept of SLB as follows. +Fig. 3 takes quadrature-PSK (QPSK) as an example, where +point A stands for the desired symbol sl [u] with the required +signal-to-noise-ratio (SNR) threshold Γu,l of the u-th user, i.e., +−→ +OA = +� +σ2 +C,uΓu,lsl [u], and point D is the received noise- +free signal, i.e., −→ +OD = ˜yu [l] = hH +u ΦRGw [u]. The CI region +refers to a polyhedron bounded by hyperplanes parallel to +decision boundaries of the constellation, which is depicted +as blue-shaded area in Fig. 3. The key of SLB is to enforce +the received signal located in the CI region, which means the +received signal is pushed away from decision boundaries and +the SNR is guaranteed to be no less than the SNR threshold +Γu,l. To mathematically depict the SLB constraint, we project +point D into the direction of −→ +OA at point C, and extend −→ +CD +to intersect with the nearest boundary of CI region at point B. +Consequently, one of the criteria that specifies the location of +−→ +OD in the CI region is +|−→ +CD| +|−→ +AC| += +��ℑ +� +hH +u ΦRGw [l] e∠(su[l])��� +ℜ +� +hH +u ΦRGw [l] e∠(su[l])� +− +� +σ2 +C,uΓu,l +≤ tan Ω, +(6) +where Ω = π/M is half of the angular range of the CI resign. +Remark 2. In this work, we adopt SLB instead of con- +ventional block-level beamforing (BLB) due to the following +two reasons: 1) By adopting SLB technology in our con- +sidered DFRC system, we directly design transmit waveform +W ∈ CNT×L for L time slots. However, the BLB in the +D +B +� � +� +� +u +l +y +� +� +C +Received +Symbol +� � +� +� +2 +, +u +c +u l +l +y +� � +� +� +� +� � +u l +y� +CI�Region +2 +, +c +u l +� � +O +� +Imag +Real +A +Fig. 3. Description of the CI region for a QPSK symbol. +same scenario requires the design of the transmit beamformer +Wl ∈ CNT×U, ∀l for all data symbols and time slots due +to the linear mapping, which results in an increasing com- +putational complexity [38]. 2) BLB regards the MUI as a +harmful component and suppresses the MUI to guarantee +communication QoS. However, the SLB utilizes the MUI to +enhance the communication QoS, which provides additional +design flexibility in DFRC [21]. +C. Radar Model +To improve the sensing performance of the BD-RIS aided +DFRC system, as shown in Fig. 1, we adopt a novel sensor- +at-RIS architecture [20], where the radar receiving sensors +are installed adjacent to the BD-RIS to collect the echo +signals. This architecture greatly reduces the multi-step path- +loss compared with the sensor-at-DFBS architecture [21]–[23]. +Moreover, we consider a scenario where the radar receiver +attempts to detect K targets in the presence of Q strong +clutters. Specifically, the k-th target of interest is characterized +by angle ϕk and time delay τ k +T , respectively, while the q-th +clutter is characterized by angle ϑq and delay τq +C, respectively1. +The backscattered signal at the radar receiver after down +conversion is thus [39]–[41] +r (t) = +K +� +k=1 +L +� +l=1 +αkA (ϕk) ΦTGw [l] rect +� +t − l∆t − τk +T +� ++ +Q +� +q=1 +L +� +l=1 +βqA (ϑq) ΦTGw [l] rect (t − l∆t − τq +C) ++ nr (t) , +(7) +where αk and βq, respectively, denote the propagation co- +efficient for the k-th target and q-th clutter consisting of +radar cross section (RCS) and channel propagation effects +with E(|αk|2) += +ζ2 +k +and E(|βq|2) += +ξ2 +q. A (ϕ) += +aR (ϕ) aH +T (ϕ) ∈ CNR×NS is the effective radar channel, +where aT (ϕ) = +1 +√NS [1, · · · , ej 2π +λ d(NS−1) sin ϕ]T and aR (ϕ) = +1 +√NR [1, · · · , ej 2π +λ d(NR−1) sin ϕ]T denote the the transmit and +1In this paper, we assume the targets and clutters are slowly moving or stay +still, whose Doppler frequencies equal to zeros. + +5 +receive steering vector, respectively, with d and λ being +element spacing and wavelength. nr (t) denotes AWGN. +Then, we select the first target echo as the reference and +sample the received signal r (t) at fs = 1/∆t, yielding the +following received baseband signal +R = +K +� +k=1 +αkA (ϕk) ΦTGWJrk +T +� +�� +� +Target Echos ++ +Q +� +q=1 +βqA (ϑq) ΦTGWJrq +C +� +�� +� +Clutter Returns ++ Nr, +(8) +where Jr = [0L×r, IL, 0L×(Lobs−L−r)] ∈ CL×Lobs is the shift +matrix with Lobs = L + {maxk rk +T} − {mink rk +T} being the +receiver observation length, rk +T = ⌊(τ k +T − {min˜k τ ˜k +T })fs⌋ the +rang ring of the k-th target, and rq +C = ⌊(τ q +C − {mink τ k +T })fs⌋ +the rang ring of the q-th clutter. Nr = [nr [1] , · · · , nr [L]] ∈ +CNR×L +is +the +Gaussian +noise +matrix +with +nr [l] +∼ +CN +� +0, σ2 +RINR +� +, ∀l. +Finally, by performing the matched filter Uk ∈ CNR×Lobs +to the k-th target at radar receiver, the k-th target detection +problem can formulated as a binary hypothesis test [39]–[41]: + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +Hk +1 : αkUH +k A (ϕk) ΦTGWJrk +T +(9a) ++ +K +� +p=1,p̸=k +αpUH +k A (ϕp) ΦTGWJrp +T ++ +Q +� +q=1 +βqUH +k A (ϑq) ΦTGWJrq +C + Nr, +(9b) +Hk +0 : +K +� +p=1,p̸=k +αpUH +k A (ϕp) ΦTGWJrp +T +(9c) ++ +Q +� +q=1 +βqUH +k A (ϑq) ΦTGWJrq +C + Nr. +(9d) +According to the above binary hypothesis test (9), the detection +probability P k +D of the k-th target can be evaluated as [41] +P k +D = Q +�� +2SCNRk, +� +−2 ln (Pfa) +� +, +(10) +where Q (·, ·) is the Marcum Q-function of order 1, Pfa is the +false alarm probability, and the radar output SCNR of the k-th +target after the matched filtering is given by +SCNRk(W, ΦT, Uk) = ς−1 +k |Tr(αkUH +k A(ϕk)ΦTGWJrk +T )| +2, +(11) +where ςk += �K +p=1,p̸=k |Tr(αpUH +p A (ϕp) ΦTGWJrp +T )| +2 + +�Q +q=1 |Tr(βqUHA (ϑq) ΦTGWJrq +C)| +2 + σ2 +R ∥Uk∥2 +F , ∀k. +III. MAX-MIN FAIRNESS FOR BD-RIS AIDED DFRC +In this section, we first formulate the joint design problem +for BD-RIS aided DFRC, followed by a general algorithm. +Finally, we propose an initialization scheme and analyze the +computational complexity of the proposed algorithm. +A. Problem Formulation +Given that (10) is strictly increasing in SCNRk, for a +specified value of false alarm probability Pfa, improving +the detection probability P k +D of the k-th target is equivalent +to maximize the radar output SCNR of the k-th target. +Moreover, for multiple target detection cases, beamforming +design usually aims to improve the detection probability for +all targets, especially for the weakest targets. Therefore, to +improve the overall target detection probability and guarantee +target detection fairness, we propose to maximize the minimal +radar output SCNR among the K targets by jointly designing +the transmit beamformer W, the BD-RIS matrices {ΦT, ΦR}, +and radar receiver filters {Uk}∀k, subject to communication +QoS constraints, transmit power constraint, and BD-RIS con- +straints. The joint design problem is thus formulated as2 +P1 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +max +W,ΦT,ΦR,{Uk} +� +min +∀k SCNRk (W, ΦT, Uk) +� +(12a) +s.t. +���ℑ +� +˜hH +u w [l] +���� +ℜ +� +˜hH +u w [l] +� +− +� +σ2 +C,uΓu,l +≤ tan Ω, (12b) +∥W∥2 +F = E, +(12c) +ΦT = BlkDiag (ΦT,1, · · · , ΦT,G) , +(12d) +ΦR = BlkDiag (ΦR,1, · · · , ΦR,G) , +(12e) +ΦH +T,gΦT,g + ΦH +R,gΦR,g = ING, ∀g, +(12f) +where ˜hH +u += +hH +u ΦRG is the equivalent channel for +DFBS→DB-RIS→ u-th user and E is the transmit power. +Problem P1 is a challenging non-convex problem. Partic- +ularly, the non-convexity stems from the complicated frac- +tional SCNR expression in the objective and highly coupled +optimization variables. To simplify the joint design, in the +following subsection, we propose a series of transformations +and an ADMM based framework to decouple problem (12) +into multiple more tractable sub-problems. +B. Overview of Proposed Joint Design Framework +To facilitate the joint design, we propose to re-arrange the +SCNR (11) into explicit and compact forms. By defining uk = +Vec(Uk), w = Vec(W) and φT = Vec (ΦT), and applying +basic vectorization properties [42], the SCNR in (11) shares +the following three equivalent expressions +SCNRk (W, ΦT, Uk) = +uH +k ΨT,kuk +uH +k (ΨC,k + σ2 +RINRL) uk +, +(13a) += +wHΥT,kw +wHΥC,kw + σ2 +R ∥Uk∥2 +F +, +(13b) += +φH +T ΞT,kφT +φH +T ΞC,kφT + σ2 +R ∥Uk∥2 +F +, +(13c) +where +ΨT,k =ζ2 +k +� ¯ +MT (k, ΦT) +� +wwH� ¯ +MT (k, ΦT) +�H, +2Based on the discussion in Remark 1, herein we focus on the design when +the BD-RIS has CW-GC architecture, which is a general case including both +CW-SC and CW-FC cases. + +6 +ΨC,k = +K +� +p=1,p̸=k +ζ2 +p +� ¯ +MT (p, ΦT) +� +wwH� ¯ +MT (p, ΦT) +�H ++ +Q +� +q=1 +ξ2 +q +� ¯ +MC (q, ΦT) +� +wwH� ¯ +MC (q, ΦT) +�H, +ΥT,k =ζ2 +k +� ¯ +MT (k, ΦT) +�HukuH +k +� ¯ +MT (k, ΦT) +� +, +ΥC,k = +K +� +p=1,p̸=k +ζ2 +p +� ¯ +MT (p, ΦT) +�HukuH +k +� ¯ +MT (p, ΦT) +� ++ +Q +� +q=1 +ξ2 +q +� ¯ +MC (q, ΦT) +�HukuH +k +� ¯ +MC (q, ΦT) +� +, +ΞT,k =ζ2 +k +� +˜ +MT (k, W) +�H +ukuH +k +� +˜ +MT (k, W) +� +, +ΞC,k = +K +� +p=1,p̸=k +ζ2 +p +� +˜ +MT (p, W) +�H +ukuH +k +� +˜ +MT (p, W) +� ++ +Q +� +q=1 +ξ2 +q +� +˜ +MC (q, W) +�H +ukuH +k +� +˜ +MC (q, W) +� +, +with ¯ +MT (k, ΦT) = JT +rk +T ⊗ (A (ϕk) ΦTG), ¯ +MC(q, ΦT) = +JT +rq +C ⊗ (A (ϑq) ΦTG), ˜ +MT(k, W) = (JT +rk +T WT GT ) ⊗ A(ϕk), +˜ +MC(q, W) = (JT +rq +CWT GT ) ⊗ A(ϑk). +Based on the above derivations, the objective in problem P1 +is more tractable with respect to uk, w, or φT. However, it is +still difficult to find the solution to P1 due to non-convex and +coupled constraints (12b), (12c), and (12f). To tackle constraint +(12f), we first define Φg = [ΦH +T,g, ΦH +R,g]H and rewrite (12f) as +ΦH +g Φg = IM. Then, we introduce auxiliary variables Θg = +[ΘH +T,g, ΘH +R,g]H = Φg and decouple constraint (12f) into two +separate constraints by adding the equality, which yields the +following problem: +P2 + + + + + + + + + + + + + +max +{Uk},W,{Φg},{Θg} +� +min +∀k SCNRk (W, ΦT, Uk) +� +(15a) +s.t. +(12b), (12c), (12d), (12e), +(15b) +ΘH +g Θg = IM, ∀g, +(15c) +Φg = Θg, ∀g. +(15d) +Problem P2 is a typical multi-variable optimization, which +could be solved based on the ADMM framework using block +coordinate descent (BCD) methods. To facilitate ADMM, we +place the equality constraints Φg = Θg, ∀g into the objective +function, and obtain the augmented Lagrangian (AL) as +L ({Uk} , W, {Φg} , {Θg}) = −{min +∀k SCNRk (W, ΦT, Uk)} ++ +G +� +g=1 +ℜ +� +Tr +� +ΛH +g (Φg − Θg) +�� ++ ̺ +2 +G +� +g=1 +∥Φg − Θg∥2 +F , +(16) +where Λg ∈ C2M×M, ∀g are dual variables associated with +Φg = Θg, and ̺ ≥ 0 is the corresponding penalty parameter. +Replacing the original objective function with AL function +(16), we obtain the AL minimization problem as +P2 +AL +� +min +{Uk},W,{Φg},{Θg} L ({Uk} , W, {Φg} , {Θg}) (17a) +s.t. +(12b)-(12e), (15c). +(17b) +Now, the ADMM framework is constructed as follows, where +the superscript of notations refers to the iteration index: +Un+1 +k += arg min +Uk L +� +{Uk} , Wn, +� +Φn +g +� +, +� +Θn +g +�� +(18a) +Wn+1 = arg min +W L +�� +Un+1 +k +� +, W, +� +Φn +g +� +, +� +Θn +g +�� +s.t. (12b), (12c). +(18b) +� +Φn+1 +g +� += arg min +Φg L +�� +Un+1 +k +� +, Wn+1, {Φg} , +� +Θn +g +�� +s.t. (12b), (12d), (12e). +(18c) +� +Θn+1 +g +� += arg min +Θg L +�� +Un+1 +k +� +, Wn+1, +� +Φn+1 +g +� +, {Θg} +� +s.t. (15c), +(18d) +Λn+1 +g += Λn +g + ̺ +� +Φn+1 +g +− Θn+1 +g +� +. +(18e) +Variables (18a) to (18e) are successively updated by solving +corresponding sub-problems until some stopping conditions +are reached. In the following subsection3, we will elaborate +on the solutions to sub-problems (18a) to (18d). +C. Solution to Sub-problems +1) Sub-problem w.r.t Uk: Given other variables, the opti- +mization problem for updating Uk can be expressed as +P2 +AL,{Uk} +� +max +{Uk} +� +min +∀k +uH +k ΨT,kuk +uH +k (ΨC,k + σ2 +RINRL) uk +� +. +(19) +It can be observed that P2 +AL,{Uk} is an unconstrained optimiza- +tion problem and has K separable objective functions, each of +which has the following form +max +Uk +uH +k ΨT,kuk +uH +k (ΨC,k + σ2 +RINRL) uk +, ∀k. +(20) +Problem (20) is a classical generalized fractional quadratic +optimization problem, whose optimal solution can be obtained +by taking the generalized eigenvalue decomposition as [39] +uk = EIG +�� +ΨCN,k + σ2 +RINRL +�−1 × ΨT,k +� +, ∀k. +(21) +where EIG (·) represents the eigenvector operator. +2) Sub-problem w.r.t W: Given other variables, the opti- +mization problem for updating W can be expressed as +P2 +AL,W + + + + + + + + + + + + + + + + + + + +max +W +� +min +∀k +wHΥT,kw +wHΥC,kw + σ2 +R ∥Uk∥2 +F +� +(22a) +s.t. +���ℑ +� +˜hH +u w [l] +���� +ℜ +� +˜hH +u w [l] +� +− +� +σ2 +C,uΓu,l +≤ tan Ω, (22b) +∥W∥2 +F = E. +(22c) +Problem P2 +AL,W is hard to settle due to the non-smooth +objective function and complicated non-convex constraints. +To simplify the design, we first equivalently transform the +3When introducing solutions to sub-problems, we omit the superscript of +notations for conciseness unless otherwise stated. + +7 +objective into a smooth form by introducing an auxiliary +variable γ, which yields the following problem +P2−1 +AL,W + + + + + + + + + + + + + + + +max +W,γ +γ +(23a) +s.t. min +∀k +wHΥT,kw +wHΥC,kw + σ2 +R ∥Uk∥2 +F +≥ γ, +(23b) +γ ≥ 0, +(23c) +(22b), (22c). +(23d) +Then, we deal with constraints (23b), (22b), and (22c) step- +by-step detailed as follows. +Step 1: Majorization minimization (MM) to (23b). We first +rewrite constraint (23b) as +wHΥC,kw − wHΥT,kw +γ ++ σ2 +R ∥Uk∥2 +F ≤ 0, ∀k, +(24) +where the second term is a composite function with both +w and γ. To simplify the joint design of problem (27), we +perform MM and propose the following lemma. +Lemma 1. Assume Υ is positive definite and γ > 0. A +majorizer of f (w, γ) = wHΥw +γ +is +f (w, γ; wn, γn) = 2ℜ +� +(wn)HΥw +� +γn +− γ (wn)HΥwn +(γn)2 +. +Proof: Please refer to Appendix A. +Using Lemma 1, we conduct the majorization on constraint +(24) at point (wn, γn), yielding +wHΥC,kw − 2ℜ +� +(wn)HΥT,kw +� +γn ++ γ (wn)HΥT,kwn +(γn)2 ++ σ2 +R ∥Uk∥2 +F ≤ 0, ∀k, +(25) +where γn is computed by +γn = min +∀k +(wn)HΥT,kwn +(wn)HΥC,kwn + σ2 +R ∥Uk∥2 +F +. +(26) +Step 2: Reformulation to (22b). After some algebraic ma- +nipulations, we rewrite (22b) as +(22b) ⇔ + + + +ℜ +�¯hH +u,1 (ΦR) w [l] +� +≥ +� +σ2 +C,uΓu,l sin Ω, +(27a) +ℜ +�¯hH +u,2 (ΦR) w [l] +� +≥ +� +σ2 +C,uΓu,l sin Ω, +(27b) +where ¯hu,1 (ΦR) = GHΦH +R hu(sin Ω + e π +2 cos Ω)e−∠(su[l]) +and ¯hu,2 (ΦR) = GHΦH +R hu(sin Ω − e π +2 cos Ω)e−∠(su[l]). +Step 3: Simplification to (22c). We first scale the equality +constraint (22c) as E−ǫ ≤ ∥W∥2 +F ≤ E+ǫ, where ǫ ≥ 0 is an +auxiliary variable whose value approaches to zero. It is easy +to notice that the right-hand side ∥W∥2 +F ≤ E + ǫ is convex, +while the left-hand side E − ǫ ≤ ∥W∥2 +F is non-convex. To +convexify the non-convex part, we perform MM and transform +(22c) as two convex constraints +� +∥W∥2 +F − E − ǫ ≤ 0, +(28a) +2ℜ {Tr (WnW)} − ∥Wn∥2 +F − E + ǫ ≥ 0. +(28b) +Replacing non-convex contraints in (23) with (25), (27) and +(28) based on Steps 1-3, and penalizing the slack variable ǫ +into the objective function, we minimize the following problem +P2−2 +AL,W + + + + + +min +W,γ,ǫ −γ + κǫ +(29a) +s.t. γ ≥ 0, ǫ ≥ 0, +(29b) +(25), (27a), (27b), (28a), (28b) , +(29c) +where κ ≥ 0 represents the penalty parameter to scale the +impact of the penalty term. Problem P2−2 +AL,W is a convex +second-order cone programming (SOCP) problem and can be +globally solved by the interior point method (IPM). +3) Sub-problem w.r.t {Φg}: Given other variables, the sub- +problem for updating {Φg} is +P2 +AL,{Φg} + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +min +Φg − +� +min +∀k +φH +T ΞT,kφT +φH +T ΞC,kφT + σ2 +R ∥Uk∥2 +F +� ++ +G +� +g=1 +ℜ +� +Tr +� +ΛH +g (Φg − Θg) +�� ++̺ +2 +G +� +g=1 +∥Φg − Θg∥2 +F +(30a) +s.t. +���ℑ +� +˜hH +u w [l] +���� +ℜ +� +˜hH +u w [l] +� +− +� +σ2 +C,uΓu,l +≤ tan Ω, (30b) +ΦT = BlkDiag (ΦT,1, · · · , ΦT,G) , +(30c) +ΦR = BlkDiag (ΦR,1, · · · , ΦR,G) , +(30d) +where ΦR and ΦT are separable in both objective and con- +straints, and thus can be designed in parallel as follows. +Solution to ΦR: The problem regarding ΦR is +P2 +AL,ΦR + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +min +ΦR +G +� +g=1 +ℜ +� +Tr +� +ΛH +R,g (ΦR,g − ΘR,g) +�� ++̺ +2 +G +� +g=1 +∥ΦR,g − ΘR,g∥2 +F +(31a) +s.t. +��ℑ +� +Tr +� ¯Hu,lΦR +���� +ℜ +� +Tr +� ¯Hu,lΦR +�� +− +� +σ2 +C,uΓu,l +≤ tan Ω, (31b) +ΦR = BlkDiag (ΦR,1, · · · , ΦR,G) , +(31c) +where ΛR,g is extracted from the last M rows of Λg, ¯Hu,l = +e∠(su[l])Gw [l] hH +u . The difficulty of solving problem (31) +comes from constraints (31b) and (31c), which can be tackled +based on the following matrix arrangements. Specifically, we +partition ¯Hu,l as +¯Hu,l = + + +¯H11 +u,l +· · · +¯H1G +u,l +... +... +... +¯HG1 +u,l +· · · +¯HGG +u,l + + , ∀u, l, +(32) +where ¯Hij +u,l ∈ CM×M. By defining ˜Hu,l = [ ¯H11 +u,l, · · · , ¯HGG +u,l ], +and re-arranging (31c) as ˜ΦR = [ΦR,1, · · · , ΦR,G], constraints +(31b) and (31c) are merged into the following constraint +���ℑ +� +Tr +� +˜Hu,l ˜ΦR +����� +ℜ +� +Tr +� +˜Hu,l ˜ΦR +�� +− +� +σ2 +C,uΓu,l +≤ tan Ω, ∀u, l, +(33a) + +8 +⇔ + + + +ℜ +� +Tr +� +ˆHu,l,1 ˜ΦR +�� +≥ +� +σ2 +C,uΓu,l sin Ω, +ℜ +� +Tr +� +ˆHu,l,2 ˜ΦR +�� +≥ +� +σ2 +C,uΓu,l sin Ω, +∀u, l, (33b) +where ˆHu,l,1 = +˜Hu,l +� +sin Ω + e− π +2 cos Ω +� +and ˆHu,l,2 = +˜Hu,l +� +sin Ω − e− π +2 cos Ω +� +. This brings the following opti- +mization problem +P2−1 +AL,˜ΦR + + + + + + + + + +min +˜ΦR +ℜ +� +Tr +� +˜ΛH +R +� +˜ΦR − ˜ΘR +��� ++̺ +2∥˜ΦR − ˜ΘR∥2 +F +(34a) +s.t. +(33b), +(34b) +where +˜ΘR += +[ΘR,1, · · · , ΘR,G] +and +˜ΛR += +[ΛH +R,1, · · · , ΛH +R,G]H. Problem P2−1 +AL,˜ΦR is a quadratic program +(QP) with linear constraints and can be efficiently tackled +via many existing optimization tools, such as the active set +method and the primal-dual subgradient method [43]. +Solution to ΦT: The problem regarding ΦT is +P2 +AL,ΦT + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +min +ΦT − +� +min +∀k +φH +T ΞT,kφT +φH +T ΞC,kφT + σ2 +R ∥Uk∥2 +F +� ++ +G +� +g=1 +ℜ +� +Tr +� +ΛH +T,g (ΦT,g − ΘT,g) +�� ++̺ +2 +G +� +g=1 +∥ΦT,g − ΘT,g∥2 +F +(35a) +s.t. ΦT = BlkDiag (ΦT,1, · · · , ΦT,G) , +(35b) +Similarly, we re-organize P2 +AL,ΦT into a concise form as +P2−1 +AL,˜ΦT + + + + + + + + + + + + + + + + + + + +min +˜ΦT,η +−η + ℜ +� +Tr +� +˜ΛH +T +� +˜ΦT − ˜ΘT +��� ++̺ +2 +���˜ΦT − ˜ΘT +��� +2 +F +(36a) +s.t. min +∀k +˜φH +T ˜ΞT,k ˜φT +˜φH +T ˜ΞC,k ˜φT + σ2 +R ∥Uk∥2 +F +≥ η, +(36b) +η ≥ 0, +(36c) +where ˜ΦT = [ΦT,1, · · · , ΦT,G], ˜ΘT = [ΘT,1, · · · , ΘT,G], +˜ΛT = [ΛH +T,1, · · · , ΛH +T,G]H with ΛT,g extracted from the first +M rows of Λg, and ˜φT = Vec(˜ΦT). ˜ΞT,k = KGΞT,kKH +G +and ˜ΞC,k = KGΞC,kKH +G, where KG = BlkDiag([IM ⊗ +[0M,(g−1)M, IM, 0M,(G−g)M]]G +g=1) ∈ {0, 1}MNS×N 2 +S denotes +the linear mapping matrix. Using Lemma 1 to simplify +constraint (36b), we have +P2−2 +AL,˜ΦT + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +min +˜ΦT,η +−η + ℜ +� +Tr +� +˜ΛH +T +� +˜ΦT − ˜ΘT +��� ++̺ +2 +���˜ΦT − ˜ΘT +��� +2 +F +(37a) +s.t. ˜φH +T ˜ΞC,k ˜φT − +2ℜ +� +( ˜φn +T)H ˜ΞT,k ˜φT +� +ηn ++η +2ℜ +� +( ˜φn +T)H ˜ΞT,k ˜φn +T +� +(ηn)2 ++σ2 +R ∥Uk∥2 +F ≤ 0, ∀k, +(37b) +η ≥ 0. +(37c) +Algorithm 1 Max-Min Fairness for BD-RIS Aided DFRC. +Input: hu, ∀u, G, ̺ and system parameters. +1: Initialize +� +U0 +k +� +, W0, Φ0 +T, and Φ0 +R. +2: Set n = 1. +3: repeat +4: +Calculate radar receive filters {Un +k} by (21) in parallel. +5: +Update transmit waveform Wn by solving (29). +6: +Compute BD-RIS matrix Φn +R by solving (34). +7: +Update BD-RIS matrix Φn +T by solving (37). +8: +Obtain auxiliary variables +� +Θn +g +� +by Theorem 1. +9: +Update dual variables +� +Λn +g +� +by (18e). +10: +n = n + 1. +11: until convergence. +12: Return {Un +k}, Wn, Φn +T and Φn +R. +Output: {U⋆ +k} = {Un +k}, W⋆ = Wn, Φ⋆ +T = Φn +T, Φ⋆ +R = Φn +R. +Problem P2−2 +AL,˜ΦT is a convex SOCP and can be solved by IPM. +4) Sub-problem w.r.t {Θg}: Given the other variables, the +sub-problem for updating {Θg} is +P2 +AL,{Θg} + + + + + + + + + + + + + + + + + + + +min +Φg +G +� +g=1 +ℜ +� +Tr +� +ΛH +g (Φg − Θg) +�� ++̺ +2 +G +� +g=1 +∥Φg − Θg∥2 +F +(38a) +s.t. ΘH +g Θg = IM, ∀g, +(38b) +Problem P2 +AL,{Θg} can be split into G sub-problems, each of +which has the following form +P2−1 +AL,Θg + + + + + + + + + +min +Φg ℜ +� +Tr +� +ΛH +g (Φg − Θg) +�� ++̺ +2 ∥Φg − Θg∥2 +F +(39a) +s.t. ΘH +g Θg = IM. +(39b) +Now, the remaining challenge of solving problem P2 +AL,Θg lies +in the unitary constraint (39b). The unitary constraint (39b) +forms a 2M dimensional complex Stiefel manifold [44], which +can be approximately solved via manifold based algorithms, +e.g., Riemannian conjugate gradient (RCG) and Riemannian +trust regions (RTR). However, the iterative procedure of man- +ifold methods might cause a lot of computational burdens. +To speed-up the design, we provide a closed-form solution of +problem P2 +AL,Θg in the following theorem. +Theorem 1. With the unitary constraint (39b), the optimal +solution for Θg is given by +Θg = Bg [IM×M, 0M×M] DH +g +(40) +where BgΣgDH +g = Λg + ̺Φg is the singular value decom- +position (SVD) of Λg + ̺Φg. +Proof: Please refer to Appendix B. +Based on the above derivations, the procedure of the above +ADMM based algorithm is summarized in Algorithm 1. + +9 +D. Initialization Scheme +Given that the ADMM procedure is usually sensitive to +initial values, we present a 2-step initialization strategy to +accelerate the convergence. +Step1: Since it is not that straightforward to quickly find +proper ΦT and ΦR, we randomly generate ΦT and ΦR, which +satisfy the BD-RIS constraints. +Step2: With initialized ΦR, we obtain the cascaded channel +˜hH +u (ΦR) = hH +u ΦRG for the communication link. To provide +a feasible and “good” initial point satisfying the constraint +(12b), we initialize the transmit waveform W by solving the +following QoS-constrained problem +max +W,Γ +Γ +s.t. +ℜ +�¯hH +u,1 (ΦR) w [l] +� +≥ +� +σ2 +C,uΓ sin Ω, ∀u, l, +ℜ +�¯hH +u,2 (ΦR) w [l] +� +≥ +� +σ2 +C,uΓ sin Ω, ∀u, l, +∥W∥2 +F ≤ E, +(41) +which is a convex problem and can be efficiently solved by +many numerical approaches [43]. +E. Complexity Analysis +We provide a broad complexity analysis for Algorithms 1, +which is summarized as follows +1) Initialization: The main computational complexity of +this stage comes from step 2 by solving the SOCP problem +(41) with IPM, which requires approximately O +� +N 3 +TL3� +. +2) ADMM: This stage includes the iterative design of the +radar receive filters Uk, transmit beamformer W, BD-RIS +coefficients (ΦT, ΦR) and auxiliary variable {Θg}. Updating +radar receive filters Uk requires O +� +KN 3 +R +� +. Solving problem +(29) for updating W with IPM method needs complexity +O +� +N 3 +TL3� +. The complexity of updating BD-RIS coefficients +(ΦT, ΦR) can be upper bounded by O +� +GN 3 +S +� +. Using Theo- +rem 1 to update auxiliary variable {Θg} requires complexity +of O +� +GM 3� +. Therefore, the overall complexity of the ADMM +framework is O(N0(KN 3 +R + N 3 +TL3 + GN 3 +S + GM 3)), where +N0 denotes the maximum number of iterations. +IV. PERFORMANCE EVALUATION +In this section, we provided extensive simulation results to +validate the effectiveness of the proposed algorithm and the +performance of the proposed BD-RIS aided DFRC system. +A. System Setup +We assume that the DFBS equipped with NT = 8 antennas +transmits QPSK symbols (M = 4) to U = 4 downlink users +and detects K = 3 targets with the assistance of a BD-RIS +having NS = 16 cells. The radar sensing receiver colocated +with the BD-RIS has NR = 8 receive elements. The code +length is L = 16 and the power budget at the DFBS is +set as E = 10 W. The noise power at the users and radar +sensing receiver are set as σ2 +C,u = σ2 +R = −100 dBm, ∀u. +The communication QoS threshold is set the same for all +users, i.e., Γu,l = Γ, ∀u, l. In addition, the distance-dependent +TABLE I +INFORMATION OF K TARGETS. +Target Index +Range (m) +Azimuth (◦) +RCS (dB) +Target 1 +10 +30 +5 +Target 2 +14 +0 +8 +Target 3 +19 +-20 +10 +TABLE II +INFORMATION OF Q CLUTTERS. +No. of clutters +Range (m) +Azimuth (◦) +RCS (dB) +5 +15 +[20:2:28] +25 +4 +20 +[-3:2:3] +25 +9 +[6:1:14] +10 +25 +5 +[16:1:20] +-30 +25 +path loss is modeled as η (d) = ℵ (d/d0)−ℓ, where ℵ = +−30 dB denotes the signal attenuation at the reference distance +d0 = 1 m, and ℓ represents the path loss exponent. We +set the path loss exponents for the DFBS→BD-RIS, BD- +RIS→user, BD-RIS→target, and BD-RIS→clutter as 2.2, 2.2, +2, and 2, respectively. The DFBS and BD-RIS are located as +(−20 m, 0 m) and (0 m, 0 m), respectively, which results in +the distance between DFBS and BD-RIS as dBR = 20 m. +The U users are randomly located at reflective side with +the same distance dRU = 16 m. The DFBS→BD-RIS and +BD-RIS→user channels are assumed to follow the Rician +fading model with the Rician factor being 3 dB. For the radar +function, we assume K = 3 targets and 4 groups (Q = 23) +of strong clutters are located in the transmissive side, whose +detailed information is presented in Tables I and II. Moreover, +we assume the range resolution as ∆d = 1 m, which indicates +the radar sampling rate fs = 150 MHz. Combining Table I and +the path loss model, the ratio of the propagation coefficients +of the three radar targets is ζ2 +1 : ζ2 +2 : ζ2 +3 ≈ 3.2 : 1.6 : 0.7 +[17]–[19], [21], indicating that target 3 is the weakest target. +B. Benchmark Schemes +For comparison, we consider the following two benchmark +schemes in the simulations. +1) Benchmark 1: The radar-only case is selected as the up- +per bound of the radar performance. We obtain this benchmark +by changing the BD-RIS into transmissive mode and removing +the downlink users, where the resultant problem can be tackled +by modifying the proposed algorithm. +2) Benchmark 2: We consider a doulbe-RIS case where +one diagonal RIS working on the reflective mode while +another +working +on +the +transmissive +mode +are +adja- +cently placed to achieve full-space coverage [30]. This +baseline +is +a +special +case +of +BD-RIS +with +CW-SC +where ΦT += Diag([φT,1, · · · , φT, NS +2 ], 01× NS +2 ) and ΦR += +Diag(01× NS +2 , [φR,1, · · · , φR, NS +2 ]). Therefore, we can obtain this +benchmark by modifying the proposed algorithm. +C. Simulation Results +1) Convergence Performance: In Fig. 4, we investigate +the convergence of the proposed Algorithm 1 for different + +10 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +Number of Iteration +0 +5 +10 +15 +20 +Radar Output SCNR (dB) +CW-FC, Target 1 +CW-FC, Target 2 +CW-FC, Target 3 +CW-GC, Target 1 +CW-GC, Target 2 +CW-GC, Target 3 +CW-SC, Target 1 +CW-SC, Target 2 +CW-SC, Target 3 +(a) +0 +50 +100 +150 +200 +250 +300 +Number of Iteration +-5 +0 +5 +10 +15 +20 +Radar Output SCNR (dB) +CW-FC, Target 1 +CW-FC, Target 2 +CW-FC, Target 3 +CW-GC, Target 1 +CW-GC, Target 2 +CW-GC, Target 3 +CW-SC, Target 1 +CW-SC, Target 2 +CW-SC, Target 3 +(b) +Fig. 4. Radar output SCNR versus the number of iterations. (a) communica- +tion threshold Γ = 0 dB, (b) communication threshold Γ = 15 dB. +BD-RIS architectures. It can be observed that the proposed +algorithm quickly converges to a stationary point. Specifically, +after several iterations, all targets have nearly the same SCNR +value, demonstrating that our algorithm can achieve fairness +for multiple targets. Moreover, the CW-FC architecture enjoys +faster convergence than other architectures under the same +communication threshold. At the same time, the CW-SC re- +quires nearly twice as many iterations of CW-FC to converge. +For the same architecture, the proposed algorithm with a large +communication threshold Γ needs more iterations to converge. +This is due to the fact that if the intended communication +threshold Γ is higher, fewer degrees of freedom (DoFs) in the +optimization problem can be used. +2) System Performance with Varying Parameters: In Fig. +5, we study the minimum radar output SCNR versus the +communication threshold Γ for different architectures. As ex- +pected, the radar output SCNR monotonically decreases with +Γ. This is because when the intended Γ is higher, less resource +can be used to maximize the radar SCNR, which indicates +that there is a trade-off between communication QoS and +radar output SCNR. Meanwhile, the proposed algorithm with +different architectures outperform the conventional RIS, which +validates the advantage of deploying BD-RIS. In addition, the +output SCNR gap between CW-FC/GC and CW-SC becomes +large with increasing communication QoS requirement, which +indicates that the advantage of CW-FC/GC BD-RIS is more +prominent in high communication QoS requirement scenarios. +Fig. 6 displays the minimum radar output SCNR as a +0 +5 +10 +15 +20 +Communication QoS Threshold (dB) +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +Minimum Radar Output SCNR (dB) +Radar Only, CW-FC +BD-RIS, CW-FC +BD-RIS, CW-GC +BD-RIS, CW-SC +Double-RIS, CW-SC +Fig. 5. Minimum radar output SCNR versus the communication threshold Γ +for different architecture. +10 +20 +30 +40 +50 +Transmit Power (W) +5 +10 +15 +20 +25 +Minimum Radar Output SCNR (dB) +Radar Only, CW-FC +BD-RIS, CW-FC +BD-RIS, CW-GC +BD-RIS, CW-SC +Double-RIS, CW-SC +Fig. 6. +Minimum radar output SCNR versus the transmit power E wit +communication threshold Γ = 15 dB for different architectures. +function of transmit power E under different architectures. +It can be observed that the output SCNR for all schemes +grows with the increase of transmit power E. Meanwhile, +the growth of SCNR becomes slow when the transmit power +is substantially large for all considered architectures. This +is because we can improve transmit power to boost system +performance to some degree, but excessive power will not +improve performance further. Moreover, the slope variation of +the BD-RIS scheme with CW-FC/GC/SC architectures is more +significant than its competitors, indicating that CW-FC/GC/SC +architectures are more sensitive to power budget. +In Fig. 7, we present the minimum radar SCNR versus +the number of groups G with different numbers of BD-RIS +cells. We observe that with the same number of groups, the +radar output SCNR increases with the increasing number of +BD-RIS cells. The performance enhancement comes from +the additional DoF of passive beamforming induced by the +increasing number of cells, and the joint design of transmit +waveform, the BD-RIS with more general constraints, and the + +11 +1 +2 +4 +8 +12 +16 +20 24 +32 +40 +Number of groups, G +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +Minimum Radar Output SCNR (dB) +CW-SC +CW-FC +Fig. 7. Minimum radar output SCNR versus the number of groups G with +different BD-RIS cells NS and communication threshold Γ = 15 dB. +-80 +-60 +-40 +-20 +0 +20 +40 +60 +80 +Angle (Degree) +-40 +-35 +-30 +-25 +-20 +-15 +-10 +-5 +0 +Normalized Transmit Beampattern (dB) +Target 1 +Target 2 +Target 3 +Radar Only, CW-FC +BD-RIS, CW-FC +BD-RIS, CW-GC +BD-RIS, CW-SC +(a) +-80 +-60 +-40 +-20 +0 +20 +40 +60 +80 +Angle (Degree) +-40 +-35 +-30 +-25 +-20 +-15 +-10 +-5 +0 +Normalized Transmit Beampattern (dB) +Target 1 +Target 2 +Target 3 +Radar Only, CW-FC +BD-RIS, CW-FC +BD-RIS, CW-GC +BD-RIS, CW-SC +(b) +Fig. 8. +Transmit beampattern of BD-RIS obtained via proposed algorithm +for different architectures. (a) communication threshold Γ = 0 dB, (b) +communication threshold Γ = 15 dB. +matched filters, which also confirms the results in [33]. More +importantly, the slope of each carve becomes steeper with the +increasing number of groups, which indicates that the number +of non-zero elements of BD-RIS matrices plays a significant +role in increasing system performance. +3) Radar Performance: In Fig. 8, we present the transmit +beampattern obtained by the proposed algorithm. Results show +that regardless of BD-RIS architectures, the transmit power +(a) Radar-only, CW-FC +(b) BD-RIS, CW-FD +(c) BD-RIS, CW-GD +(d) BD-RIS, CW-SD +Fig. 9. The space-range beampattern behavors of the receive weights for the +target 3 detection with communication threshold Γ = 10 dB. +mainly concentrates around the three targets, which guarantees +a high SCNR output at target directions. Moreover, the BD- +RIS with CW-FC/GC architectures can focus more energy +toward targets and has a lower sidelobe than that with CW-SC +architecture, thanks to the more flexible passive beamfomring +control provided by the CW-FC/GC architectures. We also +observe that the transmit power towards target 3 is much high +than other targets. This is because, as mentioned early, target +3 is the weakest one, which needs more energy to improve +the output radar SCNR. In addition, the transmit beampattern +performance for BD-RIS with all architectures gets worse +with larger communication QoS thresholds, which confirms +the conclusion in Fig. 5. +Fig. 9 shows the space-range beampattern of the designed +waveform when BD-RIS has different architectures, where the +beampattern of the k-th target is computed as P k +R (θ, l) = +|Tr{(U⋆ +k)H A (θ) ΦTGW⋆Jrl}|2 [39]–[41]. Without loss of +generality, we take target 3 (k = 3) as an example to illustrate +the space-range behavior of the designed waveform. Results +show that the space-range beampattern can form a mainlobe +at the location of the target k = 3 (green circle), but achieve +null points at the locations of the other non-of-interest targets +(red circles) and strong clutter sources (black rectangles) for +all proposed architectures. This phenomenon can be explained +as follows: i) To detect target k, the other targets are regarded +as interference. ii) BD-RIS with more general architectures +can provide more DoFs to resist strong clutters. +V. CONCLUSION +This paper considers the use of BD-RIS in the DFRC system +in the presence of multiple targets and strong clutters. We +start by reviewing the BD-RIS architectures, and deriving +the communication and radar models. Our objective is to +maximize the minimum radar output SCNR subject to the +constraints of communication QoS, BD-RIS coefficients, and +power budget. Then, a general algorithm utilizing the ADMM + +-30 +-40 +-50 +-600.8 +ncy (sino) +0.6 +0.4-70 +-80 +-90 +-100 +-110 +120 +5 +30Normalized Spatial freque +0.2 +0 +0.2 +0.4 +-0.6 +-0.8 +1 +5 +10 +15 +20 +Range (m)-30 +-40 +-50 +-600.8 +ncy (sino) +0.6 +0.4-70 +-80 +-90 +-100 +-110 +120 +5 +30Normalized Spatial freque +0.2 +0 +-0.2 +0.4 +0.6 +-0.8 +-1 +5 +10 +15 +20 +Range (m)-30 +-40 +-50 +-600.8 +ncy(sino) +0.6 +0.4-70 +-80 +-90 +-100 +-110 +120 +5 +30Normalized Spatial freque +0.2 +0 +-0.2 +0.4 +0.6 +-0.8 +-1 +5 +10 +15 +20 +Range (m)-30 +-40 +-50 +-600.8 +ncy (sino) +0.6 +0.4-70 +dB +-80 +-90 +-100 +-110 +-120 +5 +30Normalized Spatial freque +0.2 +0 +0.2 +0.4 +0.6 +-0.8 +-1 +5 +10 +15 +20 +Range (m)12 +approach is developed to solve the resulting complicated non- +convex max-min optimization problem. Finally, simulation +results demonstrate the effectiveness of the proposed design +algorithm, and the superiority of employing the BD-RIS in +DFRC systems in terms of enhancing both communication and +radar performance. Based on this initial work, there are many +issues worth studying for future research on BD-RIS aided +DFRC, such as wideband waveform design, the scenarios for +target estimation, as well as exploring the application of multi- +sector BD-RIS in DFRC systems. +APPENDIX A +PROOF OF LEMMA 1 +Given that f (w, γ) = wHΥw +γ +is jointly concave in w and +γ when Υ ⪰ 0 and γ ≥ 0 [43], the first order approximation +of f (w, γ), denoted by f (w, γ; wn, γn), is a majorizer of +f (w, γ) at the point (wn, γn), which is +f (w, γ; wn, γn) += f (wn, γn) + ( ∂f +∂w|w=wn)T (w − wn) +(42a) ++ ( ∂f +∂w∗ |w=(wn)∗)T (w − (wn)∗) ++ (∂f +∂γ |γ=γn)T (γ − γn) + ( ∂f +∂γ∗ |γ=(γn)∗)T (γ − (γn)∗) += (wn)HΥwn +γn ++ 2ℜ + + + +� +2Υwn +γn +(wn)HΥwn +(γn)2 +�H � +w − wn +γ − γn +� + + += 2ℜ +� +(wn)HAw +� +γn +− γ (wn)HΥwn +(γn)2 +. +The proof is thereby completed. +APPENDIX B +PROOF OF THEOREM 1 +We start by rewriting objective (39a) as [43] +ℜ +� +Tr +� +ΛH +g (Φg − Θg) +�� ++ ̺ +2 ∥Φg − Θg∥2 +F += −ℜ +� +Tr +� +ΘH +g (Λg + ̺Φg) +�� ++ ̺ +2 ∥Φg∥2 +F + ̺M +� +�� +� +constant +. +Then, problem P2−1 +AL,Θg can be symplified as +max +Φg +ℜ +� +Tr +� +ΘH +g (Λg + ̺Φg) +�� +s.t. 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Sepulchre, “Optimization algorithms +on matrix manifolds,” in Optimization Algorithms on Matrix Manifolds. +Princeton University Press, 2009. + diff --git a/2dE1T4oBgHgl3EQflgSY/content/tmp_files/load_file.txt b/2dE1T4oBgHgl3EQflgSY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0eee2437911b86ce924f6bff2d8ff6371c427454 --- /dev/null +++ b/2dE1T4oBgHgl3EQflgSY/content/tmp_files/load_file.txt @@ -0,0 +1,1103 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf,len=1102 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='03286v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='SP] 9 Jan 2023 1 A Dual-Function Radar-Communication System Empowered by Beyond Diagonal Reconfigurable Intelligent Surface Bowen Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Student Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Hongyu Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Student Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Ziyang Cheng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Shanpu Shen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' and Bruno Clerckx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Fellow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' IEEE Abstract—This work focuses on the use of reconfigurable intelligent surface (RIS) in dual-function radar-communication (DFRC) systems to improve communication capacity and sensing precision,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' and enhance coverage for both functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In contrast to most of the existing RIS aided DFRC works where the RIS is modeled as a diagonal phase shift matrix and can only reflect signals to half space, we propose a novel beyond diagonal RIS (BD-RIS) aided DFRC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Specifically, the proposed BD-RIS supports the hybrid reflecting and transmitting mode, and is com- patible with flexible single/group/fully-connected architectures, enabling the system to realize full-space coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To achieve the expected benefits, we jointly optimize the transmit waveform, the BD-RIS coefficients, and sensing receive filters, by maximizing the minimum signal-to-clutter-plus-noise ratio for fair target detection, subject to the constraints of the communication quality of service, different BD-RIS architectures and power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To solve the non-convex and non-smooth max-min problem, a general solution based on the alternating direction method of multipliers is provided for all considered BD-RIS architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Numerical simulations validate the efficacy of the proposed algorithm and show the superiority of the BD-RIS aided DFRC system in terms of both communication and sensing compared to conventional RIS aided DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Index Terms—Beyond diagonal reconfigurable intelligent sur- faces, dual-function radar-communication, full-space coverage, max-min optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' INTRODUCTION In recent years, spectrum resources are becoming increas- ingly limited and valuable due to the exponential growth of services in wireless communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Meanwhile, radar sys- tems are competing for the same scarce sources, which moti- vates the emergence of the dual-function radar-communication (DFRC) technology to achieve spectrum sharing between communication and radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In DFRC systems, communication and radar functionalities are integrated on a common platform, which brings the benefit of enhanced spectrum efficiency while (Corresponding author: Ziyang Cheng, Shanpu Shen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Wang and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Cheng are with the School of Information & Communica- tion Engineering, University of Electronic Science and Technology of China, Chengdu, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (email: B W Wang@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='com, zycheng@uestc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Li is with the Department of Electrical & Electronic Engineering, Impe- rial College London, London SW7 2AZ, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (email: c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='li21@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Shen is with the Department of Electronic and Computer Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong (email: sshenaa@connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='hk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Clerckx is with the Department of Electrical & Electronic Engineering, Imperial College London, London, SW7 2AZ, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' and with Silicon Austria Labs (SAL), Graz A-8010, Austria (email: b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='clerckx@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' reducing power consumption and hardware costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Therefore, DFRC is envisioned to play an important role in emerging environment-aware applications [1], such as vehicular net- works, environmental monitoring, and smart houses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Due to the benefits of DFRC, plenty of technical efforts have been devoted to designing DFRC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The design method- ology can be roughly divided into three categories: radar- centric design [2]–[4], communication-centric design [5]–[7], and joint waveform design [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Radar-centric approaches utilize the radar waveform as the information carrier, where the communication symbols are embedded in conventional radar signals, such as linear frequency modulation [2] and frequency hopping [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' On the other hand, communication- centric approaches realize the radar sensing tasks by modifying existing communication protocols [5] and waveforms [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In contrast to the first two categories [2]–[7], the DFRC waveforms can be jointly designed to provide more design freedoms so as to enhance both functionalities [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Despite the above works [2]–[9] achieve satisfactory sensing and communication performance, one limitation is that they rely on the line-of-sight (LoS) links between the base station (BS) and communication users/sensing targets, which however yields the following two issues in practice: 1) The LoS link toward sensing targets or communication users can be easily blocked by obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2) The LoS channels may suffer from severe path loss especially for high frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To overcome these issues, a promising technology named reconfigurable intelligent surface (RIS) [10]–[13] can be lever- aged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Specifically, RIS consists of numerous passive reconfig- urable scattering elements with low hardware cost and power consumption [10]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' By properly placing and adjusting the RIS, it can establish virtual non-LOS (NLoS) links to “bypass” obstacles, and therefore compensate for the path loss and enhance system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Due to its advantages, RIS has been investigated for communications [14]–[16] and sensing [17]–[20] fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Furthermore, RIS has been explored in various DFRC systems [21]–[27] to enhance both the communication and sensing performance, which are classified into the following two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The first category assumes LoS links exist from BS to users and targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In this category, the RIS is used to compensate for the propagation loss and to improve the performance [21]–[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The second category focuses on the scenario where either communication users or sensing targets are blocked by barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In this category, RIS 2 is utilized to establish a NLoS link to bypass the barriers and thus enable DFRC [24]–[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The limitation of the aforementioned works [21]–[27] is that they assume the RIS can only reflect signals towards the same side as the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In this case, both communication users and sensing targets should be located at the same side of RIS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=', within the same half-space, which limits the coverage and beam control flexibility of the RIS enabled DFRC sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To address this limitation, a novel hybrid transmissive and reflective RIS, namely simultaneously transmitting and reflecting RIS (STAR-RIS) [28] or intelligent omni-surface [29], is proposed to support signal reflection and transmission and thus extend the coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The integration of STAR- RIS and DFRC is first studied in [30], where the system is designed by minimizing the Cram´er-Rao bound (CRB) for radar target estimation subject to communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Then, a STAR-RIS is deployed at the vehicle to improve both sensing and communication performance [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Nevertheless, the achievable performance of STAR-RIS aided DFRC in [30], [31] is limited by the simple architecture of STAR-RIS without fully exploiting the architecture of RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To enhance the performance of RIS, a novel branch, namely beyond diagonal RIS (BD-RIS) [32]–[35], is proposed by exploring different architectures/modes of RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' BD-RIS with group/fully-connected architectures under the reflective mode is first proposed in [32], which provides more controllable scattering matrices than conventional RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Then, the hybrid reflective and transmissive BD-RIS is proposed in [33] to achieve full-space coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' It is proved that STAR-RIS is essentially a particular instance of two-port group-connected reconfigurable impedance network when each two antenna ports are connected to each other, namely cell-wise single- connected (CW-SC) architecture in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' More general cell- wise group/fully-connected (CW-GC/FC) architectures are also proposed based on the flexible connections among more antenna ports, which achieves better performance than STAR- RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Furthermore, a multi-sector BD-RIS is proposed in [35], which not only achieves full-space coverage but also provides higher performance gain than hybrid BD-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Due to the benefits of BD-RIS, in this paper, we propose to adopt BD-RIS in DFRC systems to achieve full-space cov- erage and better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To the best of our knowledge, adopting BD-RIS in DFRC has not been investigated in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In addition, in contrast to [30], [31] which ignore the signal-dependent clutters, we consider a more general and practical multi-target detection scenario with the presence of multiple clutters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The main contributions of this work are summarized as follows: Proposing BD-RIS aided DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We propose a BD-RIS aided DFRC system, which consists of a BD-RIS enabling the full-space coverage, multiple users, and multiple sensing targets corrupted by multiple clutters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The BD-RIS divides the space into two sides and establishes virtual NLoS links for communication and sensing, where the dual-function BS (DFBS) performs communication tasks in one half space and sensing tasks in another side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To avoid multi-step path loss, we implement the radar sensing receiver on the BD-RIS for multi-target detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Formulating Max-min fairness problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We formulate the optimization problem to jointly design the transmit wave- form at the DFBS, the reflective and transmissive beamforming at the BD-RIS, and matched filters at the radar sensing receiver, to maximize the minimum radar output signal-to- clutter-plus-noise ratio (SCNR), subject to the communication quality of service (QoS) requirement for downlink communi- cations, the transmit power constraint at the DFBS, and the BD-RIS constraints with different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Developing joint design framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The joint design of BD-RIS aided DFRC is challenging due to the complicated and non-smooth objective, and newly introduced non-convex constraints of BD-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To overcome these difficulties, we propose to decouple the BD-RIS constraints by the alternat- ing direction method of multipliers (ADMM) framework so that the resulting sub-problems are reformulated into easily handled forms and iteratively solved until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Providing insights and numerical validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We provide simulation results to illustrate the performance improvement achieved by BD-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' It is shown that benefiting from the high flexibility of BD-RIS, and the joint design of transmit waveform, BD-RIS, and the matched filters, the CW-GC/FC BD-RISs can achieve higher radar SCNR than CW-SC (STAR- RIS) ones under the same communication requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' It is also shown the BD-RIS can substantially improve the performance and coverage compared to the conventional RIS, which shows the high flexibility of BD-RIS in manipulating the incident signal for enhancing the DFRC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Organization: Section II presents the system model of the proposed BD-RIS aided DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Section III formulates the max-min fairness problem and provides a joint design algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Section IV evaluates the performance of the pro- posed algorithm and compares different BD-RIS architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Section V concludes this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Notation: Scalars, vectors and matrices are denoted by stan- dard lowercase letter a, lower case boldface letter a and upper case boldface letter A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Cn and Cm×n denote the n-dimensional complex-valued vector space and m × n complex-valued matrix space, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (·)T , (·)H, and (·)−1 denote the transpose, conjugate-transpose operations, and inversion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ℜ{·} and ℑ{·} denote the real and imaginary part of a complex number, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ∥ · ∥F and | · | denote the Frobenius norm and magnitude, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Diag(·) denotes a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' BlkDiag(·) denotes a block matrix such that the main-diagonal blocks are matrices and all off-diagonal blocks are zero matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' IL indicates an L × L identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' \uf6be denotes imaginary unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ∠(·) represent the phase values of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Tr(·) denotes the summation of diagonal elements of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ⌊·⌋ is the round-down operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' SYSTEM MODEL As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 1, we consider a DFRC system, where an NT-antenna DFBS simultaneously sends communication symbols to U single-antenna users and detects K targets in the presence of Q strong clutters with the assistance of an NS-cell BD-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The BD-RIS adopts the hybrid transmissive and reflective mode, which divides the whole space into two 3 Target Target Clutter Clutter Transmissive Area for Radar Cell 1 BD-RIS Target 1 Target K Clutter 1 Clutter Q Reflective Area for Communication Transmissive Area for Radar User User DFBS Reflective Area for Communication RIS elements Sensor elements User 1 User NU DFBS NT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Illustration of a BD-RIS aided DFRC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' half areas, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=', the transmissive and reflective areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The DFBS provides communication services at the reflective area while performing radar sensing at the transmissive area aided by BD- RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The radar sensing receiver with NR antennas is installed adjacent to the BD-RIS to collect target echos and conduct target detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In the following subsections, we will review the modeling of BD-RIS with different architectures, and establish the communication and radar models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' BD-RIS Architecture Model According to [33], the hybrid reflective and transmissive mode is essentially based on the group-connected reconfig- urable impedance network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Specifically, each two antenna ports are connected to each other, constructing one cell as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Within each cell, two antennas with uni- directional radiation pattern are back to back placed such that each antenna covers half space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Mathematically, the BD-RIS with hybrid reflective and transmissive mode is characterized by two matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=', ΦR ∈ CNS×NS and ΦT ∈ CNS×NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Depending on the inter-cell connection strategies, the BD-RIS can be categorized into the following three architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 1) CW-SC BD-RIS Architecture: As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2(a), we provide a simple example of CW-SC BD-RIS with 2 cells, from which we can observe that different RIS cells are not connected to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Therefore, matrices ΦT, ΦR are all restricted to be diagonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=', ΦT = Diag(φT,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' , φT,NS) and ΦR = Diag(φR,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' , φR,NS), and satisfy |φT,i|2 + |φR,i|2 = 1, ∀i = 1, · · · , NS, (1) which conforms to the STAR-RIS constraints, indicating that the STAR-RIS is a special case of BD-RIS with CW-SC architecture [28], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2) CW-FC BD-RIS Architecture: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2(b) depicts an exam- ple of CW-FC BD-RIS with 2 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In contrast to CW-SC case, all cells of the CW-FC BD-RIS are connected to each other through reconfigurable impedance components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Accordingly, ΦT, ΦR are all full matrices satisfying ΦH T ΦT + ΦH R ΦR = INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (2) 3) CW-GC BD-RIS Architecture: As a balance between the above two extreme cases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' CW-GC divides all cells into several Cell� User� User� BD�RIS DFBS Target� Target� Clutter� Clutter� Reflective�Area�for� Communication Transmissive�Area� for�Radar Antenna�3 Antenna�1 Antenna�4 Z3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 Z3 Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='3 Z1 Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='3 Z2 Z4 Antenna�2 2�Cell�CW�FC�BD�RIS (b) Cell�1 Cell�2 Antenna�4 Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 Z2 Z4 Antenna�2 2�Cell�CW�SC�BD�RIS� (a) Cell�2 Antenna�5 Antenna�1 Antenna�6 Z5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 Z5 Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='5 Z1 Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='5 Z2 Z6 Antenna�2 4�Cell�CW�GC�BD�RIS Antenna�7 Antenna�3 Antenna�8 Z7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 Z7 Z3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='7 Z3 Z4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 Z3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 Z3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 Z4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='7 Z4 Z8 Antenna�4 Group�1 Group�2 (c) Cell�1 Cell�2 Cell�3 Cell�4 Antenna�3 Antenna�1 Z3 Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='3 Z1 Cell�1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Examples of (a) CW-SC BD-RIS, (b) CW-FC BD-RIS, and (c) CW- GC BD-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' groups and cells in each group adopt the the fully-connected architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Depending on the group division strategies, there are plenty of CW-GS architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' For simplicity, here we consider the case where NS cells of the BD-RIS are uniformly divided into G groups and each group has the same size M = NS/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' For ease of understanding, an example of a 4-cell BD- RIS with CW-GC architecture having 2 groups is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Hence, the model for CW-GC BD-RIS can be expressed as ΦT = BlkDiag(ΦT,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' , ΦT,G), ΦR = BlkDiag(ΦR,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' , ΦR,G), ΦH T,gΦT,g + ΦH R,gΦR,g = IM, ∀g = 1, · · · , G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (3) where ΦT,g ∈ CM×M and ΦR,g ∈ CM×M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The CW-GC architecture of BD-RIS is a general case, which becomes the CW-SC architecture (STAR-RIS) with a simple circuit when G = NS, and the CW-FC architecture achieving the best performance as G = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This means that CW-SC and CW-FC architectures are special cases of CW-GC architecture and the beam control flexibility/ability of CW-GC BD-RIS can be improved by decreasing G, but at the expense of increasing circuit complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Communication Model In this paper, we consider a standard multiuser multiple input single output (MISO) downlink scenario, where the DFBS provides communication services to the reflective area aided by the BD-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We assume the direct links between the DFBS and downlink users are blocked and the channel state information (CSI) is available at the DFBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The data symbol vector sl = [sl [1] , · · · , sl [U]]T ∈ CU contains the overall U data symbols in the l-th time slot, which are assumed to 4 be drawn from a standard M order phase-shift keying (M- PSK) modulation constellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Furthermore, the data symbol vector sl is mapped to the transmit waveform w [l] ∈ CNT at the DFBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Accordingly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' the received signal of the u-th user at symbol time t is yu (t) = e\uf6be2πfct L � l=1 hH u ΦRGw [l] rect (t − l∆t) + nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='u (t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (4) where fc is the carrier frequency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' L is the number of time slots during one transmission duration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' G ∈ CNS×NT and hu ∈ CNS stand for the channel coefficients of the communication links DFBS→BD-RIS and BD-RIS→u-th user,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ∆t stands for symbol duration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' rect (t) is the rectangle window function that takes the value 1 for t ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ∆t] and 0 otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' and nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='u (t) is the additive white Gaussian noise (AWGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' By down converting the signal into baseband and sampling received signal yu (t) at the rate fs = 1/∆t within the symbol duration, the discrete baseband signal at the l-th time slot is yu [l] = hH u ΦRGw [l] + nc,u [l] , (5) where nc,u [l] is the AWGN with zero mean and variance σ2 C,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In this work, we adopt the recently emerged symbol level beamforming (SLB) technology for communication in DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Specifically, SLB technology utilizes the constructive inter- ference (CI), which is defined as the multi-user interference (MUI) that pushes the received symbols away from the detec- tion thresholds of the modulation constellation, to enhance the communication QoS while reducing BER [36], [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Here we briefly review the concept of SLB as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 3 takes quadrature-PSK (QPSK) as an example, where point A stands for the desired symbol sl [u] with the required signal-to-noise-ratio (SNR) threshold Γu,l of the u-th user, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=', −→ OA = � σ2 C,uΓu,lsl [u], and point D is the received noise- free signal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=', −→ OD = ˜yu [l] = hH u ΦRGw [u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The CI region refers to a polyhedron bounded by hyperplanes parallel to decision boundaries of the constellation, which is depicted as blue-shaded area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The key of SLB is to enforce the received signal located in the CI region, which means the received signal is pushed away from decision boundaries and the SNR is guaranteed to be no less than the SNR threshold Γu,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To mathematically depict the SLB constraint, we project point D into the direction of −→ OA at point C, and extend −→ CD to intersect with the nearest boundary of CI region at point B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Consequently, one of the criteria that specifies the location of −→ OD in the CI region is |−→ CD| |−→ AC| = ��ℑ � hH u ΦRGw [l] e\uf6be∠(su[l])��� ℜ � hH u ΦRGw [l] e\uf6be∠(su[l])� − � σ2 C,uΓu,l ≤ tan Ω, (6) where Ω = π/M is half of the angular range of the CI resign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In this work, we adopt SLB instead of con- ventional block-level beamforing (BLB) due to the following two reasons: 1) By adopting SLB technology in our con- sidered DFRC system, we directly design transmit waveform W ∈ CNT×L for L time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' However, the BLB in the D B � � � � u l y � � C Received Symbol � � � � 2 , u c u l l y � � � � � � � u l y� CI�Region 2 , c u l � � O � Imag Real A Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Description of the CI region for a QPSK symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' same scenario requires the design of the transmit beamformer Wl ∈ CNT×U, ∀l for all data symbols and time slots due to the linear mapping, which results in an increasing com- putational complexity [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2) BLB regards the MUI as a harmful component and suppresses the MUI to guarantee communication QoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' However, the SLB utilizes the MUI to enhance the communication QoS, which provides additional design flexibility in DFRC [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Radar Model To improve the sensing performance of the BD-RIS aided DFRC system, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 1, we adopt a novel sensor- at-RIS architecture [20], where the radar receiving sensors are installed adjacent to the BD-RIS to collect the echo signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This architecture greatly reduces the multi-step path- loss compared with the sensor-at-DFBS architecture [21]–[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Moreover, we consider a scenario where the radar receiver attempts to detect K targets in the presence of Q strong clutters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Specifically, the k-th target of interest is characterized by angle ϕk and time delay τ k T , respectively, while the q-th clutter is characterized by angle ϑq and delay τq C, respectively1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The backscattered signal at the radar receiver after down conversion is thus [39]–[41] r (t) = K � k=1 L � l=1 αkA (ϕk) ΦTGw [l] rect � t − l∆t − τk T � + Q � q=1 L � l=1 βqA (ϑq) ΦTGw [l] rect (t − l∆t − τq C) + nr (t) , (7) where αk and βq, respectively, denote the propagation co- efficient for the k-th target and q-th clutter consisting of radar cross section (RCS) and channel propagation effects with E(|αk|2) = ζ2 k and E(|βq|2) = ξ2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' A (ϕ) = aR (ϕ) aH T (ϕ) ∈ CNR×NS is the effective radar channel, where aT (ϕ) = 1 √NS [1, · · · , ej 2π λ d(NS−1) sin ϕ]T and aR (ϕ) = 1 √NR [1, · · · , ej 2π λ d(NR−1) sin ϕ]T denote the the transmit and 1In this paper, we assume the targets and clutters are slowly moving or stay still, whose Doppler frequencies equal to zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 5 receive steering vector, respectively, with d and λ being element spacing and wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' nr (t) denotes AWGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' we select the first target echo as the reference and sample the received signal r (t) at fs = 1/∆t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' yielding the following received baseband signal R = K � k=1 αkA (ϕk) ΦTGWJrk T � �� � Target Echos + Q � q=1 βqA (ϑq) ΦTGWJrq C � �� � Clutter Returns + Nr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (8) where Jr = [0L×r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' IL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 0L×(Lobs−L−r)] ∈ CL×Lobs is the shift matrix with Lobs = L + {maxk rk T} − {mink rk T} being the receiver observation length,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' rk T = ⌊(τ k T − {min˜k τ ˜k T })fs⌋ the rang ring of the k-th target,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' and rq C = ⌊(τ q C − {mink τ k T })fs⌋ the rang ring of the q-th clutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Nr = [nr [1] , · · · , nr [L]] ∈ CNR×L is the Gaussian noise matrix with nr [l] ∼ CN � 0, σ2 RINR � , ∀l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' by performing the matched filter Uk ∈ CNR×Lobs to the k-th target at radar receiver,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' the k-th target detection problem can formulated as a binary hypothesis test [39]–[41]: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 Hk 1 : αkUH k A (ϕk) ΦTGWJrk T (9a) + K � p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='p̸=k αpUH k A (ϕp) ΦTGWJrp T + Q � q=1 βqUH k A (ϑq) ΦTGWJrq C + Nr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (9b) Hk 0 : K � p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='p̸=k αpUH k A (ϕp) ΦTGWJrp T (9c) + Q � q=1 βqUH k A (ϑq) ΦTGWJrq C + Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (9d) According to the above binary hypothesis test (9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' the detection probability P k D of the k-th target can be evaluated as [41] P k D = Q �� 2SCNRk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' � −2 ln (Pfa) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (10) where Q (·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ·) is the Marcum Q-function of order 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Pfa is the false alarm probability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' and the radar output SCNR of the k-th target after the matched filtering is given by SCNRk(W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Uk) = ς−1 k |Tr(αkUH k A(ϕk)ΦTGWJrk T )| 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (11) where ςk = �K p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='p̸=k |Tr(αpUH p A (ϕp) ΦTGWJrp T )| 2 + �Q q=1 |Tr(βqUHA (ϑq) ΦTGWJrq C)| 2 + σ2 R ∥Uk∥2 F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' MAX-MIN FAIRNESS FOR BD-RIS AIDED DFRC In this section, we first formulate the joint design problem for BD-RIS aided DFRC, followed by a general algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Finally, we propose an initialization scheme and analyze the computational complexity of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Problem Formulation Given that (10) is strictly increasing in SCNRk, for a specified value of false alarm probability Pfa, improving the detection probability P k D of the k-th target is equivalent to maximize the radar output SCNR of the k-th target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Moreover, for multiple target detection cases, beamforming design usually aims to improve the detection probability for all targets, especially for the weakest targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Therefore, to improve the overall target detection probability and guarantee target detection fairness, we propose to maximize the minimal radar output SCNR among the K targets by jointly designing the transmit beamformer W, the BD-RIS matrices {ΦT, ΦR}, and radar receiver filters {Uk}∀k, subject to communication QoS constraints, transmit power constraint, and BD-RIS con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The joint design problem is thus formulated as2 P1 \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 max W,ΦT,ΦR,{Uk} � min ∀k SCNRk (W, ΦT, Uk) � (12a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ���ℑ � ˜hH u w [l] ���� ℜ � ˜hH u w [l] � − � σ2 C,uΓu,l ≤ tan Ω, (12b) ∥W∥2 F = E, (12c) ΦT = BlkDiag (ΦT,1, · · · , ΦT,G) , (12d) ΦR = BlkDiag (ΦR,1, · · · , ΦR,G) , (12e) ΦH T,gΦT,g + ΦH R,gΦR,g = ING, ∀g, (12f) where ˜hH u = hH u ΦRG is the equivalent channel for DFBS→DB-RIS→ u-th user and E is the transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Problem P1 is a challenging non-convex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Partic- ularly, the non-convexity stems from the complicated frac- tional SCNR expression in the objective and highly coupled optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To simplify the joint design, in the following subsection, we propose a series of transformations and an ADMM based framework to decouple problem (12) into multiple more tractable sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Overview of Proposed Joint Design Framework To facilitate the joint design, we propose to re-arrange the SCNR (11) into explicit and compact forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' By defining uk = Vec(Uk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' w = Vec(W) and φT = Vec (ΦT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' and applying basic vectorization properties [42],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' the SCNR in (11) shares the following three equivalent expressions SCNRk (W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Uk) = uH k ΨT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='kuk uH k (ΨC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='k + σ2 RINRL) uk ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (13a) = wHΥT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='kw wHΥC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='kw + σ2 R ∥Uk∥2 F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (13b) = φH T ΞT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='kφT φH T ΞC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='kφT + σ2 R ∥Uk∥2 F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (13c) where ΨT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='k =ζ2 k � ¯ MT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) � wwH� ¯ MT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) �H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2Based on the discussion in Remark 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' herein we focus on the design when the BD-RIS has CW-GC architecture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' which is a general case including both CW-SC and CW-FC cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 6 ΨC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='k = K � p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='p̸=k ζ2 p � ¯ MT (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) � wwH� ¯ MT (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) �H + Q � q=1 ξ2 q � ¯ MC (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) � wwH� ¯ MC (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) �H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΥT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='k =ζ2 k � ¯ MT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) �HukuH k � ¯ MT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΥC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='k = K � p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='p̸=k ζ2 p � ¯ MT (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) �HukuH k � ¯ MT (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) � + Q � q=1 ξ2 q � ¯ MC (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) �HukuH k � ¯ MC (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΞT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='k =ζ2 k � ˜ MT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' W) �H ukuH k � ˜ MT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' W) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΞC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='k = K � p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='p̸=k ζ2 p � ˜ MT (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' W) �H ukuH k � ˜ MT (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' W) � + Q � q=1 ξ2 q � ˜ MC (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' W) �H ukuH k � ˜ MC (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' W) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' with ¯ MT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) = JT rk T ⊗ (A (ϕk) ΦTG),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ¯ MC(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) = JT rq C ⊗ (A (ϑq) ΦTG),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ˜ MT(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' W) = (JT rk T WT GT ) ⊗ A(ϕk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ˜ MC(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' W) = (JT rq CWT GT ) ⊗ A(ϑk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Based on the above derivations, the objective in problem P1 is more tractable with respect to uk, w, or φT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' However, it is still difficult to find the solution to P1 due to non-convex and coupled constraints (12b), (12c), and (12f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To tackle constraint (12f), we first define Φg = [ΦH T,g, ΦH R,g]H and rewrite (12f) as ΦH g Φg = IM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Then, we introduce auxiliary variables Θg = [ΘH T,g, ΘH R,g]H = Φg and decouple constraint (12f) into two separate constraints by adding the equality, which yields the following problem: P2 \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 max {Uk},W,{Φg},{Θg} � min ∀k SCNRk (W, ΦT, Uk) � (15a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (12b), (12c), (12d), (12e), (15b) ΘH g Θg = IM, ∀g, (15c) Φg = Θg, ∀g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (15d) Problem P2 is a typical multi-variable optimization, which could be solved based on the ADMM framework using block coordinate descent (BCD) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To facilitate ADMM, we place the equality constraints Φg = Θg, ∀g into the objective function, and obtain the augmented Lagrangian (AL) as L ({Uk} , W, {Φg} , {Θg}) = −{min ∀k SCNRk (W, ΦT, Uk)} + G � g=1 ℜ � Tr � ΛH g (Φg − Θg) �� + ̺ 2 G � g=1 ∥Φg − Θg∥2 F , (16) where Λg ∈ C2M×M, ∀g are dual variables associated with Φg = Θg, and ̺ ≥ 0 is the corresponding penalty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Replacing the original objective function with AL function (16), we obtain the AL minimization problem as P2 AL � min {Uk},W,{Φg},{Θg} L ({Uk} , W, {Φg} , {Θg}) (17a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (12b)-(12e), (15c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (17b) Now, the ADMM framework is constructed as follows, where the superscript of notations refers to the iteration index: Un+1 k = arg min Uk L � {Uk} , Wn, � Φn g � , � Θn g �� (18a) Wn+1 = arg min W L �� Un+1 k � , W, � Φn g � , � Θn g �� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (12b), (12c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (18b) � Φn+1 g � = arg min Φg L �� Un+1 k � , Wn+1, {Φg} , � Θn g �� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (12b), (12d), (12e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (18c) � Θn+1 g � = arg min Θg L �� Un+1 k � , Wn+1, � Φn+1 g � , {Θg} � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (15c), (18d) Λn+1 g = Λn g + ̺ � Φn+1 g − Θn+1 g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (18e) Variables (18a) to (18e) are successively updated by solving corresponding sub-problems until some stopping conditions are reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In the following subsection3, we will elaborate on the solutions to sub-problems (18a) to (18d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Solution to Sub-problems 1) Sub-problem w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t Uk: Given other variables, the opti- mization problem for updating Uk can be expressed as P2 AL,{Uk} � max {Uk} � min ∀k uH k ΨT,kuk uH k (ΨC,k + σ2 RINRL) uk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (19) It can be observed that P2 AL,{Uk} is an unconstrained optimiza- tion problem and has K separable objective functions, each of which has the following form max Uk uH k ΨT,kuk uH k (ΨC,k + σ2 RINRL) uk , ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (20) Problem (20) is a classical generalized fractional quadratic optimization problem, whose optimal solution can be obtained by taking the generalized eigenvalue decomposition as [39] uk = EIG �� ΨCN,k + σ2 RINRL �−1 × ΨT,k � , ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (21) where EIG (·) represents the eigenvector operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2) Sub-problem w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t W: Given other variables, the opti- mization problem for updating W can be expressed as P2 AL,W \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 max W � min ∀k wHΥT,kw wHΥC,kw + σ2 R ∥Uk∥2 F � (22a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ���ℑ � ˜hH u w [l] ���� ℜ � ˜hH u w [l] � − � σ2 C,uΓu,l ≤ tan Ω, (22b) ∥W∥2 F = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (22c) Problem P2 AL,W is hard to settle due to the non-smooth objective function and complicated non-convex constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To simplify the design, we first equivalently transform the 3When introducing solutions to sub-problems, we omit the superscript of notations for conciseness unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 7 objective into a smooth form by introducing an auxiliary variable γ, which yields the following problem P2−1 AL,W \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 max W,γ γ (23a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' min ∀k wHΥT,kw wHΥC,kw + σ2 R ∥Uk∥2 F ≥ γ, (23b) γ ≥ 0, (23c) (22b), (22c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (23d) Then, we deal with constraints (23b), (22b), and (22c) step- by-step detailed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Step 1: Majorization minimization (MM) to (23b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We first rewrite constraint (23b) as wHΥC,kw − wHΥT,kw γ + σ2 R ∥Uk∥2 F ≤ 0, ∀k, (24) where the second term is a composite function with both w and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To simplify the joint design of problem (27), we perform MM and propose the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Assume Υ is positive definite and γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' A majorizer of f (w, γ) = wHΥw γ is f (w, γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' wn, γn) = 2ℜ � (wn)HΥw � γn − γ (wn)HΥwn (γn)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Proof: Please refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Using Lemma 1, we conduct the majorization on constraint (24) at point (wn, γn), yielding wHΥC,kw − 2ℜ � (wn)HΥT,kw � γn + γ (wn)HΥT,kwn (γn)2 + σ2 R ∥Uk∥2 F ≤ 0, ∀k, (25) where γn is computed by γn = min ∀k (wn)HΥT,kwn (wn)HΥC,kwn + σ2 R ∥Uk∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (26) Step 2: Reformulation to (22b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' After some algebraic ma- nipulations, we rewrite (22b) as (22b) ⇔ \uf8f1 \uf8f2 \uf8f3 ℜ �¯hH u,1 (ΦR) w [l] � ≥ � σ2 C,uΓu,l sin Ω, (27a) ℜ �¯hH u,2 (ΦR) w [l] � ≥ � σ2 C,uΓu,l sin Ω, (27b) where ¯hu,1 (ΦR) = GHΦH R hu(sin Ω + e\uf6be π 2 cos Ω)e−\uf6be∠(su[l]) and ¯hu,2 (ΦR) = GHΦH R hu(sin Ω − e\uf6be π 2 cos Ω)e−\uf6be∠(su[l]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Step 3: Simplification to (22c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We first scale the equality constraint (22c) as E−ǫ ≤ ∥W∥2 F ≤ E+ǫ, where ǫ ≥ 0 is an auxiliary variable whose value approaches to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' It is easy to notice that the right-hand side ∥W∥2 F ≤ E + ǫ is convex, while the left-hand side E − ǫ ≤ ∥W∥2 F is non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To convexify the non-convex part, we perform MM and transform (22c) as two convex constraints � ∥W∥2 F − E − ǫ ≤ 0, (28a) 2ℜ {Tr (WnW)} − ∥Wn∥2 F − E + ǫ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (28b) Replacing non-convex contraints in (23) with (25), (27) and (28) based on Steps 1-3, and penalizing the slack variable ǫ into the objective function, we minimize the following problem P2−2 AL,W \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 min W,γ,ǫ −γ + κǫ (29a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' γ ≥ 0, ǫ ≥ 0, (29b) (25), (27a), (27b), (28a), (28b) , (29c) where κ ≥ 0 represents the penalty parameter to scale the impact of the penalty term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Problem P2−2 AL,W is a convex second-order cone programming (SOCP) problem and can be globally solved by the interior point method (IPM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 3) Sub-problem w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t {Φg}: Given other variables, the sub- problem for updating {Φg} is P2 AL,{Φg} \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 min Φg − � min ∀k φH T ΞT,kφT φH T ΞC,kφT + σ2 R ∥Uk∥2 F � + G � g=1 ℜ � Tr � ΛH g (Φg − Θg) �� +̺ 2 G � g=1 ∥Φg − Θg∥2 F (30a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ���ℑ � ˜hH u w [l] ���� ℜ � ˜hH u w [l] � − � σ2 C,uΓu,l ≤ tan Ω, (30b) ΦT = BlkDiag (ΦT,1, · · · , ΦT,G) , (30c) ΦR = BlkDiag (ΦR,1, · · · , ΦR,G) , (30d) where ΦR and ΦT are separable in both objective and con- straints, and thus can be designed in parallel as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Solution to ΦR: The problem regarding ΦR is P2 AL,ΦR \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 min ΦR G � g=1 ℜ � Tr � ΛH R,g (ΦR,g − ΘR,g) �� +̺ 2 G � g=1 ∥ΦR,g − ΘR,g∥2 F (31a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ��ℑ � Tr � ¯Hu,lΦR ���� ℜ � Tr � ¯Hu,lΦR �� − � σ2 C,uΓu,l ≤ tan Ω, (31b) ΦR = BlkDiag (ΦR,1, · · · , ΦR,G) , (31c) where ΛR,g is extracted from the last M rows of Λg, ¯Hu,l = e\uf6be∠(su[l])Gw [l] hH u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The difficulty of solving problem (31) comes from constraints (31b) and (31c), which can be tackled based on the following matrix arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Specifically, we partition ¯Hu,l as ¯Hu,l = \uf8ee \uf8ef\uf8f0 ¯H11 u,l · · ¯H1G u,l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ¯HG1 u,l · · ¯HGG u,l \uf8f9 \uf8fa\uf8fb , ∀u, l, (32) where ¯Hij u,l ∈ CM×M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' By defining ˜Hu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l = [ ¯H11 u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ¯HGG u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' and re-arranging (31c) as ˜ΦR = [ΦR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='G],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' constraints (31b) and (31c) are merged into the following constraint ���ℑ � Tr � ˜Hu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l ˜ΦR ����� ℜ � Tr � ˜Hu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l ˜ΦR �� − � σ2 C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='uΓu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l ≤ tan Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ∀u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (33a) 8 ⇔ \uf8f1 \uf8f2 \uf8f3 ℜ � Tr � ˆHu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='1 ˜ΦR �� ≥ � σ2 C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='uΓu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l sin Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ℜ � Tr � ˆHu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 ˜ΦR �� ≥ � σ2 C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='uΓu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l sin Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ∀u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (33b) where ˆHu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='1 = ˜Hu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l � sin Ω + e−\uf6be π 2 cos Ω � and ˆHu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 = ˜Hu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l � sin Ω − e−\uf6be π 2 cos Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This brings the following opti- mization problem P2−1 AL,˜ΦR \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 min ˜ΦR ℜ � Tr � ˜ΛH R � ˜ΦR − ˜ΘR ��� +̺ 2∥˜ΦR − ˜ΘR∥2 F (34a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (33b), (34b) where ˜ΘR = [ΘR,1, · · · , ΘR,G] and ˜ΛR = [ΛH R,1, · · · , ΛH R,G]H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Problem P2−1 AL,˜ΦR is a quadratic program (QP) with linear constraints and can be efficiently tackled via many existing optimization tools, such as the active set method and the primal-dual subgradient method [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Solution to ΦT: The problem regarding ΦT is P2 AL,ΦT \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 min ΦT − � min ∀k φH T ΞT,kφT φH T ΞC,kφT + σ2 R ∥Uk∥2 F � + G � g=1 ℜ � Tr � ΛH T,g (ΦT,g − ΘT,g) �� +̺ 2 G � g=1 ∥ΦT,g − ΘT,g∥2 F (35a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT = BlkDiag (ΦT,1, · · · , ΦT,G) , (35b) Similarly, we re-organize P2 AL,ΦT into a concise form as P2−1 AL,˜ΦT \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 min ˜ΦT,η −η + ℜ � Tr � ˜ΛH T � ˜ΦT − ˜ΘT ��� +̺ 2 ���˜ΦT − ˜ΘT ��� 2 F (36a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' min ∀k ˜φH T ˜ΞT,k ˜φT ˜φH T ˜ΞC,k ˜φT + σ2 R ∥Uk∥2 F ≥ η, (36b) η ≥ 0, (36c) where ˜ΦT = [ΦT,1, · · · , ΦT,G], ˜ΘT = [ΘT,1, · · · , ΘT,G], ˜ΛT = [ΛH T,1, · · · , ΛH T,G]H with ΛT,g extracted from the first M rows of Λg, and ˜φT = Vec(˜ΦT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ˜ΞT,k = KGΞT,kKH G and ˜ΞC,k = KGΞC,kKH G, where KG = BlkDiag([IM ⊗ [0M,(g−1)M, IM, 0M,(G−g)M]]G g=1) ∈ {0, 1}MNS×N 2 S denotes the linear mapping matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Using Lemma 1 to simplify constraint (36b), we have P2−2 AL,˜ΦT \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 min ˜ΦT,η −η + ℜ � Tr � ˜ΛH T � ˜ΦT − ˜ΘT ��� +̺ 2 ���˜ΦT − ˜ΘT ��� 2 F (37a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ˜φH T ˜ΞC,k ˜φT − 2ℜ � ( ˜φn T)H ˜ΞT,k ˜φT � ηn +η 2ℜ � ( ˜φn T)H ˜ΞT,k ˜φn T � (ηn)2 +σ2 R ∥Uk∥2 F ≤ 0, ∀k, (37b) η ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (37c) Algorithm 1 Max-Min Fairness for BD-RIS Aided DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Input: hu, ∀u, G, ̺ and system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 1: Initialize � U0 k � , W0, Φ0 T, and Φ0 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2: Set n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 3: repeat 4: Calculate radar receive filters {Un k} by (21) in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 5: Update transmit waveform Wn by solving (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 6: Compute BD-RIS matrix Φn R by solving (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 7: Update BD-RIS matrix Φn T by solving (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 8: Obtain auxiliary variables � Θn g � by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 9: Update dual variables � Λn g � by (18e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 10: n = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 11: until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 12: Return {Un k}, Wn, Φn T and Φn R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Output: {U⋆ k} = {Un k}, W⋆ = Wn, Φ⋆ T = Φn T, Φ⋆ R = Φn R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Problem P2−2 AL,˜ΦT is a convex SOCP and can be solved by IPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 4) Sub-problem w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t {Θg}: Given the other variables, the sub-problem for updating {Θg} is P2 AL,{Θg} \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 min Φg G � g=1 ℜ � Tr � ΛH g (Φg − Θg) �� +̺ 2 G � g=1 ∥Φg − Θg∥2 F (38a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΘH g Θg = IM, ∀g, (38b) Problem P2 AL,{Θg} can be split into G sub-problems, each of which has the following form P2−1 AL,Θg \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 min Φg ℜ � Tr � ΛH g (Φg − Θg) �� +̺ 2 ∥Φg − Θg∥2 F (39a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΘH g Θg = IM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (39b) Now, the remaining challenge of solving problem P2 AL,Θg lies in the unitary constraint (39b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The unitary constraint (39b) forms a 2M dimensional complex Stiefel manifold [44], which can be approximately solved via manifold based algorithms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=', Riemannian conjugate gradient (RCG) and Riemannian trust regions (RTR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' However, the iterative procedure of man- ifold methods might cause a lot of computational burdens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To speed-up the design, we provide a closed-form solution of problem P2 AL,Θg in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' With the unitary constraint (39b), the optimal solution for Θg is given by Θg = Bg [IM×M, 0M×M] DH g (40) where BgΣgDH g = Λg + ̺Φg is the singular value decom- position (SVD) of Λg + ̺Φg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Proof: Please refer to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Based on the above derivations, the procedure of the above ADMM based algorithm is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 9 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Initialization Scheme Given that the ADMM procedure is usually sensitive to initial values, we present a 2-step initialization strategy to accelerate the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Step1: Since it is not that straightforward to quickly find proper ΦT and ΦR, we randomly generate ΦT and ΦR, which satisfy the BD-RIS constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Step2: With initialized ΦR, we obtain the cascaded channel ˜hH u (ΦR) = hH u ΦRG for the communication link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To provide a feasible and “good” initial point satisfying the constraint (12b), we initialize the transmit waveform W by solving the following QoS-constrained problem max W,Γ Γ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ℜ �¯hH u,1 (ΦR) w [l] � ≥ � σ2 C,uΓ sin Ω, ∀u, l, ℜ �¯hH u,2 (ΦR) w [l] � ≥ � σ2 C,uΓ sin Ω, ∀u, l, ∥W∥2 F ≤ E, (41) which is a convex problem and can be efficiently solved by many numerical approaches [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Complexity Analysis We provide a broad complexity analysis for Algorithms 1, which is summarized as follows 1) Initialization: The main computational complexity of this stage comes from step 2 by solving the SOCP problem (41) with IPM, which requires approximately O � N 3 TL3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2) ADMM: This stage includes the iterative design of the radar receive filters Uk, transmit beamformer W, BD-RIS coefficients (ΦT, ΦR) and auxiliary variable {Θg}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Updating radar receive filters Uk requires O � KN 3 R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Solving problem (29) for updating W with IPM method needs complexity O � N 3 TL3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The complexity of updating BD-RIS coefficients (ΦT, ΦR) can be upper bounded by O � GN 3 S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Using Theo- rem 1 to update auxiliary variable {Θg} requires complexity of O � GM 3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Therefore, the overall complexity of the ADMM framework is O(N0(KN 3 R + N 3 TL3 + GN 3 S + GM 3)), where N0 denotes the maximum number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' PERFORMANCE EVALUATION In this section, we provided extensive simulation results to validate the effectiveness of the proposed algorithm and the performance of the proposed BD-RIS aided DFRC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' System Setup We assume that the DFBS equipped with NT = 8 antennas transmits QPSK symbols (M = 4) to U = 4 downlink users and detects K = 3 targets with the assistance of a BD-RIS having NS = 16 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The radar sensing receiver colocated with the BD-RIS has NR = 8 receive elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The code length is L = 16 and the power budget at the DFBS is set as E = 10 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The noise power at the users and radar sensing receiver are set as σ2 C,u = σ2 R = −100 dBm, ∀u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The communication QoS threshold is set the same for all users, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=', Γu,l = Γ, ∀u, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In addition, the distance-dependent TABLE I INFORMATION OF K TARGETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target Index Range (m) Azimuth (◦) RCS (dB) Target 1 10 30 5 Target 2 14 0 8 Target 3 19 20 10 TABLE II INFORMATION OF Q CLUTTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' of clutters Range (m) Azimuth (◦) RCS (dB) 5 15 [20:2:28] 25 4 20 [-3:2:3] 25 9 [6:1:14] 10 25 5 [16:1:20] 30 25 path loss is modeled as η (d) = ℵ (d/d0)−ℓ, where ℵ = −30 dB denotes the signal attenuation at the reference distance d0 = 1 m, and ℓ represents the path loss exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We set the path loss exponents for the DFBS→BD-RIS, BD- RIS→user, BD-RIS→target, and BD-RIS→clutter as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2, 2, and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The DFBS and BD-RIS are located as (−20 m, 0 m) and (0 m, 0 m), respectively, which results in the distance between DFBS and BD-RIS as dBR = 20 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The U users are randomly located at reflective side with the same distance dRU = 16 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The DFBS→BD-RIS and BD-RIS→user channels are assumed to follow the Rician fading model with the Rician factor being 3 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' For the radar function, we assume K = 3 targets and 4 groups (Q = 23) of strong clutters are located in the transmissive side, whose detailed information is presented in Tables I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Moreover, we assume the range resolution as ∆d = 1 m, which indicates the radar sampling rate fs = 150 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Combining Table I and the path loss model, the ratio of the propagation coefficients of the three radar targets is ζ2 1 : ζ2 2 : ζ2 3 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='7 [17]–[19], [21], indicating that target 3 is the weakest target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Benchmark Schemes For comparison, we consider the following two benchmark schemes in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 1) Benchmark 1: The radar-only case is selected as the up- per bound of the radar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We obtain this benchmark by changing the BD-RIS into transmissive mode and removing the downlink users, where the resultant problem can be tackled by modifying the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2) Benchmark 2: We consider a doulbe-RIS case where one diagonal RIS working on the reflective mode while another working on the transmissive mode are adja- cently placed to achieve full-space coverage [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This baseline is a special case of BD-RIS with CW-SC where ΦT = Diag([φT,1, · · · , φT, NS 2 ], 01× NS 2 ) and ΦR = Diag(01× NS 2 , [φR,1, · · · , φR, NS 2 ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Therefore, we can obtain this benchmark by modifying the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Simulation Results 1) Convergence Performance: In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' we investigate the convergence of the proposed Algorithm 1 for different 10 0 20 40 60 80 100 120 140 160 180 200 Number of Iteration 0 5 10 15 20 Radar Output SCNR (dB) CW-FC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 1 CW-FC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 2 CW-FC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 3 CW-GC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 1 CW-GC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 2 CW-GC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 3 CW-SC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 1 CW-SC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 2 CW-SC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 3 (a) 0 50 100 150 200 250 300 Number of Iteration 5 0 5 10 15 20 Radar Output SCNR (dB) CW-FC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 1 CW-FC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 2 CW-FC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 3 CW-GC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 1 CW-GC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 2 CW-GC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 3 CW-SC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 1 CW-SC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 2 CW-SC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 3 (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Radar output SCNR versus the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (a) communica- tion threshold Γ = 0 dB, (b) communication threshold Γ = 15 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' BD-RIS architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' It can be observed that the proposed algorithm quickly converges to a stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Specifically, after several iterations, all targets have nearly the same SCNR value, demonstrating that our algorithm can achieve fairness for multiple targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Moreover, the CW-FC architecture enjoys faster convergence than other architectures under the same communication threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' At the same time, the CW-SC re- quires nearly twice as many iterations of CW-FC to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' For the same architecture, the proposed algorithm with a large communication threshold Γ needs more iterations to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This is due to the fact that if the intended communication threshold Γ is higher, fewer degrees of freedom (DoFs) in the optimization problem can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2) System Performance with Varying Parameters: In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 5, we study the minimum radar output SCNR versus the communication threshold Γ for different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' As ex- pected, the radar output SCNR monotonically decreases with Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This is because when the intended Γ is higher, less resource can be used to maximize the radar SCNR, which indicates that there is a trade-off between communication QoS and radar output SCNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Meanwhile, the proposed algorithm with different architectures outperform the conventional RIS, which validates the advantage of deploying BD-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In addition, the output SCNR gap between CW-FC/GC and CW-SC becomes large with increasing communication QoS requirement, which indicates that the advantage of CW-FC/GC BD-RIS is more prominent in high communication QoS requirement scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 6 displays the minimum radar output SCNR as a 0 5 10 15 20 Communication QoS Threshold (dB) 2 4 6 8 10 12 14 16 18 20 Minimum Radar Output SCNR (dB) Radar Only, CW-FC BD-RIS, CW-FC BD-RIS, CW-GC BD-RIS, CW-SC Double-RIS, CW-SC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Minimum radar output SCNR versus the communication threshold Γ for different architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 10 20 30 40 50 Transmit Power (W) 5 10 15 20 25 Minimum Radar Output SCNR (dB) Radar Only, CW-FC BD-RIS, CW-FC BD-RIS, CW-GC BD-RIS, CW-SC Double-RIS, CW-SC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Minimum radar output SCNR versus the transmit power E wit communication threshold Γ = 15 dB for different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' function of transmit power E under different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' It can be observed that the output SCNR for all schemes grows with the increase of transmit power E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Meanwhile, the growth of SCNR becomes slow when the transmit power is substantially large for all considered architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This is because we can improve transmit power to boost system performance to some degree, but excessive power will not improve performance further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Moreover, the slope variation of the BD-RIS scheme with CW-FC/GC/SC architectures is more significant than its competitors, indicating that CW-FC/GC/SC architectures are more sensitive to power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 7, we present the minimum radar SCNR versus the number of groups G with different numbers of BD-RIS cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We observe that with the same number of groups, the radar output SCNR increases with the increasing number of BD-RIS cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The performance enhancement comes from the additional DoF of passive beamforming induced by the increasing number of cells, and the joint design of transmit waveform, the BD-RIS with more general constraints, and the 11 1 2 4 8 12 16 20 24 32 40 Number of groups, G 0 2 4 6 8 10 12 14 16 18 20 Minimum Radar Output SCNR (dB) CW-SC CW-FC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Minimum radar output SCNR versus the number of groups G with different BD-RIS cells NS and communication threshold Γ = 15 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 80 60 40 20 0 20 40 60 80 Angle (Degree) 40 35 30 25 20 15 10 5 0 Normalized Transmit Beampattern (dB) Target 1 Target 2 Target 3 Radar Only, CW-FC BD-RIS, CW-FC BD-RIS, CW-GC BD-RIS, CW-SC (a) 80 60 40 20 0 20 40 60 80 Angle (Degree) 40 35 30 25 20 15 10 5 0 Normalized Transmit Beampattern (dB) Target 1 Target 2 Target 3 Radar Only, CW-FC BD-RIS, CW-FC BD-RIS, CW-GC BD-RIS, CW-SC (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Transmit beampattern of BD-RIS obtained via proposed algorithm for different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (a) communication threshold Γ = 0 dB, (b) communication threshold Γ = 15 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' matched filters, which also confirms the results in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' More importantly, the slope of each carve becomes steeper with the increasing number of groups, which indicates that the number of non-zero elements of BD-RIS matrices plays a significant role in increasing system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 3) Radar Performance: In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 8, we present the transmit beampattern obtained by the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Results show that regardless of BD-RIS architectures, the transmit power (a) Radar-only, CW-FC (b) BD-RIS, CW-FD (c) BD-RIS, CW-GD (d) BD-RIS, CW-SD Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The space-range beampattern behavors of the receive weights for the target 3 detection with communication threshold Γ = 10 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' mainly concentrates around the three targets, which guarantees a high SCNR output at target directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Moreover, the BD- RIS with CW-FC/GC architectures can focus more energy toward targets and has a lower sidelobe than that with CW-SC architecture, thanks to the more flexible passive beamfomring control provided by the CW-FC/GC architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We also observe that the transmit power towards target 3 is much high than other targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This is because, as mentioned early, target 3 is the weakest one, which needs more energy to improve the output radar SCNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In addition, the transmit beampattern performance for BD-RIS with all architectures gets worse with larger communication QoS thresholds, which confirms the conclusion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 9 shows the space-range beampattern of the designed waveform when BD-RIS has different architectures, where the beampattern of the k-th target is computed as P k R (θ, l) = |Tr{(U⋆ k)H A (θ) ΦTGW⋆Jrl}|2 [39]–[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Without loss of generality, we take target 3 (k = 3) as an example to illustrate the space-range behavior of the designed waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Results show that the space-range beampattern can form a mainlobe at the location of the target k = 3 (green circle), but achieve null points at the locations of the other non-of-interest targets (red circles) and strong clutter sources (black rectangles) for all proposed architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This phenomenon can be explained as follows: i) To detect target k, the other targets are regarded as interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ii) BD-RIS with more general architectures can provide more DoFs to resist strong clutters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' CONCLUSION This paper considers the use of BD-RIS in the DFRC system in the presence of multiple targets and strong clutters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We start by reviewing the BD-RIS architectures, and deriving the communication and radar models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Our objective is to maximize the minimum radar output SCNR subject to the constraints of communication QoS, BD-RIS coefficients, and power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Then, a general algorithm utilizing the ADMM 30 40 50 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 ncy (sino) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4-70 80 90 100 110 120 5 30Normalized Spatial freque 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 1 5 10 15 20 Range (m)-30 40 50 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 ncy (sino) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4-70 80 90 100 110 120 5 30Normalized Spatial freque 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 1 5 10 15 20 Range (m)-30 40 50 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 ncy(sino) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4-70 80 90 100 110 120 5 30Normalized Spatial freque 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 1 5 10 15 20 Range (m)-30 40 50 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 ncy (sino) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4-70 dB 80 90 100 110 120 5 30Normalized Spatial freque 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 1 5 10 15 20 Range (m)12 approach is developed to solve the resulting complicated non- convex max-min optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Finally, simulation results demonstrate the effectiveness of the proposed design algorithm, and the superiority of employing the BD-RIS in DFRC systems in terms of enhancing both communication and radar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Based on this initial work, there are many issues worth studying for future research on BD-RIS aided DFRC, such as wideband waveform design, the scenarios for target estimation, as well as exploring the application of multi- sector BD-RIS in DFRC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' APPENDIX A PROOF OF LEMMA 1 Given that f (w, γ) = wHΥw γ is jointly concave in w and γ when Υ ⪰ 0 and γ ≥ 0 [43], the first order approximation of f (w, γ), denoted by f (w, γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' wn, γn), is a majorizer of f (w, γ) at the point (wn, γn), which is f (w, γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' wn, γn) = f (wn, γn) + ( ∂f ∂w|w=wn)T (w − wn) (42a) + ( ∂f ∂w∗ |w=(wn)∗)T (w − (wn)∗) + (∂f ∂γ |γ=γn)T (γ − γn) + ( ∂f ∂γ∗ |γ=(γn)∗)T (γ − (γn)∗) = (wn)HΥwn γn + 2ℜ \uf8f1 \uf8f2 \uf8f3 � 2Υwn γn (wn)HΥwn (γn)2 �H � w − wn γ − γn �\uf8fc \uf8fd \uf8fe = 2ℜ � (wn)HAw � γn − γ (wn)HΥwn (γn)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The proof is thereby completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' APPENDIX B PROOF OF THEOREM 1 We start by rewriting objective (39a) as [43] ℜ � Tr � ΛH g (Φg − Θg) �� + ̺ 2 ∥Φg − Θg∥2 F = −ℜ � Tr � ΘH g (Λg + ̺Φg) �� + ̺ 2 ∥Φg∥2 F + ̺M � �� � constant .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Then, problem P2−1 AL,Θg can be symplified as max Φg ℜ � Tr � ΘH g (Λg + ̺Φg) �� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΘH g Θg = IM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (43) Performing SVD to Λg + ̺Φg as BgΣgDH g = Λg +̺Φg, we can re-arrange the objective of (43) as ℜ � Tr � ΘH g (Λg + ̺Φg) �� = ℜ {Tr (ΣgZg)} = M � i=1 Σg [i, i] Zg [i, i] , (44) where Zg = DH g ΘH g Bg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (44) achieves its maximum when Zg = IM×2M, yielding the optimal solution Θg = Bg [IM×M, 0M×M] DH g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The proof is thus completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 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In this paper we consider (polynomial) solution spaces for +the symplectic Dirac operator (with a focus on 1-homogeneous solu- +tions). This space forms an infinite-dimensional representation space for +the symplectic Lie algebra sp(2m). Because so(m) ⊂ sp(2m), this leads +to a branching problem which generalises the classical Fischer decom- +position in harmonic analysis. Due to the infinite nature of the solution +spaces for the symplectic Dirac operators, this is a non-trivial question: +both the summands appearing in the decomposition and their explicit +embedding factors will be determined in terms of a suitable Mickelsson- +Zhelobenko algebra. +Mathematics Subject Classification (2010). Primary 15A66, 17B10; +Secondary 00A00. +Keywords. Branching, Symplectic Dirac operator, Mickelsson–Zhelobenko +algebra, simplicial harmonics. +1. Introduction +The Dirac operator is a first-order differential operator acting on spinor- +valued functions which factorises the Laplace operator ∆ on Rm. It was +originally introduced by Dirac in a famous attempt to factorise the wave op- +erator, hence obtaining a relativistically invariant version of the Schr¨odinger +equation. Since then, this operator has played a crucial role in mathemati- +cal domains such as representation theory and Clifford analysis. The latter +is a multidimensional function theory which is often described as a refine- +ment of harmonic analysis, and a generalisation of complex analysis. It is +centred around a generalisation of the operator introduced by Dirac (his +operator /∂ is defined in 4 dimensions), and can be seen as a contraction +between the generators ek for a Clifford algebra (acting as endomorphisms +on so-called spinors) and corresponding partial derivatives ∂xk. To be more +precise, introducing the Clifford algebra by means of the defining relations + +2 +David Eelbode and Guner Muarem +{ea, eb} = eaeb + ebea = −2δab (with 1 ≤ a, b ≤ m) the Dirac operator is +given by +∂x = +� +e1 +. . . +em +� +Idm + + + +x1 +... +xm + + + = +m +� +j=1 +ej∂xj , +whereby the (m × m)−identity matrix Idm has been added to explain what +is meant by the ‘contraction’. Null-solutions for ∂x are called monogenics, +and can be seen as generalisations of holomorphic functions. One often starts +with the study of k-homogeneous polynomial solutions for the Dirac operator, +which belong to the space Mk(Rm, S), where S stands for the aforementioned +spinor space. +An obvious generalisation of the operator ∂x can be obtained by using +another matrix than Idm when contracting algebraic generators with partial +derivatives. An important example is the symplectic Dirac operator, which +is introduced on a symplectic space rather than an orthogonal space (see for +example the work of Habermann [5]). This operator, denoted by Ds, is de- +fined as a contraction between generators for a symplectic Clifford algebra +and partial derivatives, using a skew-symmetric matrix Ω0 (rather than Idm). +The symplectic Clifford algebra generators satisfy the Heisenberg relations +[∂zj, zk] = δjk (the symplectic analogue of the Clifford relations for the gener- +ators ek from above). Note that the symbols zj stand for real variables here, +they are chosen because the sets of (real) variables xj and yj will also appear +in this paper. In sharp contrast to the orthogonal case, the symplectic Clifford +algebra is no longer finite-dimensional. This trend continues, in the sense that +the associated symplectic spinor space S∞ +0 also becomes infinite-dimensional. +In this paper, we study infinite-dimensional spaces defined in terms of +solutions for the symplectic Dirac operator (generalised monogenics). These +spaces can be defined algebraically +S∞ +k = Ms +k(R2m, S∞ +0 ) := Pk(R2m, C) ⊠ S∞ +0 +(k ∈ N). +Here ⊠ denotes the Cartan product of the sp(2m)-representations Pk(R2m, C), +the kth-symmetric power of the fundamental vector representation (modelled +by polynomials), and the symplectic spinor space S∞ +0 (also referred to as the +Segal-Shale-Weil representation). These spaces contain k-homogeneous S∞ +0 - +valued solutions for the symplectic Dirac operator. The behaviour of these +spaces as representations for sp(2m) is known (see e.g. [1] and the references +therein), but in this paper we will look at these spaces as orthogonal repre- +sentation spaces. This is motivated by the fact that so(m) ⊂ sp(2m), which +means that we are dealing with a branching problem. +In general, a branching problem can be described as follows: given a rep- +resentation ρ of a Lie algebra g and a subalgebra h, we would like to under- +stand how the representation ρ behaves as a h-representation. This restricted +representation ρ|h will no longer be irreducible, but will decompose into h- +irreducible representations. A branching rule then describes the irreducible + +Branching symplectic monogenics using a M–Z algebra +3 +pieces which will occur, together with their multiplicities. For the symplec- +tic spinors (i.e. for the space S∞ +0 ), this gives the Fischer decomposition in +harmonic analysis, which means that the branching problem for S∞ +k leads to +generalisations thereof. To describe the branching of the infinite-dimensional +symplectic representation space S∞ +k +under the inclusion so(m) ⊂ sp(2m), +we will make use of a quadratic algebra which is known as a Mickelson- +Zhelobenko algebra (see [9] for the general construction and properties). +2. The symplectic Dirac operator and monogenics +We will work with the symplectic space R2m and coordinates (x, y) equipped +with the canonical symplectic form ω0 = �m +j=1 dxj ∧ dyj. The matrix repre- +sentation of the symplectic form is given by +Ω0 = +� +0 +Idm +−Idm +0 +� +. +The group consisting of all invertible linear transformations preserving this +non-degenerate skew-symmetric bilinear form is called the symplectic group +and is formally defined as follows: +Sp(2m, R) = {M ∈ GL(2m, R) | M T Ω0M = Ω0}. +This is a non-compact group of dimension 2m2+m. Its (real) Lie algebra will +be denoted by sp(2m, R). In the orthogonal case, the spin group determined +by the sequence +1 → Z2 → Spin(m) → SO(m) → 1 +plays a crucial role concerning the invariance of the Dirac operator ∂x and +the definition of the spinors S. In the symplectic case, this role is played by +the metaplectic group Mp(2m, R) fixed by the exact sequence +1 → Z2 → Mp(2m, R) → Sp(2m, R) → 1. +Despite the analogies, there are some fundamental differences: +(i) First of all, the group SO(m) is compact, whereas Sp(2m, R) is not. This +has important consequences for the representation theory. As a matter +of fact, the metaplectic group is not a matrix group and does not admit +(faithful) finite-dimensional representations. +(ii) The orthogonal spinors S can be realised as a maximal left ideal in the +Clifford algebra, but this is not the case for the symplectic spinors. The +latter are often modelled as smooth vectors in the infinite-dimensional +Segal-Shale-Weil representation (see [7] and the references therein). One +can also identify the symplectic spinor space S∞ +0 with the space P(Rm, C) +of polynomials in the variables (z1, . . . , zm) ∈ Rm, which is the approach +we will use in this paper. +Definition 2.1. Let (V, ω) be a symplectic vector space. The symplectic +Clifford algebra Cls(V, ω) is defined as the quotient algebra of the tensor +algebra T (V ) of V by the two-sided ideal Iω := {v ⊗ u − u ⊗ v + ω(v, u) : + +4 +David Eelbode and Guner Muarem +u, v ∈ V }. In other words Cls(V, ω) := T (V )/Iω is the algebra generated by +V in terms of the relation [v, u] = −ω(v, u), where we have omitted the tensor +product symbols. +Definition 2.2. Denote by ⟨u, v⟩ := �m +k=1 ukvk the canonical inner product +on Rm (where we allow partial derivatives to appear as coefficients, see the +operators below). We then define the following operators acting on polyno- +mial functions in P(R3m, C): +(i) The symplectic Dirac operator Ds = ⟨z, ∂y⟩ − ⟨∂x, ∂z⟩. +(ii) The adjoint operator Xs = ⟨y, ∂z⟩+⟨x, z⟩ with respect to the symplectic +Fischer product (see Section 5 of [2] for more details). +(iii) The Euler operator E = �m +j=1(xj∂xj + yj∂yj) = Ex + Ey measuring the +degree of homogeneity in the base variables (x, y) ∈ R2m. +Note that some authors use the notation ⟨∇x, ∇y⟩ for an expression such as +� +k ∂xk∂yk, but we will use the Dirac operator symbol here instead of the +nabla operator. +Lemma 2.3. The three operators X = +√ +2Ds, Y = +√ +2Xs and their commu- +tator H = [X, Y ] = −2(Ex + Ey + m) give rise to a copy of the Lie algebra +sl(2). +One now easily sees that the symplectic Dirac operator is nothing more than +the contraction between the Weyl algebra generators (zk, ∂zk) with the vector +fields (∂xk, ∂yk) for k = 1, . . . , m using the canonical symplectic form Ω0. +Definition 2.4. The space of k-homogeneous symplectic monogenics is de- +fined by S∞ +k := ker(Ds)∩ +� +Pk(R2m, C) ⊗ P(Rm, C) +� +, where the space P(Rm, C) +in the vector variable z ∈ Rm plays the role of the symplectic spinor space +S∞ +0 . +Note that as an sp(2m, R)-module, S∞ +k is reducible and decomposes into two +irreducible parts: S∞ +k = S∞ +k,+ ⊕ S∞ +k,− with highest weights +S∞ +k,+ ←→ +� +k − 1 +2, −1 +2, . . . , −1 +2 +� +and +S∞ +k,+ ←→ +� +k − 1 +2, −1 +2, . . . , −3 +2 +� +. +These weight entries are fixed by the Cartan algebra h = Alg(Xjj : 1 ≤ j ≤ +m), where the elements Xjj are defined in the lemma below. In this paper, we +will omit the parity signs and work with S∞ +k as a notation which incorporates +both the positive and negative spinors (in our model, this will correspond to +even or odd in the variable z ∈ Rm, see below, so it is always easy to ‘decom- +pose’ into irreducible components when necessary). +The three operators from Lemma 2.3 can be proven to be invariant under the +action of the symplectic Lie algebra, in the sense that they commute with +the following generators (see also Lemma 3.3 in [3]): + +Branching symplectic monogenics using a M–Z algebra +5 +Lemma 2.5. The symplectic Lie algebra sp(2m) has the following realisation +on the space of symplectic spinor-valued polynomials P(R2m, C) ⊗ S∞ +0 : + + + + + + + + + + + + + + + +Xjk = xj∂xk − yk∂yj − (zk∂zj + 1 +2δjk) +1 ≤ j, k ≤ m +Yjk = xj∂yk + xk∂yj − ∂zj∂zk +1 ≤ j < k ≤ m +Zjk = yj∂xk + yk∂xj + zjzk +1 ≤ j < k ≤ m +Yjj = xj∂yj − 1 +2∂2 +zj +1 ≤ j ≤ m +Zjj = yj∂xj + 1 +2z2 +j +1 ≤ j ≤ m +(2.1) +The branching rule for S∞ +0 , when considering it as a representation space for +the orthogonal Lie algebra so(m) ⊂ sp(2m), leads to the Fischer decomposi- +tion for C-valued polynomials in the variable z ∈ Rm (see below). Note that +so(m) is generated by the operators Xjk −Xkj for 1 ≤ j < k ≤ m, giving rise +to the well-known angular operators ubiquitous in quantum mechanics (often +denoted by Lab with 1 ≤ a < b ≤ m). In our previous paper [3], we therefore +tackled the next case k = 1 as this is a natural generalisation of said Fischer +decomposition. The main problem with our branching rule (Theorem 5.6 in +[3]) is the fact that these so(m)-spaces appear with infinite multiplicities, +which are not always easy to keep track of. Therefore the main goal of this +paper is to show that one can organise these in an algebraic framework which +extends to other values for k too, using a certain quadratic algebra. +3. Simplicial harmonics in three vector variables +In this section we describe a generalisation of harmonic polynomials, in three +vector variables. This will be done in terms of a solution space for a ‘natural’ +collection of so(m)-invariant differential operators. The corresponding Howe +dual pair will be useful for the branching problem addressed above. For the +sake of completeness, we recall the following basic definition: +Definition 3.1. A function f(x) on Rm is called harmonic if ∆f(x) = 0. The +k-homogeneous harmonics are defined as Hk(Rm, C) := Pk(Rm, C) ∩ ker(∆). +These spaces define irreducible representations for so(m) with highest weight +(k, 0, . . . , 0) for all k ∈ Z+. +It is well-known that the space of k-homogeneous polynomials Pk(Rm, C) is +reducible as an so(m)-module (see for example [4]) and decomposes into har- +monic polynomials. In fact, the decomposition of the full space of polynomials +is known as the aforementioned Fischer decomposition, given by +P(Rm, C) = +∞ +� +k=0 +Pk(Rm, C) = +∞ +� +k=0 +∞ +� +p=0 +|z|2pHk(Rm, C). +This can all be generalised to the case of several vector variables (sometimes +also called ‘a matrix variable’): for any highest weight for so(m) there is a +(polynomial) model in terms of simplicial harmonics (or monogenics for the +half-integer representations). We refer to [8] for more details. In this paper, + +6 +David Eelbode and Guner Muarem +we will consider these spaces for so(m)-weights characterised by three inte- +gers (a, b, c) where a ≥ b ≥ c ≥ 0. Also note that trailing zeros in the weight +notation will be omitted from now on, so for instance (k, 0, . . . , 0) will be writ- +ten as (k). First of all, we consider homogeneous polynomials Pa,b,c(z; x, y) +in three vector variables (z; x, y) ∈ R3m. Here we use the notation (z; x, y) +to stress the difference between the variable z (the spinor variable, refer- +ring to an element in S∞ +0 ) from the other two variables (x, y) ∈ R2m, which +are ‘ordinary’ variables. The parameters (a, b, c) then refer to the degrees of +homogeneity in (z; x, y). These polynomials carry the regular representation +of the orthogonal group (or the derived so(m)-action in terms of angular +momentum operators Lab from above). +We further introduce the Weyl algebra in three vector variables as the +algebra generated by the variables and their corresponding derivatives: +W(R3m, C) := Alg(xα, yβ, zγ, ∂xδ, ∂yε, ∂zζ) with α, β, γ, δ, ε, ζ ∈ {1, . . ., m} . +Just like in the case of the classical Fischer decomposition, where the Lie +algebra sl(2) appears as a Howe dual partner, there is a Lie algebra appearing +here. To be precise, it is the Lie algebra sp(6) = g−2⊕g0⊕g+2, with parabolic +subalgebra p := g−2 ⊕ g0 and Levi subalgebra g0 ∼= gl(3). The subspaces g±2 +contain six ‘pure’ operators each (i.e. only variables, acting as a multiplication +operator, or only derivatives). More specifically, the subspaces are spanned +by the following SO(m)-invariant operators: +g−2 := span(∆x, ∆y, ∆z, ⟨∂x, ∂y⟩, ⟨∂y, ∂z⟩, ⟨∂x, ∂z⟩) +g0 := span(⟨x, ∂y⟩, ⟨y, ∂x⟩, ⟨x, ∂z⟩, ⟨z, ∂x⟩, ⟨y, ∂z⟩, ⟨z, ∂y⟩, Ex, Ey, Ez) +g+2 := span(|x|2, |y|2, |z|2, ⟨x, y⟩, ⟨y, z⟩, ⟨x, z⟩) +Definition 3.2. The space of Howe harmonics of degree (a, b, c) in the vari- +ables (z, x, y) is defined as H∗ +a,b,c(R3m, C) := Pa,b,c(R3m, C) ∩ ker(g−2). +In what follows the notation ker(A1, . . . , An) stands for ker(A1)∩. . .∩ker(An), +so ker(g−2) means that simplicial harmonics are annihilated by all (pure dif- +ferential) operators in sp(6). As a representation space for so(m), the spaces +H∗ +a,b,c are not irreducible. In order to obtain an irreducible (sub)space, we +have to impose extra conditions. +Definition 3.3. The vector space of simplicial harmonics of degree (a, b, c) +in the variables (z, x, y) is defined by means of +Ha,b,c(R3m, C) := H∗ +a,b,c(R3m, C) ∩ ker +� +⟨z, ∂x⟩, ⟨z, ∂y⟩, ⟨x, ∂y⟩ +� +. +As was shown in [8], this defines an irreducible representation space for so(m) +with highest weight (a, b, c), where the dominant weight condition a ≥ b ≥ c +must hold. This now leads to the following generalisation of the result above +(the Fisher decompostion in three vector variables): + +Branching symplectic monogenics using a M–Z algebra +7 +Theorem 3.4. The space P(R3m, C) of complex-valued polynomials in three +vector variables (in Rm) has a multiplicity-free decomposition under the ac- +tion of sp(6) × SO(m) by means of: +P(R3m, C) ∼= +� +a≥b≥c +V∞ +a,b,c ⊗ Ha,b,c(R3m, C), +where we used the dominant weight condition in the summation. The notation +V∞ +a,b,c hereby refers to a Verma module (see for example [6]) for sp(6). +4. The Mickelsson-Zhelobenko algebra (general setup) +We have now introduced 21 differential operators giving rise to a realisation +of the Lie algebra sp(6) inside the Weyl algebra (on 3 vector variables in +Rm). In this section we construct a related algebra, the so-called Mickelsson- +Zhelobenko algebra (also called transvector or step algebra) Z. Let g be +a Lie algebra and let s ⊂ g be a reductive subalgebra. We then have the +decomposition g = s ⊕ t, where t carries an s-action for the commutator (i.e. +[s, t] ⊂ t). For s we then fix a triangular decomposition s = s− ⊕ h ⊕ s+, +where s± consists of the positive (resp. negative roots) with respect to the +Cartan subalgebra h ⊂ s. We then also define a left ideal J ⊂ U(g) in the +universal enveloping algebra U(g) by means of U(g)s+. This allows us to +define a certain subalgebra of U(g) which is known as the normaliser: +Norm(J) := {u ∈ U(g) | Ju ⊂ J}. +The crucial point is that J is a two-sided ideal of Norm(J), which allows us +two define the quotient algebra S(g, s) = Norm(J)/J which is known as the +Mickelsson algebra. +In a last step of the construction, we consider an extension of U(g) to a +suitable localisation U′(g) given by +U′(g) = U′(g) ⊗U(h) Frac(U(h)) , +where Frac(U(h)) is the field of fractions in the (universal enveloping algebra +of the) Cartan algebra. The ideal J′ can be introduced for this extension +too (in a completely similar way) and the corresponding quotient algebra +Z(g, s) := Norm(J′)/J′ is the Mickelsson-Zhelobenko algebra. These two al- +gebras are naturally identified, since one has that +Z(g, s) = S(g, s) ⊗U(h) Frac(U(h)) . +Note that this algebra is sometimes referred to as a ‘transvector algebra’, +which is what we will often use in what follows. +5. The Mickelsson-Zhelobenko algebra Z(sp(6), so(4)) +We will now define a specific example of the construction from above, which +will help us to understand how the branching of S∞ +k +works. First of all, we +note the following: + +8 +David Eelbode and Guner Muarem +Lemma 5.1. The three (orthogonally invariant) operators +L := ⟨x, ∂y⟩ − 1 +2∆z +R := ⟨y, ∂x⟩ + 1 +2|z|2 +E := Ey − Ex + Ez + n +2 +give rise to yet another copy of the Lie algebra sl(2). This Lie algebra com- +mutes with the Lie algebra sl(2) ∼= Alg(Ds, Xs). +This thus means that we have now obtained a specific realisation for the Lie +algebra so(4) ∼= Alg(Ds, Xs) ⊕ Alg(L, R) ∼= sl(2) ⊕ sl(2) which appears as a +subalgebra of sp(6). This algebra will play the role of s from Section 4. Let +us therefore consider the lowest weight vectors in so(4): +Y1 = Ds = ⟨z, ∂y⟩ − ⟨∂z, ∂x⟩ +and +Y2 = L = ⟨x, ∂y⟩ − 1 +2∆z . +We will focus on the solutions of both lowest weight vectors, i.e. ker(Ds, L). +Note that the operators in sp(6) do not necessarily act as endomorphisms +on this space, but the transvector framework allows us to ‘replace’ these +operators by (related) transvector algebra generators which do act as endo- +morphisms. We start with proving the reductiveness of the algebra so(4) in +sp(6). +Lemma 5.2. The Lie algebra so(4) is reductive in sp(6). +Proof. We need to show that sp(6) decomposes as so(4) + t, where the sub- +space t carries an action of so(4). For that purpose we introduce the following +15 (linearly independent) differential operators: +∆x +⟨z, ∂x⟩ +⟨y, ∂x⟩ − |z|2 +⟨y, z⟩ +|y|2 +⟨∂x, ∂y⟩ +⟨z, ∂y⟩ + ⟨∂z, ∂x⟩ +Ex − Ey + 2Ez + m +⟨x, z⟩ − ⟨y, ∂z⟩ +⟨x, y⟩ +∆y +⟨∂y, ∂z⟩ +⟨x, ∂y⟩ + ∆z +⟨x, ∂z⟩ +|x|2 +It is now a straightforward computation to check that for each of these op- +erators the commutator with one of the operators in so(4) is again a linear +combination of the operators above. +□ +In order to construct the generators for the algebra Z(g, s) with g = sp(6) +and s = so(4), we need the following: +Definition 5.3. The extremal projector for the Lie algebra sl(2) = Alg(X, Y, H) +is the idempotent operator π given by the (formal) expression +π := 1 + +∞ +� +j=1 +(−1)j +j! +Γ(H + 2) +Γ(H + 2 + j)Y jXj . +(5.1) +This operator satisfies Xπ = πY = 0 and π2 = π. +Note that this operator is defined on the extension U′(sl(2)) of the universal +enveloping algebra defined earlier, so that formal series containing the oper- +ator H in the denominator are well-defined (in practice it will always reduce +to a finite summation). + +Branching symplectic monogenics using a M–Z algebra +9 +Lemma 5.4. The extremal projector πso(4) is given by the product of the +extremal projectors for the Lie algebras sl(2), i.e. πso(4) = πDsπL = πLπDs +(the operator appearing as an index here refers to the realisation for sl(2) +that was used). +Proof. This is due to the fact that the two copies of sl(2) commute. +□ +The operator πso(4) is thus explicitly given by + +1 + +∞ +� +j=1 +(−1)j +j! +Γ(E + 2) +Γ(E + 2 + j)Xj +sDj +s + + + +1 + +∞ +� +j=1 +(−1)j +j! +Γ(E + 2) +Γ(E + 2 + j)RjLj + + +and satisfies Dsπso(4) = Lπso(4) = 0 = πso(4)Xs = πso(4)R. This means that +we now have a natural object that can be used to project polynomials on the +intersection of the kernel of the operators Ds and L. +The 15 operators in t ⊂ sp(6) as such do not preserve this kernel space +(as these operators do not necessarily commute with Ds and L), but their +projections will belong to End(ker(Ds, L)). In what follows we will use the +notation Qa,b, where a ∈ {±2, 0} and b ∈ {±4, ±2, 0}, to denote the operators +in t (see Lemma 5.2, and the scheme below). For each operator Qa,b we then +also define an associated operator Pa,b := πso(4)Qa,b. For instance P4,−2 = +πso(4)|y|2. +The P-operators will then be used to define the generators for our +transvector algebra. The diagram below should then be seen as the analogue +of the 15 operators Qa,b given above, grouped into a 5 × 3 rectangle, where +each operator α ∈ t carries a label. The meaning of the labels (a, b) comes +from the observation that t ∼= V4 ⊗ V2 as a representation for sl(2) ⊕ sl(2), +with Vn the standard notation for the irreducible representation of dimension +(n+1). Given an operator α ∈ t, the numbers a and b can thus be retrieved as +eigenvalues for the commutator action of the Cartan elements in so(4). Note +that the projection operator so(4) commutes with these Cartan elements (i.e. +the operators Qa,b and Pa,b indeed carry the same labels). +−2 +0 +2 +−4 +− 2 +0 +2 +4 +Despite the fact that Z(sp(6), so(4)) is not a Lie algebra, we have organised +these operators in such a way that the notions of ‘positive’ and ‘negative’ +roots can be used. To be more precise: black dots (resp. grey dots) refer +to negative (resp. positive) operators, and the white dot plays the role of +a ‘Cartan element’ (this analogy will come in handy below). The 7 black + +10 +David Eelbode and Guner Muarem +dots (resp. 7 grey dots) will be referred to as operators in ρ− (resp. in ρ+). +Together with the operator P0,0 we then get the set +GZ = {Pa,b : a ∈ {±2, 0}, b ∈ {±4, ±2, 0}}, +containing all the generators for the transvector algebra Z(sp(6), so(4)). +Due to a general result by Zhelobenko, these generators then satisfy +quadratic relations (i.e. different from the classical Lie brackets). In the next +theorem, we will relate the spaces Ha,b,c(R3m, C) introduced in Definition +3.3 to the space of polynomial solutions for the symplectic Dirac operator +Ds, the lowering operator L and the negative ‘roots’ ρ− which we have just +introduced (i.e. the operators Pa,b corresponding to black dots). +Theorem 5.5. The solutions for the operators Ds and L and the negative +roots ρ− ⊂ GZ which are homogeneous of degree (a, b, c) in the variables +(z, x, y) are precisely given by the simplicial harmonics Ha,b,c(R3m, C). In +other words, we have: +Pa,b,c(R3m, C) ∩ ker(Ds, L, ρ−) = Ha,b,c(R3m, C). +Proof. The idea behind this proof is a recursive argument, where the or- +dering on the black dots will be from left to right and from bottom to +top in the rectangular scheme above (in terms of labels this means that +(2, −4) > (0, −4) > (2, −2), as an example). The reason for doing so is the +following: the commutators [L, Qa,b] and [Ds, Qa,b] give an operator situated +below or to the left of the operator Qa,b we started from. Up to a con- +stant, these operators are equal to Qa+2,b and Qa,b−2 respectively (or trivial +whenever the parameters a and b are not in the correct range). This means +that combinations of the form LQa,b and DsQab act trivially on functions +H(z; x, y) in the kernel of L and Ds, provided we know that also Qa+2,b and +Qa,b−2 act trivially. Given the fact that each operator Pa,b ∈ ρ− is of the +form +Pa,b = +� +1 + O1L +�� +1 + O2Ds +� +Qa,b , +where Oj is a short-hand notation for the correction terms coming from +the extremal projection operator (which, unless this operator reduces to the +identity operator, always contains either an operator L or Ds at the right). +The upshot of our recursive scheme is that once we know that Qa+2,b and +Qa,b−2 act trivially, this immediately tells us that Pa,bH = 0 ⇒ Qa,bH = 0. +Because P2,−4H = 0 and P2,−4 = Q−2,4 = ∆y, we can immediately conclude +that the following operators will then act trivially: +∆y +⟨∂x, ∂y⟩ +∆x +⟨∂y, ∂z⟩ +⟨z, ∂y⟩ + ⟨∂x, ∂z⟩ +⟨z, ∂x⟩ +⟨x, ∂y⟩ + ∆z . +In order to be simplicial harmonic, H(z; x, y) should belong to the kernel of +9 operators in sp(6) (see Definition 3.3), but it is straightforward to see that +one can reproduce these operators as commutators of the 7 operators on the +previous line. For example: ∆x(⟨x, ∂y⟩ + ∆z)H = 0 leads to ∆zH = 0, since +⟨∂x, ∂y⟩H = 0 (and so on). +□ + +Branching symplectic monogenics using a M–Z algebra +11 +6. Application: branching symplectic monogenics +We will now use the operators Pa,b to explicitly describe the branching of the +k-homogeneous symplectic monogenics S∞ +k . By this we mean that it will give +us a systematic way to define the ‘embedding factors’ realising the isomorphic +copy of those spaces in S∞ +k . To do so, we will make an analogy again: one can +consider the asssociative algebra U(Z), the ‘universal enveloping algebra’ of +Z(sp(6), so(4)). The meaning should be clear here: it is a tensor algebra � V +(with V the span of GZ-generators as an underlying vector space) modulo +the ideal spanned ‘by the quadratic relations’ in the transvector algebra. We +will refer to elements in this algebra as ‘words’ in ‘an alphabet’ that can be +ordered. This statement, which should thus be seen as an analogue of the +Poincar´e–Birkhoff–Witt theorem (PBW theorem), requires a proof but we +will not do this in the present paper. As a matter of fact, the general case +k ∈ Z+ will be treated in an upcoming (longer) paper, in the present article +we will focus on the case k = 1 as a guiding example. +The main idea is the following: imposing the lexicographic ordering on +the labels (a, b) will dictate the position of our letters in the alphabet (from +left to right), with e.g. (4, 0) > (4, −2) > (2, 2). Letting such a word acting as +an operator on simplicial harmonics Ha,b,c(z; x, y), it should be clear (in view +of the previous theorem) that only the ‘letters’ corresponding to grey dots +in the scheme will play a role (the white dot acts as a constant, whereas the +black dots act trivially). Considering the fact that the total degree of ‘a word’ +in x and y should not exceed k = 1, we can only use the operators Pa,b from +the third and fourth column in our example. Note that once the operator +Pab has been chosen (i.e. +the ‘word’ in front of the simplicial harmonics), +the degree (a, b, c) of these polynomials Ha,b,c(z; x, y) is automatically fixed +too: the total degree in z and (x, y) is then equal to k and 1 respectively. So, +when the ‘word’ is homogeneous of degree one in (x, y) we get contributions +of the form P0,0Ha,1,0 and P2,0Ha,1,0. Whereas when the chosen ‘word’ is +homogeneous of degree zero we get P−2,2Ha,0,0, P0,2Ha,0,0 and P2,2Ha,0,0. +Finally, we note that we can still act with the raising operator R ∈ sl(2) on +each of the polynomials from above (i.e. a suitable projection operator acting +on a suitable space of simplicial harmonics) to arrive at a direct sum of +Verma modules which can be embedded into S∞ +1 . This is based on the trivial +albeit crucial observation that [R, Ds] = 0, so that acting with R preserves +symplectic monogenic solutions. This means that we have now resolved the +branching problem for k = 1 in a completely different way. Resulting in the +decomposition +S∞ +1 +� +sp(2m) +so(m) +∼= +� +a≥1 +∞ +� +ℓ=0 +Rℓ(Ha,1 ⊕ P2,0Ha,1) +⊕ +� +a≥0 +∞ +� +ℓ=0 +Rℓ(P−2,2Ha ⊕ P−2,0Ha ⊕ P−2,−2Ha). + +12 +David Eelbode and Guner Muarem +Summarising the idea behind this decomposition, we thus claim that S∞ +k can +be decomposed under the joint action of +so(m) × sl(2) × Z(sp(6), so(4)), +whereby the final decomposition will contain summands of the form +Rp � +U(ρ+)Ha,b,c +� +for suitable ‘words’ in the algebra U(ρ+) and suitable spaces of simplicial +harmonics. +Acknowledgments +The author G.M. was supported by the FWO-EoS project G0H4518N. +References +[1] F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis. Research Notes in +Mathematics 76, Pitman, London, 1982. +[2] H. De Bie, M. Hol´ıkov´a, P. Somberg, Basic aspects of symplectic Clifford analysis +for the symplectic Dirac operator. Advances in Applied Clifford Algebras 27(2) +(2017), 1103–1132. +[3] D. Eelbode, G. Muarem, The Orthogonal Branching Problem for Symplectic +Monogenics. Advances in Applied Clifford Algebras 33(3) (2022). +[4] J. Gilbert, M. Murray, Clifford Algebras and Dirac Operators in Harmonic Anal- +ysis. Cambridge University Press, 1991 +[5] K. Habermann, L. Habermann, Introduction to Symplectic Dirac Operators. In +Lecture Notes in Mathematics, Springer Berlin Heidelberg, 2006. +[6] R. Howe, Remarks on classical invariant theory. Transactions of the American +Mathematical Society 33(2) (1989), 539—570 +[7] P. Robinson, J. Rawnsley, The Metaplectic Representation, Mpc Structures and +Geometric Quantization. Memoirs of the A.M.S. vol. 81, no. 410, 1989. +[8] P. Van Lancker, F. Sommen, D. Constales, Models for irreducible representations +of Spin(m). Advances in Applied Clifford Algebras 11 (2001), 271–289. +[9] D. Zhelobenko, Extremal projectors and generalised Mickelsson algebras over +reductive Lie algebras. In Mathematics of the USSR 33(1) (1989), 85—100. +David Eelbode +Department of Mathematics +University of Antwerp +Middelheimlaan 1 +2020 Antwerp, Belgium +e-mail: david.eelbode@uantwerpen.be +Guner Muarem +Department of Mathematics +University of Antwerp +Middelheimlaan 1 +2020 Antwerp, Belgium +e-mail: guner.muarem@uantwerpen.be + diff --git a/2tE4T4oBgHgl3EQfawwa/content/tmp_files/load_file.txt b/2tE4T4oBgHgl3EQfawwa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..61931359b724f9de04252f194d3b6ac963b629c9 --- /dev/null +++ b/2tE4T4oBgHgl3EQfawwa/content/tmp_files/load_file.txt @@ -0,0 +1,339 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf,len=338 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='05066v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='RT] 12 Jan 2023 Branching symplectic monogenics using a Mickelsson–Zhelobenko algebra David Eelbode and Guner Muarem Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In this paper we consider (polynomial) solution spaces for the symplectic Dirac operator (with a focus on 1-homogeneous solu- tions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This space forms an infinite-dimensional representation space for the symplectic Lie algebra sp(2m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Because so(m) ⊂ sp(2m), this leads to a branching problem which generalises the classical Fischer decom- position in harmonic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Due to the infinite nature of the solution spaces for the symplectic Dirac operators, this is a non-trivial question: both the summands appearing in the decomposition and their explicit embedding factors will be determined in terms of a suitable Mickelsson- Zhelobenko algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Mathematics Subject Classification (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Primary 15A66, 17B10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Secondary 00A00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Branching, Symplectic Dirac operator, Mickelsson–Zhelobenko algebra, simplicial harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Introduction The Dirac operator is a first-order differential operator acting on spinor- valued functions which factorises the Laplace operator ∆ on Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' It was originally introduced by Dirac in a famous attempt to factorise the wave op- erator, hence obtaining a relativistically invariant version of the Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Since then, this operator has played a crucial role in mathemati- cal domains such as representation theory and Clifford analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The latter is a multidimensional function theory which is often described as a refine- ment of harmonic analysis, and a generalisation of complex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' It is centred around a generalisation of the operator introduced by Dirac (his operator /∂ is defined in 4 dimensions), and can be seen as a contraction between the generators ek for a Clifford algebra (acting as endomorphisms on so-called spinors) and corresponding partial derivatives ∂xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' To be more precise, introducing the Clifford algebra by means of the defining relations 2 David Eelbode and Guner Muarem {ea, eb} = eaeb + ebea = −2δab (with 1 ≤ a, b ≤ m) the Dirac operator is given by ∂x = � e1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' em � Idm \uf8eb \uf8ec \uf8ed x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' xm \uf8f6 \uf8f7 \uf8f8 = m � j=1 ej∂xj , whereby the (m × m)−identity matrix Idm has been added to explain what is meant by the ‘contraction’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Null-solutions for ∂x are called monogenics, and can be seen as generalisations of holomorphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' One often starts with the study of k-homogeneous polynomial solutions for the Dirac operator, which belong to the space Mk(Rm, S), where S stands for the aforementioned spinor space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' An obvious generalisation of the operator ∂x can be obtained by using another matrix than Idm when contracting algebraic generators with partial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' An important example is the symplectic Dirac operator, which is introduced on a symplectic space rather than an orthogonal space (see for example the work of Habermann [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This operator, denoted by Ds, is de- fined as a contraction between generators for a symplectic Clifford algebra and partial derivatives, using a skew-symmetric matrix Ω0 (rather than Idm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The symplectic Clifford algebra generators satisfy the Heisenberg relations [∂zj, zk] = δjk (the symplectic analogue of the Clifford relations for the gener- ators ek from above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Note that the symbols zj stand for real variables here, they are chosen because the sets of (real) variables xj and yj will also appear in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In sharp contrast to the orthogonal case, the symplectic Clifford algebra is no longer finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This trend continues, in the sense that the associated symplectic spinor space S∞ 0 also becomes infinite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In this paper, we study infinite-dimensional spaces defined in terms of solutions for the symplectic Dirac operator (generalised monogenics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' These spaces can be defined algebraically S∞ k = Ms k(R2m, S∞ 0 ) := Pk(R2m, C) ⊠ S∞ 0 (k ∈ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Here ⊠ denotes the Cartan product of the sp(2m)-representations Pk(R2m, C), the kth-symmetric power of the fundamental vector representation (modelled by polynomials), and the symplectic spinor space S∞ 0 (also referred to as the Segal-Shale-Weil representation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' These spaces contain k-homogeneous S∞ 0 - valued solutions for the symplectic Dirac operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The behaviour of these spaces as representations for sp(2m) is known (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' [1] and the references therein), but in this paper we will look at these spaces as orthogonal repre- sentation spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This is motivated by the fact that so(m) ⊂ sp(2m), which means that we are dealing with a branching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In general, a branching problem can be described as follows: given a rep- resentation ρ of a Lie algebra g and a subalgebra h, we would like to under- stand how the representation ρ behaves as a h-representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This restricted representation ρ|h will no longer be irreducible, but will decompose into h- irreducible representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' A branching rule then describes the irreducible Branching symplectic monogenics using a M–Z algebra 3 pieces which will occur, together with their multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' For the symplec- tic spinors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' for the space S∞ 0 ), this gives the Fischer decomposition in harmonic analysis, which means that the branching problem for S∞ k leads to generalisations thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' To describe the branching of the infinite-dimensional symplectic representation space S∞ k under the inclusion so(m) ⊂ sp(2m), we will make use of a quadratic algebra which is known as a Mickelson- Zhelobenko algebra (see [9] for the general construction and properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The symplectic Dirac operator and monogenics We will work with the symplectic space R2m and coordinates (x, y) equipped with the canonical symplectic form ω0 = �m j=1 dxj ∧ dyj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The matrix repre- sentation of the symplectic form is given by Ω0 = � 0 Idm −Idm 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The group consisting of all invertible linear transformations preserving this non-degenerate skew-symmetric bilinear form is called the symplectic group and is formally defined as follows: Sp(2m, R) = {M ∈ GL(2m, R) | M T Ω0M = Ω0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This is a non-compact group of dimension 2m2+m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Its (real) Lie algebra will be denoted by sp(2m, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In the orthogonal case, the spin group determined by the sequence 1 → Z2 → Spin(m) → SO(m) → 1 plays a crucial role concerning the invariance of the Dirac operator ∂x and the definition of the spinors S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In the symplectic case, this role is played by the metaplectic group Mp(2m, R) fixed by the exact sequence 1 → Z2 → Mp(2m, R) → Sp(2m, R) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Despite the analogies, there are some fundamental differences: (i) First of all, the group SO(m) is compact, whereas Sp(2m, R) is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This has important consequences for the representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' As a matter of fact, the metaplectic group is not a matrix group and does not admit (faithful) finite-dimensional representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' (ii) The orthogonal spinors S can be realised as a maximal left ideal in the Clifford algebra, but this is not the case for the symplectic spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The latter are often modelled as smooth vectors in the infinite-dimensional Segal-Shale-Weil representation (see [7] and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' One can also identify the symplectic spinor space S∞ 0 with the space P(Rm, C) of polynomials in the variables (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' , zm) ∈ Rm, which is the approach we will use in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Let (V, ω) be a symplectic vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The symplectic Clifford algebra Cls(V, ω) is defined as the quotient algebra of the tensor algebra T (V ) of V by the two-sided ideal Iω := {v ⊗ u − u ⊗ v + ω(v, u) : 4 David Eelbode and Guner Muarem u, v ∈ V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In other words Cls(V, ω) := T (V )/Iω is the algebra generated by V in terms of the relation [v, u] = −ω(v, u), where we have omitted the tensor product symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Denote by ⟨u, v⟩ := �m k=1 ukvk the canonical inner product on Rm (where we allow partial derivatives to appear as coefficients, see the operators below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' We then define the following operators acting on polyno- mial functions in P(R3m, C): (i) The symplectic Dirac operator Ds = ⟨z, ∂y⟩ − ⟨∂x, ∂z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' (ii) The adjoint operator Xs = ⟨y, ∂z⟩+⟨x, z⟩ with respect to the symplectic Fischer product (see Section 5 of [2] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' (iii) The Euler operator E = �m j=1(xj∂xj + yj∂yj) = Ex + Ey measuring the degree of homogeneity in the base variables (x, y) ∈ R2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Note that some authors use the notation ⟨∇x, ∇y⟩ for an expression such as � k ∂xk∂yk, but we will use the Dirac operator symbol here instead of the nabla operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The three operators X = √ 2Ds, Y = √ 2Xs and their commu- tator H = [X, Y ] = −2(Ex + Ey + m) give rise to a copy of the Lie algebra sl(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' One now easily sees that the symplectic Dirac operator is nothing more than the contraction between the Weyl algebra generators (zk, ∂zk) with the vector fields (∂xk, ∂yk) for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' , m using the canonical symplectic form Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The space of k-homogeneous symplectic monogenics is de- fined by S∞ k := ker(Ds)∩ � Pk(R2m, C) ⊗ P(Rm, C) � , where the space P(Rm, C) in the vector variable z ∈ Rm plays the role of the symplectic spinor space S∞ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Note that as an sp(2m, R)-module, S∞ k is reducible and decomposes into two irreducible parts: S∞ k = S∞ k,+ ⊕ S∞ k,− with highest weights S∞ k,+ ←→ � k − 1 2, −1 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' , −1 2 � and S∞ k,+ ←→ � k − 1 2, −1 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' , −3 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' These weight entries are fixed by the Cartan algebra h = Alg(Xjj : 1 ≤ j ≤ m), where the elements Xjj are defined in the lemma below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In this paper, we will omit the parity signs and work with S∞ k as a notation which incorporates both the positive and negative spinors (in our model, this will correspond to even or odd in the variable z ∈ Rm, see below, so it is always easy to ‘decom- pose’ into irreducible components when necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The three operators from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='3 can be proven to be invariant under the action of the symplectic Lie algebra, in the sense that they commute with the following generators (see also Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='3 in [3]): Branching symplectic monogenics using a M–Z algebra 5 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The symplectic Lie algebra sp(2m) has the following realisation on the space of symplectic spinor-valued polynomials P(R2m, C) ⊗ S∞ 0 : \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 Xjk = xj∂xk − yk∂yj − (zk∂zj + 1 2δjk) 1 ≤ j, k ≤ m Yjk = xj∂yk + xk∂yj − ∂zj∂zk 1 ≤ j < k ≤ m Zjk = yj∂xk + yk∂xj + zjzk 1 ≤ j < k ≤ m Yjj = xj∂yj − 1 2∂2 zj 1 ≤ j ≤ m Zjj = yj∂xj + 1 2z2 j 1 ≤ j ≤ m (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='1) The branching rule for S∞ 0 , when considering it as a representation space for the orthogonal Lie algebra so(m) ⊂ sp(2m), leads to the Fischer decomposi- tion for C-valued polynomials in the variable z ∈ Rm (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Note that so(m) is generated by the operators Xjk −Xkj for 1 ≤ j < k ≤ m, giving rise to the well-known angular operators ubiquitous in quantum mechanics (often denoted by Lab with 1 ≤ a < b ≤ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In our previous paper [3], we therefore tackled the next case k = 1 as this is a natural generalisation of said Fischer decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The main problem with our branching rule (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='6 in [3]) is the fact that these so(m)-spaces appear with infinite multiplicities, which are not always easy to keep track of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Therefore the main goal of this paper is to show that one can organise these in an algebraic framework which extends to other values for k too, using a certain quadratic algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Simplicial harmonics in three vector variables In this section we describe a generalisation of harmonic polynomials, in three vector variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This will be done in terms of a solution space for a ‘natural’ collection of so(m)-invariant differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The corresponding Howe dual pair will be useful for the branching problem addressed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' For the sake of completeness, we recall the following basic definition: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' A function f(x) on Rm is called harmonic if ∆f(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The k-homogeneous harmonics are defined as Hk(Rm, C) := Pk(Rm, C) ∩ ker(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' These spaces define irreducible representations for so(m) with highest weight (k, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' , 0) for all k ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' It is well-known that the space of k-homogeneous polynomials Pk(Rm, C) is reducible as an so(m)-module (see for example [4]) and decomposes into har- monic polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In fact, the decomposition of the full space of polynomials is known as the aforementioned Fischer decomposition, given by P(Rm, C) = ∞ � k=0 Pk(Rm, C) = ∞ � k=0 ∞ � p=0 |z|2pHk(Rm, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This can all be generalised to the case of several vector variables (sometimes also called ‘a matrix variable’): for any highest weight for so(m) there is a (polynomial) model in terms of simplicial harmonics (or monogenics for the half-integer representations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' We refer to [8] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In this paper, 6 David Eelbode and Guner Muarem we will consider these spaces for so(m)-weights characterised by three inte- gers (a, b, c) where a ≥ b ≥ c ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Also note that trailing zeros in the weight notation will be omitted from now on, so for instance (k, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' , 0) will be writ- ten as (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' First of all, we consider homogeneous polynomials Pa,b,c(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' x, y) in three vector variables (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' x, y) ∈ R3m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Here we use the notation (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' x, y) to stress the difference between the variable z (the spinor variable, refer- ring to an element in S∞ 0 ) from the other two variables (x, y) ∈ R2m, which are ‘ordinary’ variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The parameters (a, b, c) then refer to the degrees of homogeneity in (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' These polynomials carry the regular representation of the orthogonal group (or the derived so(m)-action in terms of angular momentum operators Lab from above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' We further introduce the Weyl algebra in three vector variables as the algebra generated by the variables and their corresponding derivatives: W(R3m, C) := Alg(xα, yβ, zγ, ∂xδ, ∂yε, ∂zζ) with α, β, γ, δ, ε, ζ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=', m} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Just like in the case of the classical Fischer decomposition, where the Lie algebra sl(2) appears as a Howe dual partner, there is a Lie algebra appearing here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' To be precise, it is the Lie algebra sp(6) = g−2⊕g0⊕g+2, with parabolic subalgebra p := g−2 ⊕ g0 and Levi subalgebra g0 ∼= gl(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The subspaces g±2 contain six ‘pure’ operators each (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' only variables, acting as a multiplication operator, or only derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' More specifically, the subspaces are spanned by the following SO(m)-invariant operators: g−2 := span(∆x, ∆y, ∆z, ⟨∂x, ∂y⟩, ⟨∂y, ∂z⟩, ⟨∂x, ∂z⟩) g0 := span(⟨x, ∂y⟩, ⟨y, ∂x⟩, ⟨x, ∂z⟩, ⟨z, ∂x⟩, ⟨y, ∂z⟩, ⟨z, ∂y⟩, Ex, Ey, Ez) g+2 := span(|x|2, |y|2, |z|2, ⟨x, y⟩, ⟨y, z⟩, ⟨x, z⟩) Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The space of Howe harmonics of degree (a, b, c) in the vari- ables (z, x, y) is defined as H∗ a,b,c(R3m, C) := Pa,b,c(R3m, C) ∩ ker(g−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In what follows the notation ker(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' , An) stands for ker(A1)∩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='∩ker(An), so ker(g−2) means that simplicial harmonics are annihilated by all (pure dif- ferential) operators in sp(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' As a representation space for so(m), the spaces H∗ a,b,c are not irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In order to obtain an irreducible (sub)space, we have to impose extra conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The vector space of simplicial harmonics of degree (a, b, c) in the variables (z, x, y) is defined by means of Ha,b,c(R3m, C) := H∗ a,b,c(R3m, C) ∩ ker � ⟨z, ∂x⟩, ⟨z, ∂y⟩, ⟨x, ∂y⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' As was shown in [8], this defines an irreducible representation space for so(m) with highest weight (a, b, c), where the dominant weight condition a ≥ b ≥ c must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This now leads to the following generalisation of the result above (the Fisher decompostion in three vector variables): Branching symplectic monogenics using a M–Z algebra 7 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The space P(R3m, C) of complex-valued polynomials in three vector variables (in Rm) has a multiplicity-free decomposition under the ac- tion of sp(6) × SO(m) by means of: P(R3m, C) ∼= � a≥b≥c V∞ a,b,c ⊗ Ha,b,c(R3m, C), where we used the dominant weight condition in the summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The notation V∞ a,b,c hereby refers to a Verma module (see for example [6]) for sp(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The Mickelsson-Zhelobenko algebra (general setup) We have now introduced 21 differential operators giving rise to a realisation of the Lie algebra sp(6) inside the Weyl algebra (on 3 vector variables in Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In this section we construct a related algebra, the so-called Mickelsson- Zhelobenko algebra (also called transvector or step algebra) Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Let g be a Lie algebra and let s ⊂ g be a reductive subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' We then have the decomposition g = s ⊕ t, where t carries an s-action for the commutator (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' [s, t] ⊂ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' For s we then fix a triangular decomposition s = s− ⊕ h ⊕ s+, where s± consists of the positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' negative roots) with respect to the Cartan subalgebra h ⊂ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' We then also define a left ideal J ⊂ U(g) in the universal enveloping algebra U(g) by means of U(g)s+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This allows us to define a certain subalgebra of U(g) which is known as the normaliser: Norm(J) := {u ∈ U(g) | Ju ⊂ J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The crucial point is that J is a two-sided ideal of Norm(J), which allows us two define the quotient algebra S(g, s) = Norm(J)/J which is known as the Mickelsson algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In a last step of the construction, we consider an extension of U(g) to a suitable localisation U′(g) given by U′(g) = U′(g) ⊗U(h) Frac(U(h)) , where Frac(U(h)) is the field of fractions in the (universal enveloping algebra of the) Cartan algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The ideal J′ can be introduced for this extension too (in a completely similar way) and the corresponding quotient algebra Z(g, s) := Norm(J′)/J′ is the Mickelsson-Zhelobenko algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' These two al- gebras are naturally identified, since one has that Z(g, s) = S(g, s) ⊗U(h) Frac(U(h)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Note that this algebra is sometimes referred to as a ‘transvector algebra’, which is what we will often use in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The Mickelsson-Zhelobenko algebra Z(sp(6), so(4)) We will now define a specific example of the construction from above, which will help us to understand how the branching of S∞ k works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' First of all, we note the following: 8 David Eelbode and Guner Muarem Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The three (orthogonally invariant) operators L := ⟨x, ∂y⟩ − 1 2∆z R := ⟨y, ∂x⟩ + 1 2|z|2 E := Ey − Ex + Ez + n 2 give rise to yet another copy of the Lie algebra sl(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This Lie algebra com- mutes with the Lie algebra sl(2) ∼= Alg(Ds, Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This thus means that we have now obtained a specific realisation for the Lie algebra so(4) ∼= Alg(Ds, Xs) ⊕ Alg(L, R) ∼= sl(2) ⊕ sl(2) which appears as a subalgebra of sp(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This algebra will play the role of s from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Let us therefore consider the lowest weight vectors in so(4): Y1 = Ds = ⟨z, ∂y⟩ − ⟨∂z, ∂x⟩ and Y2 = L = ⟨x, ∂y⟩ − 1 2∆z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' We will focus on the solutions of both lowest weight vectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' ker(Ds, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Note that the operators in sp(6) do not necessarily act as endomorphisms on this space, but the transvector framework allows us to ‘replace’ these operators by (related) transvector algebra generators which do act as endo- morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' We start with proving the reductiveness of the algebra so(4) in sp(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The Lie algebra so(4) is reductive in sp(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' We need to show that sp(6) decomposes as so(4) + t, where the sub- space t carries an action of so(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' For that purpose we introduce the following 15 (linearly independent) differential operators: ∆x ⟨z, ∂x⟩ ⟨y, ∂x⟩ − |z|2 ⟨y, z⟩ |y|2 ⟨∂x, ∂y⟩ ⟨z, ∂y⟩ + ⟨∂z, ∂x⟩ Ex − Ey + 2Ez + m ⟨x, z⟩ − ⟨y, ∂z⟩ ⟨x, y⟩ ∆y ⟨∂y, ∂z⟩ ⟨x, ∂y⟩ + ∆z ⟨x, ∂z⟩ |x|2 It is now a straightforward computation to check that for each of these op- erators the commutator with one of the operators in so(4) is again a linear combination of the operators above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' □ In order to construct the generators for the algebra Z(g, s) with g = sp(6) and s = so(4), we need the following: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The extremal projector for the Lie algebra sl(2) = Alg(X, Y, H) is the idempotent operator π given by the (formal) expression π := 1 + ∞ � j=1 (−1)j j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Γ(H + 2) Γ(H + 2 + j)Y jXj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='1) This operator satisfies Xπ = πY = 0 and π2 = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Note that this operator is defined on the extension U′(sl(2)) of the universal enveloping algebra defined earlier, so that formal series containing the oper- ator H in the denominator are well-defined (in practice it will always reduce to a finite summation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Branching symplectic monogenics using a M–Z algebra 9 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The extremal projector πso(4) is given by the product of the extremal projectors for the Lie algebras sl(2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' πso(4) = πDsπL = πLπDs (the operator appearing as an index here refers to the realisation for sl(2) that was used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This is due to the fact that the two copies of sl(2) commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' □ The operator πso(4) is thus explicitly given by \uf8eb \uf8ed1 + ∞ � j=1 (−1)j j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Γ(E + 2) Γ(E + 2 + j)Xj sDj s \uf8f6 \uf8f8 \uf8eb \uf8ed1 + ∞ � j=1 (−1)j j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Γ(E + 2) Γ(E + 2 + j)RjLj \uf8f6 \uf8f8 and satisfies Dsπso(4) = Lπso(4) = 0 = πso(4)Xs = πso(4)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This means that we now have a natural object that can be used to project polynomials on the intersection of the kernel of the operators Ds and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The 15 operators in t ⊂ sp(6) as such do not preserve this kernel space (as these operators do not necessarily commute with Ds and L), but their projections will belong to End(ker(Ds, L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In what follows we will use the notation Qa,b, where a ∈ {±2, 0} and b ∈ {±4, ±2, 0}, to denote the operators in t (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='2, and the scheme below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' For each operator Qa,b we then also define an associated operator Pa,b := πso(4)Qa,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' For instance P4,−2 = πso(4)|y|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The P-operators will then be used to define the generators for our transvector algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The diagram below should then be seen as the analogue of the 15 operators Qa,b given above, grouped into a 5 × 3 rectangle, where each operator α ∈ t carries a label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The meaning of the labels (a, b) comes from the observation that t ∼= V4 ⊗ V2 as a representation for sl(2) ⊕ sl(2), with Vn the standard notation for the irreducible representation of dimension (n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Given an operator α ∈ t, the numbers a and b can thus be retrieved as eigenvalues for the commutator action of the Cartan elements in so(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Note that the projection operator so(4) commutes with these Cartan elements (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' the operators Qa,b and Pa,b indeed carry the same labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' −2 0 2 −4 − 2 0 2 4 Despite the fact that Z(sp(6), so(4)) is not a Lie algebra, we have organised these operators in such a way that the notions of ‘positive’ and ‘negative’ roots can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' To be more precise: black dots (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' grey dots) refer to negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' positive) operators, and the white dot plays the role of a ‘Cartan element’ (this analogy will come in handy below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The 7 black 10 David Eelbode and Guner Muarem dots (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' 7 grey dots) will be referred to as operators in ρ− (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' in ρ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Together with the operator P0,0 we then get the set GZ = {Pa,b : a ∈ {±2, 0}, b ∈ {±4, ±2, 0}}, containing all the generators for the transvector algebra Z(sp(6), so(4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Due to a general result by Zhelobenko, these generators then satisfy quadratic relations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' different from the classical Lie brackets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In the next theorem, we will relate the spaces Ha,b,c(R3m, C) introduced in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='3 to the space of polynomial solutions for the symplectic Dirac operator Ds, the lowering operator L and the negative ‘roots’ ρ− which we have just introduced (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' the operators Pa,b corresponding to black dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The solutions for the operators Ds and L and the negative roots ρ− ⊂ GZ which are homogeneous of degree (a, b, c) in the variables (z, x, y) are precisely given by the simplicial harmonics Ha,b,c(R3m, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In other words, we have: Pa,b,c(R3m, C) ∩ ker(Ds, L, ρ−) = Ha,b,c(R3m, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The idea behind this proof is a recursive argument, where the or- dering on the black dots will be from left to right and from bottom to top in the rectangular scheme above (in terms of labels this means that (2, −4) > (0, −4) > (2, −2), as an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The reason for doing so is the following: the commutators [L, Qa,b] and [Ds, Qa,b] give an operator situated below or to the left of the operator Qa,b we started from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Up to a con- stant, these operators are equal to Qa+2,b and Qa,b−2 respectively (or trivial whenever the parameters a and b are not in the correct range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This means that combinations of the form LQa,b and DsQab act trivially on functions H(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' x, y) in the kernel of L and Ds, provided we know that also Qa+2,b and Qa,b−2 act trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Given the fact that each operator Pa,b ∈ ρ− is of the form Pa,b = � 1 + O1L �� 1 + O2Ds � Qa,b , where Oj is a short-hand notation for the correction terms coming from the extremal projection operator (which, unless this operator reduces to the identity operator, always contains either an operator L or Ds at the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The upshot of our recursive scheme is that once we know that Qa+2,b and Qa,b−2 act trivially, this immediately tells us that Pa,bH = 0 ⇒ Qa,bH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Because P2,−4H = 0 and P2,−4 = Q−2,4 = ∆y, we can immediately conclude that the following operators will then act trivially: ∆y ⟨∂x, ∂y⟩ ∆x ⟨∂y, ∂z⟩ ⟨z, ∂y⟩ + ⟨∂x, ∂z⟩ ⟨z, ∂x⟩ ⟨x, ∂y⟩ + ∆z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In order to be simplicial harmonic, H(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' x, y) should belong to the kernel of 9 operators in sp(6) (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='3), but it is straightforward to see that one can reproduce these operators as commutators of the 7 operators on the previous line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' For example: ∆x(⟨x, ∂y⟩ + ∆z)H = 0 leads to ∆zH = 0, since ⟨∂x, ∂y⟩H = 0 (and so on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' □ Branching symplectic monogenics using a M–Z algebra 11 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Application: branching symplectic monogenics We will now use the operators Pa,b to explicitly describe the branching of the k-homogeneous symplectic monogenics S∞ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' By this we mean that it will give us a systematic way to define the ‘embedding factors’ realising the isomorphic copy of those spaces in S∞ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' To do so, we will make an analogy again: one can consider the asssociative algebra U(Z), the ‘universal enveloping algebra’ of Z(sp(6), so(4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The meaning should be clear here: it is a tensor algebra � V (with V the span of GZ-generators as an underlying vector space) modulo the ideal spanned ‘by the quadratic relations’ in the transvector algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' We will refer to elements in this algebra as ‘words’ in ‘an alphabet’ that can be ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This statement, which should thus be seen as an analogue of the Poincar´e–Birkhoff–Witt theorem (PBW theorem), requires a proof but we will not do this in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' As a matter of fact, the general case k ∈ Z+ will be treated in an upcoming (longer) paper, in the present article we will focus on the case k = 1 as a guiding example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' The main idea is the following: imposing the lexicographic ordering on the labels (a, b) will dictate the position of our letters in the alphabet (from left to right), with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' (4, 0) > (4, −2) > (2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Letting such a word acting as an operator on simplicial harmonics Ha,b,c(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' x, y), it should be clear (in view of the previous theorem) that only the ‘letters’ corresponding to grey dots in the scheme will play a role (the white dot acts as a constant, whereas the black dots act trivially).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Considering the fact that the total degree of ‘a word’ in x and y should not exceed k = 1, we can only use the operators Pa,b from the third and fourth column in our example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Note that once the operator Pab has been chosen (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' the ‘word’ in front of the simplicial harmonics), the degree (a, b, c) of these polynomials Ha,b,c(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' x, y) is automatically fixed too: the total degree in z and (x, y) is then equal to k and 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' So, when the ‘word’ is homogeneous of degree one in (x, y) we get contributions of the form P0,0Ha,1,0 and P2,0Ha,1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Whereas when the chosen ‘word’ is homogeneous of degree zero we get P−2,2Ha,0,0, P0,2Ha,0,0 and P2,2Ha,0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Finally, we note that we can still act with the raising operator R ∈ sl(2) on each of the polynomials from above (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' a suitable projection operator acting on a suitable space of simplicial harmonics) to arrive at a direct sum of Verma modules which can be embedded into S∞ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This is based on the trivial albeit crucial observation that [R, Ds] = 0, so that acting with R preserves symplectic monogenic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' This means that we have now resolved the branching problem for k = 1 in a completely different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Resulting in the decomposition S∞ 1 \uf8e6\uf8e6\uf8e6� sp(2m) so(m) ∼= � a≥1 ∞ � ℓ=0 Rℓ(Ha,1 ⊕ P2,0Ha,1) ⊕ � a≥0 ∞ � ℓ=0 Rℓ(P−2,2Ha ⊕ P−2,0Ha ⊕ P−2,−2Ha).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' 12 David Eelbode and Guner Muarem Summarising the idea behind this decomposition, we thus claim that S∞ k can be decomposed under the joint action of so(m) × sl(2) × Z(sp(6), so(4)), whereby the final decomposition will contain summands of the form Rp � U(ρ+)Ha,b,c � for suitable ‘words’ in the algebra U(ρ+) and suitable spaces of simplicial harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Acknowledgments The author G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' was 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Constales, Models for irreducible representations of Spin(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Advances in Applied Clifford Algebras 11 (2001), 271–289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' Zhelobenko, Extremal projectors and generalised Mickelsson algebras over reductive Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' In Mathematics of the USSR 33(1) (1989), 85—100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content=' David Eelbode Department of Mathematics University of Antwerp Middelheimlaan 1 2020 Antwerp, Belgium e-mail: david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='eelbode@uantwerpen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='be Guner Muarem Department of Mathematics University of Antwerp Middelheimlaan 1 2020 Antwerp, Belgium e-mail: guner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='muarem@uantwerpen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} +page_content='be' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'} diff --git a/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf b/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf new file mode 100644 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b/59FJT4oBgHgl3EQflCzy/content/tmp_files/2301.11582v1.pdf.txt @@ -0,0 +1,1844 @@ +Adaptive Least-Squares Methods for Convection-Dominated +Diffusion-Reaction Problems +Zhiqiang Cai∗ +Binghe Chen† +Jing Yang‡ +Abstract +This paper studies adaptive least-squares finite element methods for convection- +dominated diffusion-reaction problems. The least-squares methods are based on the +first-order system of the primal and dual variables with various ways of imposing +outflow boundary conditions. The coercivity of the homogeneous least-squares func- +tionals are established, and the a priori error estimates of the least-squares methods +are obtained in a norm that incorporates the streamline derivative. All methods have +the same convergence rate provided that meshes in the layer regions are fine enough. +To increase computational accuracy and reduce computational cost, adaptive least- +squares methods are implemented and numerical results are presented for some test +problems. +ADAPTIVE FOSLS FOR THE CONVECTION-DOMINATED PROBLEMS +1 +Introduction +Due to the small diffusion coefficient, the solution of the convection-dominated diffusion- +reaction problem develops the boundary or interior layers, i.e., narrow regions where +derivatives of the solution change dramatically. It is well known that the conventional +numerical methods do not work well on either stability or accuracy for such problems. For +example, the standard Galerkin method with continuous linear elements exhibits large +spurious oscillation in the boundary layer region. +Over the decades, many successful +numerical methods have been studied and may be roughly grouped into three categories: +the mesh-fitted approach, the operator-fitted approach, and the stabilization approach. +The mesh-fitted approach utilizes the a priori information of the solution including the +location and the width of the layer to construct a layer-fitted mesh, e.g., the Shishkin +mesh. The operator-fitted approach applies the layer-alike functions as the bases of the +approximation space. +The stabilization approach adds some stabilization term to the +∗Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907- +2067, zcai@math.purdue.edu. This work was supported in part by the National Science Foundation under +grants DMS-1217081 and DMS-1522707. +†Wells +Fargo +Corporate +& +Investment +Banking, +Charlotte, +NC +28202-4200, +binghe.chen@wellsfargo.com. +‡School of Mathematical Science, Peking University, No.5 Yiheyuan Road Haidian District, Beijing, +P.R.China 100871, yangjingmath@pku.edu.cn. +1 +arXiv:2301.11582v1 [math.NA] 27 Jan 2023 + +2 +bilinear form. For example, the well-known streamline upwind Petrov-Galerkin (SUPG) +method [21] adds the original equation tested by the convection term as the stabilization. +For a comprehensive collection of the methods, see [23] and the references therein. +Recently, least-squares methods have been intensively studied for fluid flow and elas- +ticity problems (see, e.g., [5, 7, 8, 9, 12, 14, 15, 16]). The least-squares methods minimize +certain norms of the residual of the first-order system over appropriate finite element +spaces. The method always leads to a symmetric positive definite problem, and choices +of finite element spaces for the primal and dual variables are not subject to the LBB +condition. Moreover, one striking feature of the least-squares method is that the value of +the least-squares functional at the current approximation provides an accurate estimates +of the true error. +The application of the least-squares methods to the convection-dominated diffusion- +reaction problems is still in its infancy. Reported in [17] is a new least-squares formulation +with inflow boundary conditions weakly imposed and outflow boundary conditions ultra- +weakly imposed. This formulation works well on regions away from the boundary layer, +even on coarse meshes. However, it does not resolve the boundary layer, which is the +primary interest of the problem. This phenomena is also observed in the DG method [4], +where the boundary conditions are weakly imposed. These works motivate us to treat +outflow boundary conditions in different fashions. In particular, we study least-squares +method for the convection-dominated diffusion-reaction problem with three different ways +to handle the outflow boundary conditions. The a priori error estimates of finite element +approximations based on these formulations are established. +The solution of the convection-dominated diffusion-reaction problem usually consists +of two parts: the solution of a transport problem (ϵ = 0) and the correction (i.e., the +boundary layer). To compute the first part, it is sufficient to use a coarse mesh, while it +requires a very fine mesh to resolve the boundary layer. Without the a priori information +on locations of the layers, this observation motivates the use of adaptive mesh refinement +algorithm, which has been vastly studied (see, e.g., [2, 3, 6, 13, 19, 24]). However, many a +posteriori error estimators are not suitable for the convection-dominated diffusion-reaction +problems, since they depend on the small diffusion parameter. +To design a robust a +posteriori error estimator is non-trivial. Nevertheless, for a least-squares formulation, the a +posteriori error estimator is handy, which is simply the value of the least-squares functional +at the current approximation. Since the least-squares functional has been computed when +solving the algebraic equation, there is no additional cost. Besides, the reliability and +the efficiency stem easily from the coercivity and the continuity of the bilinear form, +respectively. +In this paper, we present numerical results of adaptive mesh refinement +algorithms using the least-squares estimator. +The rest of this paper is organized as follows. In section 2, we present the convection- +dominated diffusion-reaction problem and its first-order linear system. Based on the first- +order system, three least-squares formulations are introduced and their coercivity are +established in section 3. Section 4 is a computable counterpart of the previous section, +which introduces the computable mesh dependent norms to replace the fractional norms +in the least-squares functionals. +The main objective of section 5 is to establish the a +priori error estimates. The adaptive mesh refinement algorithm and the numerical tests + +3 +are exhibited in section 6 and section 7, respectively. +1.1 +Notation +We use the standard notation and definitions for the Sobolev spaces Hs(Ω)d and Hs(∂Ω)d +for s ≥ 0. The standard associated inner products are denoted by (·, ·)s,Ω and (·, ·)s,∂Ω, +and their respective norms are denoted by ∥·∥s,Ω and ∥·∥s,∂Ω. (We suppress the superscript +d because the dependence on dimension will be clear by context. We also omit the subscript +Ω from the inner product and norm designation when there is no risk of confusion.) For +s = 0, Hs(Ω)d coincides with L2(Ω)d. In this case, the inner product and norm will be +denoted by (·, ·) and ∥ · ∥, respectively. Finally, we define some spaces +H1 +D(Ω) := {q ∈ H1(Ω) : q = 0 on ΓD}, +H1 +D±(Ω) := {q ∈ H1(Ω) : q = 0 on ΓD±}, +and +H(div; Ω) = {v ∈ L2(Ω)2 : ∇ · v ∈ L2(Ω)}, +which is a Hilbert space under the norm +∥v∥H(div; Ω) = +� +∥v∥2 + ∥∇ · v∥2� 1 +2 . +2 +The convection-diffusion-reaction problem +Let Ω be a bounded, open, connected subset in Rd (d = 2, 3) with a Lipschitz continu- +ous boundary ∂Ω. Denote by n = (n1, · · · , nd)t the outward unit vector normal to the +boundary. For a given vector-valued function β, denote by +Γ+ = {x ∈ ∂Ω : β · n(x) > 0} +and +Γ− = {x ∈ ∂Ω : β · n(x) < 0} +the outflow and inflow boundaries, respectively. +Consider the following stationary convection-dominated diffusion-reaction problem: +−ϵ ∆u + β · ∇u + c u = f +in Ω, +(2.1) +where the diffusion coefficient ϵ is a given small constant, i.e., 0 < ϵ ≪ 1; and c and +f are given scalar-valued functions. For simplicity, we consider homogeneous Dirichlet +boundary condition: +u|∂Ω = 0. +(2.2) +For the convection and reaction coefficients, we assume that: +(1) β ∈ W 1 +∞(Ω)d and c ∈ L∞(Ω) with ∥c∥∞ ≤ γ; +(2) there exists a positive constant α0 such that +0 < α0 ≤ c − 1 +2∇ · β +a.e. in Ω. +(2.3) + +4 +Introducing the dual variable +σ = −ϵ1/2∇u, +(2.1) may be rewritten as the following first-order system: +� +σ + ϵ1/2∇u += +0 +in Ω, +ϵ1/2∇ · σ + β · ∇u + c u += +f +in Ω. +(2.4) +3 +Least-squares formulations +In this section, we study three least-squares formulations based on the first-order system in +(2.4) with the inflow boundary conditions imposed strongly. These formulations differ in +how to handle the outflow boundary conditions. More specifically, the outflow boundary +conditions are treated strongly for the first one and weakly for the other two through +weighted boundary functionals. +To this end, introduce the following least-squares functionals: +G1(τ, v; f) += +∥τ + ϵ1/2 ∇v∥2 + ∥ϵ1/2 ∇ · τ + β · ∇v + c v − f∥2, +(3.1) +G2(τ, v; f) += +G1(τ, v; f) + ∥ϵ−1/2 v∥2 +1/2,Γ+, +(3.2) +and +G3(τ, v; f) += +G1(τ, v; f) + ∥v∥2 +1/2,Γ+. +(3.3) +Since ϵ is very small, the outflow boundary conditions are enforced stronger in G2 than in +G3. Let +U1 = H(div; Ω) × H1 +0(Ω) +and +U2 = U3 = H(div; Ω) × H1 +Γ−(Ω). +Then the least-squares formulations are to find (σ, u) ∈ Ui such that +Gi(σ, u; f) = +min +(τ , v)∈ Ui +Gi(τ, v; f) +(3.4) +for i = 1, 2, 3. +For any (τ, v) ∈ Ui, define the following norms: +M1(τ, v) = ∥τ∥2 + ∥v∥2 + ∥ϵ1/2 ∇v∥2, +M2(τ, v) = M1(τ, v) + ∥ϵ−1/2 v∥2 +1/2,Γ+, +and +M3(τ, v) = M1(τ, v) + ∥v∥2 +1/2,Γ+. +Below we show that the homogeneous least-squares functionals are coercive with respect +to the corresponding norms. In particular, the coercivity of the functionals G1 and G2 are +independent of the ϵ. +Theorem 3.1 (Coercivity). For all (τ, v) ∈ Ui with i = 1, 2, 3, there exist positive +constants Ci such that +Mi(τ, v) ≤ Ci Gi(τ, v; 0), +(3.5) +where C1 and C2 are independent of the ϵ and C3 is proportional to ϵ−1/2. + +5 +Proof. We provide proofs for i = 2 and 3 in detail with an emphasis on how the weight in +G2 leads to the coercivity constant independent of the ϵ. The case of i = 1 may be proved +in a similar fashion as the case of i = 2. +For all (τ, v) ∈ Ui with i = 1, 2, 3, the triangle inequality gives +∥τ∥ ≤ ∥τ + ϵ1/2 ∇v∥ + ∥ϵ1/2 ∇v∥ ≤ G1/2 +1 +(τ, v; 0) + ∥ϵ1/2 ∇v∥. +(3.6) +Hence, to show the validity of (3.5), it suffices to prove that +∥v∥2 + ∥ϵ1/2 ∇v∥2 ≤ Ci Gi(τ, v; 0) +∀ (τ, v) ∈ Ui. +(3.7) +To this end, let +I = − +� +ϵ1/2 ∇v, τ +� ++ +� +v, (c − 1 +2 ∇ · β) v +� ++ 1 +2 ∥(β · n)1/2 v∥2 +0,Γ+. +(3.8) +It follows from the definition of the outflow boundary condition and the Cauchy-Schwarz +inequality that +∥ϵ1/2 ∇v∥2 + α0 ∥v∥2 ≤ (ϵ1/2 ∇v, ϵ1/2∇v + τ) + I ≤ ∥ϵ1/2 ∇v∥ G1(τ, v; 0) + I, +which implies +∥ϵ1/2 ∇v∥2 + ∥v∥2 ≤ C (G1(σ, u; 0) + I) . +(3.9) +To bound I, first note that integration by parts and the boundary conditions imply that +(ϵ1/2 ∇v, τ) += +(v, ϵ1/2 τ · n)∂Ω − (ϵ1/2 v, ∇ · τ) = (v, ϵ1/2 τ · n)Γ+ − (ϵ1/2 v, ∇ · τ) += +(v, ϵ1/2 τ · n)Γ+ + (v, c v) − (v, ϵ1/2 ∇ · τ + β · ∇v + c v) + (v β, ∇v) +and that +(∇v, v β) += +1 +2 ∥(β · n)1/2 v∥2 +0,Γ+ − 1 +2 (v, v ∇ · β) . +Combining the above two equalities yields +I = +� +v, ϵ1/2 ∇ · τ + β · ∇v + c v +� +− (v, ϵ1/2 τ · n)Γ+. +(3.10) +By the trace theorem and the Cauchy-Schwarz inequality, we have +∥τ · n∥−1/2,Γ+ +≤ +C +� +∥τ∥ + ∥∇ · τ∥ +� +≤ +C +� +G1/2 +1 +(τ, v; 0) + ∥ϵ1/2 ∇v∥ + ϵ−1/2 ∥β · ∇v∥ + ϵ−1/2 ∥c v∥ +� +≤ +C ϵ−1/2� +G1/2 +1 +(τ, v; 0) + ∥∇v∥ + ∥v∥ +� +. +(3.11) + +6 +Let αi = 1 for i = 2 or 1/2 for i = 3. Then it follows from (3.10), the Cauchy-Schwarz +inequality, the definition of the dual norm, and (3.11) that for i = 2 and 3 +I +≤ +∥v∥ ∥ϵ1/2 ∇ · τ + β · ∇v + c v∥ + ∥ϵ1/2−αi v∥1/2,Γ+ ∥ϵαi τ · n∥−1/2,Γ+ +(3.12) +≤ +C +� +∥v∥ + ∥ϵαi τ · n∥−1/2,Γ+ +� +G1/2 +i +(τ, v; 0) +≤ +C Gi(τ, v; 0) + C +� +∥ϵαi−1/2 ∇v∥ + ∥v∥ +� +G1/2 +i +(τ, v; 0), +which, together with (3.9), implies +∥ϵ1/2 ∇v∥2 + α0 ∥v∥2 ≤ Ci Gi(τ, v; 0) +(3.13) +with C2 independent of ϵ and C3 proportional to ϵ−1/2. This completes the proof of (3.7) +and, hence, (3.5) for i = 2 and 3. +The validity of (3.5) for i = 1 may be established in a similar fashion by noticing that +the boundary term of I in (3.8) vanishes due to the boundary conditions. This completes +the proof of the theorem. +4 +Mesh-dependent least-squares functionals +For computational feasibility, in this section, we replace the 1 +2-norm in the least-squares +functionals defined in (3.2) and (3.3) by mesh-dependent L2-norms. For the simplicity +of presentation, assume that the domain Ω is a convex polygon in the two dimensional +plane. (The extension to the higher dimension is straightforward.) Let Th = {K} be a +triangulation of Ω with triangular elements K of diameter less than or equal to h. Assume +that the triangulation Th is regular and quasi-uniform (see [18]). +Denote by Eh the set of all edges of the triangulation Th. The least-squares functionals +G2 and G3 defined in (3.2) and (3.3) are modified by the following computable least-squares +functionals: +Gh +2(τ, v; f) += +G1(τ, v; f) + +� +e∈Eh∩Γ+ +h−1 +e ∥ϵ−1/2 v∥2 +0,e +(4.1) +and +Gh +3(τ, v; f) += +G1(τ, v; f) + +� +e∈Eh∩Γ+ +h−1 +e ∥v∥2 +0,e, +(4.2) +where he denotes the diameter of the edge e. +For any triangle K ∈ Th, let Pk(K) be the space of polynomials of degree less than or +equal to k on K and denote the local Raviart–Thomas space of index k on K by +RTk(K) = Pk(K)2 + +� x1 +x2 +� +Pk(K). +Then the standard H(div; Ω) conforming Raviart–Thomas space of index k [22] and the +standard (conforming) continuous piecewise polynomials of degree k + 1 are defined, re- +spectively, by +Σk +h = {τ ∈ H(div; Ω) : τ|K ∈ RTk(K), ∀ K ∈ Th}, +(4.3) +V k+1 +h += {v ∈ H1(Ω) : v ∈ Pk+1(K), ∀ K ∈ Th}. +(4.4) + +7 +These spaces have the following approximation properties: let k ≥ 0 be an integer, and +let l ∈ (0, k + 1]: +inf +τ ∈ Σk +h +∥σ − τ∥H(div; Ω) ≤ C hl (∥σ∥l + ∥∇ · σ∥l) +(4.5) +for σ ∈ Hl(Ω)2 ∩ H(div; Ω) with ∇ · σ ∈ Hl(Ω) and +inf +v∈V k+1 +h +∥u − v∥1 ≤ C hl ∥u∥l+1 +(4.6) +for u ∈ Hl+1(Ω). In the subsequent sections, based on the smoothness of σ and u, we will +choose k + 1 to be the smallest integer greater than or equal to l. Since the triangulation +Th is regular, the following inverse inequalities hold for all K ∈ Th: +∥τ∥1,K +≤ +C h−1 +K ∥τ∥K, +∀ τ ∈ RTk(K) +(4.7) +∥v∥1,K +≤ +C h−1 +K ∥v∥K, +∀ v ∈ Pk(K) +(4.8) +with positive constant C independent of hK. +Denote by Uh +i the finite dimensional subspaces of Ui: +Uh +i = +� +Σk +h × V k+1 +h +� +∩ Ui. +(4.9) +For any (τ, v) ∈ Uh +i , define the following norms: +Mh +2 (τ, v) += +M1(τ, v) + +� +e∈Eh∩Γ+ +h−1 +e ∥ϵ−1/2 v∥2 +0,e +and +Mh +3 (τ, v) += +M1(τ, v) + +� +e∈Eh∩Γ+ +h−1 +e ∥v∥2 +0,e. +Below we establish the discrete version of Theorem 3.1, i.e., the coercivity of the discrete +functionals (4.1) and (4.2) with respect to the norms defined above. For the consistence +of notation, we also let Gh +1 = G1 and Mh +1 = M1. +Theorem 4.1. For all (τ, v) ∈ Uh +i with i = 2 and 3, there exist positive constants Ci +independent of ϵ such that +Mh +i (τ, v) ≤ Ci Gh +i (τ, v; 0). +(4.10) +Proof. Similar to the argument in the proof of Theorem 3.1, in order to establish (4.10), +it suffices to show that +∥ϵ1/2 ∇v∥2 + ∥v∥2 ≤ C Gh +i (τ, v; 0) +(4.11) +for all (τ, v) ∈ Uh +i . Moreover, we have +∥ϵ1/2 ∇v∥2 + ∥v∥2 ≤ C +� +Gh +i (τ, v; 0) + I +� +(4.12) + +8 +with I defined in (3.8). +For any e ∈ Eh ∩ Γ+, let e be an edge of element K ∈ Th. It follows from the trace +theorem and the inverse inequality in (4.7) that +he ∥τ · n∥2 +0,e ≤ C he ∥τ∥2 +0,e ≤ C he ∥τ∥0,K∥τ∥1,K ≤ C ∥τ∥2 +0,K, +which, together with (3.6), implies +� +� +� +e∈Eh∩Γ+ +he ∥τ · n∥2 +0,e +� +� +1/2 +≤ C ∥τ∥ ≤ C +� +G1/2 +1 +(τ, v; 0) + ∥ϵ1/2 ∇v∥ +� +. +(4.13) +Let αi = 1 for i = 2 or 1/2 for i = 3. It follows from (3.10), the Cauchy-Schwarz +inequality, and (4.13) that +I += +� +v, ϵ1/2 ∇ · τ + β · ∇v + c v +� +− (v, ϵ1/2 τ · n)Γ+ +≤ +C +� +�∥v∥ + ϵαi +� +� +e∈Eh∩Γ+ +he ∥τ · n∥2 +0,e +�1/2 +� +� Gh +i (τ, v; 0)1/2 +≤ +C Gh +i (τ, v; 0) + C +� +∥v∥ + ∥ϵ1/2∇v∥ +� +Gh +i (τ, v; 0)1/2 +which, together with (4.12), implies the validity of (4.11) and, hence, (4.10). This com- +pletes the proof of the theorem. +Remark 4.2. Note that the coercivity constant C3 in the discrete version is no longer +depending on ϵ, that is better than the continuous version (see Theorem 3.1). +5 +Finite element approximations +The least-squares problems are to find (σ, u) ∈ Ui (i = 1, 2, 3) such that +Gh +i (σ, u; f) = +min +(τ , v)∈ Ui +Gh +i (τ, v; f). +(5.1) +The corresponding variational problems are to find (σ, u) ∈ Ui such that +ai(σ, u; τ, v) = Fi(τ, v), +∀ (τ, v) ∈ Ui, +(5.2) +where the bilinear forms ai(· ; ·) are symmetric and given by +a1(σ, u; τ, v) += +(σ + ϵ1/2 ∇u, τ + ϵ1/2 ∇v) ++(ϵ1/2 ∇ · σ + β · ∇u + c u, ϵ1/2 ∇ · τ + β · ∇v + c v), +a2(σ, u; τ, v) += +a1(σ, u; τ, v) + +� +e ∈Eh∩Γ+ +h−1 +e +ϵ−1 (u, v)0,e, +a3(σ, u; τ, v) += +a1(σ, u; τ, v) + +� +e ∈Eh∩Γ+ +h−1 +e +(u, v)0,e, + +9 +and the linear forms Fi(·) are given by +Fi(τ, v) = (f, ϵ1/2 ∇ · τ + β · ∇v + c v) +for i = 1, 2, 3. +The least-squares finite element approximations to the variational problems in (5.2) +are to find (σi +h, ui +h) ∈ Uh +i such that +ai(σi +h, ui +h; τ, v) = Fi(τ, v), +∀ (τ, v) ∈ Uh +i , +(5.3) +for i = 1, 2, 3. Taking the difference between (5.2) and (5.3) implies the following orthog- +onality: +ai(σ − σi +h, u − ui +h; τ, v) = 0, +∀ (τ, v) ∈ Uh +i . +(5.4) +In the rest of this section, we consider a stronger norm which incorporates the norm +of the streamline derivative: +|||(τ, v)|||2 +i = Mh +i (τ, v) + +� +K∈Th +δK ∥β · ∇v∥2 +K, +where δK is a positive constant to be determined. In the following lemma, we show that +Gh +i (σ, u; 0) are also elliptic with respect to these norms if the δK is appropriately chosen. +Lemma 5.1. For all K ∈ Th, assume that 0 < δK ≤ min{h2 +K/ϵ, C}, then there exist +positive constants Ci independent of ϵ such that +|||(τ, v)|||2 +i ≤ Ci Gh +i (τ, v; 0), +∀ (τ, v) ∈ Uh +i , +i = 1, 2, 3. +Proof. By Theorems 3.1 and 4.1, to prove the validity of the lemma, it suffices to show +that +� +K∈Th +δK ∥β · ∇v∥2 +K ≤ Ci Gh +i (τ, v; 0). +(5.5) +To this end, note the facts that +δK ≤ C +and +δK ϵ +h2 +K +≤ min +� +1, C ϵ +h2 +K +� +≤ C. +Now it follows from the Cauchy-Schwarz inequality and the inverse inequality in (4.7) that +� +K∈Th +δK ∥β · ∇v∥2 +K +≤ +C +� +K∈Th +δK +� +Gh +1,K (τ, v; 0) + ∥ϵ1/2 ∇ · τ∥2 +K + ∥c v∥2 +K +� +≤ +C +� +K∈Th +� +Gh +1,K (τ, v; 0) + δK ϵ +h2 +K +∥τ∥2 +K + ∥v∥2 +K +� +≤ +C +� +Gh +1 (τ, v; 0) + ∥τ∥2 + ∥v∥2� +≤ C Gh +i (τ, v; 0), +which establishes (5.5) and hence completes the proof of the lemma. + +10 +To choose δK properly, first define the local mesh P´eclet number by +PeK = ∥β∥0,∞,K hK +2 ϵ +, +then partition the triangulation Th into two subsets: +T c +h = {K ∈ Th : PeK > 1} +and +T d +h = {K ∈ Th : PeK ≤ 1}. +(5.6) +The elements in T c +h are referred to the convection-dominated elements, while the elements +in T d +h the diffusion-dominated elements. Now, the δK is chosen to be +δK = +� +� +� +� +� +� +� +� +� +� +� +2 hK +∥β∥0,∞,K +, +if K ∈ T c +h , +h2 +K +ϵ , +if K ∈ T d +h . +(5.7) +Remark 5.2. The δK defined in (5.7) satisfies the assumption in Lemma 5.1, i.e., +δK ≤ min{h2 +K/ϵ, C}. +(5.8) +Proof. Since ∥β∥0,∞,K is large comparing to hK, we have +2 hK +∥β∥0,∞,K +≤ C. +(5.9) +For any K ∈ T c +h , the fact that PeK > 1 implies +2 hK +∥β∥0,∞,K +< h2 +K +ϵ , +which, together with (5.9), yields (5.8). For any K ∈ T d +h , (5.8) is again a consequence of +the definition of δK in (5.7), the fact that PeK ≤ 1, and (5.9). +Denote by T ∂ +h the set of elements that intersect the outflow boundary nontrivially, i.e., +T ∂ +h = {K ∈ Th : meas( ¯K ∩ Γ+) > 0}. +In this paper, we assume that +T ∂ +h ⊂ T d +h . +(5.10) +For any K ∈ T d +h , the fact that PeK ≤ 1 implies +hK < +2 ϵ +∥β∥0,∞,K +. +Hence, assumption (5.10) means that the mesh size in the boundary layer region is com- +parable to the perturbation parameter ϵ. + +11 +Theorem 5.3. Let (σ, u) be the solution of (5.2). Assume that (σ, u) ∈ Hl(Ω)2×Hl+1(Ω) +and that ∇ · σ ∈ Hl(Ω). Let (σi +h, ui +h), i = 1, 2, 3, be the solution of (5.3) with k = l. +Under the assumption in (5.10), we have the following a priori error estimation: +Ci +������(σ − σi +h, u − ui +h) +������2 +i +≤ +� +K∈T c +h +h2l−1 +K +� +ϵ ∥∇ · σ∥2 +l,K + hK ∥σ∥2 +l,K + ∥u∥2 +l+1,K +� ++ +� +K∈T d +h +h2l−1 +K +� ϵ2 +hK +∥∇ · σ∥2 +l,K + hK ∥σ∥2 +l,K + ϵ +hK +∥u∥2 +l+1,K +� +, +(5.11) +where constants Ci > 0 are independent of ϵ. +Proof. We provide proof of (5.11) only for i = 2 and 3 since (5.11) may be obtained in a +similar fashion. +To this end, let σI and uI be the interpolants of σ and u, respectively, such that the +approximation properties in (4.5) and (4.6) hold and that +(∇ · (σ − σI), v) = 0, +∀ v ∈ Dh +k, +(5.12) +where Dh +k = {v ∈ L2(Ω) : v|K ∈ Pk(K) ∀ K ∈ Th} is the space of discontinuous piecewise +polynomials of degree less than or equal to k ≥ 0. Let +EI = σ − σI, +Ei +h = σI − σi +h, +eI = u − uI, +and +ei +h = uI − ui +h. +Since Ei = σ − σi +h = EI + Ei +h and ei = u − ui +h = eI + ei +h, the triangle inequality gives +������(Ei, ei) +������ +i ≤ |||(EI, eI)|||i + +������(Ei +h, ei +h) +������ +i. +(5.13) +Let αi = −1 or 0 for i = 2, 3. By approximation property (4.6) and assumption (5.10), +we have +� +e ∈Eh∩Γ+ +h−1 +e +ϵαi ∥eI∥2 +0,e ≤ C +� +K∈T ∂ +h +h2l +K ϵαi ∥u∥2 +l+1,K ≤ C +� +K∈T ∂ +h +h2l+αi +K +∥u∥2 +l+1,K. +Now, it follows from (4.5), (4.6), the trace theorem, and the fact δK ≤ C that +|||(EI, eI)|||2 +i +≤ +C +� +�∥EI∥2 + ∥eI∥2 + ∥ϵ1/2 ∇eI∥2 + +� +e∈Γ+ +h−1 +e +ϵαi ∥eI∥2 +e + +� +K∈Th +∥β · ∇eI∥2 +K +� +� +≤ +C +� +� � +K∈Th +h2l +K ∥σ∥2 +l,K + +� +K∈Th +h2l +K ∥u∥2 +l+1,K + +� +K∈T ∂ +h +h2l+αi +K +∥u∥2 +l+1,K +� +� . +(5.14) + +12 +To bound the second term of the right-hand side in (5.13), by Lemma 5.1 and orthog- +onality (5.4), we have +Ci +������(Ei +h, ei +h) +������2 +i ≤ ai(Ei +h, ei +h; Ei +h, ei +h) = ai(Ei +h, ei +h; −EI, −eI) ≡ Ii +1 + Ii +2 + Ii +3 + Ii +4, (5.15) +where +Ii +1 += +(c ei +h, −ϵ1/2 ∇ · EI − β · ∇eI − c eI) + (Ei +h + ϵ1/2 ∇ei +h, −EI − ϵ1/2 ∇eI), +Ii +2 += +(ϵ1/2 ∇ · Ei +h, −ϵ1/2 ∇ · EI − β · ∇eI − c eI), +Ii +3 += +(β · ∇ei +h, −ϵ1/2 ∇ · EI − β · ∇eI − c eI), +and +Ii +4 += +� +e ∈Eh∩Γ+ +h−1 +e +ϵαi (ei +h, −eI)0,e. +It follows from the triangle and Cauchy-Schwarz inequalities, (4.5), and (4.6) that +Ii +1 +≤ C ∥ei +h∥ +� +∥ϵ1/2∇ · EI∥ + ∥∇eI∥ + ∥eI∥ +� ++ C +� +∥Ei +h∥ + ∥ϵ1/2∇ei +h∥ +� � +∥EI∥ + ∥ϵ1/2∇eI)∥ +� +≤C +� +∥ei +h∥ + ∥Ei +h∥ + ∥ϵ1/2∇ei +h∥ +� +� +� � +K∈Th +h2l +K +� +ϵ∥∇ · σ∥2 +l,K + ∥σ∥2 +l,K + ∥u∥2 +l+1,K +� +� +� +1/2 +. (5.16) +By (5.12), the Cauchy-Schwarz and triangle inequalities, and the inverse inequality in +(4.7), we have +Ii +2 = −(ϵ1/2 ∇ · Ei +h, β · ∇eI + c eI), +≤ C +� +K∈Th +ϵ1/2 +hK +∥Ei +h∥K +� +∥∇eI∥K + ∥eI∥K +� +≤ C ∥Ei +h∥ +� +� � +K∈Th +ϵ h2l−2 +K +∥u∥2 +l+1,K +� +� +1/2 +. (5.17) +By the Cauchy-Schwarz and the triangle inequalities, I3 is bounded by +Ii +3 ≤ C +� +K∈Th +∥β · ∇ei +h∥K +� +ϵ1/2 ∥∇ · EI∥K + ∥∇eI∥K + ∥eI∥K +� +≤ +C +� +K∈Th +∥β · ∇ei +h∥K +� +ϵ1/2 hl +K ∥∇ · σ∥l,K + hl +K ∥u∥l+1,K +� +≤ C +� +� � +K∈Th +δK∥β · ∇ei +h∥2 +K +� +� +1/2� +� � +K∈Th +δ−1 +K +� +ϵ h2l +K ∥∇ · σ∥2 +l,K + h2l +K ∥u∥2 +l+1,K +� +� +� +1/2 +.(5.18) + +13 +For Ii +4, it follows from the Cauchy-Schwarz inequality and the trace theorem that +Ii +4 +≤ +C +� +� +� +e ∈Eh∩Γ+ +h−1 +e +ϵαi ∥ei +h∥2 +0,e +� +� +1/2 � +� +� +e ∈Eh∩Γ+ +h−1 +e +ϵαi ∥eI∥2 +0,e +� +� +1/2 +≤ +C +� +� +� +e ∈Eh∩Γ+ +h−1 +e +ϵαi ∥ei +h∥2 +0,e +� +� +1/2 � +� � +K∈T ∂ +h +h2l+αi +K +∥u∥2 +l+1,K +� +� +1/2 +. +(5.19) +Combining (5.15), (5.16), (5.17), (5.18), (5.19), and (5.8), we have +Ci +������(Ei +h, ei +h) +������2 +i +≤ +� +K∈Th +h2l +K∥σ∥2 +l,K + +� +K∈Th +� +1 + δ−1 +K +� +ϵ h2l +K ∥∇ · σ∥2 +l,K + +� +K∈T ∂ +h +h2l+αi +K +∥u∥2 +l+1,K ++ +� +K∈Th +� +1 + ϵ h−2 +K + δ−1 +K +� +h2l +K∥u∥2 +l+1,K +≤ +� +K∈Th +�ϵ h2l +K +δK +∥∇ · σ∥2 +l,K + h2l +K ∥σ∥2 +l,K + h2l +K +δK +∥u∥2 +l+1,K +� ++ +� +K∈T ∂ +h +h2l+αi +K +∥u∥2 +l+1,K, +which, together with the definition of δK in (5.7), implies +Ci +������(Ei +h, ei +h) +������2 +i +≤ +� +K∈T c +h +h2l−1 +K +� +ϵ ∥∇ · σ∥2 +l,K + hK ∥σ∥2 +l,K + ∥u∥2 +l+1,K +� ++ +� +K∈T d +h +h2l−1 +K +� ϵ2 +hK +∥∇ · σ∥2 +l,K + hK ∥σ∥2 +l,K + ϵ +hK +∥u∥2 +l+1,K +� +. +Now, (5.11) is a consequence of (5.13) and (5.14). This completes the proof of the theorem. +Note that the a priori error estimate in Theorem 5.3 is not optimal. This is because +the coercivity of the homogeneous least-squares functionals in Lemma 5.1 are established +in a norm that is weaker than the norm used for the continuity of the functionals. To +restore the full order of convergence, one may use piecewise polynomials of degree l + 1 to +approximate u. +Theorem 5.4. Let (σi +h, ui +h), i = 1, 2, 3, be the solution of (5.3) with Uh +i = (Σl +h×V l+1 +h +)∩ Ui. + +14 +Under the assumption of Theorem 5.3, we have the following a priori error estimation: +Ci +������(σ − σi +h, u − ui +h) +������2 +i +≤ +� +K∈T c +h +h2l +K +� +∥∇ · σ∥2 +l,K + ∥σ∥2 +l,K + hK ∥u∥2 +l+2,K +� ++ +� +K∈T d +h +h2l +K +� ϵ2 +h2 +K +∥∇ · σ∥2 +l,K + ∥σ∥2 +l,K + ϵ ∥u∥2 +l+2,K +� +, +(5.20) +where constants Ci > 0 are independent of ϵ. +Proof. The a priori error estimate in (5.20) may be obtained in a similar fashion by noting +that +∥u − uI∥1 ≤ C hl+1∥u∥l+2. +6 +Adaptive algorithm +Asymptotic analysis (see, e.g., [20]) shows that the solution of a convection-dominated +diffusion-reaction problem consists of two parts: the solution of the reduced equation +(ϵ = 0) and the correction, i.e., the boundary or interior layers. +The boundary and +interior layers are narrow regions where derivatives of the solution change dramatically. +For example, for the following problem [20]: +� +� +� +� +� +−ϵ ∆u + ∂u +∂y = f +in Ω = (0, 1)2, +u = 0 +on ∂Ω, +the exponential layer is of width O(ϵ) at y = 1, and the width of the parabolic boundary +layers is O(ϵ1/2) at both x = 0 and x = 1. +Therefore, two sets of largely different +scales exist simultaneously in the convection-dominated diffusion problem, and hence it is +difficult computationally. +On the one hand, one can apply the small scale over the entire domain, i.e., to use +uniform fine meshes. With such a fine mesh, the standard Galerkin finite element method +can also produce a good approximation. However, it is computationally inefficient due to +the small region of the boundary and/or interior layers. On the other hand, one can use +the large scale over the entire domain. If the outflow boundary conditions are imposed +strongly, the numerical solution (away from the boundary layers) will be polluted. In +contrast, if the outflow boundary conditions are imposed weakly, the boundary layers can +not be resolved (see, e.g., numerical results in [4, 17]). +Neither of the above two approaches leads to a satisfactory numerical scheme. The fail- +ure is due to the fact that these approaches ignore this intrinsic property of the convection- +dominated diffusion problem. In contrast, the Shishkin mesh is aware of and respect it. + +15 +Basically, the Shishkin mesh is a piecewise uniform mesh. In the diffusion-dominated re- +gion where the layers stand, it is a fine mesh suitable to the layer and in the convective +region, it turns to be a coarse mesh. The disadvantage of the Shishkin mesh is that it +needs the a priori information of the solution, such as the location and the width of the +layer, in order to construct a mesh of high quality. However, this information is not always +available in advance, especially, for a complex problem. +Based on the above considerations, we employ adaptive least-squares finite element +methods. The least-squares estimators are simply defined as the value of the least-squares +functionals at the current approximation. To this end, for each element K ∈ Th, denote +the local least-squares functionals by +Gh +1,K(τ, v; f) += +∥τ + ϵ1/2 ∇v∥2 +K + ∥ϵ1/2 ∇ · τ + β · ∇v + c v − f∥2 +K, +Gh +2,K(τ, v; f) += +� +� +� +� +� +Gh +1,K(τ, v; f), +if K ∩ Γ+ = ∅, +Gh +1,K(τ, v; f) + +� +e∈K∩Γ+ +h−1 +e ∥ϵ−1/2v∥2 +0, e, +otherwise, +and Gh +3,K(τ, v; f) += +� +� +� +� +� +Gh +1,K(τ, v; f), +if K ∩ Γ+ = ∅, +Gh +1,K(τ, v; f) + +� +e∈K∩Γ+ +h−1 +e ∥v∥2 +0, e, +otherwise. +Let (ˆσh +i , ˆuh +i ) be the current approximations to the solutions of (5.3) for i = 1, 2, 3. Then +the least-squares indicators are simply the square root of the value of the local least-squares +functionals at the current approximation: +ηi +K = Gh +i,K (ˆσi +h, ˆui +h; f)1/2 +(6.1) +for all K ∈ Th and for i = 1, 2, 3. The least-squares estimators are +ηi = +� +� � +K∈Th +� +ηi +K +�2 +� +� +1/2 += Gh +i (ˆσi +h, ˆui +h; f)1/2 +(6.2) +for i = 1, 2, 3. +Let (σ, u) be the solution of (5.2) and denote the true errors by +ˆEi = σ − ˆσi +h +and +ˆei = u − ˆu1 +h +for +i = 1, 2, 3. +Theorem 6.1. There exist positive constants Ce,1 and Cr,1 independent of ϵ such that +η1 +K ≤ Ce,1 +� +M1,K(ˆE1, ˆe1) + ∥β · ∇ ˆe1∥2 +K + ϵ ∥∇ · ˆE1∥2 +K +�1/2 +(6.3) +for all K ∈ T and that +M1(ˆE1, ˆe1)1/2 ≤ Cr,1 η1. +(6.4) + +16 +Proof. Since the exact solution (σ, u) satisfies (2.4), we have +� +η1 +K +�2 = Gh +1,K(ˆE1, ˆe1; 0) +and +� +η1�2 = Gh +1(ˆE1, ˆe1; 0). +which, together with the triangle inequality and Theorem 3.1, imply the efficiency and the +reliability bounds, respectively. +Theorem 6.2. There exist positive constants Ce,i independent of ϵ such that +Ce, i +� +ηi +K +�2 ≤ Mh +i,K(ˆEi, ˆei) + ∥β · ∇ˆei∥2 +K + ϵ ∥∇ · ˆEi∥2 +(6.5) +for all K ∈ T and i = 2, 3. +Proof. Let αi = −1 for i = 2 or 0 for i = 3. With the fact that (σ, u) is the exact solution +satisfying (2.4), we have +ηi(ˆσh +i , ˆuh +i )2 = Gh +i (ˆσh +i , ˆuh +i ; f) += +∥ˆσh +i + ϵ1/2 ∇ˆuh +i ∥2 + ∥ϵ1/2 ∇ · ˆσh +i + β · ∇ˆuh +i + c ˆuh +i − f∥2 + +� +e∈Eh∩Γ+ +ϵαi h−1 +e ∥ˆuh +i ∥2 +0,e += +∥ˆEi + ϵ1/2 ∇ˆei∥2 + ∥ϵ1/2 ∇ · ˆEi + β · ∇ˆei + c ˆei∥2 + +� +e∈Eh∩Γ+ +ϵαi h−1 +e ∥ˆei∥ += +Gh +i (ˆEi, ˆei; 0), +(6.6) +with which, the efficiency bound simply follows from (6.6) and the Cauchy-Schwarz in- +equality. +In the remainder of this section, we describe the standard adaptive mesh refinement +algorithm. Starting with an initial triangulation T0, a sequence of nested triangulations +{Tl} is generated through the well known AFEM-Loop: +SOLVE −→ ESTIMATE −→ MARK −→ REFINE. +The SOLVE step solves (5.3) in the finite element space corresponding to the mesh +Tl for a numerical approximation (σi +h(l), ui +h(l)) ∈ Uh +i (l), where Uh +i (l) is the finite element +space defined on Tl. Hereafter, we shall explicitly express the dependence of a quantity +on the level l by either the subscript like Tl or the variable like Uh +i (l). +The ESTIMATE step computes the indicators {ηi +K(l)} and the estimator ηi(l) defined +in (6.1) and (6.2), respectively. +The way to choose elements for refinement influences the efficiency of the adaptive +algorithm. If most of elements are marked for refinement, then it is comparable to uniform +refinement, which does not take full advantage of the adaptive algorithm and results in +redundant degrees of freedom. On the other hand, if few elements are refined, then it +requires many iterations, which undermines the efficiency of the adaptive algorithm, since +each iteration is costly. For the singularly perturbed problems, it is well known that the +indicators associated with the elements in the layer region are much larger than others. + +17 +Therefore, we MARK by the maximum algorithm, which defines the set ˆTl of marked +elements such that for all K ∈ ˆTl +ηi +K(l) ≥ θ max +K∈Tl ηi +K(l). +The REFINE step is to bisect all the triangles in ˆTl into two sub-triangles to generate +a new triangulation Tl+1. Note that some triangles in Tl \ ˆTl adjacent to triangles in ˆTl are +also refined in order to avoid hanging nodes. +In summary, the adaptive least-squares finite element algorithm can be cast as follows: +with the initial mesh T0, marking parameter θ ∈ (0, 1), and the maximal number of +iteration maxIt, for l = 0, 1, · · · , maxIt, do +(1) (σi +h(l), ui +h(l)) = SOLVE(Tl); +(2) {ηi +K(l)} = ESTIMATE(Tl, σi +h(l), ui +h(l)); +(3) ˆTl = MARK(Tl, {ηi +K(l)}); +(4) Tl+1 = REFINE(Tl, ˆTl). +7 +Numerical experiments +In this section, we conduct several numerical experiments on two model problems used by +many authors (see, e.g., [4, 17]). Both the model problems are defined in the unit square +and all numerical experiments are started with the same initial mesh, which consists of +sixteen isosceles right triangles. The marking parameter θ is chosen to be 0.6. +7.1 +Boundary layer +In this example, β = [1, 1]T , and c = 0, and the external force f is chosen such that the +exact solution is +u(x, y) = sin πx +2 + sin πy +2 +� +1 − sin πx +2 +� ++ e−1/ϵ − e−(1−x)(1−y)/ϵ +1 − e−1/ϵ +. +This solution is smooth, but develops boundary layers at x = 1 and y = 1 with width +O(ϵ). This example is suitable for testing capability of the numerical approximations on +resolving the boundary layers. +In this numerical experiment, ϵ = 10−3. Given the tolerance tol = 0.5, computation is +terminated if +ηi(l) ≤ tol. +(7.1) +Since the exact solution is available, the true error is computed and the effectivity index +is defined as follows: +eff-index := +ηi(σi +h, ui +h) +������(σ − σi +h, u − ui +h) +������ +i +. +(7.2) + +18 +Figure 1: The final meshes and the numerical solutions are, respectively, displayed in the +first and the second columns and the rows are corresponding to i = 1, 2, 3. +The final meshes are displayed in the first column of Figure 1 when the stopping criterion +(7.1) is satisfied. They clearly show that the refinements cluster around the boundary +layer area. The numerical solutions on the final meshes are depicted in the second column +of Figure 1. All the three methods successfully capture the sharp boundary layers, and +no visible oscillation appears in the numerical solutions. +Reported in Figure 2 is the + +0.9 . +0.5 +0.1 . +0.2 .. +A +: +0.6 +: -? +0.8 +0.4 +0.b +0.2 +... +0.4 +c.2 +00.5 . +0.5 . +0.4 . +0.2 +0.2 +0.6 +0.8 +0.4 +0.6 +0.2 +0.4 +c.2 +00.5 . +0.5 . +0.4 . +0.2 +5 +0.2 +0.6 +0.8 +0.4 +0.6 +0.2 +0.4 +c.2 +019 +convergence rates of the numerical solutions. The errors with the norm |||·|||i that are +used in the a priori error estimate are tracked, which converge in the order of (DoF)−1. +Moreover, the convergence rate is independent of the value of ϵ. This is also verified by +the test problem with ϵ = 10−4, where the convergence rate does not deteriorate (see the +second column of Figure 2). +Figure 2: The convergence rates corresponding to ϵ = 10−3 and 10−4 are displayed in the +first and the second columns, respectively, and the rows are corresponding to i = 1, 2, 3. + +10° +10° +10 +10° +10 +10* +10° +errEne3 +estimator +DoF-1 +effindex +10 +102 +103 +104 +105 +10° +Degree of Freedom10° +10 +00.000 +10° +10~ +10° +10 +10 +errEne +estimator +10 +DoF-1 +effindex +10° +102 +103 +104 +105 +10° +Degree of Freedom10° +102 +10° +DODO +00: +080:00 +10° +10 +10* +10° +10 +errEne +estimator +10 +DoF-1 +effindex +10 +102 +103 +104 +105 +10° +Degree of Freedom10 +10 +0000:: +80:8 +10 +10~ +10* +10 +10 +erEne +estimator +10 +DoF-1 +effindex +10 +102 +103 +104 +105 +10° +Degree of Freedom10 +10 +10 +10° +10 +10° +10 +errEne +estimator +10 +DoF-1 +effindex +0 +10 +102 +103 +10* +105 +10° +Degree of Freedom10° +10 +：*黑 +%08 +00:000 +10° +10~ +10* +10 +10 +errEne +estimator +10 +DoF-1 +effindex +10° +102 +103 +104 +105 +10° +Degree of Freedom20 +7.2 +Interior layer +In the second example, β = [1/2, +√ +3/2]T , c = 0, f = 0, and the boundary condition is +u = +� +� +� +� +� +� +� +1, +on {(x, y) : y = 0, 0 ≤ x ≤ 1}, +1, +on {(x, y) : x = 0, 0 ≤ y ≤ 1/5}, +0, +otherwise. +The exact solution of the problem remains unknown. However, it is known that, additional +to the boundary layers, the solution develops an interior layer along the line y = +√ +3 x+0.2 +due to the discontinuity at (0, 0.2) of the boundary condition. The problem is chosen to +test whether the formulations can capture the interior layers. +Figure 3 shows that all the three methods capture both the boundary and the interior +layers. Moreover, the numerical solutions do not exhibit any visible oscillation, which is +much better than the results reported in [4]. +Figure 3: Numerical solutions corresponding to i = 1, 2, 3 from left to right. +Acknowledgements +We thank Dr. Shuhao Cao for the discussion and helpful suggestions on the computation +of the test problems. +References +[1] D.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. +[2] L. Angermann, Balanced a posteriori error estimates for finite volume type dis- +cretizations of convection-dominated elliptic problems, Computing, 55:4 (1995), 305- +323. 1 +[3] M. Anisworth, A. 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Wang, Least-squares methods for incompressible newtonian +fluid flow: linear stationary problems, SIAM J. Numer. Anal., 42 (2004), 843-859. 1 +[15] Z. Cai, T. Manteuffel, and S. McCormick, First-order system least squares +for velocity-vorticity- pressure form of the Stokes equations, with application to linear +elasticity, Electron. Trans. Numer. Anal., 3 (1995), 150-159. 1 +[16] Z. Cai, T.A. Manteuffel, and S.F. McCormick, First-order system least +squares for the Stokes equations, with application to linear elasticity, SIAM J. Numer. +Anal., 34 (1997), 1727-1741. 1 +[17] H. Chen, G. Fu, J. Li, and W. Qiu, First order least squares method with weakly +imposed boundary condition for convection dominated diffusion problems. Comput. +Math. Appl, 68 (2014), 1635-1652. 1, 6, 7 +[18] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, +Amsterdam, 1978. 4 + +22 +[19] W.D¨orfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Nu- +meri. Anal., 33 (1996), 1106-1124. 1 +[20] W. Eckhaus, Asymptotic Analysis of Singular Perturbations, North-Holland, Ams- +terdam, 1979. 6 +[21] T.J.R. Hughes and A. Brooks, Streamline upwind/Petrov Galerkin formulations +for the convection dominated flows with particular emphasis on the incompressible +Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 54 (1982), 199-259. +1 +[22] P.A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order +elliptic problems, in Mathematical Aspects of Finite Element Methods, Lecture Notes +in Math. 606, I. Galligani and E. Magenes, eds., Springer, New York, 1977, 292-315. +4 +[23] H. Roos, M. Stynes, and L. Tobiska, Robust Numerical Methods for Singularly +Perturbed Differential Equations, Springer, Berlin, 2008. 1 +[24] R. Verfurth, A posteriori error estimation and adaptive mesh-refinement tech- +niques, J. Comput. Appl. Math., 50 (1994), 67-83. 1 + diff --git a/59FJT4oBgHgl3EQflCzy/content/tmp_files/load_file.txt b/59FJT4oBgHgl3EQflCzy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a586ebe71f59b33dd83f36d649de10671feabfd9 --- /dev/null +++ b/59FJT4oBgHgl3EQflCzy/content/tmp_files/load_file.txt @@ -0,0 +1,866 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf,len=865 +page_content='Adaptive Least-Squares Methods for Convection-Dominated Diffusion-Reaction Problems Zhiqiang Cai∗ Binghe Chen† Jing Yang‡ Abstract This paper studies adaptive least-squares finite element methods for convection- dominated diffusion-reaction problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The least-squares methods are based on the first-order system of the primal and dual variables with various ways of imposing outflow boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The coercivity of the homogeneous least-squares func- tionals are established, and the a priori error estimates of the least-squares methods are obtained in a norm that incorporates the streamline derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' All methods have the same convergence rate provided that meshes in the layer regions are fine enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' To increase computational accuracy and reduce computational cost, adaptive least- squares methods are implemented and numerical results are presented for some test problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' ADAPTIVE FOSLS FOR THE CONVECTION-DOMINATED PROBLEMS 1 Introduction Due to the small diffusion coefficient, the solution of the convection-dominated diffusion- reaction problem develops the boundary or interior layers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', narrow regions where derivatives of the solution change dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' It is well known that the conventional numerical methods do not work well on either stability or accuracy for such problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For example, the standard Galerkin method with continuous linear elements exhibits large spurious oscillation in the boundary layer region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Over the decades, many successful numerical methods have been studied and may be roughly grouped into three categories: the mesh-fitted approach, the operator-fitted approach, and the stabilization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The mesh-fitted approach utilizes the a priori information of the solution including the location and the width of the layer to construct a layer-fitted mesh, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', the Shishkin mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The operator-fitted approach applies the layer-alike functions as the bases of the approximation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The stabilization approach adds some stabilization term to the ∗Department of Mathematics, Purdue University, 150 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' University Street, West Lafayette, IN 47907- 2067, zcai@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This work was supported in part by the National Science Foundation under grants DMS-1217081 and DMS-1522707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' †Wells Fargo Corporate & Investment Banking, Charlotte, NC 28202-4200, binghe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='chen@wellsfargo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' ‡School of Mathematical Science, Peking University, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5 Yiheyuan Road Haidian District, Beijing, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='China 100871, yangjingmath@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11582v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='NA] 27 Jan 2023 2 bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For example, the well-known streamline upwind Petrov-Galerkin (SUPG) method [21] adds the original equation tested by the convection term as the stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For a comprehensive collection of the methods, see [23] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Recently, least-squares methods have been intensively studied for fluid flow and elas- ticity problems (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', [5, 7, 8, 9, 12, 14, 15, 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The least-squares methods minimize certain norms of the residual of the first-order system over appropriate finite element spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The method always leads to a symmetric positive definite problem, and choices of finite element spaces for the primal and dual variables are not subject to the LBB condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Moreover, one striking feature of the least-squares method is that the value of the least-squares functional at the current approximation provides an accurate estimates of the true error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The application of the least-squares methods to the convection-dominated diffusion- reaction problems is still in its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Reported in [17] is a new least-squares formulation with inflow boundary conditions weakly imposed and outflow boundary conditions ultra- weakly imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This formulation works well on regions away from the boundary layer, even on coarse meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' However, it does not resolve the boundary layer, which is the primary interest of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This phenomena is also observed in the DG method [4], where the boundary conditions are weakly imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' These works motivate us to treat outflow boundary conditions in different fashions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In particular, we study least-squares method for the convection-dominated diffusion-reaction problem with three different ways to handle the outflow boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The a priori error estimates of finite element approximations based on these formulations are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The solution of the convection-dominated diffusion-reaction problem usually consists of two parts: the solution of a transport problem (ϵ = 0) and the correction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', the boundary layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' To compute the first part, it is sufficient to use a coarse mesh, while it requires a very fine mesh to resolve the boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Without the a priori information on locations of the layers, this observation motivates the use of adaptive mesh refinement algorithm, which has been vastly studied (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', [2, 3, 6, 13, 19, 24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' However, many a posteriori error estimators are not suitable for the convection-dominated diffusion-reaction problems, since they depend on the small diffusion parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' To design a robust a posteriori error estimator is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Nevertheless, for a least-squares formulation, the a posteriori error estimator is handy, which is simply the value of the least-squares functional at the current approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Since the least-squares functional has been computed when solving the algebraic equation, there is no additional cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Besides, the reliability and the efficiency stem easily from the coercivity and the continuity of the bilinear form, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In this paper, we present numerical results of adaptive mesh refinement algorithms using the least-squares estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In section 2, we present the convection- dominated diffusion-reaction problem and its first-order linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Based on the first- order system, three least-squares formulations are introduced and their coercivity are established in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Section 4 is a computable counterpart of the previous section, which introduces the computable mesh dependent norms to replace the fractional norms in the least-squares functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The main objective of section 5 is to establish the a priori error estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The adaptive mesh refinement algorithm and the numerical tests 3 are exhibited in section 6 and section 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1 Notation We use the standard notation and definitions for the Sobolev spaces Hs(Ω)d and Hs(∂Ω)d for s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The standard associated inner products are denoted by (·, ·)s,Ω and (·, ·)s,∂Ω, and their respective norms are denoted by ∥·∥s,Ω and ∥·∥s,∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (We suppress the superscript d because the dependence on dimension will be clear by context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' We also omit the subscript Ω from the inner product and norm designation when there is no risk of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=') For s = 0, Hs(Ω)d coincides with L2(Ω)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In this case, the inner product and norm will be denoted by (·, ·) and ∥ · ∥, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Finally, we define some spaces H1 D(Ω) := {q ∈ H1(Ω) : q = 0 on ΓD}, H1 D±(Ω) := {q ∈ H1(Ω) : q = 0 on ΓD±}, and H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ω) = {v ∈ L2(Ω)2 : ∇ · v ∈ L2(Ω)}, which is a Hilbert space under the norm ∥v∥H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ω) = � ∥v∥2 + ∥∇ · v∥2� 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 2 The convection-diffusion-reaction problem Let Ω be a bounded, open, connected subset in Rd (d = 2, 3) with a Lipschitz continu- ous boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Denote by n = (n1, · · · , nd)t the outward unit vector normal to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For a given vector-valued function β, denote by Γ+ = {x ∈ ∂Ω : β · n(x) > 0} and Γ− = {x ∈ ∂Ω : β · n(x) < 0} the outflow and inflow boundaries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Consider the following stationary convection-dominated diffusion-reaction problem: −ϵ ∆u + β · ∇u + c u = f in Ω, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) where the diffusion coefficient ϵ is a given small constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', 0 < ϵ ≪ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' and c and f are given scalar-valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For simplicity, we consider homogeneous Dirichlet boundary condition: u|∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) For the convection and reaction coefficients, we assume that: (1) β ∈ W 1 ∞(Ω)d and c ∈ L∞(Ω) with ∥c∥∞ ≤ γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (2) there exists a positive constant α0 such that 0 < α0 ≤ c − 1 2∇ · β a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) 4 Introducing the dual variable σ = −ϵ1/2∇u, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) may be rewritten as the following first-order system: � σ + ϵ1/2∇u = 0 in Ω, ϵ1/2∇ · σ + β · ∇u + c u = f in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4) 3 Least-squares formulations In this section, we study three least-squares formulations based on the first-order system in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4) with the inflow boundary conditions imposed strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' These formulations differ in how to handle the outflow boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' More specifically, the outflow boundary conditions are treated strongly for the first one and weakly for the other two through weighted boundary functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' To this end, introduce the following least-squares functionals: G1(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = ∥τ + ϵ1/2 ∇v∥2 + ∥ϵ1/2 ∇ · τ + β · ∇v + c v − f∥2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) G2(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = G1(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) + ∥ϵ−1/2 v∥2 1/2,Γ+, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) and G3(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = G1(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) + ∥v∥2 1/2,Γ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) Since ϵ is very small, the outflow boundary conditions are enforced stronger in G2 than in G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Let U1 = H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ω) × H1 0(Ω) and U2 = U3 = H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ω) × H1 Γ−(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Then the least-squares formulations are to find (σ, u) ∈ Ui such that Gi(σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = min (τ , v)∈ Ui Gi(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4) for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For any (τ, v) ∈ Ui, define the following norms: M1(τ, v) = ∥τ∥2 + ∥v∥2 + ∥ϵ1/2 ∇v∥2, M2(τ, v) = M1(τ, v) + ∥ϵ−1/2 v∥2 1/2,Γ+, and M3(τ, v) = M1(τ, v) + ∥v∥2 1/2,Γ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Below we show that the homogeneous least-squares functionals are coercive with respect to the corresponding norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In particular, the coercivity of the functionals G1 and G2 are independent of the ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1 (Coercivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For all (τ, v) ∈ Ui with i = 1, 2, 3, there exist positive constants Ci such that Mi(τ, v) ≤ Ci Gi(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5) where C1 and C2 are independent of the ϵ and C3 is proportional to ϵ−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' We provide proofs for i = 2 and 3 in detail with an emphasis on how the weight in G2 leads to the coercivity constant independent of the ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The case of i = 1 may be proved in a similar fashion as the case of i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For all (τ, v) ∈ Ui with i = 1, 2, 3, the triangle inequality gives ∥τ∥ ≤ ∥τ + ϵ1/2 ∇v∥ + ∥ϵ1/2 ∇v∥ ≤ G1/2 1 (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + ∥ϵ1/2 ∇v∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6) Hence, to show the validity of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5), it suffices to prove that ∥v∥2 + ∥ϵ1/2 ∇v∥2 ≤ Ci Gi(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) ∀ (τ, v) ∈ Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7) To this end, let I = − � ϵ1/2 ∇v, τ � + � v, (c − 1 2 ∇ · β) v � + 1 2 ∥(β · n)1/2 v∥2 0,Γ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8) It follows from the definition of the outflow boundary condition and the Cauchy-Schwarz inequality that ∥ϵ1/2 ∇v∥2 + α0 ∥v∥2 ≤ (ϵ1/2 ∇v, ϵ1/2∇v + τ) + I ≤ ∥ϵ1/2 ∇v∥ G1(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + I, which implies ∥ϵ1/2 ∇v∥2 + ∥v∥2 ≤ C (G1(σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='9) To bound I, first note that integration by parts and the boundary conditions imply that (ϵ1/2 ∇v, τ) = (v, ϵ1/2 τ · n)∂Ω − (ϵ1/2 v, ∇ · τ) = (v, ϵ1/2 τ · n)Γ+ − (ϵ1/2 v, ∇ · τ) = (v, ϵ1/2 τ · n)Γ+ + (v, c v) − (v, ϵ1/2 ∇ · τ + β · ∇v + c v) + (v β, ∇v) and that (∇v, v β) = 1 2 ∥(β · n)1/2 v∥2 0,Γ+ − 1 2 (v, v ∇ · β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Combining the above two equalities yields I = � v, ϵ1/2 ∇ · τ + β · ∇v + c v � − (v, ϵ1/2 τ · n)Γ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10) By the trace theorem and the Cauchy-Schwarz inequality, we have ∥τ · n∥−1/2,Γ+ ≤ C � ∥τ∥ + ∥∇ · τ∥ � ≤ C � G1/2 1 (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + ∥ϵ1/2 ∇v∥ + ϵ−1/2 ∥β · ∇v∥ + ϵ−1/2 ∥c v∥ � ≤ C ϵ−1/2� G1/2 1 (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + ∥∇v∥ + ∥v∥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11) 6 Let αi = 1 for i = 2 or 1/2 for i = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Then it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10), the Cauchy-Schwarz inequality, the definition of the dual norm, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11) that for i = 2 and 3 I ≤ ∥v∥ ∥ϵ1/2 ∇ · τ + β · ∇v + c v∥ + ∥ϵ1/2−αi v∥1/2,Γ+ ∥ϵαi τ · n∥−1/2,Γ+ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='12) ≤ C � ∥v∥ + ∥ϵαi τ · n∥−1/2,Γ+ � G1/2 i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) ≤ C Gi(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + C � ∥ϵαi−1/2 ∇v∥ + ∥v∥ � G1/2 i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0), which, together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='9), implies ∥ϵ1/2 ∇v∥2 + α0 ∥v∥2 ≤ Ci Gi(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='13) with C2 independent of ϵ and C3 proportional to ϵ−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This completes the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7) and, hence, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5) for i = 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The validity of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5) for i = 1 may be established in a similar fashion by noticing that the boundary term of I in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8) vanishes due to the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 4 Mesh-dependent least-squares functionals For computational feasibility, in this section, we replace the 1 2-norm in the least-squares functionals defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) by mesh-dependent L2-norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For the simplicity of presentation, assume that the domain Ω is a convex polygon in the two dimensional plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (The extension to the higher dimension is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=') Let Th = {K} be a triangulation of Ω with triangular elements K of diameter less than or equal to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Assume that the triangulation Th is regular and quasi-uniform (see [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Denote by Eh the set of all edges of the triangulation Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The least-squares functionals G2 and G3 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) are modified by the following computable least-squares functionals: Gh 2(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = G1(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) + � e∈Eh∩Γ+ h−1 e ∥ϵ−1/2 v∥2 0,e (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) and Gh 3(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = G1(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) + � e∈Eh∩Γ+ h−1 e ∥v∥2 0,e, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) where he denotes the diameter of the edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For any triangle K ∈ Th, let Pk(K) be the space of polynomials of degree less than or equal to k on K and denote the local Raviart–Thomas space of index k on K by RTk(K) = Pk(K)2 + � x1 x2 � Pk(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Then the standard H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ω) conforming Raviart–Thomas space of index k [22] and the standard (conforming) continuous piecewise polynomials of degree k + 1 are defined, re- spectively, by Σk h = {τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ω) : τ|K ∈ RTk(K), ∀ K ∈ Th}, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) V k+1 h = {v ∈ H1(Ω) : v ∈ Pk+1(K), ∀ K ∈ Th}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4) 7 These spaces have the following approximation properties: let k ≥ 0 be an integer, and let l ∈ (0, k + 1]: inf τ ∈ Σk h ∥σ − τ∥H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ω) ≤ C hl (∥σ∥l + ∥∇ · σ∥l) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5) for σ ∈ Hl(Ω)2 ∩ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ω) with ∇ · σ ∈ Hl(Ω) and inf v∈V k+1 h ∥u − v∥1 ≤ C hl ∥u∥l+1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6) for u ∈ Hl+1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In the subsequent sections, based on the smoothness of σ and u, we will choose k + 1 to be the smallest integer greater than or equal to l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Since the triangulation Th is regular, the following inverse inequalities hold for all K ∈ Th: ∥τ∥1,K ≤ C h−1 K ∥τ∥K, ∀ τ ∈ RTk(K) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7) ∥v∥1,K ≤ C h−1 K ∥v∥K, ∀ v ∈ Pk(K) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8) with positive constant C independent of hK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Denote by Uh i the finite dimensional subspaces of Ui: Uh i = � Σk h × V k+1 h � ∩ Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='9) For any (τ, v) ∈ Uh i , define the following norms: Mh 2 (τ, v) = M1(τ, v) + � e∈Eh∩Γ+ h−1 e ∥ϵ−1/2 v∥2 0,e and Mh 3 (τ, v) = M1(τ, v) + � e∈Eh∩Γ+ h−1 e ∥v∥2 0,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Below we establish the discrete version of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', the coercivity of the discrete functionals (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) with respect to the norms defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For the consistence of notation, we also let Gh 1 = G1 and Mh 1 = M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For all (τ, v) ∈ Uh i with i = 2 and 3, there exist positive constants Ci independent of ϵ such that Mh i (τ, v) ≤ Ci Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Similar to the argument in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1, in order to establish (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10), it suffices to show that ∥ϵ1/2 ∇v∥2 + ∥v∥2 ≤ C Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11) for all (τ, v) ∈ Uh i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Moreover, we have ∥ϵ1/2 ∇v∥2 + ∥v∥2 ≤ C � Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + I � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='12) 8 with I defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For any e ∈ Eh ∩ Γ+, let e be an edge of element K ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' It follows from the trace theorem and the inverse inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7) that he ∥τ · n∥2 0,e ≤ C he ∥τ∥2 0,e ≤ C he ∥τ∥0,K∥τ∥1,K ≤ C ∥τ∥2 0,K, which, together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6), implies � � � e∈Eh∩Γ+ he ∥τ · n∥2 0,e � � 1/2 ≤ C ∥τ∥ ≤ C � G1/2 1 (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + ∥ϵ1/2 ∇v∥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='13) Let αi = 1 for i = 2 or 1/2 for i = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10), the Cauchy-Schwarz inequality, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='13) that I = � v, ϵ1/2 ∇ · τ + β · ∇v + c v � − (v, ϵ1/2 τ · n)Γ+ ≤ C � �∥v∥ + ϵαi � � e∈Eh∩Γ+ he ∥τ · n∥2 0,e �1/2 � � Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0)1/2 ≤ C Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + C � ∥v∥ + ∥ϵ1/2∇v∥ � Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0)1/2 which, together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='12), implies the validity of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11) and, hence, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This com- pletes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Note that the coercivity constant C3 in the discrete version is no longer depending on ϵ, that is better than the continuous version (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 5 Finite element approximations The least-squares problems are to find (σ, u) ∈ Ui (i = 1, 2, 3) such that Gh i (σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = min (τ , v)∈ Ui Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) The corresponding variational problems are to find (σ, u) ∈ Ui such that ai(σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' τ, v) = Fi(τ, v), ∀ (τ, v) ∈ Ui, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) where the bilinear forms ai(· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' ·) are symmetric and given by a1(σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' τ, v) = (σ + ϵ1/2 ∇u, τ + ϵ1/2 ∇v) +(ϵ1/2 ∇ · σ + β · ∇u + c u, ϵ1/2 ∇ · τ + β · ∇v + c v), a2(σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' τ, v) = a1(σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' τ, v) + � e ∈Eh∩Γ+ h−1 e ϵ−1 (u, v)0,e, a3(σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' τ, v) = a1(σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' τ, v) + � e ∈Eh∩Γ+ h−1 e (u, v)0,e, 9 and the linear forms Fi(·) are given by Fi(τ, v) = (f, ϵ1/2 ∇ · τ + β · ∇v + c v) for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The least-squares finite element approximations to the variational problems in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) are to find (σi h, ui h) ∈ Uh i such that ai(σi h, ui h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' τ, v) = Fi(τ, v), ∀ (τ, v) ∈ Uh i , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Taking the difference between (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) implies the following orthog- onality: ai(σ − σi h, u − ui h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' τ, v) = 0, ∀ (τ, v) ∈ Uh i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4) In the rest of this section, we consider a stronger norm which incorporates the norm of the streamline derivative: |||(τ, v)|||2 i = Mh i (τ, v) + � K∈Th δK ∥β · ∇v∥2 K, where δK is a positive constant to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In the following lemma, we show that Gh i (σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) are also elliptic with respect to these norms if the δK is appropriately chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For all K ∈ Th, assume that 0 < δK ≤ min{h2 K/ϵ, C}, then there exist positive constants Ci independent of ϵ such that |||(τ, v)|||2 i ≤ Ci Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0), ∀ (τ, v) ∈ Uh i , i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' By Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1, to prove the validity of the lemma, it suffices to show that � K∈Th δK ∥β · ∇v∥2 K ≤ Ci Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5) To this end, note the facts that δK ≤ C and δK ϵ h2 K ≤ min � 1, C ϵ h2 K � ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Now it follows from the Cauchy-Schwarz inequality and the inverse inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7) that � K∈Th δK ∥β · ∇v∥2 K ≤ C � K∈Th δK � Gh 1,K (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + ∥ϵ1/2 ∇ · τ∥2 K + ∥c v∥2 K � ≤ C � K∈Th � Gh 1,K (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + δK ϵ h2 K ∥τ∥2 K + ∥v∥2 K � ≤ C � Gh 1 (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + ∥τ∥2 + ∥v∥2� ≤ C Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0), which establishes (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5) and hence completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 10 To choose δK properly, first define the local mesh P´eclet number by PeK = ∥β∥0,∞,K hK 2 ϵ , then partition the triangulation Th into two subsets: T c h = {K ∈ Th : PeK > 1} and T d h = {K ∈ Th : PeK ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6) The elements in T c h are referred to the convection-dominated elements, while the elements in T d h the diffusion-dominated elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Now, the δK is chosen to be δK = � � � � � � � � � � � 2 hK ∥β∥0,∞,K , if K ∈ T c h , h2 K ϵ , if K ∈ T d h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7) Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The δK defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7) satisfies the assumption in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', δK ≤ min{h2 K/ϵ, C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Since ∥β∥0,∞,K is large comparing to hK, we have 2 hK ∥β∥0,∞,K ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='9) For any K ∈ T c h , the fact that PeK > 1 implies 2 hK ∥β∥0,∞,K < h2 K ϵ , which, together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='9), yields (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For any K ∈ T d h , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8) is again a consequence of the definition of δK in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7), the fact that PeK ≤ 1, and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Denote by T ∂ h the set of elements that intersect the outflow boundary nontrivially, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', T ∂ h = {K ∈ Th : meas( ¯K ∩ Γ+) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In this paper, we assume that T ∂ h ⊂ T d h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10) For any K ∈ T d h , the fact that PeK ≤ 1 implies hK < 2 ϵ ∥β∥0,∞,K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Hence, assumption (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10) means that the mesh size in the boundary layer region is com- parable to the perturbation parameter ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 11 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Let (σ, u) be the solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Assume that (σ, u) ∈ Hl(Ω)2×Hl+1(Ω) and that ∇ · σ ∈ Hl(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Let (σi h, ui h), i = 1, 2, 3, be the solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) with k = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Under the assumption in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10), we have the following a priori error estimation: Ci ������(σ − σi h, u − ui h) ������2 i ≤ � K∈T c h h2l−1 K � ϵ ∥∇ · σ∥2 l,K + hK ∥σ∥2 l,K + ∥u∥2 l+1,K � + � K∈T d h h2l−1 K � ϵ2 hK ∥∇ · σ∥2 l,K + hK ∥σ∥2 l,K + ϵ hK ∥u∥2 l+1,K � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11) where constants Ci > 0 are independent of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' We provide proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11) only for i = 2 and 3 since (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11) may be obtained in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' To this end, let σI and uI be the interpolants of σ and u, respectively, such that the approximation properties in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6) hold and that (∇ · (σ − σI), v) = 0, ∀ v ∈ Dh k, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='12) where Dh k = {v ∈ L2(Ω) : v|K ∈ Pk(K) ∀ K ∈ Th} is the space of discontinuous piecewise polynomials of degree less than or equal to k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Let EI = σ − σI, Ei h = σI − σi h, eI = u − uI, and ei h = uI − ui h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Since Ei = σ − σi h = EI + Ei h and ei = u − ui h = eI + ei h, the triangle inequality gives ������(Ei, ei) ������ i ≤ |||(EI, eI)|||i + ������(Ei h, ei h) ������ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='13) Let αi = −1 or 0 for i = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' By approximation property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6) and assumption (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10), we have � e ∈Eh∩Γ+ h−1 e ϵαi ∥eI∥2 0,e ≤ C � K∈T ∂ h h2l K ϵαi ∥u∥2 l+1,K ≤ C � K∈T ∂ h h2l+αi K ∥u∥2 l+1,K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Now, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6), the trace theorem, and the fact δK ≤ C that |||(EI, eI)|||2 i ≤ C � �∥EI∥2 + ∥eI∥2 + ∥ϵ1/2 ∇eI∥2 + � e∈Γ+ h−1 e ϵαi ∥eI∥2 e + � K∈Th ∥β · ∇eI∥2 K � � ≤ C � � � K∈Th h2l K ∥σ∥2 l,K + � K∈Th h2l K ∥u∥2 l+1,K + � K∈T ∂ h h2l+αi K ∥u∥2 l+1,K � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='14) 12 To bound the second term of the right-hand side in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='13), by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1 and orthog- onality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4), we have Ci ������(Ei h, ei h) ������2 i ≤ ai(Ei h, ei h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ei h, ei h) = ai(Ei h, ei h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' −EI, −eI) ≡ Ii 1 + Ii 2 + Ii 3 + Ii 4, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='15) where Ii 1 = (c ei h, −ϵ1/2 ∇ · EI − β · ∇eI − c eI) + (Ei h + ϵ1/2 ∇ei h, −EI − ϵ1/2 ∇eI), Ii 2 = (ϵ1/2 ∇ · Ei h, −ϵ1/2 ∇ · EI − β · ∇eI − c eI), Ii 3 = (β · ∇ei h, −ϵ1/2 ∇ · EI − β · ∇eI − c eI), and Ii 4 = � e ∈Eh∩Γ+ h−1 e ϵαi (ei h, −eI)0,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' It follows from the triangle and Cauchy-Schwarz inequalities, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6) that Ii 1 ≤ C ∥ei h∥ � ∥ϵ1/2∇ · EI∥ + ∥∇eI∥ + ∥eI∥ � + C � ∥Ei h∥ + ∥ϵ1/2∇ei h∥ � � ∥EI∥ + ∥ϵ1/2∇eI)∥ � ≤C � ∥ei h∥ + ∥Ei h∥ + ∥ϵ1/2∇ei h∥ � � � � K∈Th h2l K � ϵ∥∇ · σ∥2 l,K + ∥σ∥2 l,K + ∥u∥2 l+1,K � � � 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='16) By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='12), the Cauchy-Schwarz and triangle inequalities, and the inverse inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7), we have Ii 2 = −(ϵ1/2 ∇ · Ei h, β · ∇eI + c eI), ≤ C � K∈Th ϵ1/2 hK ∥Ei h∥K � ∥∇eI∥K + ∥eI∥K � ≤ C ∥Ei h∥ � � � K∈Th ϵ h2l−2 K ∥u∥2 l+1,K � � 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='17) By the Cauchy-Schwarz and the triangle inequalities, I3 is bounded by Ii 3 ≤ C � K∈Th ∥β · ∇ei h∥K � ϵ1/2 ∥∇ · EI∥K + ∥∇eI∥K + ∥eI∥K � ≤ C � K∈Th ∥β · ∇ei h∥K � ϵ1/2 hl K ∥∇ · σ∥l,K + hl K ∥u∥l+1,K � ≤ C � � � K∈Th δK∥β · ∇ei h∥2 K � � 1/2� � � K∈Th δ−1 K � ϵ h2l K ∥∇ · σ∥2 l,K + h2l K ∥u∥2 l+1,K � � � 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='18) 13 For Ii 4, it follows from the Cauchy-Schwarz inequality and the trace theorem that Ii 4 ≤ C � � � e ∈Eh∩Γ+ h−1 e ϵαi ∥ei h∥2 0,e � � 1/2 � � � e ∈Eh∩Γ+ h−1 e ϵαi ∥eI∥2 0,e � � 1/2 ≤ C � � � e ∈Eh∩Γ+ h−1 e ϵαi ∥ei h∥2 0,e � � 1/2 � � � K∈T ∂ h h2l+αi K ∥u∥2 l+1,K � � 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='19) Combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='15), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='16), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='17), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='18), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='19), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8), we have Ci ������(Ei h, ei h) ������2 i ≤ � K∈Th h2l K∥σ∥2 l,K + � K∈Th � 1 + δ−1 K � ϵ h2l K ∥∇ · σ∥2 l,K + � K∈T ∂ h h2l+αi K ∥u∥2 l+1,K + � K∈Th � 1 + ϵ h−2 K + δ−1 K � h2l K∥u∥2 l+1,K ≤ � K∈Th �ϵ h2l K δK ∥∇ · σ∥2 l,K + h2l K ∥σ∥2 l,K + h2l K δK ∥u∥2 l+1,K � + � K∈T ∂ h h2l+αi K ∥u∥2 l+1,K, which, together with the definition of δK in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7), implies Ci ������(Ei h, ei h) ������2 i ≤ � K∈T c h h2l−1 K � ϵ ∥∇ · σ∥2 l,K + hK ∥σ∥2 l,K + ∥u∥2 l+1,K � + � K∈T d h h2l−1 K � ϵ2 hK ∥∇ · σ∥2 l,K + hK ∥σ∥2 l,K + ϵ hK ∥u∥2 l+1,K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Now, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11) is a consequence of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='13) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Note that the a priori error estimate in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3 is not optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This is because the coercivity of the homogeneous least-squares functionals in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1 are established in a norm that is weaker than the norm used for the continuity of the functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' To restore the full order of convergence, one may use piecewise polynomials of degree l + 1 to approximate u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Let (σi h, ui h), i = 1, 2, 3, be the solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) with Uh i = (Σl h×V l+1 h )∩ Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 14 Under the assumption of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3, we have the following a priori error estimation: Ci ������(σ − σi h, u − ui h) ������2 i ≤ � K∈T c h h2l K � ∥∇ · σ∥2 l,K + ∥σ∥2 l,K + hK ∥u∥2 l+2,K � + � K∈T d h h2l K � ϵ2 h2 K ∥∇ · σ∥2 l,K + ∥σ∥2 l,K + ϵ ∥u∥2 l+2,K � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='20) where constants Ci > 0 are independent of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The a priori error estimate in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='20) may be obtained in a similar fashion by noting that ∥u − uI∥1 ≤ C hl+1∥u∥l+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 6 Adaptive algorithm Asymptotic analysis (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', [20]) shows that the solution of a convection-dominated diffusion-reaction problem consists of two parts: the solution of the reduced equation (ϵ = 0) and the correction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', the boundary or interior layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The boundary and interior layers are narrow regions where derivatives of the solution change dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For example, for the following problem [20]: � � � � � −ϵ ∆u + ∂u ∂y = f in Ω = (0, 1)2, u = 0 on ∂Ω, the exponential layer is of width O(ϵ) at y = 1, and the width of the parabolic boundary layers is O(ϵ1/2) at both x = 0 and x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Therefore, two sets of largely different scales exist simultaneously in the convection-dominated diffusion problem, and hence it is difficult computationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' On the one hand, one can apply the small scale over the entire domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', to use uniform fine meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' With such a fine mesh, the standard Galerkin finite element method can also produce a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' However, it is computationally inefficient due to the small region of the boundary and/or interior layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' On the other hand, one can use the large scale over the entire domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' If the outflow boundary conditions are imposed strongly, the numerical solution (away from the boundary layers) will be polluted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In contrast, if the outflow boundary conditions are imposed weakly, the boundary layers can not be resolved (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', numerical results in [4, 17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Neither of the above two approaches leads to a satisfactory numerical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The fail- ure is due to the fact that these approaches ignore this intrinsic property of the convection- dominated diffusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In contrast, the Shishkin mesh is aware of and respect it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 15 Basically, the Shishkin mesh is a piecewise uniform mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In the diffusion-dominated re- gion where the layers stand, it is a fine mesh suitable to the layer and in the convective region, it turns to be a coarse mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The disadvantage of the Shishkin mesh is that it needs the a priori information of the solution, such as the location and the width of the layer, in order to construct a mesh of high quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' However, this information is not always available in advance, especially, for a complex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Based on the above considerations, we employ adaptive least-squares finite element methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The least-squares estimators are simply defined as the value of the least-squares functionals at the current approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' To this end, for each element K ∈ Th, denote the local least-squares functionals by Gh 1,K(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = ∥τ + ϵ1/2 ∇v∥2 K + ∥ϵ1/2 ∇ · τ + β · ∇v + c v − f∥2 K, Gh 2,K(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = � � � � � Gh 1,K(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f), if K ∩ Γ+ = ∅, Gh 1,K(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) + � e∈K∩Γ+ h−1 e ∥ϵ−1/2v∥2 0, e, otherwise, and Gh 3,K(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = � � � � � Gh 1,K(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f), if K ∩ Γ+ = ∅, Gh 1,K(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) + � e∈K∩Γ+ h−1 e ∥v∥2 0, e, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Let (ˆσh i , ˆuh i ) be the current approximations to the solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Then the least-squares indicators are simply the square root of the value of the local least-squares functionals at the current approximation: ηi K = Gh i,K (ˆσi h, ˆui h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f)1/2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) for all K ∈ Th and for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The least-squares estimators are ηi = � � � K∈Th � ηi K �2 � � 1/2 = Gh i (ˆσi h, ˆui h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f)1/2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Let (σ, u) be the solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) and denote the true errors by ˆEi = σ − ˆσi h and ˆei = u − ˆu1 h for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' There exist positive constants Ce,1 and Cr,1 independent of ϵ such that η1 K ≤ Ce,1 � M1,K(ˆE1, ˆe1) + ∥β · ∇ ˆe1∥2 K + ϵ ∥∇ · ˆE1∥2 K �1/2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) for all K ∈ T and that M1(ˆE1, ˆe1)1/2 ≤ Cr,1 η1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4) 16 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Since the exact solution (σ, u) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4), we have � η1 K �2 = Gh 1,K(ˆE1, ˆe1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) and � η1�2 = Gh 1(ˆE1, ˆe1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' which, together with the triangle inequality and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1, imply the efficiency and the reliability bounds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' There exist positive constants Ce,i independent of ϵ such that Ce, i � ηi K �2 ≤ Mh i,K(ˆEi, ˆei) + ∥β · ∇ˆei∥2 K + ϵ ∥∇ · ˆEi∥2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5) for all K ∈ T and i = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Let αi = −1 for i = 2 or 0 for i = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' With the fact that (σ, u) is the exact solution satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4), we have ηi(ˆσh i , ˆuh i )2 = Gh i (ˆσh i , ˆuh i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = ∥ˆσh i + ϵ1/2 ∇ˆuh i ∥2 + ∥ϵ1/2 ∇ · ˆσh i + β · ∇ˆuh i + c ˆuh i − f∥2 + � e∈Eh∩Γ+ ϵαi h−1 e ∥ˆuh i ∥2 0,e = ∥ˆEi + ϵ1/2 ∇ˆei∥2 + ∥ϵ1/2 ∇ · ˆEi + β · ∇ˆei + c ˆei∥2 + � e∈Eh∩Γ+ ϵαi h−1 e ∥ˆei∥ = Gh i (ˆEi, ˆei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6) with which, the efficiency bound simply follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6) and the Cauchy-Schwarz in- equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In the remainder of this section, we describe the standard adaptive mesh refinement algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Starting with an initial triangulation T0, a sequence of nested triangulations {Tl} is generated through the well known AFEM-Loop: SOLVE −→ ESTIMATE −→ MARK −→ REFINE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The SOLVE step solves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) in the finite element space corresponding to the mesh Tl for a numerical approximation (σi h(l), ui h(l)) ∈ Uh i (l), where Uh i (l) is the finite element space defined on Tl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Hereafter, we shall explicitly express the dependence of a quantity on the level l by either the subscript like Tl or the variable like Uh i (l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The ESTIMATE step computes the indicators {ηi K(l)} and the estimator ηi(l) defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The way to choose elements for refinement influences the efficiency of the adaptive algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' If most of elements are marked for refinement, then it is comparable to uniform refinement, which does not take full advantage of the adaptive algorithm and results in redundant degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' On the other hand, if few elements are refined, then it requires many iterations, which undermines the efficiency of the adaptive algorithm, since each iteration is costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For the singularly perturbed problems, it is well known that the indicators associated with the elements in the layer region are much larger than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 17 Therefore, we MARK by the maximum algorithm, which defines the set ˆTl of marked elements such that for all K ∈ ˆTl ηi K(l) ≥ θ max K∈Tl ηi K(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The REFINE step is to bisect all the triangles in ˆTl into two sub-triangles to generate a new triangulation Tl+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Note that some triangles in Tl \\ ˆTl adjacent to triangles in ˆTl are also refined in order to avoid hanging nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In summary, the adaptive least-squares finite element algorithm can be cast as follows: with the initial mesh T0, marking parameter θ ∈ (0, 1), and the maximal number of iteration maxIt, for l = 0, 1, · · · , maxIt, do (1) (σi h(l), ui h(l)) = SOLVE(Tl);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (2) {ηi K(l)} = ESTIMATE(Tl, σi h(l), ui h(l));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (3) ˆTl = MARK(Tl, {ηi K(l)});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (4) Tl+1 = REFINE(Tl, ˆTl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 7 Numerical experiments In this section, we conduct several numerical experiments on two model problems used by many authors (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', [4, 17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Both the model problems are defined in the unit square and all numerical experiments are started with the same initial mesh, which consists of sixteen isosceles right triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The marking parameter θ is chosen to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1 Boundary layer In this example, β = [1, 1]T , and c = 0, and the external force f is chosen such that the exact solution is u(x, y) = sin πx 2 + sin πy 2 � 1 − sin πx 2 � + e−1/ϵ − e−(1−x)(1−y)/ϵ 1 − e−1/ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This solution is smooth, but develops boundary layers at x = 1 and y = 1 with width O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This example is suitable for testing capability of the numerical approximations on resolving the boundary layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In this numerical experiment, ϵ = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Given the tolerance tol = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5, computation is terminated if ηi(l) ≤ tol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) Since the exact solution is available, the true error is computed and the effectivity index is defined as follows: eff-index := ηi(σi h, ui h) ������(σ − σi h, u − ui h) ������ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) 18 Figure 1: The final meshes and the numerical solutions are, respectively, displayed in the first and the second columns and the rows are corresponding to i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The final meshes are displayed in the first column of Figure 1 when the stopping criterion (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' They clearly show that the refinements cluster around the boundary layer area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The numerical solutions on the final meshes are depicted in the second column of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' All the three methods successfully capture the sharp boundary layers, and no visible oscillation appears in the numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Reported in Figure 2 is the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='. A : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6 : -?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2 019 convergence rates of the numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The errors with the norm |||·|||i that are used in the a priori error estimate are tracked, which converge in the order of (DoF)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Moreover, the convergence rate is independent of the value of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This is also verified by the test problem with ϵ = 10−4, where the convergence rate does not deteriorate (see the second column of Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Figure 2: The convergence rates corresponding to ϵ = 10−3 and 10−4 are displayed in the first and the second columns, respectively, and the rows are corresponding to i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 10° 10° 10 10° 10 10* 10° errEne3 estimator DoF-1 effindex 10 102 103 104 105 10° Degree of Freedom10° 10 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10° ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10~ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10° ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='errEne ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10° ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='Degree of Freedom20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2 Interior layer In the second example, β = [1/2, √ 3/2]T , c = 0, f = 0, and the boundary condition is u = � � � � � � � 1, on {(x, y) : y = 0, 0 ≤ x ≤ 1}, 1, on {(x, y) : x = 0, 0 ≤ y ≤ 1/5}, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The exact solution of the problem remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' However, it is known that, additional to the boundary layers, the solution develops an interior layer along the line y = √ 3 x+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2 due to the discontinuity at (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) of the boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The problem is chosen to test whether the formulations can capture the interior layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Figure 3 shows that all the three methods capture both the boundary and the interior layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Moreover, the numerical solutions do not exhibit any visible oscillation, which is much better than the results reported in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Figure 3: Numerical solutions corresponding to i = 1, 2, 3 from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Acknowledgements We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Shuhao Cao for the discussion and helpful suggestions on the computation of the test problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Adams, Sobolev Spaces, Academic Press, New York, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Angermann, Balanced a posteriori error estimates for finite volume type dis- cretizations of convection-dominated elliptic problems, Computing, 55:4 (1995), 305- 323.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Fluids, 73:9 (2013), 765-790.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='9 .' 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b/8NE2T4oBgHgl3EQfPgby/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c1a53f00c959035760d5b9c6e34a83c1b79cdb99c032aa48bd075381701db0e3 +size 190389 diff --git a/8dE2T4oBgHgl3EQfPwZL/content/tmp_files/2301.03762v1.pdf.txt b/8dE2T4oBgHgl3EQfPwZL/content/tmp_files/2301.03762v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b24db9fdf717a0915838f1ec8df1cb44c50fb8a8 --- /dev/null +++ b/8dE2T4oBgHgl3EQfPwZL/content/tmp_files/2301.03762v1.pdf.txt @@ -0,0 +1,1435 @@ +arXiv:2301.03762v1 [math.AG] 10 Jan 2023 +REGULAR SEMISIMPLE HESSENBERG VARIETIES WITH COHOMOLOGY RINGS +GENERATED IN DEGREE TWO +MIKIYA MASUDA AND TAKASHI SATO +Abstract. A regular semisimple Hessenberg variety Hess(S, h) is a smooth subvariety of the flag variety +determined by a square matrix S with distinct eigenvalues and a Hessenberg function h. The cohomology +ring H∗(Hess(S, h)) is independent of the choice of S and is not explicitly described except for a few cases. +In this paper, we characterize the Hessenberg function h such that H∗(Hess(S, h)) is generated in degree two +as a ring. It turns out that such h is what is called a (double) lollipop. +1. Introduction +The flag variety Fl(n) consists of nested sequences of linear subspaces in the complex vector space Cn: +Fl(n) = {V• = (V1 ⊂ V2 ⊂ · · · ⊂ Vn = Cn) | dimC Vi = i +(∀i ∈ [n] = {1, 2, . . ., n})}. +A Hessenberg function h: [n] → [n] is a monotonically non-decreasing function satisfying h(j) ≥ j for any +j ∈ [n]. We often express a Hessenberg function h as a vector (h(1), . . . , h(n)) by listing the values of h. +Given an n × n matrix A and a Hessenberg function h, a Hessenberg variety Hess(A, h) is defined as +Hess(A, h) = {V• ∈ Fl(n) | AVi ⊂ Vh(i) +(∀i ∈ [n])} +where the matrix A is regarded as a linear operator on Cn. Note that Hess(A, h) = Fl(n) if h = (n, . . . , n). +The family of Hessenberg varieties Hess(A, h) contains important varieties such as Springer fibers (A is +nilpotent and h = (1, 2, . . ., n)), Peterson varieties (A is regular nilpotent and h = (2, 3, . . . , n, n)), and +permutohedral varieties (A is regular semisimple and h = (2, 3, . . . , n, n)), which are toric varieties with +permutohedra as moment polytopes. +Among n × n matrices, regular semisimple ones S (i.e. matrices S having distinct eigenvalues) are generic +and Hess(S, h) is called a regular semisimple Hessenberg variety. The regular semisimple Hessenberg variety +Hess(S, h) has nice properties. For instance, it is smooth and its cohomology H∗(Hess(S, h)) becomes a +module over the symmetric group Sn on [n] by Tymoczko’s dot action [20]. Remarkably, the solution of +Shareshian–Wachs conjecture [18] by Brosnan and Chow [5] (and Guay-Paquet [10]) connected H∗(Hess(S, h)) +as an Sn-module and chromatic symmetric functions on certain graphs. This opened a way to prove the +famous Stanley–Stembridge conjecture in graph theory through the geometry or topology of Hessenberg +varieties and motivated us to study H∗(Hess(S, h)). Note that H∗(Hess(S, h)) (indeed the diffeomorphism +type of Hess(S, h)) is independent of the choice of S. We write the regular semisimple Hessenberg variety +Hess(S, h) as X(h) for brevity since our concern in this paper is its cohomology ring. +The Sn-module structure on H∗(X(h)) is determined in some cases (e.g. [12]). In particular, that on +H2(X(h)) was explicitly described by Chow [7] combinatorially (through the theorem by Brosnan-Chow +mentioned above) and by Cho-Hong-Lee [6] geometrically. Motivated by their works, Ayzenberg and the +authors [4] reproved their results by giving explicit additive generators of H2(X(h)) in terms of GKM theory. +The ring structure on H∗(X(h)) is not explicitly described except for a few cases. Remember that X(h) +for h = (n, . . . , n) is the flag variety Fl(n) and H∗(Fl(n)) is generated in degree 2 as a ring. Moreover, X(h) +for h = (2, 3, . . . , n, n) is the permutohedral variety and H∗(X(h)) is also generated in degree 2 as a ring. On +the other hand, for h = (h(1), n, . . . , n) with h(1) arbitrary, a result of [2] shows that H∗(X(h)) is generated +Date: January 11, 2023. +2020 Mathematics Subject Classification. Primary: 57S12, Secondary: 14M15. +Key words and phrases. Hessenberg variety, torus action, GKM theory, equivariant cohomology, lollipop. +1 + +2 +M. MASUDA AND T. SATO +in degree 2 as a ring if and only if h(1) = 2 or n, where X(h) = Fl(n) for the latter case. Therefore, it is +natural to ask when H∗(X(h)) is generated in degree 2 as a ring. The answer is the following, which is our +main result in this paper. +Theorem 1.1. Assume that h(j) ≥ j + 1 for j ∈ [n − 1]. Then H∗(X(h)) is generated in degree 2 as a ring +if and only if h is of the following form (1.1) for some 1 ≤ a < b ≤ n, +(1.1) +h(j) = + + + + + +a + 1 +(1 ≤ j ≤ a) +j + 1 +(a < j < b) +n +(b ≤ j ≤ n). +Remark 1.1. +(1) X(h) is connected if and only if h(j) ≥ j + 1 for any j ∈ [n − 1]. When X(h) is not +connected, each connected component of X(h) is a product of smaller regular semisimple Hessenberg +varieties. +(2) X(h) is the flag variety Fl(n) when (a, b) = (n − 1, n) and is the permutohedral variety when (a, b) = +(1, n). +(3) We will give an explicit presentation of the ring structure on H∗(X(h)) for h of the form (1.1) in a +forthcoming paper [17]. +We can visualize a Hessenberg function h by drawing a configuration of the shaded boxes on a square +grid of size n × n, which consists of boxes in the i-th row and the j-th column satisfying i ≤ h(j). Since +h(j) ≥ j for any j ∈ [n], the essential part is the shaded boxes below the diagonal. For example, Figure +1 below is the configurations of two Hessenberg functions h of the form (1.1) with n = 11: one is h = +(2, 3, 4, 5, 6, 7, 11, 11, 11, 11) where (a, b) = (1, 7) and the other is h = (4, 4, 4, 5, 6, 7, 11, 11, 11, 11) where +(a, b) = (3, 7). We often identify a Hessenberg function h with its configuration. +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +❅❅ +Figure 1. The configurations for h = (2, 3, 4, 5, 6, 7, 11, 11, 11, 11) and h = (4, 4, 4, 5, 6, 7, 11, 11, 11, 11) +The chromatic symmetric functions and LLT polynomials associated with h of the form (1.1) are studied +from the viewpoint of combinatorics in [8, 13], and when a = 1 or b = n, the corresponding Hessenberg +functions +h = (2, 3, . . . , b, n, . . ., n) +or +(a + 1, . . . , a + 1, a + 2, . . . , n − 1, n, n) +are called lollipops in those papers, so the Hessenberg function of the form (1.1) may be called a double +lollipop. +The paper is organized as follows. +In Section 2, we review GKM theory to compute the equivariant +cohomology of X(h). +We prove the “only if” part in Theorem 1.1 in Section 3. +Indeed, we consider a +Morse-Bott function fh on X(h), where the inverse image of the minimum or maximum value of fh is a +regular semisimple Hessenberg variety X(h′) with h′ of size one less than that of h. Then a property of the + +REGULAR SEMISIMPLE HESSENBERG VARIETIES +3 +Morse-Bott function fh shows the surjectivity of the restriction map H∗(X(h); Q) → H∗(X(h′); Q), and this +enables us to use an inductive argument to prove the “only if” part. In Section 4, we prove the “if” part in +Theorem 1.1 by applying the method developed in [2, 9] together with the explicit generators of H2(X(h)) +obtained in our previous work [4]. +2. Regular semisimple Hessenberg varieties +We first recall some properties of a regular semisimple Hessenberg variety X(h). +Theorem 2.1 ([14]). +(1) X(h) is smooth. +(2) dimC X(h) = �n +j=1(h(j) − j). +(3) X(h) is connected if and only if h(j) ≥ j + 1 for ∀j ∈ [n − 1]. +(4) Hodd(X(h)) = 0 and the 2k-th Betti number of X(h) is given by +#{w ∈ Sn | ℓh(w) = k} +where +(2.1) +ℓh(w) = #{1 ≤ j < i ≤ n | w(j) > w(i), i ≤ h(j)}. +For calculation of the cohomology ring of X(h), we use equivariant cohomology which we shall explain. We +assume that the matrix S in X(h) = Hess(S, h) is a diagonal matrix. Let T be an algebraic torus consisting +of diagonal matrices in the general linear group GLn(C). The linear action of T on Cn induces an action on +the flag variety Fl(n) and preserves X(h) since S commutes with T . The fixed point sets of the T -actions on +X(h) and Fl(n) consist of all permutation flags, that is, +(2.2) +X(h)T = Fl(n)T ∼= Sn. +Since T can naturally be identified with (C∗)n, the classifying space BT of T is B(C∗)n = (CP ∞)n. +Let pi : T → C∗ be the projection on the i-th diagonal component of T and ti = p∗ +i (t) ∈ H2(BT ) where +p∗ +i : H∗(BC∗) → H∗(BT ) and t ∈ H2(BC∗) is the first Chern class of the tautological line bundle over +BC∗ = CP ∞. Then +(2.3) +H∗(BT ) = Z[t1, . . . , tn]. +The equivariant cohomology of the T -variety X(h) is defined as +H∗ +T (X(h)) := H∗(ET ×T X(h)) +where ET is the total space of the universal principal T -bundle ET → BT and ET ×T X(h) is the orbit +space of the product ET × X(h) by the diagonal T -action. The projection ET × X(h) → ET on the first +factor induces a fibration +X(h) +ρ−→ ET ×T X(h) +π−→ BT. +Since Hodd(X(h)) = 0 as in Theorem 2.1 and Hodd(BT ) = 0, the Serre spectral sequence of the fibration above +collapses. It implies that ρ∗ : H∗ +T (X(h)) → H∗(X(h)) is surjective and induces a graded ring isomorphism +(2.4) +H∗(X(h)) ∼= H∗ +T (X(h))/(π∗(t1), . . . , π∗(tn)) +by (2.3). Therefore, one can find the ring structure on H∗(X(h)) through H∗ +T (X(h)). +Since Hodd(X(h)) = 0, it follows from the localization theorem that the homomorphism +(2.5) +H∗ +T (X(h)) → H∗ +T (X(h)T ) = +� +w∈Sn +H∗ +T (w) = +� +w∈Sn +Z[t1, . . . , tn] = Map(Sn, Z[t1, . . . , tn]) +induced from the inclusion map X(h)T → X(h) is injective, where X(h)T is identified with Sn by (2.2) and +Map(P, Q) denotes the set of all maps from P to Q. The T -variety X(h) is what is called a GKM manifold +and the image of the homomorphism in (2.5) is described in [20] as follows; +(2.6) +{f ∈ Map(Sn, Z[t1, . . . , tn]) | f(w) − f(w(i, j)) ∈ (tw(i) − tw(j)), for ∀w ∈ Sn, j < i ≤ h(j)}, + +4 +M. MASUDA AND T. SATO +where (i, j) denotes the transposition interchanging i and j. We note that the image of π∗(ti) ∈ π∗(H∗(BT )) ⊂ +H∗ +T (X(h)) by the homomorphism in (2.5) is the constant function ti ∈ Map(Sn, Z[t1, . . . , tn]). +Guillemin and Zara [11] assigned a labeled graph to a GKM manifold. The labeled graph of X(h) is as +follows. The vertex set is the fixed point set X(h)T = Sn. There is an edge between vertices w and v if and +only if v = w(i, j) for some j ≤ i ≤ h(j), and the edge between w and w(i, j) is labeled by tw(i) − tw(j) up to +sign. +Example 2.1. Let n = 3. For h = (2, 3, 3) and h′ = (3, 3, 3), the labeled graphs of X(h) and X(h′) are +drawn in Figure 2, where we use the one-line notation for each vertex. +❞ +❞ +❞ +❞ +❞ +❞ +❍ +❍ +❍ +❍ +❍ +❍ +✟✟✟ +✟✟✟ +✟ +✟ +✟ +✟ +✟ +✟ +123 +321 +132 +312 +213 +231 +X(h) +❞ +❞ +❞ +❞ +❞ +❞ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +✟✟✟ +✟✟✟ +✟ +✟ +✟ +✟ +✟ +✟ +✟ +✟ +✟ +✟ +✟ +✟ +✟ +✟ +✟ +✟ +✟ +✟ +✟ +✟ +123 +321 +132 +312 +213 +231 +X(h′) = Fl(3) +labels +: t1 − t2 +: t2 − t3 +: t1 − t3 +Figure 2. The labeled graphs of X(h) and X(h′) +In general, labeled graphs and their graph cohomologies are defined as follows. +Definition 2.2. Let R be a ring. A labeled graph (Γ, α) consists of a graph Γ = (V, E) and a labeling +α: E → R. The graph cohomology of a labeled graph (Γ, α) is defined as +H∗(Γ, α) = {f ∈ Map(V, R) | f(w) − f(v) ∈ (α(e)) for ∀e = wv ∈ E}. +The graph cohomology H∗(Γ, α) is a subring of Map(V, R) with the coordinate-wise sum and multiplication. +Note that we may ignore the signs of the labels α(e) since (α(e)) = (−α(e)). +The observation above shows that the graph cohomology of the labeled graph of X(h) coincides with +H∗ +T (X(h)). +Sending ti to tσ(i) for σ ∈ Sn and i ∈ [n] induces an action of Sn on Z[t1, . . . , tn]. Then, the module +Map(Sn, Z[t1, . . . , tn]) becomes an Sn-module under what is called the dot action defined by +(σ · f)(w) := σ(f(σ−1w)). +As easily checked, the graph cohomology of X(h) is invariant under the dot action and H∗ +T (X(h)) becomes +a module over Sn. +Moreover, since the action of Sn preserves the ideal (π∗(t1), . . . , π∗(tn)), the action +descends to H∗(X(h)) and H∗(X(h)) also becomes an module over Sn. +Obviously, constant functions in Map(Sn, Z[t1, . . . , tn]) satisfy the condition in (2.6). +They are ele- +ments corresponding to π∗(H∗(BT )) ⊂ H∗ +T (X(h)). +Below are three types of elements xi, yj,k, and τA +in Map(Sn, Z[t1, . . . , tn]) which satisfy the condition in (2.6), so they are in H∗ +T (X(h)). Let +⊥(h) : = {j ∈ [n − 1] | h(j − 1) = h(j) = j + 1} +L(h) : = {j ∈ [n − 1] | h(j − 1) = j and h(j) = j + 1} +(2.7) +where we understand h(0) = 1. +Definition 2.3. +(1) For i ∈ [n], xi(w) := tw(i). +(2) For j ∈ [n − 1] with j ∈ ⊥(h) and k ∈ [n], +yj,k(w) := +� +tk − tw(j+1) +(if k ∈ {w(1), . . . , w(j)}) +0 +(otherwise). + +REGULAR SEMISIMPLE HESSENBERG VARIETIES +5 +(3) For A ⊂ [n] with |A| ∈ L(h) +τA(w) := +� +tw(|A|) − tw(|A|+1) +(if {w(1), . . . , w(|A|)} = A) +0 +(otherwise). +The cohomological degrees of the elements xk, yj,k, τA are two. One can easily check that the dot actions +of σ ∈ Sn on these elements are given as follows: +(2.8) +σ · xk = xk, +σ · yj,k = yj,σ(k), +σ · τA = τσ(A). +Remark 2.1. Here is a geometrical meaning of xk’s (regarded as elements in H2(X(h)) through the isomor- +phism (2.4)). There is a nested sequence of tautological vector bundles over the flag variety Fl(n): +F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fn = Fl(n) × Cn +where +Fk := {(V•, v) ∈ Fl(n) × Cn | v ∈ Vk} +and +V• = ({0} = V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vn = Cn). +Then xk (k ∈ [n]) is the image of the first Chern class of the quotient line bundle Fk/Fk−1 over Fl(n) by the +homomorphism +ι∗ : H∗(Fl(n)) → H∗(X(h)) +induced from the inclusion map ι: X(h) → Fl(n). The dot action on H∗(Fl(n)) is trivial, so the image of ι∗ +must be contained in the ring of invariants H∗(X(h))Sn. In fact, it follows from [1, Theorems A and B] that +the image of ι∗ agrees with H∗(X(h))Sn when tensoring with Q and +(2.9) +H∗(X(h))Sn ⊗ Q = Q[x1, . . . , xn]/(fh(1),1, . . . , fh(n),n) +where +(2.10) +fh(j),j = +j +� +k=1 + +xk +h(j) +� +ℓ=j+1 +(xk − xℓ) + + . +In particular, the Hilbert series of H∗(X(h))Sn is given by +(2.11) +Hilb(H∗(X(h))Sn, √q) = +n−1 +� +j=1 +[h(j) − j]q +where the Hilbert series of a graded algebra A = �∞ +r=0 Ar over Z is defined as +Hilb(A, q) := +∞ +� +r=0 +(rankZAr)qr. +Through the isomorphism (2.4), the elements xk, yj,k, τA determine elements in H2(X(h)), denoted by the +same notation. +Theorem 2.4 ([4, Theorem 5.1]). The elements +{xk, yj,k, τA | k ∈ [n], j ∈ ⊥(h)\{n − 1}, A ⊂ [n] with |A| ∈ L(h)\{n − 1}} +generate H2(X(h)) with relations +(1) �n +k=1 xk = 0, +(2) �n +k=1 yj,k = (x1 + · · · + xj) − jxj+1 for j ∈ ⊥(h)\{n − 1}, +(3) � +|A|=j τA = xj − xj+1 for j ∈ L(h)\{n − 1}. +Remark 2.2 (see Subsection 6.2 in [4] for more details). The element yj,k is defined by looking at the j-th +column of the configuration associated to the Hessenberg function h. Similarly, one can define an element +y∗ +i,k of H∗ +T (Hess(S, h)) by looking at the i-th row of the configuration as follows. For i ∈ [n], we define +h∗(i) := min{j ∈ [n] | h(j) ≥ i}, + +6 +M. MASUDA AND T. SATO +so that the shaded boxes in the i-th row and under the diagonal in the configuration associated to h are at +positions (i, ℓ) (h∗(i) ≤ ℓ < i). When h∗(i) = i − 1, we define +(2.12) +y∗ +i,k(w) := +� +tk − tw(i−1) +(k ∈ {w(i), . . . , w(n)}) +0 +(otherwise). +One can see that y∗ +i.k is in H2 +T (Hess(S, h)) and we may replace yj,k’s for j ∈ ⊥(h)\{n − 1} in the generating +set in Theorem 2.4 by y∗ +i,k’s for i ≥ 3 such that h∗(i) = h∗(i + 1) = i − 1. +Example 2.2. When h = (4, 4, 4, 5, 6, 7, 11, 11, 11, 11) in Figure 1 (i.e. (a, b) = (3, 7)), we have +⊥(h) = {3, 10}, +L(h) = {4, 5, 6}, +so Theorem 2.4 says that H2(X(h)) is generated by +xk (k ∈ [11]), +y3,k (k ∈ [11]), +τA for A ⊂ [11] with |A| = 4, 5 or 6. +Moreover, it follows from Remark 2.2 that y3,k above may be replaced by y∗ +8,k. +3. Necessity +In this section, we study a necessary condition on h for H∗(X(h)) to be generated in degree 2 as a ring. +3.1. Moment maps. Let µ: Fl(n) → Rn be the standard moment map on the flag variety Fl(n). Its image is +the permutohedron Πn obtained as the convex hull of the orbits of (1, 2, . . . , n) by permuting its coordinates. +Indeed, if ew (w ∈ Sn) denotes the permutation flag associated with w, then we have +µ(ew) = (w−1(1), . . . , w−1(n)) ∈ Rn +(see [16, Lemma 3.1] for example). Let +(3.1) +Sr +n := {w ∈ Sn | w(r) = n}. +Then µ(Sr +n) is the set of all vertices of Πn whose n-th coordinate is r. Therefore the projection +πn : Πn → R, +πn(x1, . . . , xn) = xn +on the n-th coordinate takes minimum on S1 +n and maximum on Sn +n. The composition of µ and πn +(3.2) +f := πn ◦ µ: Fl(n) → R +is the moment map induced from the following S1-action on Cn +(3.3) +(z1, . . . , zn) → (z1, . . . , zn−1, gzn) +(g ∈ S1 ⊂ C), +and it is a Morse-Bott function. +Let hj be the Hessenberg function obtained by removing all the boxes in the j-th row and all the boxes +in the j-th column from its configuration (see Figure 3). To be precise, hj is given as follows. +hj(i) = + + + + + +h(i) +(i < j, h(i) < j) +h(i) − 1 +(i < j, h(i) ≥ j) +h(i + 1) − 1 +(i ≥ j) + +REGULAR SEMISIMPLE HESSENBERG VARIETIES +7 +j-th row → +↓ +j-th column +h +❀ +remove +← +տ +↑ +❀ +hj +Figure 3. The configuration corresponding to hj. +The following is a key lemma in our argument. +Lemma 3.1. The restriction maps +H∗(X(h); Q) → H∗(X(h1); Q), +H∗(X(h); Q) → H∗(X(hn); Q) +are surjective. +Proof. Let fh be the map f in (3.2) restricted to X(h), which is also a Morse-Bott function. The inverse +image of the minimum value under fh is X(h1), so it follows from [19, Lemma 3.1] that the restriction map +(3.4) +H∗ +S1(X(h); Q) → H∗ +S1(X(h1); Q) +is surjective, where the S1-action on X(h) is the induced one from the S1-action defined in (3.3). Since the +S1-action on X(h1) is trivial, we have H∗ +S1(X(h1); Q) = H∗(BS1; Q)⊗ H∗(X(h1); Q) and hence the forgetful +map H∗ +S1(X(h1); Q) → H∗(X(h1); Q) is surjective. Therefore, the surjectivity of (3.4) implies the surjectivity +of the restriction map +H∗(X(h); Q) → H∗(X(h1); Q) +in ordinary cohomology. The same argument applied to −fh proves the statement for X(hn). +✷ +Remark 3.1. The surjectivity of the above restriction maps (even with Z coefficients) can also be verified by +GKM theory as follows. Recall that the inclusion of the fixed point set induces an injective homomorphism +H∗ +T (X(h)) → H∗ +T (X(h)T ) ∼= Map(Sn, H∗(BT )). The equivariant cohomology H∗ +T (X(h)) has an H∗(BT )- +module basis {σw,h | w ∈ Sn} (see [6, Definition 2.9 and Proposition 2.11]). It corresponds to a natural +paving and then it is a ‘flow-up basis.’ +Note that any element of Sn +n = Sn−1 is not greater than any +element of Sn \ Sn +n. +The restriction of {σw,h | w ∈ Sn +n} onto X(hn), that is, its restriction onto the +fixed point set Sn +n = X(hn)T as elements of Map(Sn, H∗(BT )), is a flow-up basis of H∗ +T (X(hn)). Hence +H∗ +T (X(h)) → H∗ +T (X(hn)) is surjective, and then H∗(X(h)) → H∗(X(hn)) is also surjective. The surjectivity +of H∗(X(h)) → H∗(X(h1)) can be verified by a similar argument. +Given a Hessenberg function h, we obtain a smaller Hessenberg function by removing the first column and +row or the last column and row repeatedly, i.e. by taking h1 or hn repeatedly. We call it a minor of h. The +following corollary follows from Lemma 3.1. +Corollary 3.2. Let h′ be a minor of h. +If H∗(X(h); Q) is generated in degree 2 as a ring, then so is +H∗(X(h′); Q). +An easy argument shows that h being of the form (1.1) can be rephrased as follows. +Proposition 3.3. The Hessenberg function h is of the form (1.1) if and only if h has neither +(α, β, . . . , β), (β − 1, . . . , β − 1, β, . . . , β +� +�� +� +α +) for 3 ≤ α < β, nor (2, γ − 1, . . . , γ − 1, γ, γ) for γ ≥ 5 +as its minor. + +8 +M. MASUDA AND T. SATO +Recall that if h† denotes the Hessenberg function obtained by flipping the configuration of h along the +anti-diagonal, then X(h†) ∼= X(h) as varieties. Therefore +X((α, β, . . . , β)) ∼= X((β − 1, . . . , β − 1, β, . . . , β +� +�� +� +α +)). +Here, we know that H∗(X((α, β, . . . , β)); Q) is not generated in degree 2 for 3 ≤ α < β by [2, Theorem 4.3]. +Thus, it suffices to treat the last case in Proposition 3.3, which we shall discuss in the next subsection. +3.2. The case h = (2, n − 1, . . . , n − 1, n, n). In this subsection we prove the following proposition. +Proposition 3.4. H∗(X(h); Q) is not generated in degree 2 when h = (2, n − 1, . . . , n − 1, n, n) for n ≥ 5. +Some computation is involved in the proof of this proposition but the idea of the proof is simple. We +compute the Poincar´e polynomial of X(h) using Theorem 2.1(4). On the other hand, using explicit generators +of H2(X(h)) by [4], we compute an upper bound of the Hilbert series of the subring of H∗(X(h)) generated +by H2(X(h)). Then it turns out that the latter is strictly smaller than the former at a certain degree. +3.2.1. Poincar´e polynomial of X(h). The following proposition, which easily follows from Theorem 2.1(4), +enables us to compute the Poincar´e polynomial of X(h) inductively. +Proposition 3.5 ([4, Proposition 3.1]). +(3.5) +Poin(X(h), √q) = +n +� +j=1 +qh(j)−j Poin(X(hj), √q). +Using the proposition above, the Poincar´e polynomial of X(h) is explicitly computed as follows when +h = (h(1), n, . . . , n). +Proposition 3.6 ([2]). When h = (h(1), n . . . , n), we have +(3.6) +Poin(X(h), √q) = [h(1)]q[n − 1]q! + (n − 1)qh(1)−1[n − h(1)]q[n − 2]q!, +where +[m]q = 1 − qm +1 − q , +[m]q! = [1]q[2]q · · · [m]q = +m +� +j=1 +1 − qj +1 − q . +Now, let h = (2, n − 1, . . . , n − 1, n, n) and set +Pn(q) := Poin(X(h), √q). +Lemma 3.7. For n ≥ 5, the following recurrence formula holds +Pn(q) = (1 + q)2[n − 2]q! + (n − 2)(q + q2)[n − 3]q[n − 3]q! ++ (n − 1)(q + qn−3) {(1 + q)[n − 3]q! + (n − 3)q[n − 4]q[n − 4]q!} ++ (q + q2 + · · · + qn−4)Pn−1(q). +Proof. Let Fn(q) denote the right-hand side of (3.6) with h(1) = 2, that is, +(3.7) +Fn(q) := (1 + q)[n − 1]q! + (n − 1)q[n − 2]q[n − 2]q!. +Then we have +Poin(X(h1), √q) = Poin(X(hn), √q) = Fn−1(q) +Poin(X(h2), √q) = Poin(X(hn−1), √q) = (n − 1)Fn−2(q) +Poin(X(hj), √q) = Pn−1(q) +(3 ≤ j ≤ n − 2), + +REGULAR SEMISIMPLE HESSENBERG VARIETIES +9 +where we note that X(h2) consists of n − 1 copies of Fl(n − 2). Hence, by (3.5), we have +Pn(q) = qFn−1(q) + (n − 1)qn−3Fn−2(q) + (qn−4 + · · · + q)Pn−1(q) ++ (n − 1)qFn−2(q) + Fn−1(q) += (1 + q)Fn−1(q) + (n − 1)(q + qn−3)Fn−2(q) + (q + · · · + qn−4)Pn−1(q). +Combining this equation with (3.7), we obtain the desired equation. +✷ +Lemma 3.8. For n ≥ 4, let +Qn(q) = (1 + 2nq + n(n − 1)q2)[n − 2]q! + n(n − 3) +2 +qn−3. +Then we have +Pn(q) ≡ Qn(q) +mod (qn−2). +In other words, Pn(q) and Qn(q) coincide up to degree n − 3. +Proof. We prove the lemma by induction on n. When n = 4, we have +P4(q) = 1 + 11q + 11q2 + q3, +Q4(q) = 1 + 11q + 20q2 + 12q3, +and the lemma is true for n = 4. +Let n be given and suppose that the lemma is true for n − 1, that is, +(3.8) +Pn−1(q) ≡ Qn−1(q) +mod (qn−3). +Hereafter, in this proof, all congruences will be taken modulo qn−2 unless otherwise stated. Since we have +(q + q2)[n − 3]q[n − 3]q! ≡ (q + q2)[n − 2]q! +q2[n − 4]q[n − 4]q! ≡ q2[n − 3]q!, +the recurrence formula in Lemma 3.7 reduces to the following congruence relation: +Pn(q) ≡ (1 + nq + (n − 1)q2)[n − 2]q! + (n − 1)(q + (n − 2)q2)[n − 3]q! ++ (n − 1)qn−3 + (q + · · · + qn−4)Pn−1(q). +(3.9) +It follows from (3.8) and the definition of Qn that the sum of the last two terms above becomes as follows. +(n − 1)qn−3 + (q + · · · + qn−4)Pn−1(q) +≡ (n − 1)qn−3 + +� +1 + (2n − 2)q + (n − 1)(n − 2)q2� +(q + · · · + qn−4)[n − 3]q! + (n − 1)(n − 4) +2 +qn−3 += +� +1 − q + (n − 1)q(1 − q) + nq + (n − 1)2q2� +(q + · · · + qn−4)[n − 3]q! + (n − 1)(n − 2) +2 +qn−3 +≡ +� +q − qn−3 + (n − 1)q2 + (nq + (n − 1)2q2)(q + · · · + qn−4) +� +[n − 3]q! + (n − 1)(n − 2) +2 +qn−3 +≡ +� +q + (n − 1)q2 + (nq + (n − 1)2q2)(q + · · · + qn−4) +� +[n − 3]q! + n(n − 3) +2 +qn−3 +By substituting it to (3.9), we obtain +Pn(q) ≡ (1 + nq + (n − 1)q2)[n − 2]q! ++ +� +(nq + (n − 1)2q2) + (nq + (n − 1)2q2)(q + · · · + qn−4) +� +[n − 3]q! + n(n − 3) +2 +qn−3 +≡ (1 + nq + (n − 1)q2)[n − 2]q! + (nq + (n − 1)2q2)[n − 2]q! + n(n − 3) +2 +qn−3 += (1 + 2nq + n(n − 1)q2)[n − 2]q! + n(n − 3) +2 +qn−3 += Qn(q). + +10 +M. MASUDA AND T. SATO +This completes the induction step and the lemma has been proved. +✷ +3.2.2. Hilbert series of the subring generated by H2(X(h)). When h = (2, n − 1, . . ., n − 1, n, n) for n ≥ 5, we +first observe H2(X(h)). By (2.7), we have +⊥(h) = {n − 2}, +L(h) = {1, n − 1}. +Therefore, it follows from Theorem 2.4 that H2(X(h)) is generated by the following elements +(3.10) +xk, +yk := yn−2,k, +τk := τ{k} +(k ∈ [n]), +where +xk(w) = tw(k), +yk(w) = yn−2,k(w) = +� +tk − tw(n−1) +(if k ∈ {w(1), . . . , w(n − 2)}) +0 +(otherwise), +τk(w) = τ{k}(w) = +� +tw(1) − tw(2) +(if k = w(1)) +0 +(otherwise) +(3.11) +for w ∈ Sn by Definition 2.3, and +(3.12) +n +� +k=1 +yk = x1 + · · · + xn−2 − (n − 2)xn−1, +n +� +k=1 +τk = x1 − x2 +by Theorem 2.4. We also have +σ · xk = xk, +σ · yk = yσ(k), +σ · τk = τσ(k) +for σ ∈ Sn by (2.8). +To make the following argument clearer, we introduce elements ρk for k ∈ [n] defined by +(3.13) +ρk(w) := +� +tw(n−1) − tw(n) +(if k = w(n)) +0 +(otherwise). +Similarly to τk, the ρk satisfies the condition (2.6) so that it defines an element of H2 +T (X(h)) and H2(X(h)) +and +(3.14) +n +� +k=1 +ρk = xn−1 − xn, +σ · ρk = ρσ(k) +for σ ∈ Sn. +An elementary check shows that +(yk − yℓ)(w) − (ρk − ρℓ)(w) = tk − tℓ +(k, ℓ ∈ [n], w ∈ Sn) +and hence yk − yℓ = ρk − ρℓ in H2(X(h)). Moreover, �n +k=1 yk and �n +k=1 ρk are both linear polynomials in +xi’s by (3.12) and (3.14), so we may replace yk’s in the generating set (3.10) by ρk’s. Namely H2(X(h)) is +generated by +xk, +τk, +ρk +(k ∈ [n]) +with relations +(3.15) +n +� +k=1 +xk = 0, +n +� +k=1 +τk = x1 − x2, +n +� +k=1 +ρk = xn−1 − xn, +and the actions of σ ∈ Sn on those generators are given by +(3.16) +σ · xk = xk, +σ · τk = τσ(k), +σ · ρk = ρσ(k). + +REGULAR SEMISIMPLE HESSENBERG VARIETIES +11 +Our purpose is to find a sharp upper bound of the Hilbert series of the subring R(h) of H∗(X(h)) generated +by H2(X(h)). Let A(h) be the subring of H∗(X(h)) generated by xk’s and we regard R(h) as a module over +A(h). It follows from (3.11) and (3.13) that +τkτℓ = +� +(x1 − x2)τk +(k = ℓ) +0 +(k ̸= ℓ), +ρkρℓ = +� +(xn−1 − xn)ρk +(k = ℓ) +0 +(k ̸= ℓ), +τkρk = 0. +Therefore, R(h) is generated by 1, τk, ρk (k ∈ [n]), and τiρj (i ̸= j ∈ [n]) as a module over A(h). The subring +A(h) itself is a submodule of R(h) over A(h). We consider three other submodules of R(h) over A(h): +B(h) :={ +n +� +k=1 +bkτk | bk ∈ A(h), +n +� +k=1 +bk = 0}, +C(h) :={ +n +� +k=1 +ckρk | ck ∈ A(h), +n +� +k=1 +ck = 0}, +D(h) :={ +� +1≤i,j≤n +dijτiρj | dij ∈ A(h), +n +� +j=1 +dij = 0 for i ∈ [n], +n +� +i=1 +dij = 0 for j ∈ [n]} +(3.17) +where dkk = 0 for k ∈ [n]. Note that A(h)⊗Q agrees with the ring of invariants H∗(X(h); Q)Sn as mentioned +in Remark 2.1. +Lemma 3.9. R(h) is additively generated by A(h), B(h), C(h), and D(h) when tensoring with Q. +Proof. Since H∗(X(h)) is generated by 1, τk, ρk (k ∈ [n]), and τiρj (i ̸= j ∈ [n]) as a module over A(h), it +suffices to show that any element of the form +(3.18) +n +� +k=1 +bkτk + +n +� +k=1 +ckρk + +� +1≤i,j≤n +dijτiρj +(bk, ck, dij ∈ A(h), dkk = 0) +can be expressed as a sum of elements in A(h), B(h), C(h), and D(h) when tensoring with Q. +Step 1. Set b := �n +k=1 bk and c := �n +k=1 ck. Since �n +k=1 τk = x1 −x2 and �n +k=1 ρk = xn−1 −xn by (3.15), +we have +n +� +k=1 +bkτk + +n +� +k=1 +ckρk = +n +� +k=1 +� +bk − b +n +� +τk + b +n(x1 − x2) + +n +� +k=1 +� +ck − c +n +� +ρk + c +n(xn−1 − xn). +Here the two sums at the right hand side above respectively belong to B(h) ⊗ Q and C(h) ⊗ Q, and the +remaining two terms belong to A(h) ⊗ Q. +Step 2. As for the last term in (3.18), since �n +i=1 τi = x1 − x2, we have +� +1≤i,j≤n +dijτiρj = +n +� +j=1 +� n +� +i=1 +� +dij − dj +n +� +τi +� +ρj + +n +� +j=1 +dj +n (x1 − x2)ρj += +� +1≤i,j≤n +˜dijτiρj + +n +� +j=1 +dj +n (x1 − x2)ρj +(3.19) +where +dj := +n +� +i=1 +dij +and +˜dij := dij − dj +n . +The last sum in (3.19) is a sum of elements in A(h) ⊗ Q and C(h) ⊗ Q by Step 1. We shall show that the +sum � +1≤i,j≤n ˜dijτiρj in (3.19) is a sum of elements in A(h) ⊗ Q, B(h) ⊗ Q, and D(h) ⊗ Q. We note that +(3.20) +n +� +i=1 +˜dij = +n +� +i=1 +� +dij − dj +n +� += +n +� +i=1 +dij − dj = 0 + +12 +M. MASUDA AND T. SATO +and set +(3.21) +˜di := +n +� +j=1 +˜dij. +Since �n +j=1 ρj = xn−1 − xn, we have +(3.22) +� +1≤i,j≤n +˜dijτiρj = +n +� +i=1 + + +n +� +j=1 +� +˜dij − +˜di +n +� +ρj + + τi + +n +� +i=1 +˜di +n (xn−1 − xn)τi. +Here the second sum at the right hand side of (3.22) is a sum of elements in A(h) ⊗ Q and B(h) ⊗ Q by Step +1. As for the coefficients ˜dij − +˜di +n of τiρj in the first sum at the right hand side of (3.22), it follows from +(3.20) and (3.21) that we have +n +� +i=1 +� +˜dij − +˜di +n +� += +n +� +i=1 +˜dij − 1 +n +n +� +i=1 +˜di = − 1 +n +n +� +i=1 +n +� +j=1 +˜dij = − +n +� +j=1 +� n +� +i=1 +˜dij +� += 0, +n +� +j=1 +� +˜dij − +˜di +n +� += +n +� +j=1 +˜dij − ˜di = 0. +Thus, the first sum at the right hand side of (3.22) belongs to D(h) ⊗ Q. This completes the proof of the +lemma. +✷ +We shall calculate upper bounds of the Hilbert series of A(h), B(h), C(h), and D(h). +Hilbert series of A(h). Since A(h) ⊗ Q = H∗(X(h))Sn ⊗ Q and h = (2, n − 1, . . . , n − 1, n, n) in our case, +it follows from (2.11) that +(3.23) +Hilb(A(h), √q) = +n−1 +� +j=1 +[h(j) − j]q = (1 + q)2[n − 2]q!. +Hilbert series of B(h). It follows from(3.11) that (x1 − tk)τk vanishes at every w ∈ Sn, so we have +(3.24) +(x1 − tk)τk = 0 +in H∗ +T (X(h)) +and hence +x1τk = 0 +in H∗(X(h)). +Therefore, B(h) is indeed a module over A(h)/(x1). Here +A(h)/(x1) ⊗ Q = A(h1) ⊗ Q +by (2.9) and (2.10). Since h1 = (n − 2, . . . , n − 2, n − 1, n − 1), it follows from (2.11) that +Hilb(A(h)/(x1), √q) = +n−2 +� +j=1 +[h1(j) − j]q = (1 + q)[n − 2]q!. +Since B(h) is a module over A(h)/(x1) generated by τi − τi+1 (i ∈ [n − 1]) and the cohomological degrees of +τk’s are two, we obtain an upper bound of Hilb(B(h), q) as follows: +(3.25) +Hilb(B(h), √q) ≤ (n − 1)q Hilb(A(h)/(x1), √q) = (n − 1)(q + q2)[n − 2]q!. +Here �∞ +i=0 aiqi ≤ �∞ +i=0 biqi (ai, bi ∈ Z) means that ai ≤ bi for all i’s. +Hilbert series of C(h). To f ∈ Map(Sn, Z[t1, . . . , tn]) we associate f ∨ ∈ Map(Sn, Z[t1, . . . , tn]) defined by +f ∨(w) := f(ww0) +for w ∈ Sn, +where w0 denotes the longest element in Sn, i.e. w0 = n n − 1 · · · 2 1 in one-line notation. This defines an +involution on Map(Sn, Z[t1, . . . , tn]) and one can easily check that +x∨ +k = xn−k+1, +τ ∨ +k = −ρk, +ρ∨ +k = −τk + +REGULAR SEMISIMPLE HESSENBERG VARIETIES +13 +from (3.11) and (3.13). Hence the involution gives an isomorphism between B(h) and C(h), and the same +inequality as (3.25) holds for C(h), i.e. +(3.26) +Hilb(C(h), √q) ≤ (n − 1)(q + q2)[n − 2]q!. +Hilbert series of D(h). We have x1τk = 0 by (3.24). Similarly we have xnρk = 0 since (x1τk)∨ = −xnρk. +(The fact xnρk = 0 also follows from the definition (3.11) and (3.13) of xk and ρk.) Therefore, D(h) is indeed +a module over A(h)/(x1, xn). +As mentioned in Remark 2.1, A(h) ⊗ Q = H∗(X(h))Sn ⊗ Q and it is the image of the restriction map +ι∗ : H∗(Fl(n)) → H∗(X(h)). Therefore, A(h)/(x1, xn) is the image of the restriction map from H∗(Fl(n−2)) +and hence +Hilb(A(h)/(x1, xn), √q) ≤ [n − 2]q!. +(In fact, the equality holds above.) There are 2n relations among dij (i ̸= j) in the definition (3.17) of D(h), +but one relation can be obtained from the other 2n−1 relations because �n +i=1 +��n +j=1 dij +� += �n +j=1 (�n +i=1 dij). +Moreover, there are n(n − 1) number of dij’s and the cohomological degree of τiρj is four. Thus +(3.27) +Hilb(D(h), √q) ≤ Hilb(A(h)/(x1, xn), √q) {n(n − 1) − (2n − 1)} q2 ≤ (n2 − 3n + 1)q2[n − 2]q!. +Proof of Proposition 3.4. It follows from Lemma 3.9, (3.23), (3.25), (3.26), and (3.27) that +Hilb(R(h), √q) ≤ (1 + q)2[n − 2]q! + 2(n − 1)(q + q2)[n − 2]q! + (n2 − 3n + 1)q2[n − 2]q! += (1 + 2nq + n(n − 1)q2)[n − 2]q!. +The coefficient of qn−3 in the last term above is less than that of Pn(q) in Lemma 3.8 by n(n − 3)/2, proving +the proposition. +✷ +4. Sufficiency +The purpose of this section is to prove the following proposition, which implies the sufficiency of Theorem +1.1. +Proposition 4.1. When h is of the form (1.1), the equivariant cohomology H∗ +T (X(h)) is generated in degree +2 as an algebra over H∗(BT ). +By Theorem 2.1(3), X(h) is not connected when h(k) = k for some 1 ≤ k ≤ n − 1. In this case, a flag +V• = (V0 ⊂ V1 ⊂ · · · ⊂ Vn) ∈ X(h) is of the form Vk = ⟨ei1, ei2, . . . , eik⟩ for some {i1, . . . , ik} ⊂ [n], where ei +is the i-th standard basis vector of Cn. Therefore, decomposing V• into two flags (V0 ⊂ V1 ⊂ · · · ⊂ Vk) and +(V ′ +0 ⊂ V ′ +1 ⊂ · · · ⊂ V ′ +n−k), where V ′ +i = Vk+i/Vk, one can see that X(h) is the disjoint union of +�n +k +� +copies of +X(h1) × X(h2), where h1 and h2 are the Hessenberg function obtained by restricting h onto intervals [k] and +[k + 1, n], respectively. Each copy corresponds to the choice of a k-subset {i1, . . . , ik} ⊂ [n]. To be precise, +h2 : [n − k] → [n − k] is given by shift−1 +k +◦ h ◦ shiftk, where shiftk : [n − k] → [k + 1, n] shifts integers by k. +Suppose h is of the form (1.1) and 1 ≤ r ≤ n. Then +X(hr) is not connected ⇐⇒ a + 1 ≤ r ≤ b +by Theorem 2.1(3) and that hr is also of the form (1.1) when r < a + 1 or r > b. When a + 1 ≤ r ≤ b, each +connected component of X(hr) is isomorphic to X(h1) × X(h2) and both h1 and h2 are of the form (1.1). +Let Γ(Sn, h) denote the labeled graph of X(h). Recall that H∗ +T (X(h)) ∼= H∗(Γ(Sn, h)). For the subset +Sr +n ⊂ Sn in (3.1), let Γ(Sr +n, h) be the induced labeled subgraph of Γ(Sn, h) on the subset Sr +n of vertices, +and let Γ0(Sr +n, h) denote a connected component of Γ(Sr +n, h). +Lemma 4.2. When h is of the form (1.1), the restriction map H2(Γ(Sn, h)) → H2(Γ0(Sr +n, h)) is surjective. +We admit the lemma and complete the proof of Proposition 4.1. +Before that, we shall observe that +Γ0(Sr +n, h) is essentially a connected component of a labeled graph of X(hr). Indeed, for 1 ≤ r ≤ n, let cr be +the cyclic permutation (r r + 1 r + 2 · · · n) and +ϕr : Γ0(Sr +n, h) → Γ0(Sn−1, hr) + +14 +M. MASUDA AND T. SATO +a graph isomorphism defined by ϕr(w) = wcr for w ∈ Sr +n. When i, j ̸= r, the (i, j)-th box in the configuration +for h corresponds to the (c−1 +r (i), c−1 +r (j))-th box in the configuration for hr (see Figure 3). In particular, +v = w(i, j) corresponds to vcr = wcr(c−1 +r (i), c−1 +r (j)) and the edges between these vertices have the same +label tw(i) − tw(j). Therefore, ϕr induces an isomorphism +(4.1) +ϕ∗ +r : H∗(Γ0(Sn−1, hr)) +∼ += +−→ H∗(Γ0(Sr +n, h)) +of graded algebras over H∗(BT ). +Proof of Proposition 4.1. Recall that H∗ +T (X(h)) ∼= H∗(Γ(Sn, h)). We prove the proposition by induction on +n. Let 1 ≤ r ≤ n. For any z ∈ H∗(Γ(Sn, h)) that vanishes on �r−1 +j=1 Sj +n, it is sufficient to show the existence +of a polynomial f in elements of H2(Γ(Sn, h)) such that z − f vanishes on �r +j=1 Sj +n. Then the induction on +r proves the proposition. We shall show the existence of f by division into cases according to the value of r. +Case 1. The case 1 ≤ r ≤ a. In this case, Γ(Sr +n, h) is connected. We note that z vanishes on �r−1 +j=1 Sj +n +and this implies that z(w) for w ∈ Sr +n decomposes as follows: +(4.2) +z(w) = + + +r−1 +� +j=1 +(tw(j) − tn) + + g(w), +g ∈ H∗(Γ(Sr +n, h)). +Indeed, for w ∈ Sr +n, we have w(r) = n and w(j, r) ∈ Sj +n. If j ≤ r − 1, then there is an edge in the graph +Γ(Sn, h) between the vertices w and w(j, r). The label on the edge is tw(j) −tw(r) = tw(j) −tn and z vanishes +at w(j, r) ∈ Sj +n (j ≤ r − 1) by assumption. Therefore z(w) is divisible by the product in the big parenthesis +in (4.2) and g ∈ Map(Sr +n, H∗(BT )). Furthermore, one can easily check that the g is indeed in H∗(Γ(Sr +n, h)) +since z is in H∗(Γ(Sn, h)). +Since H∗(Γ(Sr +n, h)) ∼= H∗(Γ(Sn−1, hr)) by (4.1), g is a polynomial in elements of H2(Γ(Sr +n, h)) by induc- +tion on n. Moreover, by Lemma 4.2, there is a polynomial ˜g in H2(Γ(Sn, h)) which coincides with g on Sr +n. +On the other hand, �r−1 +j=1(xj −tn) coincides with the product in (4.2) on Sr +n since xj(w) = tw(j) by definition +of xj, and vanishes on �r−1 +j=1 Sj +n since xj(w) = tw(j) = tn for w ∈ Sj +n. Therefore, +� �r−1 +j=1(xj − tn) +� +˜g coincides +with the element z on �r +j=1 Sj +n. Thus +� �r−1 +j=1(xj − tn) +� +˜g is a desired polynomial f. +Case 2. The case r = a + 1. Similarly to Case 1, z(w) for w ∈ Sa+1 +n +decomposes as follows: +(4.3) +z(w) = + + +a +� +j=1 +(tw(j) − tn) + + g(w), +g ∈ H∗(Γ(Sa+1 +n +, h)). +Note that Γ(Sa+1 +n +, h) is not connected. Two vertices v, w ∈ Sa+1 +n +lie in the same connected component if +and only if +{v(1), . . . , v(a)} = {w(1), . . . , w(a)} ⊂ [n − 1]. +For K := {k1, . . . , ka} ⊂ [n − 1], we consider the element ρK defined by +ρK = +a +� +j=1 +ya,kj, +where +ya,k(w) = +� +tk − tw(a+1) +(k ∈ {w(1), . . . , w(a)}) +0 +(k /∈ {w(1), . . . , w(a)}) +by definition. Therefore, since w(a + 1) = n for w ∈ Sa+1 +n +, we have +ρK(w) = +��a +j=1(tw(j) − tn) +(K = {w(1), . . . , w(a)}) +0 +(K ̸= {w(1), . . . , w(a)}). +Hence ρK coincides with the product in the big parentheses of (4.3) on the connected component +(4.4) +{w ∈ Sa+1 +n +| w([a]) = K} + +REGULAR SEMISIMPLE HESSENBERG VARIETIES +15 +and vanishes on the other components. Since n /∈ K and w(j) = n for w ∈ Sj +n, ρK also vanishes on �a +j=1 Sj +n. +On the other hand, the element g in (4.3) restricted to the connected component (4.4) is obtained as the +restriction of a polynomial ˜gK in H2(Γ(Sn, h)) similarly to Case 1. Therefore, we obtain a desired polynomial +f as +� +K⊂[n−1], |K|=a +ρK˜gK. +Case 3. The case a + 2 ≤ r ≤ b. In this case, z(w) for w ∈ Sr +n decomposes as follows: +(4.5) +z(w) = (tw(r−1) − tw(r))g(w), +g ∈ H∗(Γ(Sr +n, h)). +Similarly to Case 2, Γ(Sr +n, h) is not connected and two vertices v, w ∈ Sr +n lie in the same connected component +if and only if +{v(1), . . . , v(r − 1)} = {w(1), . . . , w(r − 1)} ⊂ [n − 1]. +For A ⊂ [n − 1] with |A| = r − 1, we have +τA(w) = +� +tw(r−1) − tw(r) +(A = {w(1), . . . , w(r − 1)}) +0 +(A ̸= {w(1), . . . , w(r − 1)}) +by definition. Hence, τA coincides with the factor of the right-hand side of (4.5) on the connected component +{w ∈ Sr +n | w([r − 1]) = A}, and vanishes on the other connected components. Since n /∈ A and w(j) = n for +w ∈ Sj +n, τA also vanishes on �r−1 +j=1 Sj +n. Therefore, similarly to Case 2, we obtain a desired polynomial f as +� +A⊂[n−1], |A|=r−1 +τA˜gA, where ˜gA is a polynomial in H2(Γ(Sn, h)). +Case 4. The case b + 1 ≤ r ≤ n. In this case, z(w) for w ∈ Sr +n decomposes as follows: +(4.6) +z(w) = + + +r−1 +� +j=b +(tn − tw(j)) + + g(w), +g ∈ H∗(Γ(Sr +n, h)). +Similarly to Case 1, X(hr) is connected and g is the restriction of a polynomial ˜g in H2(Γ(Sn, h)). +We consider the element y∗ +b+1,n ∈ H∗(Γ(Sn, h)) in Remark 2.2, which is defined as +y∗ +b+1,n(w) = +� +tn − tw(b) +(n ∈ {w(b + 1), . . . , w(n)}) +0 +(n /∈ {w(b + 1), . . . , w(n)}). +Then + +y∗ +b+1,n +r−1 +� +j=b+1 +(tn − xj) + + (w) = +��r−1 +j=b(tn − tw(j)) +(n ∈ {w(b + 1), . . . , w(n)}) +0 +(n /∈ {w(b + 1), . . . , w(n)}). +Hence y∗ +b+1,n +�r−1 +j=b+1(tn − xj) coincides with the product in the big parentheses of (4.6) on Sr +n, and vanishes +on �r−1 +j=1 Sj +n. Therefore, +� +y∗ +b+1,n +�r−1 +j=b+1(tn − xj) +� +˜g is a desired polynomial f. +✷ +Finally we give a proof of Lemma 4.2. +Proof of Lemma 4.2. It follows from Theorem 2.4 and Remark 2.2 that when h is of the form (1.1), the +elements in +{xi, ya,k, τA, ti | i, k ∈ [n], A ⊂ [n], a + 1 ≤ |A| < b} +span H2(Γ(Sn, h)). Through the isomorphism (4.1), one can find generators of H2(Γ0(Sr +n, h)) which corre- +spond to the generators of H2(Γ0(Sn−1, hr)). They are given as restrictions of +xi for i ∈ [n], i ̸= r, +ti for i ∈ [n], +and the following elements in H2(Γ(Sn, h)). +Case 1. When 1 ≤ r ≤ a, +ya,k for k ∈ [n − 1], +τA⊔{n} for A ⊂ [n − 1], a ≤ |A| < b − 1. + +16 +M. MASUDA AND T. SATO +Case 2. When r = a + 1, for a connected component Γ0(Sa+1 +n +, h) which contains σ ∈ Sa+1 +n +; +τB⊔σ([a+1]) for B ⊂ σ([n]\[a + 1]), 1 ≤ |B| < b − (a + 1). +Case 3. When a + 1 < r ≤ b, for a connected component Γ0(Sr +n, h) which contains σ ∈ Sr +n; +ya,k for k ∈ [n − 1], +τA for A ⊂ σ([r − 1]), a + 1 ≤ |A| < r − 1, +τB⊔σ([r]) for B ⊂ σ([n] \ [r]), 1 ≤ |B| < b − r. +Case 4. When b < r ≤ n, +ya,k for k ∈ [n − 1], +τA for A ⊂ [n − 1], a + 1 ≤ |A| < b. +This proves the lemma. +✷ +Acknowledgment. +We thank Yunhyung Cho for his help on moment map. Masuda was supported in part by JSPS Grant- +in-Aid for Scientific Research 22K03292 and a HSE University Basic Research Program. This work was +partly supported by Osaka Central Advanced Mathematical Institute (MEXT Joint Usage/Research Center +on Mathematics and Theoretical Physics JPMXP0619217849). +References +[1] H. Abe, M. Harada, T. Horiguchi, and M. Masuda, The cohomology rings of regular nilpotent Hessenberg varieties in Lie +type A, Int. Math. Res. Not. IMRN (2019), no. 17, 5316–5388. +[2] H. Abe, T. Horiguchi, and M. Masuda, The cohomology rings of regular semisimple Hessenberg varieties for h = +(h(1), n, ..., n), J. Comb. 10.1 (2019), pp. 27–59. +[3] T. Abe, T. Horiguchi, M. Masuda, S. Murai, and T. Sato, Hessenberg varieties and hyperplane arrangements, J. f¨ur die +Reine und Angew. Math. 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Sato, The cohomology ring of a regular semisimple Hessenberg variety of double lollipop type, in +preparation. +[18] J. Shareshian and M. L. Wachs, Chromatic quasisymmetric functions, Adv. Math. 295 (2016), 497–551. +[19] S. Tolman and J. Weitsman, The cohomology rings of symplectic quotients, Comm. Anal. Geom. 11 (2003), 751–773. +[20] J. Tymoczko, Permutation actions on equivariant cohomology of flag varieties, Toric topology, 365–384, Contemp. Math., +460, Amer. Math. Soc., Providence, RI, 2008. +Osaka Metropolitan University Advanced Mathematical Institute, Sumiyoshi-ku, Osaka 558-8585, Japan. +Email address: mikiyamsd@gmail.com +Osaka Metropolitan University Advanced Mathematical Institute, Sumiyoshi-ku, Osaka 558-8585, Japan. +Email address: 00tkshst00@gmail.com + diff --git a/8dE2T4oBgHgl3EQfPwZL/content/tmp_files/load_file.txt b/8dE2T4oBgHgl3EQfPwZL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..41c3c1fb1147e25028ab923a1838178fe4036e38 --- /dev/null +++ b/8dE2T4oBgHgl3EQfPwZL/content/tmp_files/load_file.txt @@ -0,0 +1,975 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf,len=974 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='03762v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='AG] 10 Jan 2023 REGULAR SEMISIMPLE HESSENBERG VARIETIES WITH COHOMOLOGY RINGS GENERATED IN DEGREE TWO MIKIYA MASUDA AND TAKASHI SATO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' A regular semisimple Hessenberg variety Hess(S, h) is a smooth subvariety of the flag variety determined by a square matrix S with distinct eigenvalues and a Hessenberg function h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The cohomology ring H∗(Hess(S, h)) is independent of the choice of S and is not explicitly described except for a few cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In this paper, we characterize the Hessenberg function h such that H∗(Hess(S, h)) is generated in degree two as a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' It turns out that such h is what is called a (double) lollipop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Introduction The flag variety Fl(n) consists of nested sequences of linear subspaces in the complex vector space Cn: Fl(n) = {V• = (V1 ⊂ V2 ⊂ · · · ⊂ Vn = Cn) | dimC Vi = i (∀i ∈ [n] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=', n})}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' A Hessenberg function h: [n] → [n] is a monotonically non-decreasing function satisfying h(j) ≥ j for any j ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We often express a Hessenberg function h as a vector (h(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , h(n)) by listing the values of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Given an n × n matrix A and a Hessenberg function h, a Hessenberg variety Hess(A, h) is defined as Hess(A, h) = {V• ∈ Fl(n) | AVi ⊂ Vh(i) (∀i ∈ [n])} where the matrix A is regarded as a linear operator on Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Note that Hess(A, h) = Fl(n) if h = (n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The family of Hessenberg varieties Hess(A, h) contains important varieties such as Springer fibers (A is nilpotent and h = (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=', n)), Peterson varieties (A is regular nilpotent and h = (2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n, n)), and permutohedral varieties (A is regular semisimple and h = (2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n, n)), which are toric varieties with permutohedra as moment polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Among n × n matrices, regular semisimple ones S (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' matrices S having distinct eigenvalues) are generic and Hess(S, h) is called a regular semisimple Hessenberg variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The regular semisimple Hessenberg variety Hess(S, h) has nice properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For instance, it is smooth and its cohomology H∗(Hess(S, h)) becomes a module over the symmetric group Sn on [n] by Tymoczko’s dot action [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Remarkably, the solution of Shareshian–Wachs conjecture [18] by Brosnan and Chow [5] (and Guay-Paquet [10]) connected H∗(Hess(S, h)) as an Sn-module and chromatic symmetric functions on certain graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' This opened a way to prove the famous Stanley–Stembridge conjecture in graph theory through the geometry or topology of Hessenberg varieties and motivated us to study H∗(Hess(S, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Note that H∗(Hess(S, h)) (indeed the diffeomorphism type of Hess(S, h)) is independent of the choice of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We write the regular semisimple Hessenberg variety Hess(S, h) as X(h) for brevity since our concern in this paper is its cohomology ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The Sn-module structure on H∗(X(h)) is determined in some cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In particular, that on H2(X(h)) was explicitly described by Chow [7] combinatorially (through the theorem by Brosnan-Chow mentioned above) and by Cho-Hong-Lee [6] geometrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Motivated by their works, Ayzenberg and the authors [4] reproved their results by giving explicit additive generators of H2(X(h)) in terms of GKM theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The ring structure on H∗(X(h)) is not explicitly described except for a few cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Remember that X(h) for h = (n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n) is the flag variety Fl(n) and H∗(Fl(n)) is generated in degree 2 as a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Moreover, X(h) for h = (2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n, n) is the permutohedral variety and H∗(X(h)) is also generated in degree 2 as a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' On the other hand, for h = (h(1), n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n) with h(1) arbitrary, a result of [2] shows that H∗(X(h)) is generated Date: January 11, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Primary: 57S12, Secondary: 14M15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hessenberg variety, torus action, GKM theory, equivariant cohomology, lollipop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 1 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' MASUDA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' SATO in degree 2 as a ring if and only if h(1) = 2 or n, where X(h) = Fl(n) for the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, it is natural to ask when H∗(X(h)) is generated in degree 2 as a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The answer is the following, which is our main result in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Assume that h(j) ≥ j + 1 for j ∈ [n − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then H∗(X(h)) is generated in degree 2 as a ring if and only if h is of the following form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) for some 1 ≤ a < b ≤ n, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) h(j) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 a + 1 (1 ≤ j ≤ a) j + 1 (a < j < b) n (b ≤ j ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (1) X(h) is connected if and only if h(j) ≥ j + 1 for any j ∈ [n − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When X(h) is not connected, each connected component of X(h) is a product of smaller regular semisimple Hessenberg varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (2) X(h) is the flag variety Fl(n) when (a, b) = (n − 1, n) and is the permutohedral variety when (a, b) = (1, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (3) We will give an explicit presentation of the ring structure on H∗(X(h)) for h of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) in a forthcoming paper [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We can visualize a Hessenberg function h by drawing a configuration of the shaded boxes on a square grid of size n × n, which consists of boxes in the i-th row and the j-th column satisfying i ≤ h(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since h(j) ≥ j for any j ∈ [n], the essential part is the shaded boxes below the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For example, Figure 1 below is the configurations of two Hessenberg functions h of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) with n = 11: one is h = (2, 3, 4, 5, 6, 7, 11, 11, 11, 11) where (a, b) = (1, 7) and the other is h = (4, 4, 4, 5, 6, 7, 11, 11, 11, 11) where (a, b) = (3, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We often identify a Hessenberg function h with its configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The configurations for h = (2, 3, 4, 5, 6, 7, 11, 11, 11, 11) and h = (4, 4, 4, 5, 6, 7, 11, 11, 11, 11) The chromatic symmetric functions and LLT polynomials associated with h of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) are studied from the viewpoint of combinatorics in [8, 13], and when a = 1 or b = n, the corresponding Hessenberg functions h = (2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , b, n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=', n) or (a + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , a + 1, a + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n − 1, n, n) are called lollipops in those papers, so the Hessenberg function of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) may be called a double lollipop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In Section 2, we review GKM theory to compute the equivariant cohomology of X(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We prove the “only if” part in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Indeed, we consider a Morse-Bott function fh on X(h), where the inverse image of the minimum or maximum value of fh is a regular semisimple Hessenberg variety X(h′) with h′ of size one less than that of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then a property of the REGULAR SEMISIMPLE HESSENBERG VARIETIES 3 Morse-Bott function fh shows the surjectivity of the restriction map H∗(X(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) → H∗(X(h′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q), and this enables us to use an inductive argument to prove the “only if” part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In Section 4, we prove the “if” part in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1 by applying the method developed in [2, 9] together with the explicit generators of H2(X(h)) obtained in our previous work [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Regular semisimple Hessenberg varieties We first recall some properties of a regular semisimple Hessenberg variety X(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1 ([14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (1) X(h) is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (2) dimC X(h) = �n j=1(h(j) − j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (3) X(h) is connected if and only if h(j) ≥ j + 1 for ∀j ∈ [n − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (4) Hodd(X(h)) = 0 and the 2k-th Betti number of X(h) is given by #{w ∈ Sn | ℓh(w) = k} where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) ℓh(w) = #{1 ≤ j < i ≤ n | w(j) > w(i), i ≤ h(j)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For calculation of the cohomology ring of X(h), we use equivariant cohomology which we shall explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We assume that the matrix S in X(h) = Hess(S, h) is a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let T be an algebraic torus consisting of diagonal matrices in the general linear group GLn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The linear action of T on Cn induces an action on the flag variety Fl(n) and preserves X(h) since S commutes with T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The fixed point sets of the T -actions on X(h) and Fl(n) consist of all permutation flags, that is, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2) X(h)T = Fl(n)T ∼= Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since T can naturally be identified with (C∗)n, the classifying space BT of T is B(C∗)n = (CP ∞)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let pi : T → C∗ be the projection on the i-th diagonal component of T and ti = p∗ i (t) ∈ H2(BT ) where p∗ i : H∗(BC∗) → H∗(BT ) and t ∈ H2(BC∗) is the first Chern class of the tautological line bundle over BC∗ = CP ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3) H∗(BT ) = Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The equivariant cohomology of the T -variety X(h) is defined as H∗ T (X(h)) := H∗(ET ×T X(h)) where ET is the total space of the universal principal T -bundle ET → BT and ET ×T X(h) is the orbit space of the product ET × X(h) by the diagonal T -action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The projection ET × X(h) → ET on the first factor induces a fibration X(h) ρ−→ ET ×T X(h) π−→ BT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since Hodd(X(h)) = 0 as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1 and Hodd(BT ) = 0, the Serre spectral sequence of the fibration above collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' It implies that ρ∗ : H∗ T (X(h)) → H∗(X(h)) is surjective and induces a graded ring isomorphism (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4) H∗(X(h)) ∼= H∗ T (X(h))/(π∗(t1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , π∗(tn)) by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, one can find the ring structure on H∗(X(h)) through H∗ T (X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since Hodd(X(h)) = 0, it follows from the localization theorem that the homomorphism (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='5) H∗ T (X(h)) → H∗ T (X(h)T ) = � w∈Sn H∗ T (w) = � w∈Sn Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn] = Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]) induced from the inclusion map X(h)T → X(h) is injective, where X(h)T is identified with Sn by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2) and Map(P, Q) denotes the set of all maps from P to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The T -variety X(h) is what is called a GKM manifold and the image of the homomorphism in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='5) is described in [20] as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6) {f ∈ Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]) | f(w) − f(w(i, j)) ∈ (tw(i) − tw(j)), for ∀w ∈ Sn, j < i ≤ h(j)}, 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' MASUDA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' SATO where (i, j) denotes the transposition interchanging i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We note that the image of π∗(ti) ∈ π∗(H∗(BT )) ⊂ H∗ T (X(h)) by the homomorphism in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='5) is the constant function ti ∈ Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Guillemin and Zara [11] assigned a labeled graph to a GKM manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The labeled graph of X(h) is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The vertex set is the fixed point set X(h)T = Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' There is an edge between vertices w and v if and only if v = w(i, j) for some j ≤ i ≤ h(j), and the edge between w and w(i, j) is labeled by tw(i) − tw(j) up to sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For h = (2, 3, 3) and h′ = (3, 3, 3), the labeled graphs of X(h) and X(h′) are drawn in Figure 2, where we use the one-line notation for each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ❞ ❞ ❞ ❞ ❞ ❞ ❍ ❍ ❍ ❍ ❍ ❍ ✟✟✟ ✟✟✟ ✟ ✟ ✟ ✟ ✟ ✟ 123 321 132 312 213 231 X(h) ❞ ❞ ❞ ❞ ❞ ❞ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ✟✟✟ ✟✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ 123 321 132 312 213 231 X(h′) = Fl(3) labels : t1 − t2 : t2 − t3 : t1 − t3 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The labeled graphs of X(h) and X(h′) In general, labeled graphs and their graph cohomologies are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let R be a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' A labeled graph (Γ, α) consists of a graph Γ = (V, E) and a labeling α: E → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The graph cohomology of a labeled graph (Γ, α) is defined as H∗(Γ, α) = {f ∈ Map(V, R) | f(w) − f(v) ∈ (α(e)) for ∀e = wv ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The graph cohomology H∗(Γ, α) is a subring of Map(V, R) with the coordinate-wise sum and multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Note that we may ignore the signs of the labels α(e) since (α(e)) = (−α(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The observation above shows that the graph cohomology of the labeled graph of X(h) coincides with H∗ T (X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Sending ti to tσ(i) for σ ∈ Sn and i ∈ [n] induces an action of Sn on Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then, the module Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]) becomes an Sn-module under what is called the dot action defined by (σ · f)(w) := σ(f(σ−1w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' As easily checked, the graph cohomology of X(h) is invariant under the dot action and H∗ T (X(h)) becomes a module over Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Moreover, since the action of Sn preserves the ideal (π∗(t1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , π∗(tn)), the action descends to H∗(X(h)) and H∗(X(h)) also becomes an module over Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Obviously, constant functions in Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]) satisfy the condition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' They are ele- ments corresponding to π∗(H∗(BT )) ⊂ H∗ T (X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Below are three types of elements xi, yj,k, and τA in Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]) which satisfy the condition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6), so they are in H∗ T (X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let ⊥(h) : = {j ∈ [n − 1] | h(j − 1) = h(j) = j + 1} L(h) : = {j ∈ [n − 1] | h(j − 1) = j and h(j) = j + 1} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='7) where we understand h(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (1) For i ∈ [n], xi(w) := tw(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (2) For j ∈ [n − 1] with j ∈ ⊥(h) and k ∈ [n], yj,k(w) := � tk − tw(j+1) (if k ∈ {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(j)}) 0 (otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' REGULAR SEMISIMPLE HESSENBERG VARIETIES 5 (3) For A ⊂ [n] with |A| ∈ L(h) τA(w) := � tw(|A|) − tw(|A|+1) (if {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(|A|)} = A) 0 (otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The cohomological degrees of the elements xk, yj,k, τA are two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' One can easily check that the dot actions of σ ∈ Sn on these elements are given as follows: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='8) σ · xk = xk, σ · yj,k = yj,σ(k), σ · τA = τσ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Here is a geometrical meaning of xk’s (regarded as elements in H2(X(h)) through the isomor- phism (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' There is a nested sequence of tautological vector bundles over the flag variety Fl(n): F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fn = Fl(n) × Cn where Fk := {(V•, v) ∈ Fl(n) × Cn | v ∈ Vk} and V• = ({0} = V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vn = Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then xk (k ∈ [n]) is the image of the first Chern class of the quotient line bundle Fk/Fk−1 over Fl(n) by the homomorphism ι∗ : H∗(Fl(n)) → H∗(X(h)) induced from the inclusion map ι: X(h) → Fl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The dot action on H∗(Fl(n)) is trivial, so the image of ι∗ must be contained in the ring of invariants H∗(X(h))Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In fact, it follows from [1, Theorems A and B] that the image of ι∗ agrees with H∗(X(h))Sn when tensoring with Q and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='9) H∗(X(h))Sn ⊗ Q = Q[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , xn]/(fh(1),1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , fh(n),n) where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='10) fh(j),j = j � k=1 \uf8eb \uf8edxk h(j) � ℓ=j+1 (xk − xℓ) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In particular, the Hilbert series of H∗(X(h))Sn is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11) Hilb(H∗(X(h))Sn, √q) = n−1 � j=1 [h(j) − j]q where the Hilbert series of a graded algebra A = �∞ r=0 Ar over Z is defined as Hilb(A, q) := ∞ � r=0 (rankZAr)qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Through the isomorphism (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4), the elements xk, yj,k, τA determine elements in H2(X(h)), denoted by the same notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4 ([4, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The elements {xk, yj,k, τA | k ∈ [n], j ∈ ⊥(h)\\{n − 1}, A ⊂ [n] with |A| ∈ L(h)\\{n − 1}} generate H2(X(h)) with relations (1) �n k=1 xk = 0, (2) �n k=1 yj,k = (x1 + · · · + xj) − jxj+1 for j ∈ ⊥(h)\\{n − 1}, (3) � |A|=j τA = xj − xj+1 for j ∈ L(h)\\{n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2 (see Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2 in [4] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The element yj,k is defined by looking at the j-th column of the configuration associated to the Hessenberg function h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Similarly, one can define an element y∗ i,k of H∗ T (Hess(S, h)) by looking at the i-th row of the configuration as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For i ∈ [n], we define h∗(i) := min{j ∈ [n] | h(j) ≥ i}, 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' MASUDA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' SATO so that the shaded boxes in the i-th row and under the diagonal in the configuration associated to h are at positions (i, ℓ) (h∗(i) ≤ ℓ < i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When h∗(i) = i − 1, we define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='12) y∗ i,k(w) := � tk − tw(i−1) (k ∈ {w(i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(n)}) 0 (otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' One can see that y∗ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='k is in H2 T (Hess(S, h)) and we may replace yj,k’s for j ∈ ⊥(h)\\{n − 1} in the generating set in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4 by y∗ i,k’s for i ≥ 3 such that h∗(i) = h∗(i + 1) = i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When h = (4, 4, 4, 5, 6, 7, 11, 11, 11, 11) in Figure 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (a, b) = (3, 7)), we have ⊥(h) = {3, 10}, L(h) = {4, 5, 6}, so Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4 says that H2(X(h)) is generated by xk (k ∈ [11]), y3,k (k ∈ [11]), τA for A ⊂ [11] with |A| = 4, 5 or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Moreover, it follows from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2 that y3,k above may be replaced by y∗ 8,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Necessity In this section, we study a necessary condition on h for H∗(X(h)) to be generated in degree 2 as a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Moment maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let µ: Fl(n) → Rn be the standard moment map on the flag variety Fl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Its image is the permutohedron Πn obtained as the convex hull of the orbits of (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n) by permuting its coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Indeed, if ew (w ∈ Sn) denotes the permutation flag associated with w, then we have µ(ew) = (w−1(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w−1(n)) ∈ Rn (see [16, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1] for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) Sr n := {w ∈ Sn | w(r) = n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then µ(Sr n) is the set of all vertices of Πn whose n-th coordinate is r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore the projection πn : Πn → R, πn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , xn) = xn on the n-th coordinate takes minimum on S1 n and maximum on Sn n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The composition of µ and πn (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2) f := πn ◦ µ: Fl(n) → R is the moment map induced from the following S1-action on Cn (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3) (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , zn) → (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , zn−1, gzn) (g ∈ S1 ⊂ C), and it is a Morse-Bott function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let hj be the Hessenberg function obtained by removing all the boxes in the j-th row and all the boxes in the j-th column from its configuration (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' To be precise, hj is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' hj(i) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 h(i) (i < j, h(i) < j) h(i) − 1 (i < j, h(i) ≥ j) h(i + 1) − 1 (i ≥ j) REGULAR SEMISIMPLE HESSENBERG VARIETIES 7 j-th row → ↓ j-th column h ❀ remove ← տ ↑ ❀ hj Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The configuration corresponding to hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The following is a key lemma in our argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The restriction maps H∗(X(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) → H∗(X(h1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q), H∗(X(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) → H∗(X(hn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) are surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let fh be the map f in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2) restricted to X(h), which is also a Morse-Bott function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The inverse image of the minimum value under fh is X(h1), so it follows from [19, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1] that the restriction map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4) H∗ S1(X(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) → H∗ S1(X(h1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) is surjective, where the S1-action on X(h) is the induced one from the S1-action defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since the S1-action on X(h1) is trivial, we have H∗ S1(X(h1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) = H∗(BS1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q)⊗ H∗(X(h1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) and hence the forgetful map H∗ S1(X(h1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) → H∗(X(h1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, the surjectivity of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4) implies the surjectivity of the restriction map H∗(X(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) → H∗(X(h1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) in ordinary cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The same argument applied to −fh proves the statement for X(hn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ✷ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The surjectivity of the above restriction maps (even with Z coefficients) can also be verified by GKM theory as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Recall that the inclusion of the fixed point set induces an injective homomorphism H∗ T (X(h)) → H∗ T (X(h)T ) ∼= Map(Sn, H∗(BT )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The equivariant cohomology H∗ T (X(h)) has an H∗(BT )- module basis {σw,h | w ∈ Sn} (see [6, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='9 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' It corresponds to a natural paving and then it is a ‘flow-up basis.’ Note that any element of Sn n = Sn−1 is not greater than any element of Sn \\ Sn n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The restriction of {σw,h | w ∈ Sn n} onto X(hn), that is, its restriction onto the fixed point set Sn n = X(hn)T as elements of Map(Sn, H∗(BT )), is a flow-up basis of H∗ T (X(hn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hence H∗ T (X(h)) → H∗ T (X(hn)) is surjective, and then H∗(X(h)) → H∗(X(hn)) is also surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The surjectivity of H∗(X(h)) → H∗(X(h1)) can be verified by a similar argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Given a Hessenberg function h, we obtain a smaller Hessenberg function by removing the first column and row or the last column and row repeatedly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' by taking h1 or hn repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We call it a minor of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The following corollary follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let h′ be a minor of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' If H∗(X(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) is generated in degree 2 as a ring, then so is H∗(X(h′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' An easy argument shows that h being of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) can be rephrased as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The Hessenberg function h is of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) if and only if h has neither (α, β, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , β), (β − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , β − 1, β, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , β � �� � α ) for 3 ≤ α < β, nor (2, γ − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , γ − 1, γ, γ) for γ ≥ 5 as its minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' MASUDA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' SATO Recall that if h† denotes the Hessenberg function obtained by flipping the configuration of h along the anti-diagonal, then X(h†) ∼= X(h) as varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore X((α, β, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , β)) ∼= X((β − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , β − 1, β, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , β � �� � α )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Here, we know that H∗(X((α, β, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , β));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) is not generated in degree 2 for 3 ≤ α < β by [2, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Thus, it suffices to treat the last case in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3, which we shall discuss in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The case h = (2, n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n − 1, n, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In this subsection we prove the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' H∗(X(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) is not generated in degree 2 when h = (2, n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n − 1, n, n) for n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Some computation is involved in the proof of this proposition but the idea of the proof is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We compute the Poincar´e polynomial of X(h) using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' On the other hand, using explicit generators of H2(X(h)) by [4], we compute an upper bound of the Hilbert series of the subring of H∗(X(h)) generated by H2(X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then it turns out that the latter is strictly smaller than the former at a certain degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Poincar´e polynomial of X(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The following proposition, which easily follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1(4), enables us to compute the Poincar´e polynomial of X(h) inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='5 ([4, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='5) Poin(X(h), √q) = n � j=1 qh(j)−j Poin(X(hj), √q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Using the proposition above, the Poincar´e polynomial of X(h) is explicitly computed as follows when h = (h(1), n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6 ([2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When h = (h(1), n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6) Poin(X(h), √q) = [h(1)]q[n − 1]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 1)qh(1)−1[n − h(1)]q[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=', where [m]q = 1 − qm 1 − q , [m]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' = [1]q[2]q · · · [m]q = m � j=1 1 − qj 1 − q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Now, let h = (2, n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n − 1, n, n) and set Pn(q) := Poin(X(h), √q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For n ≥ 5, the following recurrence formula holds Pn(q) = (1 + q)2[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 2)(q + q2)[n − 3]q[n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 1)(q + qn−3) {(1 + q)[n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 3)q[n − 4]q[n − 4]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='} + (q + q2 + · · · + qn−4)Pn−1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let Fn(q) denote the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6) with h(1) = 2, that is, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='7) Fn(q) := (1 + q)[n − 1]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 1)q[n − 2]q[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='. Then we have Poin(X(h1), √q) = Poin(X(hn), √q) = Fn−1(q) Poin(X(h2), √q) = Poin(X(hn−1), √q) = (n − 1)Fn−2(q) Poin(X(hj), √q) = Pn−1(q) (3 ≤ j ≤ n − 2), REGULAR SEMISIMPLE HESSENBERG VARIETIES 9 where we note that X(h2) consists of n − 1 copies of Fl(n − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hence, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='5), we have Pn(q) = qFn−1(q) + (n − 1)qn−3Fn−2(q) + (qn−4 + · · · + q)Pn−1(q) + (n − 1)qFn−2(q) + Fn−1(q) = (1 + q)Fn−1(q) + (n − 1)(q + qn−3)Fn−2(q) + (q + · · · + qn−4)Pn−1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Combining this equation with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='7), we obtain the desired equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ✷ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For n ≥ 4, let Qn(q) = (1 + 2nq + n(n − 1)q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + n(n − 3) 2 qn−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then we have Pn(q) ≡ Qn(q) mod (qn−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In other words, Pn(q) and Qn(q) coincide up to degree n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We prove the lemma by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When n = 4, we have P4(q) = 1 + 11q + 11q2 + q3, Q4(q) = 1 + 11q + 20q2 + 12q3, and the lemma is true for n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let n be given and suppose that the lemma is true for n − 1, that is, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='8) Pn−1(q) ≡ Qn−1(q) mod (qn−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hereafter, in this proof, all congruences will be taken modulo qn−2 unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since we have (q + q2)[n − 3]q[n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ≡ (q + q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' q2[n − 4]q[n − 4]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ≡ q2[n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=', the recurrence formula in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='7 reduces to the following congruence relation: Pn(q) ≡ (1 + nq + (n − 1)q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 1)(q + (n − 2)q2)[n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 1)qn−3 + (q + · · · + qn−4)Pn−1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='9) It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='8) and the definition of Qn that the sum of the last two terms above becomes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (n − 1)qn−3 + (q + · · · + qn−4)Pn−1(q) ≡ (n − 1)qn−3 + � 1 + (2n − 2)q + (n − 1)(n − 2)q2� (q + · · · + qn−4)[n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 1)(n − 4) 2 qn−3 = � 1 − q + (n − 1)q(1 − q) + nq + (n − 1)2q2� (q + · · · + qn−4)[n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 1)(n − 2) 2 qn−3 ≡ � q − qn−3 + (n − 1)q2 + (nq + (n − 1)2q2)(q + · · · + qn−4) � [n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 1)(n − 2) 2 qn−3 ≡ � q + (n − 1)q2 + (nq + (n − 1)2q2)(q + · · · + qn−4) � [n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + n(n − 3) 2 qn−3 By substituting it to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='9), we obtain Pn(q) ≡ (1 + nq + (n − 1)q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + � (nq + (n − 1)2q2) + (nq + (n − 1)2q2)(q + · · · + qn−4) � [n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + n(n − 3) 2 qn−3 ≡ (1 + nq + (n − 1)q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (nq + (n − 1)2q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + n(n − 3) 2 qn−3 = (1 + 2nq + n(n − 1)q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + n(n − 3) 2 qn−3 = Qn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' MASUDA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' SATO This completes the induction step and the lemma has been proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ✷ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hilbert series of the subring generated by H2(X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When h = (2, n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=', n − 1, n, n) for n ≥ 5, we first observe H2(X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='7), we have ⊥(h) = {n − 2}, L(h) = {1, n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, it follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4 that H2(X(h)) is generated by the following elements (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='10) xk, yk := yn−2,k, τk := τ{k} (k ∈ [n]), where xk(w) = tw(k), yk(w) = yn−2,k(w) = � tk − tw(n−1) (if k ∈ {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(n − 2)}) 0 (otherwise), τk(w) = τ{k}(w) = � tw(1) − tw(2) (if k = w(1)) 0 (otherwise) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11) for w ∈ Sn by Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='12) n � k=1 yk = x1 + · · · + xn−2 − (n − 2)xn−1, n � k=1 τk = x1 − x2 by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We also have σ · xk = xk, σ · yk = yσ(k), σ · τk = τσ(k) for σ ∈ Sn by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' To make the following argument clearer, we introduce elements ρk for k ∈ [n] defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='13) ρk(w) := � tw(n−1) − tw(n) (if k = w(n)) 0 (otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Similarly to τk, the ρk satisfies the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6) so that it defines an element of H2 T (X(h)) and H2(X(h)) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='14) n � k=1 ρk = xn−1 − xn, σ · ρk = ρσ(k) for σ ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' An elementary check shows that (yk − yℓ)(w) − (ρk − ρℓ)(w) = tk − tℓ (k, ℓ ∈ [n], w ∈ Sn) and hence yk − yℓ = ρk − ρℓ in H2(X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Moreover, �n k=1 yk and �n k=1 ρk are both linear polynomials in xi’s by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='14), so we may replace yk’s in the generating set (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='10) by ρk’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Namely H2(X(h)) is generated by xk, τk, ρk (k ∈ [n]) with relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='15) n � k=1 xk = 0, n � k=1 τk = x1 − x2, n � k=1 ρk = xn−1 − xn, and the actions of σ ∈ Sn on those generators are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='16) σ · xk = xk, σ · τk = τσ(k), σ · ρk = ρσ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' REGULAR SEMISIMPLE HESSENBERG VARIETIES 11 Our purpose is to find a sharp upper bound of the Hilbert series of the subring R(h) of H∗(X(h)) generated by H2(X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let A(h) be the subring of H∗(X(h)) generated by xk’s and we regard R(h) as a module over A(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='13) that τkτℓ = � (x1 − x2)τk (k = ℓ) 0 (k ̸= ℓ), ρkρℓ = � (xn−1 − xn)ρk (k = ℓ) 0 (k ̸= ℓ), τkρk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, R(h) is generated by 1, τk, ρk (k ∈ [n]), and τiρj (i ̸= j ∈ [n]) as a module over A(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The subring A(h) itself is a submodule of R(h) over A(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We consider three other submodules of R(h) over A(h): B(h) :={ n � k=1 bkτk | bk ∈ A(h), n � k=1 bk = 0}, C(h) :={ n � k=1 ckρk | ck ∈ A(h), n � k=1 ck = 0}, D(h) :={ � 1≤i,j≤n dijτiρj | dij ∈ A(h), n � j=1 dij = 0 for i ∈ [n], n � i=1 dij = 0 for j ∈ [n]} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='17) where dkk = 0 for k ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Note that A(h)⊗Q agrees with the ring of invariants H∗(X(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q)Sn as mentioned in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' R(h) is additively generated by A(h), B(h), C(h), and D(h) when tensoring with Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since H∗(X(h)) is generated by 1, τk, ρk (k ∈ [n]), and τiρj (i ̸= j ∈ [n]) as a module over A(h), it suffices to show that any element of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='18) n � k=1 bkτk + n � k=1 ckρk + � 1≤i,j≤n dijτiρj (bk, ck, dij ∈ A(h), dkk = 0) can be expressed as a sum of elements in A(h), B(h), C(h), and D(h) when tensoring with Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Set b := �n k=1 bk and c := �n k=1 ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since �n k=1 τk = x1 −x2 and �n k=1 ρk = xn−1 −xn by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='15), we have n � k=1 bkτk + n � k=1 ckρk = n � k=1 � bk − b n � τk + b n(x1 − x2) + n � k=1 � ck − c n � ρk + c n(xn−1 − xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Here the two sums at the right hand side above respectively belong to B(h) ⊗ Q and C(h) ⊗ Q, and the remaining two terms belong to A(h) ⊗ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' As for the last term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='18), since �n i=1 τi = x1 − x2, we have � 1≤i,j≤n dijτiρj = n � j=1 � n � i=1 � dij − dj n � τi � ρj + n � j=1 dj n (x1 − x2)ρj = � 1≤i,j≤n ˜dijτiρj + n � j=1 dj n (x1 − x2)ρj (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='19) where dj := n � i=1 dij and ˜dij := dij − dj n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The last sum in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='19) is a sum of elements in A(h) ⊗ Q and C(h) ⊗ Q by Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We shall show that the sum � 1≤i,j≤n ˜dijτiρj in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='19) is a sum of elements in A(h) ⊗ Q, B(h) ⊗ Q, and D(h) ⊗ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='20) n � i=1 ˜dij = n � i=1 � dij − dj n � = n � i=1 dij − dj = 0 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' MASUDA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' SATO and set (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='21) ˜di := n � j=1 ˜dij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since �n j=1 ρj = xn−1 − xn, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='22) � 1≤i,j≤n ˜dijτiρj = n � i=1 \uf8eb \uf8ed n � j=1 � ˜dij − ˜di n � ρj \uf8f6 \uf8f8 τi + n � i=1 ˜di n (xn−1 − xn)τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Here the second sum at the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='22) is a sum of elements in A(h) ⊗ Q and B(h) ⊗ Q by Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' As for the coefficients ˜dij − ˜di n of τiρj in the first sum at the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='22), it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='20) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='21) that we have n � i=1 � ˜dij − ˜di n � = n � i=1 ˜dij − 1 n n � i=1 ˜di = − 1 n n � i=1 n � j=1 ˜dij = − n � j=1 � n � i=1 ˜dij � = 0, n � j=1 � ˜dij − ˜di n � = n � j=1 ˜dij − ˜di = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Thus, the first sum at the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='22) belongs to D(h) ⊗ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ✷ We shall calculate upper bounds of the Hilbert series of A(h), B(h), C(h), and D(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hilbert series of A(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since A(h) ⊗ Q = H∗(X(h))Sn ⊗ Q and h = (2, n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n − 1, n, n) in our case, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11) that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='23) Hilb(A(h), √q) = n−1 � j=1 [h(j) − j]q = (1 + q)2[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='. Hilbert series of B(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' It follows from(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11) that (x1 − tk)τk vanishes at every w ∈ Sn, so we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='24) (x1 − tk)τk = 0 in H∗ T (X(h)) and hence x1τk = 0 in H∗(X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, B(h) is indeed a module over A(h)/(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Here A(h)/(x1) ⊗ Q = A(h1) ⊗ Q by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since h1 = (n − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n − 2, n − 1, n − 1), it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11) that Hilb(A(h)/(x1), √q) = n−2 � j=1 [h1(j) − j]q = (1 + q)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='. Since B(h) is a module over A(h)/(x1) generated by τi − τi+1 (i ∈ [n − 1]) and the cohomological degrees of τk’s are two, we obtain an upper bound of Hilb(B(h), q) as follows: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='25) Hilb(B(h), √q) ≤ (n − 1)q Hilb(A(h)/(x1), √q) = (n − 1)(q + q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='. Here �∞ i=0 aiqi ≤ �∞ i=0 biqi (ai, bi ∈ Z) means that ai ≤ bi for all i’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hilbert series of C(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' To f ∈ Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]) we associate f ∨ ∈ Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]) defined by f ∨(w) := f(ww0) for w ∈ Sn, where w0 denotes the longest element in Sn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' w0 = n n − 1 · · · 2 1 in one-line notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' This defines an involution on Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]) and one can easily check that x∨ k = xn−k+1, τ ∨ k = −ρk, ρ∨ k = −τk REGULAR SEMISIMPLE HESSENBERG VARIETIES 13 from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hence the involution gives an isomorphism between B(h) and C(h), and the same inequality as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='25) holds for C(h), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='26) Hilb(C(h), √q) ≤ (n − 1)(q + q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='. Hilbert series of D(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We have x1τk = 0 by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Similarly we have xnρk = 0 since (x1τk)∨ = −xnρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (The fact xnρk = 0 also follows from the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='13) of xk and ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=') Therefore, D(h) is indeed a module over A(h)/(x1, xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' As mentioned in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1, A(h) ⊗ Q = H∗(X(h))Sn ⊗ Q and it is the image of the restriction map ι∗ : H∗(Fl(n)) → H∗(X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, A(h)/(x1, xn) is the image of the restriction map from H∗(Fl(n−2)) and hence Hilb(A(h)/(x1, xn), √q) ≤ [n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='. (In fact, the equality holds above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=') There are 2n relations among dij (i ̸= j) in the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='17) of D(h), but one relation can be obtained from the other 2n−1 relations because �n i=1 ��n j=1 dij � = �n j=1 (�n i=1 dij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Moreover, there are n(n − 1) number of dij’s and the cohomological degree of τiρj is four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Thus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='27) Hilb(D(h), √q) ≤ Hilb(A(h)/(x1, xn), √q) {n(n − 1) − (2n − 1)} q2 ≤ (n2 − 3n + 1)q2[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='. Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' It follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='9, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='23), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='25), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='26), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='27) that Hilb(R(h), √q) ≤ (1 + q)2[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + 2(n − 1)(q + q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n2 − 3n + 1)q2[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' = (1 + 2nq + n(n − 1)q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='. The coefficient of qn−3 in the last term above is less than that of Pn(q) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='8 by n(n − 3)/2, proving the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ✷ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Sufficiency The purpose of this section is to prove the following proposition, which implies the sufficiency of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When h is of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1), the equivariant cohomology H∗ T (X(h)) is generated in degree 2 as an algebra over H∗(BT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1(3), X(h) is not connected when h(k) = k for some 1 ≤ k ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In this case, a flag V• = (V0 ⊂ V1 ⊂ · · · ⊂ Vn) ∈ X(h) is of the form Vk = ⟨ei1, ei2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , eik⟩ for some {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , ik} ⊂ [n], where ei is the i-th standard basis vector of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, decomposing V• into two flags (V0 ⊂ V1 ⊂ · · · ⊂ Vk) and (V ′ 0 ⊂ V ′ 1 ⊂ · · · ⊂ V ′ n−k), where V ′ i = Vk+i/Vk, one can see that X(h) is the disjoint union of �n k � copies of X(h1) × X(h2), where h1 and h2 are the Hessenberg function obtained by restricting h onto intervals [k] and [k + 1, n], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Each copy corresponds to the choice of a k-subset {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , ik} ⊂ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' To be precise, h2 : [n − k] → [n − k] is given by shift−1 k h ◦ shiftk, where shiftk : [n − k] → [k + 1, n] shifts integers by k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Suppose h is of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) and 1 ≤ r ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then X(hr) is not connected ⇐⇒ a + 1 ≤ r ≤ b by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1(3) and that hr is also of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) when r < a + 1 or r > b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When a + 1 ≤ r ≤ b, each connected component of X(hr) is isomorphic to X(h1) × X(h2) and both h1 and h2 are of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let Γ(Sn, h) denote the labeled graph of X(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Recall that H∗ T (X(h)) ∼= H∗(Γ(Sn, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For the subset Sr n ⊂ Sn in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1), let Γ(Sr n, h) be the induced labeled subgraph of Γ(Sn, h) on the subset Sr n of vertices, and let Γ0(Sr n, h) denote a connected component of Γ(Sr n, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When h is of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1), the restriction map H2(Γ(Sn, h)) → H2(Γ0(Sr n, h)) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We admit the lemma and complete the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Before that, we shall observe that Γ0(Sr n, h) is essentially a connected component of a labeled graph of X(hr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Indeed, for 1 ≤ r ≤ n, let cr be the cyclic permutation (r r + 1 r + 2 · · · n) and ϕr : Γ0(Sr n, h) → Γ0(Sn−1, hr) 14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' MASUDA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' SATO a graph isomorphism defined by ϕr(w) = wcr for w ∈ Sr n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When i, j ̸= r, the (i, j)-th box in the configuration for h corresponds to the (c−1 r (i), c−1 r (j))-th box in the configuration for hr (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In particular, v = w(i, j) corresponds to vcr = wcr(c−1 r (i), c−1 r (j)) and the edges between these vertices have the same label tw(i) − tw(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, ϕr induces an isomorphism (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) ϕ∗ r : H∗(Γ0(Sn−1, hr)) ∼ = −→ H∗(Γ0(Sr n, h)) of graded algebras over H∗(BT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Recall that H∗ T (X(h)) ∼= H∗(Γ(Sn, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We prove the proposition by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let 1 ≤ r ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For any z ∈ H∗(Γ(Sn, h)) that vanishes on �r−1 j=1 Sj n, it is sufficient to show the existence of a polynomial f in elements of H2(Γ(Sn, h)) such that z − f vanishes on �r j=1 Sj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then the induction on r proves the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We shall show the existence of f by division into cases according to the value of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The case 1 ≤ r ≤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In this case, Γ(Sr n, h) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We note that z vanishes on �r−1 j=1 Sj n and this implies that z(w) for w ∈ Sr n decomposes as follows: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2) z(w) = \uf8eb \uf8ed r−1 � j=1 (tw(j) − tn) \uf8f6 \uf8f8 g(w), g ∈ H∗(Γ(Sr n, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Indeed, for w ∈ Sr n, we have w(r) = n and w(j, r) ∈ Sj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' If j ≤ r − 1, then there is an edge in the graph Γ(Sn, h) between the vertices w and w(j, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The label on the edge is tw(j) −tw(r) = tw(j) −tn and z vanishes at w(j, r) ∈ Sj n (j ≤ r − 1) by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore z(w) is divisible by the product in the big parenthesis in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2) and g ∈ Map(Sr n, H∗(BT )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Furthermore, one can easily check that the g is indeed in H∗(Γ(Sr n, h)) since z is in H∗(Γ(Sn, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since H∗(Γ(Sr n, h)) ∼= H∗(Γ(Sn−1, hr)) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1), g is a polynomial in elements of H2(Γ(Sr n, h)) by induc- tion on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Moreover, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2, there is a polynomial ˜g in H2(Γ(Sn, h)) which coincides with g on Sr n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' On the other hand, �r−1 j=1(xj −tn) coincides with the product in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2) on Sr n since xj(w) = tw(j) by definition of xj, and vanishes on �r−1 j=1 Sj n since xj(w) = tw(j) = tn for w ∈ Sj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, � �r−1 j=1(xj − tn) � ˜g coincides with the element z on �r j=1 Sj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Thus � �r−1 j=1(xj − tn) � ˜g is a desired polynomial f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The case r = a + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Similarly to Case 1, z(w) for w ∈ Sa+1 n decomposes as follows: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3) z(w) = \uf8eb \uf8ed a � j=1 (tw(j) − tn) \uf8f6 \uf8f8 g(w), g ∈ H∗(Γ(Sa+1 n , h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Note that Γ(Sa+1 n , h) is not connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Two vertices v, w ∈ Sa+1 n lie in the same connected component if and only if {v(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , v(a)} = {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(a)} ⊂ [n − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For K := {k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , ka} ⊂ [n − 1], we consider the element ρK defined by ρK = a � j=1 ya,kj, where ya,k(w) = � tk − tw(a+1) (k ∈ {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(a)}) 0 (k /∈ {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(a)}) by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, since w(a + 1) = n for w ∈ Sa+1 n , we have ρK(w) = ��a j=1(tw(j) − tn) (K = {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(a)}) 0 (K ̸= {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(a)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hence ρK coincides with the product in the big parentheses of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3) on the connected component (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4) {w ∈ Sa+1 n | w([a]) = K} REGULAR SEMISIMPLE HESSENBERG VARIETIES 15 and vanishes on the other components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since n /∈ K and w(j) = n for w ∈ Sj n, ρK also vanishes on �a j=1 Sj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' On the other hand, the element g in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3) restricted to the connected component (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4) is obtained as the restriction of a polynomial ˜gK in H2(Γ(Sn, h)) similarly to Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, we obtain a desired polynomial f as � K⊂[n−1], |K|=a ρK˜gK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The case a + 2 ≤ r ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In this case, z(w) for w ∈ Sr n decomposes as follows: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='5) z(w) = (tw(r−1) − tw(r))g(w), g ∈ H∗(Γ(Sr n, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Similarly to Case 2, Γ(Sr n, h) is not connected and two vertices v, w ∈ Sr n lie in the same connected component if and only if {v(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , v(r − 1)} = {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(r − 1)} ⊂ [n − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For A ⊂ [n − 1] with |A| = r − 1, we have τA(w) = � tw(r−1) − tw(r) (A = {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(r − 1)}) 0 (A ̸= {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(r − 1)}) by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hence, τA coincides with the factor of the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='5) on the connected component {w ∈ Sr n | w([r − 1]) = A}, and vanishes on the other connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since n /∈ A and w(j) = n for w ∈ Sj n, τA also vanishes on �r−1 j=1 Sj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, similarly to Case 2, we obtain a desired polynomial f as � A⊂[n−1], |A|=r−1 τA˜gA, where ˜gA is a polynomial in H2(Γ(Sn, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The case b + 1 ≤ r ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In this case, z(w) for w ∈ Sr n decomposes as follows: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6) z(w) = \uf8eb \uf8ed r−1 � j=b (tn − tw(j)) \uf8f6 \uf8f8 g(w), g ∈ H∗(Γ(Sr n, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Similarly to Case 1, X(hr) is connected and g is the restriction of a polynomial ˜g in H2(Γ(Sn, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We consider the element y∗ b+1,n ∈ H∗(Γ(Sn, h)) in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2, which is defined as y∗ b+1,n(w) = � tn − tw(b) (n ∈ {w(b + 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(n)}) 0 (n /∈ {w(b + 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(n)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then \uf8eb \uf8edy∗ b+1,n r−1 � j=b+1 (tn − xj) \uf8f6 \uf8f8 (w) = ��r−1 j=b(tn − tw(j)) (n ∈ {w(b + 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(n)}) 0 (n /∈ {w(b + 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(n)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hence y∗ b+1,n �r−1 j=b+1(tn − xj) coincides with the product in the big parentheses of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6) on Sr n, and vanishes on �r−1 j=1 Sj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, � y∗ b+1,n �r−1 j=b+1(tn − xj) � ˜g is a desired polynomial f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ✷ Finally we give a proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' It follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2 that when h is of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1), the elements in {xi, ya,k, τA, ti | i, k ∈ [n], A ⊂ [n], a + 1 ≤ |A| < b} span H2(Γ(Sn, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Through the isomorphism (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1), one can find generators of H2(Γ0(Sr n, h)) which corre- spond to the generators of H2(Γ0(Sn−1, hr)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' They are given as restrictions of xi for i ∈ [n], i ̸= r, ti for i ∈ [n], and the following elements in H2(Γ(Sn, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When 1 ≤ r ≤ a, ya,k for k ∈ [n − 1], τA⊔{n} for A ⊂ [n − 1], a ≤ |A| < b − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' MASUDA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' SATO Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When r = a + 1, for a connected component Γ0(Sa+1 n , h) which contains σ ∈ Sa+1 n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' τB⊔σ([a+1]) for B ⊂ σ([n]\\[a + 1]), 1 ≤ |B| < b − (a + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When a + 1 < r ≤ b, for a connected component Γ0(Sr n, h) which contains σ ∈ Sr n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ya,k for k ∈ [n − 1], τA for A ⊂ σ([r − 1]), a + 1 ≤ |A| < r − 1, τB⊔σ([r]) for B ⊂ σ([n] \\ [r]), 1 ≤ |B| < b − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When b < r ≤ n, ya,k for k ∈ [n − 1], τA for A ⊂ [n − 1], a + 1 ≤ |A| < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' This proves the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ✷ Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We thank Yunhyung Cho for his help on moment map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Masuda was supported in part by JSPS Grant- in-Aid for Scientific Research 22K03292 and a HSE University Basic Research Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' This work was partly supported by Osaka Central Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Abe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Harada, T.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=', n), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1 (2019), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 27–59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Abe, T.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=', Providence, RI, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Osaka Metropolitan University Advanced Mathematical Institute, Sumiyoshi-ku, Osaka 558-8585, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Email address: mikiyamsd@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='com Osaka Metropolitan University Advanced Mathematical Institute, Sumiyoshi-ku, Osaka 558-8585, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Email address: 00tkshst00@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} diff --git a/99E2T4oBgHgl3EQfQQZv/content/tmp_files/2301.03768v1.pdf.txt b/99E2T4oBgHgl3EQfQQZv/content/tmp_files/2301.03768v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dc38cf88abdd95d7877fb27116702881cbe3e604 --- /dev/null +++ b/99E2T4oBgHgl3EQfQQZv/content/tmp_files/2301.03768v1.pdf.txt @@ -0,0 +1,1247 @@ +Prepared for submission to JHEP +Search for lepton-flavor-violating τ decays into a +lepton and a vector meson using the full Belle data +sample +The Belle Collaboration +N. Tsuzuki +, K. Inami +, I. Adachi +, H. Aihara +, D. M. Asner +, H. Atmacan +, +T. Aushev +, R. Ayad +, V. Babu +, Sw. Banerjee +, P. Behera +, K. Belous +, +J. Bennett +, M. Bessner +, B. Bhuyan +, T. Bilka +, D. Biswas +, D. Bodrov +, +J. Borah +, A. Bozek +, M. Braˇcko +, P. Branchini +, T. E. Browder +, A. Budano +, +M. Campajola +, D. ˇCervenkov +, M.-C. Chang +, B. G. Cheon +, K. Chilikin +, +H. E. Cho +, K. Cho +, S.-J. Cho +, S.-K. Choi +, Y. Choi +, S. Choudhury +, +D. Cinabro +, J. Cochran +, S. Das +, N. Dash +, G. De Nardo +, G. De Pietro +, +R. Dhamija +, F. Di Capua +, Z. Doleˇzal +, T. V. Dong +, D. Dossett +, S. Dubey +, +D. Epifanov +, T. Ferber +, D. Ferlewicz +, B. G. Fulsom +, V. Gaur +, A. Giri +, +P. Goldenzweig +, Y. Guan +, K. Gudkova +, X. Han +, T. Hara +, K. Hayasaka +, +H. Hayashii +, M. T. Hedges +, D. Herrmann +, W.-S. Hou +, C.-L. Hsu +, +T. Iijima +, N. Ipsita +, A. Ishikawa +, R. Itoh +, M. Iwasaki +, W. W. Jacobs +, +E.-J. Jang +, S. Jia +, Y. Jin +, T. Kawasaki +, C. Kiesling +, C. H. Kim +, +D. Y. Kim +, K.-H. Kim +, Y.-K. Kim +, K. Kinoshita +, P. Kodyˇs +, T. Konno +, +A. Korobov +, S. Korpar +, E. Kovalenko +, P. Kriˇzan +, P. Krokovny +, M. Kumar +, +K. Kumara +, A. Kuzmin +, Y.-J. Kwon +, S. C. Lee +, J. Li +, L. K. Li +, Y. Li +, +J. Libby +, K. Lieret +, Y.-R. Lin +, D. Liventsev +, Y. Ma +, A. Martini +, +M. Masuda +, K. Matsuoka +, D. Matvienko +, S. K. Maurya +, F. Meier +, +M. Merola +, K. Miyabayashi +, R. Mizuk +, G. B. Mohanty +, M. Nakao +, +Z. Natkaniec +, A. Natochii +, L. Nayak +, M. Niiyama +, N. K. Nisar +, S. Nishida +, +S. Ogawa +, H. Ono +, P. Oskin +, G. Pakhlova +, T. Pang +, S. Pardi +, H. Park +, +J. Park +, S.-H. Park +, A. Passeri +, S. Paul +, T. K. Pedlar +, R. Pestotnik +, +L. E. Piilonen +, T. Podobnik +, E. Prencipe +, M. T. Prim +, A. Rostomyan +, +N. Rout +, G. Russo +, S. Sandilya +, A. Sangal +, L. Santelj +, V. Savinov +, +G. Schnell +, C. Schwanda +, Y. Seino +, K. Senyo +, M. E. Sevior +, W. Shan +, +M. Shapkin +, C. Sharma +, J.-G. Shiu +, E. Solovieva +, M. Stariˇc +, +M. Sumihama +, T. Sumiyoshi +, M. Takizawa +, U. Tamponi +, K. Tanida +, +F. Tenchini +, M. Uchida +, T. Uglov +, Y. Unno +, S. Uno +, P. Urquijo +, +Y. Ushiroda +, S. E. Vahsen +, G. Varner +, A. Vinokurova +, D. Wang +, E. Wang +, +M.-Z. Wang +, X. L. Wang +, S. Watanuki +, X. Xu +, B. D. Yabsley +, W. Yan +, +S. B. Yang +, J. Yelton +, Y. Yook +, L. Yuan +, Y. Zhai +, V. Zhilich +, +V. Zhukova +, +arXiv:2301.03768v1 [hep-ex] 10 Jan 2023 + +Abstract: Charged-lepton-flavor-violation is predicted in several new physics scenarios. +We update the analysis of τ lepton decays into a light charged lepton (ℓ = e± or µ±) and a +vector meson (V 0 = ρ0, φ, ω, K∗0, or K∗0) using 980 fb−1 of data collected with the Belle +detector at the KEKB collider. No significant excess of such signal events is observed, and +thus 90% credibility level upper limits are set on the τ → ℓV 0 branching fractions in the +range of (1.7–4.2) × 10−8. These limits are improved by 30% on average from the previous +results. +Keywords: e+–e− Experiments, Tau Physics + +Contents +1 +Introduction +1 +2 +Belle experiment +1 +3 +Reconstruction and event selection +2 +4 +Signal efficiency and background estimation +6 +5 +Results +10 +6 +Conclusion +11 +1 +Introduction +In the Standard Model, charged-lepton-flavor-violation (CLFV) is so strongly suppressed +that it is undiscoverable by current experiments. Therefore, a discovery of a CLFV event +indicates new physics (NP). Verifying various NP models requires many searches of various +CLFV modes [1]. Whereas the CLFV constraints are much more stringent for µ-to-e than +for τ through the precise measurements [2–4], we are interested in τ, the third-generation +and heaviest lepton. So-called B-anomalies, which indicate NP effects in B semileptonic +decays [5–16], also motivate the CLFV searches. +We focus on τ CLFV decays into a charged lepton (ℓ = e± or µ±) and a neutral vector +meson (V 0 = ρ0, φ, ω, K∗0, or K∗0). In refs. [17–22], the τ → µφ mode is a sensitive probe +for leptoquark models that can explain the B-anomalies.1 Some other NP models predict +branching fractions of O(10−10)–O(10−8) for τ → ℓV 0 [25–28]. +We previously searched for τ → ℓV 0 events using 854 fb−1 of Belle data, and set +90% credibility level (C.L.) upper limits on the branching fractions in the range of (1.2– +8.4) × 10−8 [29].2 This paper reports an updated search for τ → ℓV 0 using the full 980 +fb−1 Belle data set. The signal efficiency is improved through new event selection criteria +with a multivariate analysis. +2 +Belle experiment +The Belle detector is a spectrometer that covers large solid angles of the e+e− collision +events from the KEKB accelerator [30, 31]. The detector consists of a silicon vertex de- +tector, a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov +1One of the B-anomalies which motivated the models described in those references is the R(K(∗)) +anomaly reported by the LHCb experiment [23], but it disappeared in their updated analysis [24]. +2In common high energy physics usage, this credibility level has been reported as “confidence level,” +which is a frequentist-statistics term. +– 1 – + +counters, time-of-flight scintillation counters, and an electromagnetic calorimeter composed +of 8736 CsI(Tl) crystals (ECL). These devices are located inside a superconducting solenoid +coil that provides a 1.5 T magnetic field. An iron flux return located outside of the coil is +instrumented to detect K0 +L mesons and identify muons. The Belle detector is described in +detail elsewhere [32, 33]. +Of the 980 fb−1 data set, 703 fb−1 was collected at the Υ(4S) resonance, 121 fb−1 at +the Υ(5S), 89 fb−1 at an energy 60 MeV below the Υ(4S), 28 fb−1 of energy-scans above the +Υ(4S), and the remainder at and near the Υ(1–3S). Compared to the previous paper [29], +the following data sets have been added: 78 fb−1 at and near the Υ(5S), 38 fb−1 at and +near the Υ(1–3S), and 10 fb−1 at an energy 60 MeV below the Υ(4S). +The e+e− collision events in the Belle detector are simulated by the Monte Carlo (MC) +method. Signal MC events of τ → ℓV 0 are generated by a dedicated MC with KKMC and +TAUOLA [34], where τ +τ − pairs are initially produced and one of the τ’s decays into ℓV 0 +and the other decays generically. The numbers of generated signal MC events are 1.1×106 +events at the Υ(4S) resonance, 0.4 × 106 events at the Υ(5S), 0.1 × 106 events at each of +the Υ(1–3S), and 0.1 × 106 events at an energy 60 MeV below the Υ(4S). We assume a +uniform CLFV decay angle in the τ rest frame. No specific NP model is assumed in the +CLFV decay process, and the spin direction of V 0 is set randomly and independently of the +spin of the mother τ. For background MC simulations, e+e− → q¯q (q = u, d, s, c), e+e− → +τ +τ −, Bhabha, and two-photon processes are generated by EvtGen [35], KKMC [34], +BHLUMI [36], and AAFH [37], respectively. +The detector responses are simulated by +GEANT3 [38]. +3 +Reconstruction and event selection +A signal τ is reconstructed from a lepton and a neutral vector meson. We separate the +event into two hemispheres in the center-of-mass (c.m.) frame by a plane perpendicular +to the thrust vector (⃗nT ) [39, 40]. The thrust vector is obtained by maximizing the thrust +T = Σi|⃗p c.m. +i +· ⃗nT |/Σi|⃗p c.m. +i +|, where i runs over all tracks and photons, and ⃗p c.m. +i +is the +momentum in the c.m. frame. In the hemisphere that contains a τ CLFV decay (called +signal side and τsig), V 0 is reconstructed as follows: ρ0 from π+π− within the reconstructed +mass window of 0.445–1.08 GeV/c2, φ from K+K− within 1.00–1.04 GeV/c2, ω from +π+π−π0 within 0.7–0.9 GeV/c2, K∗0 from K+π− within 0.7–1.1 GeV/c2, and K∗0 from +K−π+ within 0.7–1.1 GeV/c2. In the other hemisphere (called tag side), the other τ (τtag) +is reconstructed from ℓ±νν, π±ν, π±π0ν, π±π0π0ν, or π±π∓π±ν. This τtag information +enables the suppression of background events that have no neutrinos in the tag side. +The signal τ → ℓV 0 events have a unique kinematical feature; the ℓV 0 invariant mass +(MℓV 0) is close to the τ mass and the difference of the ℓV 0 energy from the beam energy +in the c.m. frame (∆E) is close to zero. The signal events within 1.65 GeV/c2 < MℓV 0 < +1.90 GeV/c2 and |∆E| < 0.5 GeV are reconstructed in this paper. +We follow a blind +analysis approach in this search by not looking at the signal candidates in the data set +until finalizing the event selection and background estimation. The blind region is 1.75 +– 2 – + +GeV/c2 ≤ MℓV 0 < 1.81 GeV/c2 and |∆E| < 0.08 GeV for the µρ0, µφ and µK∗0(K∗0) +modes, and 1.74 GeV/c2 ≤ MℓV 0 < 1.82 GeV/c2 and |∆E| < 0.1 GeV for the other modes. +Charged tracks, photons, and π0s should satisfy the following selection criteria. Each +charged track or photon is within the fiducial volume defined by −0.866 < cos θ < 0.956, +where θ is the polar angle with respect to the direction opposite to the e+ beam in the +laboratory frame. Charged tracks come from the interaction point; the distance of the +closest point from the interaction point is less than 0.5 cm in the transverse direction and +less than 3.0 cm in the longitudinal direction. Each π0 is reconstructed from two photons +inside the same hemisphere and the photon energy (Eγ) should be larger than 0.05 GeV. +The π0 mass window is 0.12 GeV/c2 < Mγγ < 0.15 GeV/c2, corresponding to ±3σ in the π0 +mass resolution. A π0 mass-constrained fit is performed to improve the energy resolution. +After reconstructing the signal and tag τ’s, no extra charged tracks are allowed. We +count the number of photons (nγ) with Eγ larger than 0.1 GeV in the signal side, and +require nγ ≤ 3 for the ℓω mode, which includes a π0 → γγ, and nγ ≤ 1 for the other +modes. +Particle identification is effective in suppressing the main background events of three- +hadron-track to the τ → ℓV 0 signal. We use likelihood ratios for electron identification +(P(e)) [41] and muon identification (P(µ)) [42]. +The lepton identification criteria are +P(e) > 0.9 for electrons, and P(µ) > 0.95 and the momentum is larger than 0.6 GeV/c for +muons. The electron (muon) identification efficiency is 90% (75%), whereas the probability +of misidentifying a pion as an electron (muon) is 0.1% (2%). The energy loss of an electron +by bremsstrahlung is recovered by adding back the energy of every photon within 0.05 +radians from the electron track direction into the electron momentum. To suppress low- +multiplicity background events like Bhabha, ee → eeee, or ee → eeµµ, an electron veto +(P(e) < 0.9) is applied to all hadron candidate tracks. +For hadron identification, we use a binary likelihood ratio P(i|j) = Li/(Li+Lj), where +Li(j) is the likelihood of particle i (j) [43] and i (j) is π, K, or p. The kaon identification +criteria are P(K|π) > 0.6 for both charged kaons from φ decay and P(K|π) > 0.8 for the +charged kaon from K∗0 and K∗0 decays. The kaon identification efficiency is 86% (77%), +whereas the probability of misidentifying a charged pion as a kaon is 4% (2%) for the kaons +from φ (K∗0, K∗0). A kaon veto (P(K|π) < 0.6) is applied to both charged pions from +ρ0 in the signal side, and 96% of pions are retained, whereas 14% of kaons are not vetoed. +To suppress muons from kaons decaying inside the CDC (K± → µ±ν), the kaon veto is +also applied to the signal-side muon track for the hadronic tags (τtag → πν, ππ0ν, πππν, +or ππ0π0ν). For the µV 0 modes with the hadronic tags, a proton veto (P(p|K) < 0.6 and +P(p|π) < 0.6) is applied for the tag-side tracks. +The signal events have one or two neutrinos from the τtag decay. We introduce some +event selection criteria requiring one or more neutrinos in the tag side. The missing mo- +mentum due to the neutrino(s) is calculated by subtracting the vector sum of the momenta +of all tracks and photons from the sum of the beam momenta. The missing energy is also +calculated by subtracting the sum of the energy of all tracks and photons from the sum +of the beam energy. Here, extra photons that are not used for the τ reconstruction are +included. The transverse missing momentum is required to be larger than 0.5 GeV/c, and +– 3 – + +the missing energy in the c.m. frame (Ec.m. +miss) is required to be larger than 0 GeV. Events +with missing particles other than neutrinos should be rejected as background events. These +non-neutrino missing particles can arise in two ways: neutral particles pass through the +gaps between the barrel and end-cap ECLs, and any particles go outside the CDC volume. +Thus, the direction of the missing momentum is required not to point to such regions. The +missing particles should be in the tag side and hence cos θc.m. +miss−tag > 0, where θc.m. +miss−tag +is the angle between the missing momentum and the vector sum of the momenta of the +tag-side tracks and photons in the c.m. frame. The neutrino angle with respect to the τtag +momentum direction is restricted in a τtag two-body decay; thus cos θc.m. +miss−tag < 0.85 is also +applied for the ℓρ0 modes with τtag → πν. +We require features of a generic τ decay in the tag side. The invariant mass of the +particles including all photons in the tag hemisphere should be less than the τ mass (1.777 +GeV/c2). For τtag decays into ππ0ν (3πν), the reconstructed mass of those pions is required +to be 0.4 GeV/c2 < Mππ0 < 1.3 GeV/c2 (0.7 GeV/c2 < M3π < 1.7 GeV/c2), which +corresponds to the mass of ρ± (a± +1 ). +After the above event reconstruction, the background sources are the q¯q continuum +(q = u, d, s, c), generic τ +τ −, and low-multiplicity events. +The low-multiplicity events +especially contribute to the background events for eV 0 modes that have electron tracks. +We suppress the low-multiplicity events first, and then use a maltivariate analysis tool to +suppress the q¯q continuum and generic τ +τ − events. +The Bhabha events have tracks from photon conversion. To suppress these background +events for the eV 0 modes, the invariant mass of the electron and one of the tracks from the +V 0, assigned the electron-mass hypothesis, should be larger than 0.2 GeV/c2. In addition, +for the eK∗0 and eK∗0 modes, the invariant mass of the two tracks from the V 0, each +assigned the electron-mass hypothesis, is required to be larger than 0.1 GeV/c2. +This +event selection also suppresses some of the generic τ +τ − events, which have tracks from +photon conversion. +The low-multiplicity background events are still not negligible for the events with elec- +trons: τ → eV 0 or τtag → eνν. Because the missing particles of the low-multiplicity back- +ground events are the bremsstrahlung photons from the electron in the tag side, cos θc.m. +miss−tag +is close to one (Figure 1). In addition, the missing energy is small for some low-multiplicity +background events. For the µρ0 mode with τtag → eνν, cos θc.m. +miss−tag < 0.99 and Ec.m. +miss > 0.4 +GeV selection criteria are applied. For the eV 0 modes with τtag → eνν or πν, cos θc.m. +miss−tag < +0.97 is applied. For the eV 0 modes with τtag → eνν, Ec.m. +miss should be larger than 0.4, 2.0, +and 1.5 GeV for eφ, eρ0, and the other eV 0 modes, respectively. +The remaining background events are mainly from the q¯q continuum (q = u, d, s, c) +and generic τ +τ − events, which have three charged pion tracks in the signal side. We use a +two-class Boosted Decision Tree (BDT) for signal and these background classification. The +BDT library is LightGBM [44]. This BDT outputs a signal probability using the following +input variables: +• MV 0, M2 +ν , P c.m. +ν +, T, P sig +ℓ , Ehemi +tag , cos θc.m. +miss−tag +• (categorical variables) τtag decay mode, collision energy +– 4 – + +Figure 1: The cos θc.m. +miss−tag distribution of the τ → eρ0 mode with a electron tag track +after the reconstruction, particle identification, and photon conversion event suppression. +Black points with error bars are the data outside the blind region. Red solid histogram +is the signal MC. The signal MC is scaled to the number of events corresponding to 100 +times as large branching fraction as the current upper limit. The red dashed line is the +upper limit to remove the low-multiplicity events. +The low-multiplicity events cluster +around cos θc.m. +miss−tag = 1, whereas the other background events are linearly distributed in +the region of cos θc.m. +miss−tag > 0.8. +• (additional for the ℓω modes) P sig +π0 , Elow +γ +, +where MV 0 is the invariant mass of the vector meson, M2 +ν is the missing mass squared, P c.m. +ν +is the missing momentum in the c.m. frame, T is the magnitude of the thrust vector [39, 40], +P sig +ℓ +is the momentum of the lepton in the signal side, Ehemi +tag +is the energy sum of the tracks +and photons in the tag hemisphere, P sig +π0 is the momentum of π0 from ω and Elow +γ +is the +lower energy of the two photons from the π0. The variables of neutrino kinematics (M2 +ν and +P c.m. +ν +) were not used for the event selection in the previous paper [29]. They are calculated +from the momenta of the reconstructed τsig and τtag, where the energy of τsig is fixed to the +half of the beam energy in the c.m. frame. The q¯q continuum background events can be +effectively suppressed by a M2 +ν selection in the hadronic tags, involving only one neutrino +(Figure 2). +The training, validation and evaluation of the BDT are done with 40%, 10%, and +50% of the signal MC, respectively. Regarding the training and validation samples for +the background events, we utilize hadron background enhanced data that are obtained by +removing the lepton identification for the signal-side leptons but with a lepton identification +veto (P(e) ≤ 0.9 and P(µ) ≤ 0.95) for all the signal-side tracks in the data. The hadron +background enhanced data have a much larger number of events than the background data +with the nominal selection criteria, whereas both data sets are composed mainly of three +charged pions from τ decays or from continuum events. The training is done with 80% +of the hadron background enhanced data and the validation is done with 20%. During +BDT training, a weight is applied to each of the signal MC events such that the sum of +the weights is equal to the number of background events. +We monitor the area under +– 5 – + +t→μpo, electron tag +70 +BR(→μp0)=1.2×10-8 × 100 +Number of events/(0.01) +60 +data +50 +40 +30 +20 +10 +t+.+- +0.2 +0.0 +0.4 +0.6 +0.8 +1.0 +CosAc.m. +miss -tagcurve (AUC) of the Receiver Operating Characteristic curve [45] for the validation samples +during the training and choose the BDT with the best AUC score. +The event selection with the BDT output (BDT selection) is determined only by a +target signal efficiency. +The target signal efficiency is determined based on the signal +efficiency with a cut-based event selection. +In the cut-based event selection, the MV 0 +windows correspond to ±2σ of reconstructed mass distribution, and the M2 +ν windows are +set for each ℓV 0 mode and each τtag decay mode so that the expected number of background +events inside the signal region (NBG, see the next section) is approximately one or less. +The target signal efficiency with the BDT selection is set as relatively 5% larger than that +with the cut-based event selection, because we expect improvement in separating the signal +events from the background events. +The finalized BDT selection shows similar NBG to that of the cut-based event selection. +The BDT selection is not applied to the ℓφ modes because NBG in each of the two modes +is small enough. +Figure 2: The M2 +ν distribution of the τ → µρ0 mode with the hadronic tags after the +event selection except for the requirement of the BDT output. Black points with error bars +are the data outside the blind region. Red solid histogram is the signal MC. The signal MC +is scaled to the number of events corresponding to 100 times as large a branching fraction +as the current upper limit. The events constituting the upper tail of the signal distribution +originate from wrong or missing π0 in the tag side. +4 +Signal efficiency and background estimation +We define the signal region with an ellipse in the MℓV 0–∆E plane. Most of the signal +events cluster around MℓV 0 = 1.777 GeV/c2 and ∆E = 0 GeV with some correlation. The +ellipse oblateness and the rotation angle are calculated from the covariance matrix of the +signal MC distribition after the event selection. The center of the ellipse is the mean of the +distribution. The ellipse size is determined to maximize the figure-of-merit (FOM) [46], +FOM = +ε +α +2 + √NBG +, +(4.1) +– 6 – + +t→μpo, hadronic tag +300 +BR(t→μp°)=l.2×10-8 × 100 +250 +data +200 +150 +100 +50 +-2 +-1 +0 +1 +2 +M2 (GeV2/c4)where ε is the signal efficiency inside the ellipse, α is the confidence coefficient (α = 1.64 +at 90% C.L.). +Figure 3: The MℓV 0 vs. ∆E distribution of the τ → µρ0 hadron background enhanced +samples: the data (upper side), the generic τ +τ − MC (lower left) and the q¯q continuum +MC (lower right, q = u, d, s, c). The range of the ∆E axis is limited to the fitting region. +The MC sets are scaled to the data. The low-multiplicity background events are negligible +for the hadron background enhanced samples and are not shown in this figure. +We estimate NBG through interpolation from the sideband data. Here the sideband +data is a set of data passing the event selection and inside the sideband region: 1.65 GeV/c2 +< MℓV 0 < 1.9 GeV/c2 and |∆E| < 0.1 GeV outside of the blind region. The interpolation +is based on a function in the MℓV 0–∆E plane. This function is obtained by fitting the +distribution of the hadron background enhanced data within |∆E| < 0.1 GeV, and then it +is scaled to the sideband data. Figure 3 shows the distributions of the hadron background +enhanced data and MC for the µρ0 mode. The function is: +F(MℓV 0, ∆E) = f(MℓV 0) × +1 +1 + exp[ay(∆E − y0)] + cflat +0 , +(4.2) +– 7 – + +data +0.100 +100 +0.075 +0.050 +80 +△E (GeV) +0.025 +60 +0.000 +-0.025 +40 +-0.050 +20 +-0.075 +-0.100 +0 +1.65 +1.70 +1.75 +1.80 +1.85 +1.90 +Mμpo (GeV/c2)t+t- MC +0.100 +0.075 +80 +0.050 +△E (GeV) +60 +0.025 +0.000 +40 +-0.025 +-0.050 +20 +-0.075 +-0.100 +0 +1.65 +1.70 +1.75 +1.80 +1.85 +1.90 +Mμpo (GeV/c2)qq MC (q = u, d,s,c) +0.100 +8 +0.075 +7 +0.050 +6 +(GeV) +0.025 +5 +0.000 +4 +△E +-0.025 +3 +-0.050 +2 +-0.075 +1 +-0.100 +0 +1.65 +1.70 +1.75 +1.80 +1.85 +1.90 +Mμpo (GeV/c2)f(x) = +� +� +� +� +� +� +� +� +� +� x+5σ +x−5σ +g(x′) × +1 +√ +2πσexp +�−(x − x′)2 +2σ2 +� +dx′ +(V 0 = ρ0, ω) +c1(x − x0)2 + c0 +(V 0 = K∗0, K∗0) +c0 +(V 0 = φ) +g(x) = +� +� +� +� +� +c1[(x − x0)2 + k(x − x0)] + c0 +(x < x0, V 0 = ρ0) +c1(x − x0) + c0 +(x < x0, V 0 = ω) +c0 +(x ≥ x0) +(4.3) +where f(x) represents the background distribution as a function of MℓV 0; c1, c0, x0, and k +are parameters that define the shape of the function; ay represents sharpness of the sigmoid +function along the ∆E axis; y0 is the center of the sigmoid function; and cflat +0 +is a term of +flat background events in the MℓV 0–∆E plane. We define f(x) for each V 0 in eq. (4.3) and +the functions for the ℓρ0 (ℓω) modes are smeared by a Gaussian with standard deviation +(σ) of 6.6 (9.6) MeV/c2. This σ corresponds to the mass resolution that affects the edge +of the MℓV 0 distribution close to the τ mass for the τ +τ − background. The edge is broad +for the other modes owing to wrong mass assignment of fake kaons. The τ +τ − background +events for the ℓφ modes are included in c0 because they are flat along the MℓV 0 axis in +1.65 GeV/c2 < MℓV 0 < 1.9 GeV/c2. +We obtain the optimal fit parameters by a likelihood fit using MINUIT [47]. +The +following region is excluded from the fitting to avoid D+ → K−π+π+ and D+ → π+φ +background events, which cluster around the D meson mass: 1.83(1.82) GeV/c2 ≤ MℓV 0 < +1.89 GeV/c2 and ∆E < 0.04 GeV for the µK∗0 (eK∗0) and µφ (eφ) modes. +The parameters of ay, y0, k, and x0 are fixed at the fit results of the hadron background +enhanced data within |∆E| < 0.1 GeV. The fit uncertainties of these fixed parameters are +included in the systematic uncertainty of NBG. The other fit parameters correspond to +the scale factors of each background component: generic τ +τ − (c1), and continuum and +low-multiplicity background events (c0 and cflat +0 ). We fit the function floating these scale +factors (c1, c0, and cflat +0 ) to the sideband data. The same region around the D meson mass +as for the fit to the hadron background enhanced data is excluded from the fitting for the +ℓφ and ℓK∗0 modes. The functions are integrated in the elliptical signal regions to deduce +NBG, which are in the range of 0.25–0.95. +Another systematic uncertainty on NBG comes from difference of the MℓV 0–∆E dis- +tributions between the sideband data and the hadron background enhanced data within +|∆E| < 0.1 GeV. The difference mainly arises from the electron (muon) identification +fake rate, Rfake +e(µ)(P, θ), which depends on the momentum P and θ of the track. The side- +band data have a pion misidentified as a lepton, which tends to have a lower momen- +tum than the pions in the hadron background enhanced data. We evaluate a change of +NBG when the parameters—ay, y0, k, and x0—are redetermined with weighted hadron +background enhanced data, where each event is weighted by the ratio of Rfake +e(µ)(P, θ) to +1 − Rfake +e +(P, θ) − Rfake +µ +(P, θ) for the track in order to conform the MℓV 0–∆E distribution +to the one of the sideband data. The amout of change of NBG is taken as the systematic +uncertainty of NBG. +The statistical uncertainty of NBG is calculated as follows: We generate 100 sets of +– 8 – + +pseudo-data for each mode in the MℓV 0–∆E histogram. The content of each bin in the +histogram is set randomly following a Poisson distribution, with the mean taken from the +function fitted to the sideband data. We fit the function to each set of the pseudo-data to +deduce NBG, and the standard deviation of these NBG is taken as the statistical uncertainty. +The major contribution to NBG comes from the MℓV 0 flat term in eq. (4.2) (c0 and +cflat +0 ), which corresponds to the continuum or low-multiplicity background events. +The +contribution of the generic τ +τ − background events, which depends on MℓV 0, is about +one-third as large as the other background contributions. We cannot distinguish the back- +ground components of the ℓφ modes through the fit to the data, because the generic τ +τ − +background events are distributed evenly along the MℓV 0 axis. +The systematic uncertainties of the expected number of signal events are listed in +Table 1. The dominant uncertainties are from the particle identification. +The track and photon energy resolutions in the MC are corrected such that the +mass resolution of the D(∗)+ meson matches between the data and MC, where D(∗)+ → +K−π+π+(π0) is reconstructed with similar event selection criteria to the signal ones (e.g. +|∆E| < 0.5 GeV). The uncertainty of the data mass resolution propagates to the uncer- +tainties of the corrected energy resolutions. We generate two additional signal MC sets in +which the track (photon) energy resolution is different by plus and minus its uncertainty, +and take the half of the difference in the expected number of the signal events as the +systematic uncertainty. +All the uncertainties in Table 1 are summed in quadrature to yield the total systematic +uncertainties shown in Table 2. +Table 1: List of the systematic uncertainties of the expected number of signal events. The +average number of tracks (particles) in the reconstructed τ +τ − events for each signal mode +is represented as Ntrack(particle). When the uncertainty is different mode by mode, we show +the range of the uncertainty. +Source +σsyst (%) +Integrated luminosity +1.4 +ee → ττ(γ) cross section [48] +0.3 +B(φ → KK) and B(ω → πππ0) +1.2 and 0.7 +Trigger efficiency +0.2–0.9 +Tracking efficiency +0.35 × Ntrack +Electron identification efficiency +1.7 × Nelectron +Muon identification efficiency +1.8 × Nmuon +K and π identification efficiency +1.6 (ρ0), 1.8 (φ) and 1.1 (K∗0 and K∗0) +π0 efficiency +2.2 × Nπ0 +Electron veto for hadrons +0.4–1.2 +MC statistics +0.3–0.5 +Track energy resolution +0.3–1.3 +Photon energy resolution +0.0–0.4 +– 9 – + +5 +Results +Figures 4 and 5 show the observed event distributions in the MℓV 0–∆E plane. The observed +number of events in the signal region (Nobs) has no excess over NBG. +We set 90% C.L. upper limits on the branching fractions based on a Bayesian method +with the use of Markov Chain Monte Carlo [49]. +The probability density function of +the branching fraction (B(τ → ℓV 0)) is calculated assuming that Nobs follows a Poisson +distribution function whose mean value is the expected number of events (Nexp), +Nexp = L × 2σττB(τ → ℓV 0) × ε + NBG, +(5.1) +where L is the integrated luminosity (980.4 ± 13.7 fb−1), σττ is the cross section of τ-pair +production that is calculated with KKMC [48] (the weighted average of σττ at all the beam +energies is 0.916 ± 0.003 nb), and ε is the signal efficiency including the branching fraction +of the V 0. We assume that these values (L, σττ, ε, and NBG) follow a Gaussian distribution +with the width equal to the uncertainty of each value. +The upper limits on B(τ → ℓV 0) are listed in Table 2. The average of the limits is +better than that of the previous results using 854 fb−1 [29] by 30%. This is due to the +additional 15% of integrated luminosity; the addition of π±π∓π±ν and π±π0π0ν modes in +τtag reconstruction, which increases the signal efficiency by 9.6%; and the event selection +by multivariate analysis (BDT). The upper limit on B(τ → µρ0) is worse than that of +the previous result, though the expected upper limit before unblinding is better. This is +because we use the Bayesian limits instead of the Frequentist limits, which are negatively +proportional to NBG when Nobs is fixed. +Table 2: The signal efficiency (ε), the expected number of background events (NBG), +total systematic uncertainty of the expected number of signal events (σsyst), the number +of observed events in the signal region (Nobs), and the observed 90% C.L. upper limits on +the branching fraction (Bobs (10−8)). +Mode +ε (%) +NBG +σsyst (%) +Nobs +Bobs (×10−8) +τ − → µ−ρ0 +7.78 +0.95±0.20(stat.) ±0.11(syst.) +4.6 +0 +< 1.7 +τ − → e−ρ0 +8.49 +0.80±0.27(stat.) ±0.02(syst.) +4.4 +1 +< 2.2 +τ − → µ−φ +5.59 +0.47±0.15(stat.) ±0.05(syst.) +4.8 +0 +< 2.3 +τ − → e−φ +6.45 +0.38±0.21(stat.) ±0.00(syst.) +4.5 +0 +< 2.0 +τ − → µ−ω +3.27 +0.32±0.23(stat.) ±0.03(syst.) +4.8 +0 +< 3.9 +τ − → e−ω +5.41 +0.74±0.43(stat.) ±0.01(syst.) +4.5 +0 +< 2.4 +τ − → µ−K∗0 +4.52 +0.84±0.25(stat.) ±0.03(syst.) +4.3 +0 +< 2.9 +τ − → e−K∗0 +6.94 +0.54±0.21(stat.) ±0.12(syst.) +4.1 +0 +< 1.9 +τ − → µ−K∗0 +4.58 +0.58±0.17(stat.) ±0.06(syst.) +4.3 +1 +< 4.2 +τ − → e−K∗0 +7.45 +0.25±0.11(stat.) ±0.01(syst.) +4.1 +0 +< 1.7 +– 10 – + +6 +Conclusion +To conclude, we searched for lepton-flavor-violating τ decays into one lepton and one vector +meson using the full 980 fb−1 of Belle data. No statistically significant signal candidates are +observed, and the 90% C.L. upper limits on the branching fraction are in the range of (1.7– +4.2) × 10−8 for τ → µV 0 and (1.7–2.4) × 10−8 for τ → eV 0. The upper limits are improved +by 30% on average from the previous results. We achieve these improvements both with +the reconsideration of the event selection criteria and with the 126 fb−1 of additional data +set. +Acknowledgments +This work, based on data collected using the Belle detector, which was operated until +June 2010, was supported by the Ministry of Education, Culture, Sports, Science, and +Technology (MEXT) of Japan, the Japan Society for the Promotion of Science (JSPS), +and the Tau-Lepton Physics Research Center of Nagoya University; the Australian Re- +search Council including grants DP180102629, DP170102389, DP170102204, DE220100462, +DP150103061, FT130100303; Austrian Federal Ministry of Education, Science and Re- +search (FWF) and FWF Austrian Science Fund No. P 31361-N36; the National Natural +Science Foundation of China under Contracts No. 11675166, No. 11705209; No. 11975076; +No. 12135005; No. 12175041; No. 12161141008; Key Research Program of Frontier Sci- +ences, Chinese Academy of Sciences (CAS), Grant No. QYZDJ-SSW-SLH011; Project +ZR2022JQ02 supported by Shandong Provincial Natural Science Foundation; the Ministry +of Education, Youth and Sports of the Czech Republic under Contract No. LTT17020; +the Czech Science Foundation Grant No. 22-18469S; Horizon 2020 ERC Advanced Grant +No. 884719 and ERC Starting Grant No. 947006 “InterLeptons” (European Union); the +Carl Zeiss Foundation, the Deutsche Forschungsgemeinschaft, the Excellence Cluster Uni- +verse, and the VolkswagenStiftung; the Department of Atomic Energy (Project Identi- +fication No. +RTI 4002) and the Department of Science and Technology of India; the +Istituto Nazionale di Fisica Nucleare of Italy; National Research Foundation (NRF) of +Korea Grant Nos. 2016R1D1A1B02012900, 2018R1A2B3003643, 2018R1A6A1A06024970, +RS202200197659, 2019R1I1A3A01058933, 2021R1A6A1A03043957, 2021R1F1A1060423, +2021R1F1A1064008, 2021R1A4A2001897, 2022R1A2C1003993; Radiation Science Research +Institute, Foreign Large-size Research Facility Application Supporting project, the Global +Science Experimental Data Hub Center of the Korea Institute of Science and Technology +Information and KREONET/GLORIAD; the Polish Ministry of Science and Higher Ed- +ucation and the National Science Center; the Ministry of Science and Higher Education +of the Russian Federation, Agreement 14.W03.31.0026, and the HSE University Basic Re- +search Program, Moscow; University of Tabuk research grants S-1440-0321, S-0256-1438, +and S-0280-1439 (Saudi Arabia); the Slovenian Research Agency Grant Nos. J1-9124 and +P1-0135; Ikerbasque, Basque Foundation for Science, Spain; the Swiss National Science +Foundation; the Ministry of Education and the Ministry of Science and Technology of Tai- +wan; and the United States Department of Energy and the National Science Foundation. +– 11 – + +These acknowledgements are not to be interpreted as an endorsement of any statement +made by any of our institutes, funding agencies, governments, or their representatives. We +thank the KEKB group for the excellent operation of the accelerator; the KEK cryogenics +group for the efficient operation of the solenoid; and the KEK computer group and the +Pacific Northwest National Laboratory (PNNL) Environmental Molecular Sciences Labora- +tory (EMSL) computing group for strong computing support; and the National Institute of +Informatics, and Science Information NETwork 6 (SINET6) for valuable network support. +– 12 – + +(a) τ → µρ0 +(b) τ → µφ +(c) τ → µω +(d) τ → µK∗0 +(e) τ → µK∗0 +Figure 4: Observed event distributions of MℓV 0 vs. ∆E after the τ → µV 0 event selection. +Black points are the data, blue squares show the signal MC distribution with an arbitrary +normalization. The red ellipse line is the signal region. The estimation of the number of +background events is done using the data between the red horizontal lines except the blind +region. +– 13 – + +0.4 +口 +Data +0.2 +△E (GeV) +0.0 +-0.2 +-0.4 +1.65 +1.70 +1.75 +1.80 +1.85 +1.90 +Muk*0 (GeV/c2)0.4 +口 +Data +0.2 +(GeV) +0.0 +△E ( +-0.2 +-0.4 +1.65 +1.70 +1.75 +1.80 +1.85 +1.90 +Mμpo (GeV/c2)口 +0.4 +Data +0.2 +(GeV) +0.0 +△E ( +-0.2 +: +-0.4 +1.65 +1.70 +1.75 +1.80 +1.85 +1.90 +Mus (GeV/c2)3n↑ +口 +0.4 +Data +0.2 +△E (GeV) +0.0 +-0.2 +-0.4 +1.70 +1.65 +1.75 +1.80 +1.85 +1.90 +Mμw (GeV/c2)0.4 +口 +Data +0.2 +(GeV) +0.0 +△E ( +. +-0.2 +。 +.. +-0.4 +1.65 +1.70 +1.75 +1.80 +1.85 +1.90 +Muk*0 (GeV/c2)(a) τ → eρ0 +(b) τ → eφ +(c) τ → eω +(d) τ → eK∗0 +(e) τ → eK∗0 +Figure 5: Observed event distributions of MℓV 0 vs. ∆E after the τ → eV 0 event selection. +Black points are the data, blue squares show the signal MC distribution with an arbitrary +normalization. The red ellipse line is the signal region. 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Vinokurova , D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Wang , E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Wang , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Wang , X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Wang , S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Watanuki , X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Xu , B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Yabsley , W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Yan , S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Yang , J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Yelton , Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Yook , L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Yuan , Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Zhai , V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Zhilich , V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Zhukova , arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='03768v1 [hep-ex] 10 Jan 2023 Abstract: Charged-lepton-flavor-violation is predicted in several new physics scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We update the analysis of τ lepton decays into a light charged lepton (ℓ = e± or µ±) and a vector meson (V 0 = ρ0, φ, ω, K∗0, or K∗0) using 980 fb−1 of data collected with the Belle detector at the KEKB collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' No significant excess of such signal events is observed, and thus 90% credibility level upper limits are set on the τ → ℓV 0 branching fractions in the range of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='7–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2) × 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' These limits are improved by 30% on average from the previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Keywords: e+–e− Experiments, Tau Physics Contents 1 Introduction 1 2 Belle experiment 1 3 Reconstruction and event selection 2 4 Signal efficiency and background estimation 6 5 Results 10 6 Conclusion 11 1 Introduction In the Standard Model, charged-lepton-flavor-violation (CLFV) is so strongly suppressed that it is undiscoverable by current experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Therefore, a discovery of a CLFV event indicates new physics (NP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Verifying various NP models requires many searches of various CLFV modes [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Whereas the CLFV constraints are much more stringent for µ-to-e than for τ through the precise measurements [2–4], we are interested in τ, the third-generation and heaviest lepton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' So-called B-anomalies, which indicate NP effects in B semileptonic decays [5–16], also motivate the CLFV searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We focus on τ CLFV decays into a charged lepton (ℓ = e± or µ±) and a neutral vector meson (V 0 = ρ0, φ, ω, K∗0, or K∗0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' In refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' [17–22], the τ → µφ mode is a sensitive probe for leptoquark models that can explain the B-anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1 Some other NP models predict branching fractions of O(10−10)–O(10−8) for τ → ℓV 0 [25–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We previously searched for τ → ℓV 0 events using 854 fb−1 of Belle data, and set 90% credibility level (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') upper limits on the branching fractions in the range of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2– 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='4) × 10−8 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2 This paper reports an updated search for τ → ℓV 0 using the full 980 fb−1 Belle data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The signal efficiency is improved through new event selection criteria with a multivariate analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' 2 Belle experiment The Belle detector is a spectrometer that covers large solid angles of the e+e− collision events from the KEKB accelerator [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The detector consists of a silicon vertex de- tector, a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov 1One of the B-anomalies which motivated the models described in those references is the R(K(∗)) anomaly reported by the LHCb experiment [23], but it disappeared in their updated analysis [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' 2In common high energy physics usage, this credibility level has been reported as “confidence level,” which is a frequentist-statistics term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' – 1 – counters, time-of-flight scintillation counters, and an electromagnetic calorimeter composed of 8736 CsI(Tl) crystals (ECL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' These devices are located inside a superconducting solenoid coil that provides a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='5 T magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' An iron flux return located outside of the coil is instrumented to detect K0 L mesons and identify muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The Belle detector is described in detail elsewhere [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Of the 980 fb−1 data set, 703 fb−1 was collected at the Υ(4S) resonance, 121 fb−1 at the Υ(5S), 89 fb−1 at an energy 60 MeV below the Υ(4S), 28 fb−1 of energy-scans above the Υ(4S), and the remainder at and near the Υ(1–3S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Compared to the previous paper [29], the following data sets have been added: 78 fb−1 at and near the Υ(5S), 38 fb−1 at and near the Υ(1–3S), and 10 fb−1 at an energy 60 MeV below the Υ(4S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The e+e− collision events in the Belle detector are simulated by the Monte Carlo (MC) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Signal MC events of τ → ℓV 0 are generated by a dedicated MC with KKMC and TAUOLA [34], where τ +τ − pairs are initially produced and one of the τ’s decays into ℓV 0 and the other decays generically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The numbers of generated signal MC events are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1×106 events at the Υ(4S) resonance, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='4 × 106 events at the Υ(5S), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1 × 106 events at each of the Υ(1–3S), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1 × 106 events at an energy 60 MeV below the Υ(4S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We assume a uniform CLFV decay angle in the τ rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' No specific NP model is assumed in the CLFV decay process, and the spin direction of V 0 is set randomly and independently of the spin of the mother τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' For background MC simulations, e+e− → q¯q (q = u, d, s, c), e+e− → τ +τ −, Bhabha, and two-photon processes are generated by EvtGen [35], KKMC [34], BHLUMI [36], and AAFH [37], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The detector responses are simulated by GEANT3 [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' 3 Reconstruction and event selection A signal τ is reconstructed from a lepton and a neutral vector meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We separate the event into two hemispheres in the center-of-mass (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') frame by a plane perpendicular to the thrust vector (⃗nT ) [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The thrust vector is obtained by maximizing the thrust T = Σi|⃗p c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' i ⃗nT |/Σi|⃗p c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' i |, where i runs over all tracks and photons, and ⃗p c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' i is the momentum in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' In the hemisphere that contains a τ CLFV decay (called signal side and τsig), V 0 is reconstructed as follows: ρ0 from π+π− within the reconstructed mass window of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='445–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='08 GeV/c2, φ from K+K− within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='00–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='04 GeV/c2, ω from π+π−π0 within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='7–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='9 GeV/c2, K∗0 from K+π− within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='7–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1 GeV/c2, and K∗0 from K−π+ within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='7–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1 GeV/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' In the other hemisphere (called tag side), the other τ (τtag) is reconstructed from ℓ±νν, π±ν, π±π0ν, π±π0π0ν, or π±π∓π±ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' This τtag information enables the suppression of background events that have no neutrinos in the tag side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The signal τ → ℓV 0 events have a unique kinematical feature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' the ℓV 0 invariant mass (MℓV 0) is close to the τ mass and the difference of the ℓV 0 energy from the beam energy in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' frame (∆E) is close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The signal events within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='65 GeV/c2 < MℓV 0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='90 GeV/c2 and |∆E| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='5 GeV are reconstructed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We follow a blind analysis approach in this search by not looking at the signal candidates in the data set until finalizing the event selection and background estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The blind region is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='75 – 2 – GeV/c2 ≤ MℓV 0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='81 GeV/c2 and |∆E| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='08 GeV for the µρ0, µφ and µK∗0(K∗0) modes, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='74 GeV/c2 ≤ MℓV 0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='82 GeV/c2 and |∆E| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1 GeV for the other modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Charged tracks, photons, and π0s should satisfy the following selection criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Each charged track or photon is within the fiducial volume defined by −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='866 < cos θ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='956, where θ is the polar angle with respect to the direction opposite to the e+ beam in the laboratory frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Charged tracks come from the interaction point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' the distance of the closest point from the interaction point is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='5 cm in the transverse direction and less than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='0 cm in the longitudinal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Each π0 is reconstructed from two photons inside the same hemisphere and the photon energy (Eγ) should be larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='05 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The π0 mass window is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='12 GeV/c2 < Mγγ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='15 GeV/c2, corresponding to ±3σ in the π0 mass resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' A π0 mass-constrained fit is performed to improve the energy resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' After reconstructing the signal and tag τ’s, no extra charged tracks are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We count the number of photons (nγ) with Eγ larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1 GeV in the signal side, and require nγ ≤ 3 for the ℓω mode, which includes a π0 → γγ, and nγ ≤ 1 for the other modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Particle identification is effective in suppressing the main background events of three- hadron-track to the τ → ℓV 0 signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We use likelihood ratios for electron identification (P(e)) [41] and muon identification (P(µ)) [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The lepton identification criteria are P(e) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='9 for electrons, and P(µ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='95 and the momentum is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='6 GeV/c for muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The electron (muon) identification efficiency is 90% (75%), whereas the probability of misidentifying a pion as an electron (muon) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1% (2%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The energy loss of an electron by bremsstrahlung is recovered by adding back the energy of every photon within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='05 radians from the electron track direction into the electron momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' To suppress low- multiplicity background events like Bhabha, ee → eeee, or ee → eeµµ, an electron veto (P(e) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='9) is applied to all hadron candidate tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' For hadron identification, we use a binary likelihood ratio P(i|j) = Li/(Li+Lj), where Li(j) is the likelihood of particle i (j) [43] and i (j) is π, K, or p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The kaon identification criteria are P(K|π) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='6 for both charged kaons from φ decay and P(K|π) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='8 for the charged kaon from K∗0 and K∗0 decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The kaon identification efficiency is 86% (77%), whereas the probability of misidentifying a charged pion as a kaon is 4% (2%) for the kaons from φ (K∗0, K∗0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' A kaon veto (P(K|π) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='6) is applied to both charged pions from ρ0 in the signal side, and 96% of pions are retained, whereas 14% of kaons are not vetoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' To suppress muons from kaons decaying inside the CDC (K± → µ±ν), the kaon veto is also applied to the signal-side muon track for the hadronic tags (τtag → πν, ππ0ν, πππν, or ππ0π0ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' For the µV 0 modes with the hadronic tags, a proton veto (P(p|K) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='6 and P(p|π) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='6) is applied for the tag-side tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The signal events have one or two neutrinos from the τtag decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We introduce some event selection criteria requiring one or more neutrinos in the tag side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The missing mo- mentum due to the neutrino(s) is calculated by subtracting the vector sum of the momenta of all tracks and photons from the sum of the beam momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The missing energy is also calculated by subtracting the sum of the energy of all tracks and photons from the sum of the beam energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Here, extra photons that are not used for the τ reconstruction are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The transverse missing momentum is required to be larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='5 GeV/c, and – 3 – the missing energy in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' frame (Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' miss) is required to be larger than 0 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Events with missing particles other than neutrinos should be rejected as background events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' These non-neutrino missing particles can arise in two ways: neutral particles pass through the gaps between the barrel and end-cap ECLs, and any particles go outside the CDC volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Thus, the direction of the missing momentum is required not to point to such regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The missing particles should be in the tag side and hence cos θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' miss−tag > 0, where θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' miss−tag is the angle between the missing momentum and the vector sum of the momenta of the tag-side tracks and photons in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The neutrino angle with respect to the τtag momentum direction is restricted in a τtag two-body decay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' thus cos θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' miss−tag < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='85 is also applied for the ℓρ0 modes with τtag → πν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We require features of a generic τ decay in the tag side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The invariant mass of the particles including all photons in the tag hemisphere should be less than the τ mass (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='777 GeV/c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' For τtag decays into ππ0ν (3πν), the reconstructed mass of those pions is required to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='4 GeV/c2 < Mππ0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='3 GeV/c2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='7 GeV/c2 < M3π < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='7 GeV/c2), which corresponds to the mass of ρ± (a± 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' After the above event reconstruction, the background sources are the q¯q continuum (q = u, d, s, c), generic τ +τ −, and low-multiplicity events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The low-multiplicity events especially contribute to the background events for eV 0 modes that have electron tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We suppress the low-multiplicity events first, and then use a maltivariate analysis tool to suppress the q¯q continuum and generic τ +τ − events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The Bhabha events have tracks from photon conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' To suppress these background events for the eV 0 modes, the invariant mass of the electron and one of the tracks from the V 0, assigned the electron-mass hypothesis, should be larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2 GeV/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' In addition, for the eK∗0 and eK∗0 modes, the invariant mass of the two tracks from the V 0, each assigned the electron-mass hypothesis, is required to be larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1 GeV/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' This event selection also suppresses some of the generic τ +τ − events, which have tracks from photon conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The low-multiplicity background events are still not negligible for the events with elec- trons: τ → eV 0 or τtag → eνν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Because the missing particles of the low-multiplicity back- ground events are the bremsstrahlung photons from the electron in the tag side, cos θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' miss−tag is close to one (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' In addition, the missing energy is small for some low-multiplicity background events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' For the µρ0 mode with τtag → eνν, cos θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' miss−tag < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='99 and Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' miss > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='4 GeV selection criteria are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' For the eV 0 modes with τtag → eνν or πν, cos θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' miss−tag < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='97 is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' For the eV 0 modes with τtag → eνν, Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' miss should be larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='0, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='5 GeV for eφ, eρ0, and the other eV 0 modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The remaining background events are mainly from the q¯q continuum (q = u, d, s, c) and generic τ +τ − events, which have three charged pion tracks in the signal side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We use a two-class Boosted Decision Tree (BDT) for signal and these background classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The BDT library is LightGBM [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' This BDT outputs a signal probability using the following input variables: MV 0, M2 ν , P c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' ν , T, P sig ℓ , Ehemi tag , cos θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' miss−tag (categorical variables) τtag decay mode, collision energy – 4 – Figure 1: The cos θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' miss−tag distribution of the τ → eρ0 mode with a electron tag track after the reconstruction, particle identification, and photon conversion event suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Black points with error bars are the data outside the blind region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Red solid histogram is the signal MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The signal MC is scaled to the number of events corresponding to 100 times as large branching fraction as the current upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The red dashed line is the upper limit to remove the low-multiplicity events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The low-multiplicity events cluster around cos θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' miss−tag = 1, whereas the other background events are linearly distributed in the region of cos θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' miss−tag > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' (additional for the ℓω modes) P sig π0 , Elow γ , where MV 0 is the invariant mass of the vector meson, M2 ν is the missing mass squared, P c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' ν is the missing momentum in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' frame, T is the magnitude of the thrust vector [39, 40], P sig ℓ is the momentum of the lepton in the signal side, Ehemi tag is the energy sum of the tracks and photons in the tag hemisphere, P sig π0 is the momentum of π0 from ω and Elow γ is the lower energy of the two photons from the π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The variables of neutrino kinematics (M2 ν and P c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' ν ) were not used for the event selection in the previous paper [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' They are calculated from the momenta of the reconstructed τsig and τtag, where the energy of τsig is fixed to the half of the beam energy in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The q¯q continuum background events can be effectively suppressed by a M2 ν selection in the hadronic tags, involving only one neutrino (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The training, validation and evaluation of the BDT are done with 40%, 10%, and 50% of the signal MC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Regarding the training and validation samples for the background events, we utilize hadron background enhanced data that are obtained by removing the lepton identification for the signal-side leptons but with a lepton identification veto (P(e) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='9 and P(µ) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='95) for all the signal-side tracks in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The hadron background enhanced data have a much larger number of events than the background data with the nominal selection criteria, whereas both data sets are composed mainly of three charged pions from τ decays or from continuum events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The training is done with 80% of the hadron background enhanced data and the validation is done with 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' During BDT training, a weight is applied to each of the signal MC events such that the sum of the weights is equal to the number of background events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We monitor the area under – 5 – t→μpo, electron tag 70 BR(→μp0)=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2×10-8 × 100 Number of events/(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='01) 60 data 50 40 30 20 10 t+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='+- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='0 CosAc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' miss -tagcurve (AUC) of the Receiver Operating Characteristic curve [45] for the validation samples during the training and choose the BDT with the best AUC score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The event selection with the BDT output (BDT selection) is determined only by a target signal efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The target signal efficiency is determined based on the signal efficiency with a cut-based event selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' In the cut-based event selection, the MV 0 windows correspond to ±2σ of reconstructed mass distribution, and the M2 ν windows are set for each ℓV 0 mode and each τtag decay mode so that the expected number of background events inside the signal region (NBG, see the next section) is approximately one or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The target signal efficiency with the BDT selection is set as relatively 5% larger than that with the cut-based event selection, because we expect improvement in separating the signal events from the background events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The finalized BDT selection shows similar NBG to that of the cut-based event selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The BDT selection is not applied to the ℓφ modes because NBG in each of the two modes is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Figure 2: The M2 ν distribution of the τ → µρ0 mode with the hadronic tags after the event selection except for the requirement of the BDT output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Black points with error bars are the data outside the blind region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Red solid histogram is the signal MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The signal MC is scaled to the number of events corresponding to 100 times as large a branching fraction as the current upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The events constituting the upper tail of the signal distribution originate from wrong or missing π0 in the tag side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' 4 Signal efficiency and background estimation We define the signal region with an ellipse in the MℓV 0–∆E plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Most of the signal events cluster around MℓV 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='777 GeV/c2 and ∆E = 0 GeV with some correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The ellipse oblateness and the rotation angle are calculated from the covariance matrix of the signal MC distribition after the event selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The center of the ellipse is the mean of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The ellipse size is determined to maximize the figure-of-merit (FOM) [46], FOM = ε α 2 + √NBG , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1) – 6 – t→μpo, hadronic tag 300 BR(t→μp°)=l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2×10-8 × 100 250 data 200 150 100 50 2 1 0 1 2 M2 (GeV2/c4)where ε is the signal efficiency inside the ellipse, α is the confidence coefficient (α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='64 at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Figure 3: The MℓV 0 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' ∆E distribution of the τ → µρ0 hadron background enhanced samples: the data (upper side), the generic τ +τ − MC (lower left) and the q¯q continuum MC (lower right, q = u, d, s, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The range of the ∆E axis is limited to the fitting region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The MC sets are scaled to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The low-multiplicity background events are negligible for the hadron background enhanced samples and are not shown in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We estimate NBG through interpolation from the sideband data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Here the sideband data is a set of data passing the event selection and inside the sideband region: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='65 GeV/c2 < MℓV 0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='9 GeV/c2 and |∆E| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1 GeV outside of the blind region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The interpolation is based on a function in the MℓV 0–∆E plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' This function is obtained by fitting the distribution of the hadron background enhanced data within |∆E| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1 GeV, and then it is scaled to the sideband data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Figure 3 shows the distributions of the hadron background enhanced data and MC for the µρ0 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The function is: F(MℓV 0, ∆E) = f(MℓV 0) × 1 1 + exp[ay(∆E − y0)] + cflat 0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2) – 7 – data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='100 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='050 80 △E (GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='025 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='025 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='050 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='100 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='90 Mμpo (GeV/c2)t+t- MC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='075 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='050 △E (GeV) 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='025 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='90 Mμpo (GeV/c2)qq MC (q = u, d,s,c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='100 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='075 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='050 6 (GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='025 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='000 4 △E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='025 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='050 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='075 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='100 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='90 Mμpo (GeV/c2)f(x) = � � � � � � � � � � x+5σ x−5σ g(x′) × 1 √ 2πσexp �−(x − x′)2 2σ2 � dx′ (V 0 = ρ0, ω) c1(x − x0)2 + c0 (V 0 = K∗0, K∗0) c0 (V 0 = φ) g(x) = � � � � � c1[(x − x0)2 + k(x − x0)] + c0 (x < x0, V 0 = ρ0) c1(x − x0) + c0 (x < x0, V 0 = ω) c0 (x ≥ x0) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='3) where f(x) represents the background distribution as a function of MℓV 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' c1, c0, x0, and k are parameters that define the shape of the function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' ay represents sharpness of the sigmoid function along the ∆E axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' y0 is the center of the sigmoid function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' and cflat 0 is a term of flat background events in the MℓV 0–∆E plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We define f(x) for each V 0 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='3) and the functions for the ℓρ0 (ℓω) modes are smeared by a Gaussian with standard deviation (σ) of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='6 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='6) MeV/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' This σ corresponds to the mass resolution that affects the edge of the MℓV 0 distribution close to the τ mass for the τ +τ − background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The edge is broad for the other modes owing to wrong mass assignment of fake kaons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The τ +τ − background events for the ℓφ modes are included in c0 because they are flat along the MℓV 0 axis in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='65 GeV/c2 < MℓV 0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='9 GeV/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We obtain the optimal fit parameters by a likelihood fit using MINUIT [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The following region is excluded from the fitting to avoid D+ → K−π+π+ and D+ → π+φ background events, which cluster around the D meson mass: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='83(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='82) GeV/c2 ≤ MℓV 0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='89 GeV/c2 and ∆E < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='04 GeV for the µK∗0 (eK∗0) and µφ (eφ) modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The parameters of ay, y0, k, and x0 are fixed at the fit results of the hadron background enhanced data within |∆E| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The fit uncertainties of these fixed parameters are included in the systematic uncertainty of NBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The other fit parameters correspond to the scale factors of each background component: generic τ +τ − (c1), and continuum and low-multiplicity background events (c0 and cflat 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We fit the function floating these scale factors (c1, c0, and cflat 0 ) to the sideband data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The same region around the D meson mass as for the fit to the hadron background enhanced data is excluded from the fitting for the ℓφ and ℓK∗0 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The functions are integrated in the elliptical signal regions to deduce NBG, which are in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='25–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Another systematic uncertainty on NBG comes from difference of the MℓV 0–∆E dis- tributions between the sideband data and the hadron background enhanced data within |∆E| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The difference mainly arises from the electron (muon) identification fake rate, Rfake e(µ)(P, θ), which depends on the momentum P and θ of the track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The side- band data have a pion misidentified as a lepton, which tends to have a lower momen- tum than the pions in the hadron background enhanced data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We evaluate a change of NBG when the parameters—ay, y0, k, and x0—are redetermined with weighted hadron background enhanced data, where each event is weighted by the ratio of Rfake e(µ)(P, θ) to 1 − Rfake e (P, θ) − Rfake µ (P, θ) for the track in order to conform the MℓV 0–∆E distribution to the one of the sideband data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The amout of change of NBG is taken as the systematic uncertainty of NBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The statistical uncertainty of NBG is calculated as follows: We generate 100 sets of – 8 – pseudo-data for each mode in the MℓV 0–∆E histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The content of each bin in the histogram is set randomly following a Poisson distribution, with the mean taken from the function fitted to the sideband data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We fit the function to each set of the pseudo-data to deduce NBG, and the standard deviation of these NBG is taken as the statistical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The major contribution to NBG comes from the MℓV 0 flat term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2) (c0 and cflat 0 ), which corresponds to the continuum or low-multiplicity background events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The contribution of the generic τ +τ − background events, which depends on MℓV 0, is about one-third as large as the other background contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We cannot distinguish the back- ground components of the ℓφ modes through the fit to the data, because the generic τ +τ − background events are distributed evenly along the MℓV 0 axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The systematic uncertainties of the expected number of signal events are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The dominant uncertainties are from the particle identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The track and photon energy resolutions in the MC are corrected such that the mass resolution of the D(∗)+ meson matches between the data and MC, where D(∗)+ → K−π+π+(π0) is reconstructed with similar event selection criteria to the signal ones (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' |∆E| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='5 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The uncertainty of the data mass resolution propagates to the uncer- tainties of the corrected energy resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We generate two additional signal MC sets in which the track (photon) energy resolution is different by plus and minus its uncertainty, and take the half of the difference in the expected number of the signal events as the systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' All the uncertainties in Table 1 are summed in quadrature to yield the total systematic uncertainties shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Table 1: List of the systematic uncertainties of the expected number of signal events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The average number of tracks (particles) in the reconstructed τ +τ − events for each signal mode is represented as Ntrack(particle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' When the uncertainty is different mode by mode, we show the range of the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Source σsyst (%) Integrated luminosity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='4 ee → ττ(γ) cross section [48] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='3 B(φ → KK) and B(ω → πππ0) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='7 Trigger efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='9 Tracking efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='35 × Ntrack Electron identification efficiency 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='7 × Nelectron Muon identification efficiency 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='8 × Nmuon K and π identification efficiency 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='6 (ρ0), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='8 (φ) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1 (K∗0 and K∗0) π0 efficiency 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2 × Nπ0 Electron veto for hadrons 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='4–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2 MC statistics 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='3–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='5 Track energy resolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='3–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='3 Photon energy resolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='0–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='4 – 9 – 5 Results Figures 4 and 5 show the observed event distributions in the MℓV 0–∆E plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The observed number of events in the signal region (Nobs) has no excess over NBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We set 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' upper limits on the branching fractions based on a Bayesian method with the use of Markov Chain Monte Carlo [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The probability density function of the branching fraction (B(τ → ℓV 0)) is calculated assuming that Nobs follows a Poisson distribution function whose mean value is the expected number of events (Nexp), Nexp = L × 2σττB(τ → ℓV 0) × ε + NBG, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1) where L is the integrated luminosity (980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='4 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='7 fb−1), σττ is the cross section of τ-pair production that is calculated with KKMC [48] (the weighted average of σττ at all the beam energies is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='916 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='003 nb), and ε is the signal efficiency including the branching fraction of the V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We assume that these values (L, σττ, ε, and NBG) follow a Gaussian distribution with the width equal to the uncertainty of each value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The upper limits on B(τ → ℓV 0) are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The average of the limits is better than that of the previous results using 854 fb−1 [29] by 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' This is due to the additional 15% of integrated luminosity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' the addition of π±π∓π±ν and π±π0π0ν modes in τtag reconstruction, which increases the signal efficiency by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='6%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' and the event selection by multivariate analysis (BDT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The upper limit on B(τ → µρ0) is worse than that of the previous result, though the expected upper limit before unblinding is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' This is because we use the Bayesian limits instead of the Frequentist limits, which are negatively proportional to NBG when Nobs is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Table 2: The signal efficiency (ε), the expected number of background events (NBG), total systematic uncertainty of the expected number of signal events (σsyst), the number of observed events in the signal region (Nobs), and the observed 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' upper limits on the branching fraction (Bobs (10−8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Mode ε (%) NBG σsyst (%) Nobs Bobs (×10−8) τ − → µ−ρ0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='20(stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='11(syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='6 0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='7 τ − → e−ρ0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='80±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='27(stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='02(syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='4 1 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2 τ − → µ−φ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='15(stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='05(syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='8 0 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='3 τ − → e−φ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='21(stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='00(syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='5 0 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='0 τ − → µ−ω 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='32±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='23(stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='03(syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='8 0 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='9 τ − → e−ω 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='74±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='43(stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='01(syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='5 0 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='4 τ − → µ−K∗0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='25(stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='03(syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='3 0 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='9 τ − → e−K∗0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='54±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='21(stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='12(syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1 0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='9 τ − → µ−K∗0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='58±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='17(stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='06(syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='3 1 < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2 τ − → e−K∗0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='11(stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='01(syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='1 0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='7 – 10 – 6 Conclusion To conclude, we searched for lepton-flavor-violating τ decays into one lepton and one vector meson using the full 980 fb−1 of Belle data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' No statistically significant signal candidates are observed, and the 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' upper limits on the branching fraction are in the range of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='7– 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2) × 10−8 for τ → µV 0 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='7–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='4) × 10−8 for τ → eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The upper limits are improved by 30% on average from the previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We achieve these improvements both with the reconsideration of the event selection criteria and with the 126 fb−1 of additional data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Acknowledgments This work, based on data collected using the Belle detector, which was operated until June 2010, was supported by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan, the Japan Society for the Promotion of Science (JSPS), and the Tau-Lepton Physics Research Center of Nagoya University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' the Australian Re- search Council including grants DP180102629, DP170102389, DP170102204, DE220100462, DP150103061, FT130100303;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Austrian Federal Ministry of Education, Science and Re- search (FWF) and FWF Austrian Science Fund No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' P 31361-N36;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' the National Natural Science Foundation of China under Contracts No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' 11675166, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' 11705209;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' 11975076;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' 12135005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' 12175041;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' 12161141008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Key Research Program of Frontier Sci- ences, Chinese Academy of Sciences (CAS), Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' QYZDJ-SSW-SLH011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Project ZR2022JQ02 supported by Shandong Provincial Natural Science Foundation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' the Ministry of Education, Youth and Sports of the Czech Republic under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' LTT17020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' the Czech Science Foundation Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' 22-18469S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Horizon 2020 ERC Advanced Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' 884719 and ERC Starting Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' 947006 “InterLeptons” (European Union);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' the Carl Zeiss Foundation, the Deutsche Forschungsgemeinschaft, the Excellence Cluster Uni- verse, and the VolkswagenStiftung;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' the Department of Atomic Energy (Project Identi- fication No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' RTI 4002) and the Department of Science and Technology of India;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' the Istituto Nazionale di Fisica Nucleare of Italy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' National Research Foundation (NRF) of Korea Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' 2016R1D1A1B02012900, 2018R1A2B3003643, 2018R1A6A1A06024970, RS202200197659, 2019R1I1A3A01058933, 2021R1A6A1A03043957, 2021R1F1A1060423, 2021R1F1A1064008, 2021R1A4A2001897, 2022R1A2C1003993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Radiation Science Research Institute, Foreign Large-size Research Facility Application Supporting project, the Global Science Experimental Data Hub Center of the Korea Institute of Science and Technology Information and KREONET/GLORIAD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' the Polish Ministry of Science and Higher Ed- ucation and the National Science Center;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' the Ministry of Science and Higher Education of the Russian Federation, Agreement 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='W03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='0026, and the HSE University Basic Re- search Program, Moscow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' University of Tabuk research grants S-1440-0321, S-0256-1438, and S-0280-1439 (Saudi Arabia);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' the Slovenian Research Agency Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' J1-9124 and P1-0135;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Ikerbasque, Basque Foundation for Science, Spain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' the Swiss National Science Foundation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' the Ministry of Education and the Ministry of Science and Technology of Tai- wan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' and the United States Department of Energy and the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' – 11 – These acknowledgements are not to be interpreted as an endorsement of any statement made by any of our institutes, funding agencies, governments, or their representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' We thank the KEKB group for the excellent operation of the accelerator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' the KEK cryogenics group for the efficient operation of the solenoid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' and the KEK computer group and the Pacific Northwest National Laboratory (PNNL) Environmental Molecular Sciences Labora- tory (EMSL) computing group for strong computing support;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' and the National Institute of Informatics, and Science Information NETwork 6 (SINET6) for valuable network support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' – 12 – (a) τ → µρ0 (b) τ → µφ (c) τ → µω (d) τ → µK∗0 (e) τ → µK∗0 Figure 4: Observed event distributions of MℓV 0 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' ∆E after the τ → µV 0 event selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Black points are the data, blue squares show the signal MC distribution with an arbitrary normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The red ellipse line is the signal region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' The estimation of the number of background events is done using the data between the red horizontal lines except the blind region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' – 13 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='4 口 Data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2 △E (GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='90 Muk*0 (GeV/c2)0.' metadata={'source': 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Cirigliano and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Passemar, Model-discriminating power of lepton flavor violating τ decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' D 89 (2014) 095014 [1403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content='5781].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfQQZv/content/2301.03768v1.pdf'} +page_content=' [2] MEG collaboration, Search for the lepton flavour violating decay µ+ → e+γ with the full dataset of the MEG experiment, Eur.' metadata={'source': 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a/E9E1T4oBgHgl3EQf-gaE/content/tmp_files/2301.03570v1.pdf.txt b/E9E1T4oBgHgl3EQf-gaE/content/tmp_files/2301.03570v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3fa260e51fc21a53200a3c96084899675a6595a6 --- /dev/null +++ b/E9E1T4oBgHgl3EQf-gaE/content/tmp_files/2301.03570v1.pdf.txt @@ -0,0 +1,1433 @@ +Principal deuterium Hugoniot via Quantum Monte Carlo and ∆-learning +Giacomo Tenti,1, ∗ Andrea Tirelli,1, † Kousuke Nakano,1, 2, ‡ Michele Casula,3 and Sandro Sorella1, 4 +1International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy +2School of Information Science, JAIST, Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan +3Institut de Min´eralogie, de Physique des Mat´eriaux et de Cosmochimie (IMPMC), +Sorbonne Universit´e, CNRS UMR 7590, MNHN, 4 Place Jussieu, 75252 Paris, France +4Computational Materials Science Research Team, +RIKEN Center for Computational Science (R-CCS), Kobe, Hyogo 650-0047, Japan +(Dated: January 10, 2023) +We present a study of the principal deuterium Hugoniot for pressures up to 150 GPa, using +Machine Learning potentials (MLPs) trained with Quantum Monte Carlo (QMC) energies, forces +and pressures. In particular, we adopted a recently proposed workflow based on the combination +of Gaussian kernel regression and ∆-learning. +By fully taking advantage of this method, we +explicitly considered finite-temperature electrons in the dynamics, whose effects are highly relevant +for temperatures above 10 kK. The Hugoniot curve obtained by our MLPs shows an excellent +agreement with the most recent experiments, with an accuracy comparable to the best DFT +functionals. Our work demonstrates that QMC can be successfully combined with ∆-learning to +deploy reliable MLPs for complex extended systems across different thermodynamic conditions, by +keeping the QMC precision at the computational cost of a mean-field calculation. +Introduction − +The study of hydrogen under extreme +conditions has been a very active topic in condensed +matter physics. Hydrogen is the most abundant element +in the universe and the accurate knowledge of its phase +diagram at pressures of the order of hundreds of GPa is +extremely important for a variety of applications, such +as modelling the interior of stars and giant gas planets +[1–3], the inertial-confinement fusion [4], and the high- +Tc hydrogen-based superconductors [5, 6]. Nevertheless, +several properties of this system are still highly debated, +even at the qualitative level [7–10]. +One of the main reasons that hamper our full +understanding of high-pressure hydrogen is the difficulty +of reproducing extreme pressures in a laboratory. Typical +shock-wave experiments [11] make use of accelerated flyer +plates to compress a material sample in a very short time, +thus allowing to study the specimen at high temperatures +and pressures. +In particular, the set of possible end- +states that the system can reach from some given initial +conditions, also named principal Hugoniot, must satisfy +a set of equations, known as Rankine-Hugoniot (RH) +relations [12], linking the thermodynamic properties of +the final shocked state with those of the starting one. +During the years, the principal deuterium Hugoniot has +been measured for a wide range of pressures and with a +great degree of accuracy [13–20], reaching a relative error +on the density as small as 2% in recent experiments. +In this context, numerical approaches, - in particular +Ab Initio Molecular Dynamics (AIMD) simulations -, +are extremely valuable, since they are not constrained +by any experimental setup and can thus give further +insight into this part of the phase diagram [21]. +The +Hugoniot region is particularly important because of +the availability of experimental data that can be used +to benchmark different theoretical methods. +Among +them, Density Functional Theory (DFT) simulations +have been extensively used and provided excellent +results for the Hugoniot curve [22–28]. +However, +the approximations behind the particular exchange- +correlation functional often produce discrepancies across +existing DFT schemes whose accuracy varies according to +the thermodynamic conditions, making the functional- +based approach unsatisfactory. Quantum Monte Carlo +(QMC) simulations, which depend on more controllable +approximations, +have also been performed [29, 30]. +Although in principle more accurate and systematically +improvable, +these calculations have a much larger +computational cost than DFT, and they are thus +limited in system size and simulation length. Moreover, +previous QMC calculations seem to give results for +the principal Hugoniot in disagreement with the most +recent experimental data, with the possible origin of this +discrepancy being recently debated [31]. +To overcome the large computational cost of ab +initio simulations, machine learning techniques, aimed +at constructing accurate potential energy surfaces, have +become increasingly popular. Within this approach, one +uses a dataset of configurations, i.e. +the training set, +to build a machine learning potential (MLP) that is +able to reproduce energies and forces calculated with +the given target method [32]. +Unlike DFT MLPs, +the QMC ones are relatively less common, given the +larger computational cost and the consequent difficulty of +generating large datasets, usually necessary to construct +accurate MLPs. +In this work, we have successfully built a very accurate +MLP with QMC energies, forces and pressures in the +region of the deuterium Hugoniot, using the so-called ∆- +learning approach. The Hugoniot curve computed by the +MLP shows an excellent agreement with the most recent +arXiv:2301.03570v1 [cond-mat.str-el] 9 Jan 2023 + +2 +experiments, and it shares with the best DFT functionals +the same, - if not better -, accuracy. +Computational details − +In order to build an MLP +with QMC references, we employed a combination of +Gaussian Kernel Regression (GKR), Smooth Overlap +of Atomic Positions (SOAP) descriptors [33], and ∆- +learning. The same approach has been recently proposed +in Ref. 34, where it was applied to the study of high- +pressure hydrogen in similar thermodynamic conditions. +Following the ∆-learning approach, an MLP is trained +on the difference between the target method and a +usually much cheaper baseline potential. +Here, we +trained 5 different MLPs, +using Variational Monte +Carlo (VMC) and Lattice Regularized Diffusion Monte +Carlo (LRDMC) [35, 36] datapoints as targets, and +several DFT baselines, with the Perdew-Zunger Local +Density Approximation (PZ-LDA) [37], the Perdew- +Burke-Ernzerhof (PBE) [38] and the van der Waals +(vdW) -DF [39, 40] functionals. The QMC calculations +were performed using the TurboRVB package [41]. +To determine the principal Hugoniot, we made use of +the RH jump equation: +H(ρ, T) = e(ρ, T) − e0 + 1 +2(ρ−1 − ρ−1 +0 ) [p(ρ, T) + p0] = 0, +(1) +where ρ, +T, +e(ρ, T), +p(ρ, T) and ρ0, +T0, +e0, +p0 +are the density, temperature, energy per particle and +pressure of the final and initial states, respectively. In +particular, we ran a first set of NV T simulations at +several temperatures in the [4 kK : 10 kK] range, and +Wigner-Seitz radii between 1.80 Bohr and 2.28 Bohr, +corresponding to the range where the zero of H(ρ, T) was +expected. These simulations were performed considering +classical nuclei and ground-state electrons, as quantum +corrections and thermal effects have been shown to be +negligible for these temperatures [30]. At each step, the +energy, forces and pressure were calculated using the +Quantum Espresso package in its GPU accelerated +version [42–44] with the chosen functional (PBE in most +cases), and then corrected with our MLP trained on the +difference between QMC and DFT data. The resulting +dynamics has the same efficiency as a standard DFT +AIMD simulation, which is roughly 100 times faster +than the original QMC one. +The details of our QMC +simulations are reported in the Supplemental Material +(SM) [45]. +For the DFT simulations, we considered a +60 Ry plane-wave cutoff with a Projector Augmented +Wave (PAW) pseudopotential [46] and a 4 × 4 × 4 +Monkhorst-Pack k-point grid, while for the dynamics we +used a time step of 0.25 fs and a Langevin thermostat [47, +48] with damping γ = 13 ps−1. For each temperature, the +Hugoniot (ρ∗, p∗) coordinates are determined by fitting +the Hugoniot function H(ρ, T) and the pressure p(ρ, T) +with a spline function, and by numerically finding ρ∗ and +the corresponding p∗. +Within our approach, we can fully take advantage +of the ∆-learning method by estimating the effect +of thermalized electrons in our calculations. +To do +so, we considered two MLPs trained on the VMC- +LDA and LRDMC-LDA differences, respectively, and +ran simulations at temperatures T = 10 kK, 15 kK, +and 35 kK with the corrected Karasiev-Sjostrom-Dufty- +Trickey (KSDT) finite-temperature (FT) LDA functional +[49–51] as baseline, in place of the usual ground-state PZ- +LDA functional. In this way, we can include the effects of +thermally excited electrons in our MLP without changing +it, at least at the DFT level of theory. +Results and Discussion − +Fig. 1a shows our results +together with several experimental values for pressures +below 150 GPa [16, 19, 20]. We also report the principal +Hugoniot obtained by directly using the PBE baseline, +and the Coupled Electron Ion Monte Carlo (CEIMC) +results of Ref. 30 for comparison. For T = 10 kK we +show both the ground-state and FT results obtained +with the procedure described previously, while for larger +temperatures we plotted only the latter. Both the VMC +and LRDMC models seem to reproduce very accurately +the experimental points over the entire range of pressure +considered. +With respect to the most accurate data +of Ref. 19, our estimate of the relative density ρ/ρ0 at +the compressibility peak is only 1% lower for the VMC +model and 3% lower for the LRDMC model, both being +compatible within one error bar. +Our results are in +better agreement with experiments than the CEIMC ones +reported in Ref. 30, which predicts a relative density 10% +larger for the Hugoniot curve. The disagreement between +the two results seems to be due to a large difference in the +pressure estimates between the two methods, as further +discussed in the SM [45]. +Fig. 1b displays the same points in the up − Us space, +where up is the particle velocity and Us is the shock +velocity, the two being calculated using the following RH +relations: +up = +� +(p + p0)(ρ−1 +0 +− ρ−1), +Us = ρ−1 +0 +� +p + p0 +ρ−1 +0 +− ρ−1 . +The difference ∆Us between these points and the linear +fit on the gas-gun data re-analyzed in Ref. 19 is also +shown (bottom panel of Fig. 1b). Notice that the drop +in the slope of Us relative to up coincides with the +onset of the molecular-atomic (MA) transition, while the +magnitude of the ∆Us minimum relates to the position +of the relative compression peak. +In particular, the +PBE Hugoniot curve manifests a premature start of the +dissociation, while it predicts correctly the magnitude of +the compressibility maximum. +Remarkably, our QMC +results are very similar to the experimental findings not + +3 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +/ +0 +0 +20 +40 +60 +80 +100 +120 +140 +160 +Pressure (GPa) +PBE-FT +VdW-DF1 (Ref.19) +Experiments +VMC +VMC-FT +LRDMC +LRDMC-FT +VMC (Ref. 30) +RMC (Ref. 30) +(a) +15 +20 +25 +30 +35 +Shock velocity Us (km/s) +10 +15 +20 +25 +Particle velocity up (km/s) +0.0 +0.5 +1.0 +1.5 +Us (km/s) +(b) +FIG. 1: (1a) Principal Hugoniot in the density-pressure space. Red and yellow circles are the results obtained with our +MLPs trained on VMC and LRDMC datapoints, respectively, and a PBE baseline. Empty symbols refer to the results +obtained using the finite-temperature (FT) KSDT functional as baseline. Blue and pink triangles are the PBE result +calculated in this work and the VdW-DF1 result of Ref. 19 respectively. CEIMC results of Ref. 30 based on Variational +Monte Carlo (VMC) and Reptation Monte Carlo (RMC) are reported in green squares. Cyan diamonds are the experimental +results of Refs. 16, 19, and 20. Dashed-dotted lines are guides for the eye. (1b) [top panel] Hugoniot in the up–Us space. +Black-dashed line is the re-analyzed gas-gun fit reported in Ref. 19. [bottom panel] Relative shock velocity with respect to the +gas-gun fit. Only the experimental points of Ref. 19 are reported. +only for the compressibility peak but also for the shock +velocity slope. +Thus, the Hugoniot curve obtained by our MLPs +shows a much better agreement with the most recent +experiments than the PBE functional, and is close to +improved functionals, such as VdW-DF1 reported in Fig. +1, which has been proved more accurate than PBE for +high pressure hydrogen [52]. Cancellation of errors taking +place in the DFT Hugoniot [31] is less apparent in the +∆Us = ∆Us(up) relation (Fig. 1b), where the difference +between PBE and improved theories is clear. +The presence of an MA transition is also investigated +in Fig. 2, +where we report the radial distribution +function, g(r), calculated on trajectories obtained with +the LRDMC model for several temperatures at densities +close to the Hugoniot curve. The inset of Fig. 2 displays +the value of the molecular fraction m, defined as the +percentage of atoms that stay within a distance of 2 Bohr +(roughly corresponding to the first g(r) minimum after +the molecular peak) from another particle for longerthan +a characteristic time, here set to 6 fs. The results show +a distinct atomic character for T ≥ 10 kK and a clear +molecular peak at lower temperatures. +Error analysis − +To +assess +the +quality +of +our +principal Hugoniot determination, +we analyzed the +possible sources of errors in relation to our machine +learning scheme. There are three main sources of errors: +the uncertainties in the fit of H(ρ, T), the prediction error +of the MLP, and the uncertainties in the reference state +energy estimate, i.e. e0 in Eq.(1). We verified that, in our +case, the error produced by the fit is negligible compared +to the other two sources, which we will discuss next. +As mentioned before, we followed Ref. 34 to construct +our MLPs and used a GKR model based on a modified +version of the SOAP kernel [33]. +Our final dataset, +including both training and test sets, comprises 871 +configurations selected through an iterative procedure +with 128 hydrogen atoms each, where we calculated +energies, pressures and forces at the VMC and LRDMC +levels. These configurations correspond to temperatures +from 4 kK up to 35 kK and Wigner-Seitz radii from 1.80 +Bohr to 2.12 Bohr. Finite size corrections have also been +estimated using the KZK functional [53]. +Details on the training set construction and the QMC +calculations, together with the performances of all MLP +models can be found in the SM [45]. In particular we +found a final root mean square error, calculated on the +test set, of the order of 20 meV/atom for the energy, 130 +meV/˚A for the forces, and 0.1 GPa for the pressures. +At this point, it is worth to highlight some favourable + +4 +0 +1 +2 +3 +4 +5 +6 +7 +8 + r (Bohr) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 + g(r) +T = 4 kK +T = 6 kK +T = 7 kK +T = 8 kK +T = 10 kK +T = 15 kK +T = 35 kK +3.0 +3.5 +4.0 +4.5 +/ +0 +25 +50 +75 +100 +125 +150 +Pressure (GPa) +0.92 +0.58 0.43 +0.34 +0.21 +0.12 +0.03 +FIG. 2: +g(r) for several temperatures and densities close to +the principal Hugoniot, obtained using the LRDMC model. +The molecular fraction value, m, is reported in the inset, +beside each point distributed according to their +corresponding location in the density-pressure space. +features of our machine learning approach, especially in +applications where it is coupled with computationally +expensive methods such as QMC. They can be itemized +as follows: +• transferability: the total energy of the system is +expressed as a sum of local terms [32], therefore our +models are capable of making accurate predictions +on +configurations +whose +size +has +never +been +encountered in the training set. In particular, our +MLPs find their applicability to systems with an +arbitrary number of atoms N. +• efficiency and accuracy: +within the ∆-learning +framework, the machine learning task becomes +easier. +Indeed, we obtained very accurate QMC +potentials, by training models on small datasets +and, thus, by reducing the amount of calculations +needed. +Moreover, since the computational cost +of the ML inference is negligible compared to the +baseline DFT calculation, we were able to perform +QMC-driven MD simulations at the cost of a DFT +dynamics. +• overfitting prevention: using a local sparsification +technique based on the farthest point sampling +(see SM of Ref. 34), we discarded from each +configuration a possibly large fraction of the +corresponding N local environments, preventing +overfitting and allowing for an increased predictive +power of the model on unseen data. +Since +the computational cost of the predictions scales +with the size of the training set, this procedure +drastically improves the efficiency of the final +model. +We further validated the accuracy of our MLP models +by comparing the Hugoniot curve obtained using three +potentials, independently trained with the same target, +e.g. VMC, but with different baselines. In particular, +we found the results to be consistent within an error of +≲ 1% and ≲ 2% for density and pressure, respectively. +We now turn to the last source of error we identified, +i.e. +the one related to the calculation of e0 and p0. +To estimate the reference state energy and pressure, we +followed a procedure similar to Ref. 30. We performed a +path integral molecular dynamics (PIMD) simulation [54] +on a system of N = 64 deuterium atoms at a temperature +T = 22 K and density ρ0 = 0.167 g/cm3 (corresponding +to the initial conditions reported in Ref. 19), using DFT- +PBE energy and forces. Details of this simulation are +reported in the SM [45]. From the PIMD trajectory, we +extracted 170 configurations and we calculated energies +and pressures with both DFT-PBE and QMC at VMC +and LRDMC levels, adding the necessary finite size +corrections. +The reference sample was generated by +extracting atomic positions from one of the 128 beads +taken at random, belonging to de-correlated snapshots +of the trajectory. Results for e0 for the various methods +are reported in Tab. I. The reference state pressure p0 is +not reported, since it is two orders of magnitude smaller +than the shocked pressure, and thus irrelevant for the +Hugoniot determination. Also in this case, we studied +the effect of varying e0 within its confidence interval on +the Hugoniot density and pressure. Its variability within +standard deviation leads to shifts in the final principal +Hugoniot which fall in the stochastic error range of our +predictions. +To summarize, we estimated the MLP prediction error +to be the most relevant source of uncertainty for the +Hugoniot, yielding, as discussed before, an error of 1% +and 2% on the relative density and pressure, respectively, +reflected on the error bars reported in Fig. 1. Notice that +our Hugoniot curve is consistent with the experiments +even after considering the possible uncertainties. +epot (Ha/atom) e0 (Ha/atom) +PBE +-0.58217(2) +-0.58055(2) +VMC +-0.58465(3) +-0.58303(3) +LRDMC +-0.58653(2) +-0.58491(2) +TABLE I: Estimated potential (epot) and total (e0) +energies per atom of the reference state at ρ0 = 0.167 +g/cm3 and T = 22 K for different methods. + +5 +Conclusions − +In conclusion, +using our recently +proposed workflow for the construction of MLPs, we have +been able to run reliable VMC- and LRDMC-based MD +simulations and study the principal deuterium Hugoniot, +in a pressure range relevant for experiments. +The +accuracy of the MLPs employed here has been extensively +tested, supporting the validity of our calculations. The +resulting Hugoniot curve shows an excellent agreement +with the most recent measures, comparable to the best +DFT functionals and better than previous QMC results. +Moreover, within the ∆-learning framework, we have +also been able to treat FT electrons effects in a QMC- +MLP, and we have thus managed to perform accurate +simulations at higher temperatures. +The efficiency +of this approach could be further improved, e.g., by +using cheaper baseline potentials than DFT. Longer +simulations and larger systems will then be at reach. +Other many-body methods, even more expensive than +QMC, can also be used as targets for this type of +MLPs, since the required size of the dataset is at +least one order of magnitude smaller compared to other +ML approaches. +Finally, our MLPs, and in particular +those trained on LRDMC datapoints, are promising for +exploring the hydrogen phase diagram by keeping a high +level of accuracy across a wide range of thermodynamic +conditions. +Data availability − +The machine learning code used +in this work is available upon request. +Additional +information, such as datasets and detailed results of +the simulations are available at https://github.com/ +giacomotenti/QMC_hugoniot. +Acknowledgments. The computations in this work have +mainly been performed using the Fugaku supercomputer +provided by RIKEN through the HPCI System Research +Project (Project ID: hp210038 and hp220060) and +Marconi100 provided by CINECA through the ISCRA +project No. HP10BGJH1X and the SISSA three-year +agreement 2022. K.N. is also grateful for computational +resources from the facilities of Research Center for +Advanced Computing Infrastructure at Japan Advanced +Institute of Science and Technology (JAIST). +A.T. acknowledges financial support from the MIUR +Progetti di Ricerca di Rilevante Interesse Nazionale +(PRIN) +Bando +2017 +- +grant +2017BZPKSZ. +K.N. +acknowledges +a +support +from +the +JSPS +Overseas +Research Fellowships, that from Grant-in-Aid for Early- +Career Scientists Grant Number JP21K17752, and that +from Grant-in-Aid for Scientific Research(C) Grant +Number JP21K03400. +This work is supported by the +European Centre of Excellence in Exascale Computing +TREX - Targeting Real Chemical Accuracy at the +Exascale. +This project has received funding from +the European Union’s Horizon 2020 - Research and +Innovation program - under grant agreement no. 952165. +We dedicate this paper to the memory of Prof. Sandro +Sorella (SISSA), who tragically passed away during +this project, remembering him as one of the most +influential contributors to the quantum Monte Carlo +community,and in particular for deeply inspiring this +work with the development of the ab initio QMC code, +TurboRVB. +∗ gtenti@sissa.it +† atirelli@sissa.it +‡ kousuke 1123@icloud.com +[1] D. 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Rev. 137, A1441 (1965). + +Supplemental material: Principal deuterium Hugoniot via Quantum Monte +Carlo and ∆-Learning +Giacomo Tenti∗ and Andrea Tirelli† +International School for Advanced Studies (SISSA), +Via Bonomea 265, 34136 Trieste, Italy +Kousuke Nakano‡ +International School for Advanced Studies (SISSA), +Via Bonomea 265, 34136 Trieste, Italy and +School of Information Science, JAIST, +Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan +Michele Casula +Institut de Min´eralogie, de Physique des Mat´eriaux et de Cosmochimie (IMPMC), +Sorbonne Universit´e, CNRS UMR 7590, +MNHN, 4 Place Jussieu, 75252 Paris, France +Sandro Sorella +International School for Advanced Studies (SISSA), +Via Bonomea 265, 34136 Trieste, Italy and +Computational Materials Science Research Team, +RIKEN Center for Computational Science (R-CCS), Kobe, Hyogo 650-0047, Japan +(Dated: January 10, 2023) +1 +arXiv:2301.03570v1 [cond-mat.str-el] 9 Jan 2023 + +I. +COMPUTATIONAL DETAILS OF QMC CALCULATIONS +The Variational Monte Carlo (VMC) and lattice regularized diffusion Monte Carlo (LRDMC) [1] +calculations in this study were performed by TurboRVB package [2]. The package employs a +many-body WF ansatz Ψ which can be written as the product of two terms, i.e., Ψ = ΦAS × exp J , +where the term exp J and ΦAS are conventionally called Jastrow and antisymmetric parts, re- +spectively. The antisymmetric part is denoted as the Antisymmetrized Geminal Power (AGP) +that reads: +ΨAGP (r1, . . . , rN) = +ˆA +� +Φ +� +r↑ +1, r↓ +1 +� +Φ +� +r↑ +2, r↓ +2 +� +· · · Φ +� +r↑ +N/2, r↓ +N/2 +�� +, where ˆA is the an- +tisymmetrization operator, and Φ +� +r↑, r↓� +is called the paring function [3]. +The spatial part +of the geminal function is expanded over the Gaussian-type atomic orbitals: ΦAGP +� +ri, rj +� += +� +l,m,a,b f{a,l},{b,m}ψa,l (ri) ψb,m +� +r j +� +where ψa,l and ψb,m are primitive Gaussian atomic orbitals, their +indices l and m indicate different orbitals centered on atoms a and b, and i and j are coordi- +nates of spin up and down electrons, respectively, and f{a,l},{b,m} are the variational parameters. In +this study, a basis set composed of [4s2p1d] Gaussian atomic orbitals (GTOs) was employed +for the atomic orbitals of the antisymmetric part. +The pairing function can be also written +as ΦAGPn +� +ri, r j +� += �M +k=1 λkφk(ri)φk(rj) with λk > 0, where φk(r) is a molecular orbital, i.e., +φk(r) = �L +i=1 ci,kψi(r). When the paring function is expanded over M molecular orbitals where +M is equal to half of the total number of electrons (N/2), the AGP coincides with the Slater- +Determinant ansatz. In this study, we restricted ourselves to a Jastrow-Slater determinant (JSD) by +setting M = 1 +2 ·N, wherein the coefficients of atomic orbitals, i.e., ci,k, were obtained by the build-in +Density Functional theory (DFT) package (prep), and were fixed during a VMC optimization. +The Jastrow term is composed of one-body, two-body and three/four-body factors (J = J1 + +J2 + J3/4). The one-body and two-body factors are essentially used to fulfill the electron-ion and +electron-electron cusp conditions, respectively, and the three/four-body factor is employed to con- +sider further electron-electron correlations (e.g., electron-nucleus-electron). The one-body Jastrow +is decomposed into the so-called homogeneous and inhomogeneous parts, i.e., J1 = Jhom +1 ++ Jinh +1 . +The homogeneous one-body Jastrow factor is J1 +hom (r1, . . . , rN) = � +i,I +� +−(2ZI)3/4u +� +2ZI +1/4 |ri − RI| +�� +where ri are the electron positions, RI are the atomic positions with corresponding atomic number +ZI, and u (r) is a short-range function containing a variational parameter b: u (r) = b +2 +� +1 − e−r/b� +. +The inhomogeneous one-body Jastrow factor Jinh +1 +is represented as: +∗ gtenti@sissa.it +† atirelli@sissa.it +‡ knakano@sissa.it +2 + +Jinh +1 (r1, . . . , rN) = �N +i=1 +�Natom +a=1 +�� +l Ma,lχa,l (ri) +� +, where ri are the electron positions, Ra are the +atomic positions with corresponding atomic number Za, l runs over atomic orbitals χa,l (e.g., GTO) +centered on the atom a, Natom is the total number of atoms in a system, and {Ma,l} are variational +parameters. The two-body Jastrow factor is defined as: J2 (r1, . . . rN) = exp +�� +i 0 is the maximum +transmit power. The scaled signal received at BS m is +¯gm = +1 +√ηm +rH +myul +m = +1 +√ηm +rH +m +� +k∈K +¯hm,kbksul +k + rH +mnul +m +√ηm +, (8) +where rm ∈ CN is the receive beamforming vector and ηm +is a normalizing factor for cell m. To compensate for the +phase distortion introduced by complex channel responses, +the transmit scalar at device k in cell m is set to bk = +√ηm +(rH +m¯hm,k)H +|rHm¯hm,k|2 , ∀k ∈ Km, and ηm can be expressed as +ηm = P ul mink∈Km |rH +m¯hm,k|2. Then the estimated function +at BS for cell m is given as +ˆgm = ℜ{¯gm} += ℜ{gm + +1 +√ηm +rH +m +� +l̸=m +� +j∈Kl +¯hm,jbjsul +j + rH +mnul +m +√ηm +� +�� +� +eulm +} += gm + ℜ{eul +m} +(9) +3) Downlink transmission: After obtaining the estimate +ˆgm in the cell m, BS m computes G(ˆgm; y) with noisy +aggregation ˆgm, and then broadcasts the result to the associ- +ated devices in Km. And we write G(ˆgm; y) in terms of Gm +for simplify. Without loss of generality, we assume that the +transmitted signal follows the standard Gaussian distribution, +i.e., Gm ∼ CN(0, 1). The received signal at device k is +ydl +k = +� +m +¯hH +m,ktmGm + ndl +k , +(10) +where tm denotes the transmit beamforming vector at BS +m, and ndl +k ∼ CN +� +0, (σdl)2� +is the additive white Gaus- +sian noise with zero mean and variance (σdl)2 at device +k. The maximum transmit power at BS m is P dl, i.e., +E(∥Gmtm∥2) = ∥tm∥2 ≤ P dl. +To compensate for the phase distortion introduced by +complex channel responses, the receive scalar at device k +in cell m is set to rk = +(¯hH +m,ktm)H +|¯hH +m,ktm| +2 . The estimated Gm at +device k is given as +ˆGm,k = ℜ{rkydl +k } += ℜ{Gm + +(¯hH +m,ktm)H +���¯hH +m,ktm +��� +2 +� +�� +l̸=m +¯hH +l,ktlGl + ndl +k +� +� +� +�� +� +¯edl +k +} += Gm + ℜ{¯edl +k }. +(11) +Note that the uplink noise is embedded in function Gm. In +order to directly describe the effective noise, we expand Gm +to its first-order Taylor expansion as follows +ˆGm,k += Gm + ℜ{¯edl +k } += G(gm + ℜ{eul +m}; y) + ℜ{¯edl +k } += G(gm; y) + G +′(gm; y)ℜ{eul +m} + O(|ℜ{eul +m}|2) + ℜ{¯edl +k } +≈ G(gm; y) + G +′(gm; y)ℜ{eul +m} + ℜ{¯edl +k } +� +�� +� +edl +k +, +(12) +where G′(·) is the first derivative of G(·). Assume that the +noise amplitude is small, the term O(|ℜ{eul +m}|2) is neglected, +which implies the last approximation in (12). +III. CONVERGENCE ANALYSIS AND PROBLEM +FORMULATION +A. Convergence Analysis +In previous work [12], [14], [15], the convergence analysis +of the AirComp-based vertical FL process in each cell has +been established under the following assumptions. +Assumption 1 (α-strongly convexity). The function F(·) is +assumed to be α-strongly convex on Rd with constant α, +namely, for all x, y ∈ Rd, we have +F(y) ≥ F(x) + ∇F(x)T(y − x) + α +2 ∥y − x∥2 +2. + +Assumption 2 (β-smoothness). The function F(·) is assumed +to be β-smooth on Rd with constant β, namely, for all x, y ∈ +Rd, we have +F(y) ≤ F(x) + ∇F(x)T(y − x) + β +2 ∥y − x∥2 +2. +Theorem 1 (Convergence of vertical FL process). Suppose +that Assumption 1 and 2 hold, setting the learning rate to +be 0 < µ(t) ≤ +1 +β , then the expected optimality gap after T +communication rounds is upper bounded by +E +� +F(w(T ) +m ) − F(w∗ +m) +� +≤ ρT E +� +F(w(0) +m ) − F(w∗ +m) +� ++ +1 +2βL2 +T −1 +� +t=0 +ρT −t−1 � +k∈Km +� +Φ1,kE[|ℜ{eul +m}|2] + Φ2,kE[|ℜ{¯edl +k }|2] +� +, +(13) +where ρ = 1−α/β, Φ1,k = �L +i=1 ∥(Gi +m,k) +′xi +k∥2 +2 and Φ2,k = +�L +i=1 ∥xi +k∥2 +2. +Proof. Please refer to previous work [12]. +B. Problem Formulation +According to Theorem 1, the convergence optimality gap +is largely determined by the mean-squared-error (MSE) of +both gm and Gm. However, solely optimizing MSE for each +cell through AirComp may result in significant inter-cell +interference in the considered multi-cell wireless networks, +which can negatively impact the learning performance of +other cells. As such, it is necessary to carefully balance the +learning performance among various FL tasks in multiple +cells through a cooperative design. +We begin by identifying the gap region G, to be the set of +tuples (∆1, ∆2, . . . , ∆M), which represents the instantaneous +errors that cause gaps in all cells, and can be achieved simul- +taneously under specific downlink and uplink transmission +power constraints. The gap region G can be represented as +G = +� +{(∆1, ∆2, . . . , ∆M)|∆m ≥ Gapm, ∀m ∈ M}, (14) +where +Gapm = +� +k∈Km +� +Φ1,kE[|ℜ{eul +m}|2] + Φ2,kE[|ℜ{¯edl +k }|2] +� +, +(15) +E[|ℜ{eul +m}|2] = +� +l̸=m,j∈Kl +ηl|rH +m¯hm,j|2 +ηm|rH +l ¯hl,j|2 + ∥rm∥2σ2 +ul +ηm +, +E[|ℜ{¯edl +k }|2] = +� +l̸=m |¯hH +l,ktl|2 + (σdl)2 +���¯hH +m,ktm +��� +2 +. +(16) +As previously stated, in order to decrease the error-induced +gap in one cell, the gaps of other cells maybe increased. +In light of this, our objective is to find a suitable solution +that allows us to achieve the Pareto boundary of the gap +region G, so as to balance the performance of learning among +multiple cells. In this context, the Pareto optimality of a tuple +is described as follows [16]. +Here, we leverage the profiling technique [17] to char- +acterize the Pareto boundary by coordinating all BSs to +minimize the sum of Gap of all cells. Specifically, let +κ = [κ1, κ2, . . . , κM] denote a given profiling vector, which +satisfies κm ≥ 0, ∀m ∈ M, and � +m∈M κm = 1. The gap +tuple on Pareto boundary can be obtained by solving the +following problem +minimize +ζ,{rm},{tm},Θt,Θr +ζ +(17a) +s.t. +Gapm ≤ κmζ, ∀m ∈ M +(17b) +ζ ≥ 0, +(17c) +where ζ denotes the sum of the gaps of all cells. Thus, +the gap tuple can be represented as (∆1, ∆2, . . . , ∆M) = +(κ1ζ, κ2ζ, . . . , κMζ), where a smaller value of κm implies a +more stringent requirement for the gap of cell m. +Denote ζ = ζul +ζdl, where ζul and ζdl are used to quan- +tify the sum of instantaneous error-induced gaps generated by +uplink and downlink transmissions, respectively. Hence, we +rewrite problem (17) as +minimize +ζul,ζdl,{rm},{tm},Θt,Θr +ζul + ζdl +(18a) +s.t. +Gapul +m ≤ κmζul, ∀m ∈ M +(18b) +Gapdl +m ≤ κmζdl, ∀m ∈ M +(18c) +ζul ≥ 0 +(18d) +ζdl ≥ 0. +(18e) +The downlink and uplink transmissions can be decoupled in +problem (18), which allows us to separately optimize the +downlink and uplink transmission resources. +IV. OPTIMIZATION FRAMEWORK +In this section, we specify the optimization framework +for solving the uplink and downlink optimization problems, +respectively. +A. Uplink Optimization +For the uplink aggregation, the optimization problem is +minimize +ζul,{rm},Θul +ζul +(19a) +s.t. +� +l̸=m +� +j∈Kl +ηl|rH +m¯hm,j|2 +ηm|rH +l ¯hl,j|2 +(19b) ++ ∥rm∥2(σul)2 +ηm +≤ κmζul, ∀m ∈ M +(19c) +ζul ≥ 0. +(19d) +By setting optimzing varibales qi = ri/√ηi, ∀i ∈ M, the +problem can be converted to +minimize +ζul,{qm},Θul +ζul +(20a) +s.t. +� +l̸=m +� +j∈Kl +|qH +m¯hm,j|2 +|qH +l ¯hl,j|2 ++ (σul)2∥qH +m∥2 ≤ κmζul, ∀m ∈ M +(20b) +|qH +m¯hm,k|2 ≥ 1 +Pul +, ∀m, ∀k ∈ Km +(20c) +(19d). + +Then we let +|qH +m¯hm,j|2 +|qH +l ¯hl,j|2 +≤ bl,j, the optimization problem +relaxes to +minimize +ζul,{qm,b},Θul +ζul +(21a) +s.t. +� +l̸=m +� +j∈Kl +bl,j + (σul)2∥qH +m∥2 ≤ κmζul, ∀m +(21b) +|qH +m¯hm,j|2 +|qH +l ¯hl,j|2 ≤ bl,j, ∀l, j +(21c) +(19d), (20c). +However, constraint (21c) is still non-convex, then we use +the SCA method to transform (21c) into a linear con- +straint which satisfies the property of convex. Let al,j = +[ℜ(qH +l ¯hl,j), ℑ(qH +l ¯hl,j)], the corresponding approximated lin- +ear constraint is +|qH +m¯hm,j|2 +bl,j +≤ ∥al,j∥2 +≤ ∥a(t) +l,j ∥2 + 2(a(t) +l,j )T(al,j − a(t) +l,j ) +(22) +and +∥a(t) +m,k∥2 + 2(a(t) +m,k)T(am,k − a(t) +m,k) ≥ 1 +Pul +. +(23) +The origin problem (21) is then approximated as +minimize +ζul,{qm,b,a},Θul +ζul +s.t. +al,j = [ℜ(qH +l ¯hl,j), ℑ(qH +l ¯hl,j)], ∀l, j +(19d), (21b), (22), (23). +(24) +And we can observe that the above problem turns out to be +highly intractable due to the non-convexity of multiplication +between variables q and Θul. Hence, a classical alternative +optimization algorithm can be used to solve it. +B. Downlink Optimization +For the downlink dissemination, the optimization problem +can be written as +minimize +ζdl,{tm},Θdl +ζdl +(25a) +s.t. +� +k∈Km +� +l̸=m |¯hH +l,ktl|2 + (σdl)2 +���¯hH +m,ktm +��� +2 +≤ κmζdl, ∀m +(25b) +∥tm∥2 ≤ P dl, +(25c) +ζdl ≥ 0. +(25d) +By letting +� +l̸=m |¯hH +l,ktl|2+(σdl)2 +|¯hH +m,ktm| +2 +≤ dk, the optimization prob- +lem is relaxed to +minimize +ζdl,{tm},Θdl +ζdl +(26a) +s.t. +� +k∈Km +dk ≤ κmζdl, ∀m ∈ M +(26b) +� +l̸=m |¯hH +l,ktl|2 + (σdl)2 +���¯hH +m,ktm +��� +2 +≤ dk, +(26c) +(25c), (25d). +Similar as the uplink optimization, we can still convert +(26c) to linear constraints using the SCA method. By setting +cm,k = [ℜ(¯hH +m,ktm), ℑ(¯hH +m,ktm)], the relaxed problem is +given as +minimize +ζdl,{tm,c},Θdl +ζdl +(27a) +s.t. +� +l̸=m |¯hH +l,ktl|2 + (σdl)2 +dk +≤ +∥c(t) +m,k∥2 + 2(c(t) +m,k)T(cm,k − c(t) +m,k), ∀m, k +(27b) +cm,k = [ℜ(¯hH +m,ktm), ℑ(¯hH +m,ktm)], ∀m, k +(27c) +(25c), (25d), (26b). +The above problem (27) can be solved in the same way as +uplink optimization. +V. SIMULATION RESULTS +In this section, we conduct extensive numerical experi- +ments to evaluate the performance of the proposed SCA +algorithm for the STAR-RIS assisted AirComp-based vertical +FL system in multi-cell wireless network. +We consider a STAR-RIS assisted two-cell wireless vertical +FL network in a two-dimensional space, where the coordi- +nates of the BSs are (0m, 0m) and (40m, 0m), the STAR- +RIS is deployed at the edge of two cells, i.e., (20m, 0m). +And the devices in each cell are uniformly located within a +circular region centered at their corresponding BS with radius +20 meters. All channel coefficients are modeled as +h = ρ−α/2 +�� +β +1 + β hLoS + +� +1 +1 + β hNLoS +� +(28) +and vary independently over different rounds, where ρ denotes +the distance between the transmitter and the receiver, α = 2.5 +denotes the pathloss exponent, β = 5 dB represents the Ri- +cian factor, hLoS denotes the line-of-sight (LoS) component, +and hNLoS denotes the non-line-of-sight (NLoS) exponent. +In addition, the noise power are set to +� +σul�2 = +� +σdl�2 = +−10dBm. All simulation results in the following are obtained +by averaging over 100 experiments. +We first evaluate the performance of uplink aggregation +using AirComp and downlink dissemination error by consid- +ering the MSE as the metric. As shown in Fig. 1, the MSE + +5 +10 +15 +20 +25 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +(a) MSE of AirComp versus the num- +ber of elements at STAR-RIS when +N = 8 and Km = 4. +5 +10 +15 +20 +25 +0.01 +0.015 +0.02 +0.025 +0.03 +0.035 +0.04 +0.045 +(b) Downlink MSE versus the num- +ber of elements at STAR-RIS when +N = 8 and Km = 4. +Fig. 1. Performance of uplink aggregation via AirComp under +different settings. +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +(a) Training loss vs. Round +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +40 +50 +60 +70 +80 +90 +100 +Average Testing Accuracy (%) +(b) Testing accuracy vs. Round +Fig. 2. Performance of AirComp assisted Vertical FL. +decreases as the number of STAR-RIS elements increases for +both downlink and uplink transmission, indicating that STAR- +RIS can effectively enhance the signal transmission quality, +particularly when it has a large number of elements. +We further evaluate the performance of our proposed +STAR-RIS assisted vertical FL system, where Km = 4 +devices in each cell cooperatively train a regularized logistic +regression model. The number of antennas at each BS is +N = 8, and the number of elements at STAR-RIS is Q = 10. +We simulate the image classification task on Fashion-MNIST +dataset [18]. And we assume that each cell perform a different +binary classification task for simplify (0-1 in cell 1, 2-3 in +cell 2). The traditional binary cross-entropy loss function is +given as +F(w) = − 1 +L +L +� +i=1 +� +yi � +wxi� +− ln +� +1 + exp(wxi) +�� +. +The learning rate µ(t) is set to 0.01. +We consider the noiseless case as the performance upper +bound. Fig. 2 shows that our proposed STAR-RIS assisted +system converges quickly and achieves 96% testing accuracy +in inference, which is far ahead compared with the other two +cases. And it is even close to the performance upper bound. +VI. CONCLUSION +In this paper, we proposed a STAR-RIS assisted AirComp- +based vertical FL system in multi-cell networks. To be +specific, a STAR-RIS is deployed at the cell edge to facilitate +the completion of different FL tasks by each cell. The Pareto +boundary of the gap region is introduced to characterize +the trade-off of learning performance among cells. We then +formulate an optimization problem to minimize the sum of +error-induced gaps across all cells, which is then solved by +SCA-based algorithms. Our simulation results demonstrate +that the proposed STAR-RIS assisted system can significantly +improve the learning performance in both training and infer- +ence phases thanks to its powerful capability of reducing the +transmission errors. +REFERENCES +[1] K. B. Letaief, Y. Shi, J. Lu, and J. Lu, “Edge artificial intelligence for +6g: Vision, enabling technologies, and applications,” IEEE J. Sel. Areas +Commun., vol. 40, no. 1, pp. 5–36, 2022. +[2] K. B. Letaief, W. Chen, Y. Shi, J. Zhang, and Y.-J. A. Zhang, “The +roadmap to 6g: Ai empowered wireless networks,” IEEE Commun. +Mag., vol. 57, no. 8, pp. 84–90, 2019. +[3] Y. Shi, K. Yang, T. Jiang, J. Zhang, and K. B. Letaief, “Communication- +efficient edge ai: Algorithms and systems,” IEEE Commun. Surveys +Tuts., vol. 22, no. 4, pp. 2167–2191, 2020. +[4] Z. Wang, Y. Shi, Y. Zhou, H. Zhou, and N. 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Zhang, “Reconfigurable intelligent surface +enabled federated learning: A unified communication-learning design +approach,” IEEE Trans. Wireless Commun., vol. 20, no. 11, pp. 7595– +7609, 2021. +[14] Z. Wang, J. Qiu, Y. Zhou, Y. Shi, L. Fu, W. Chen, and K. B. +Letaief, “Federated learning via intelligent reflecting surface,” IEEE +Tran. Wireless Commun., vol. 21, no. 2, pp. 808–822, 2021. +[15] X. Li, K. Huang, W. Yang, S. Wang, and Z. Zhang, “On the convergence +of fedavg on non-iid data,” arXiv preprint arXiv:1907.02189, 2019. +[16] E. A. Jorswieck, E. G. Larsson, and D. Danev, “Complete characteriza- +tion of the pareto boundary for the MISO interference channel,” IEEE +Trans. Signal Process., vol. 56, no. 10, pp. 5292–5296, 2008. +[17] X. Cao, G. Zhu, J. Xu, and K. Huang, “Cooperative interference man- +agement for over-the-air computation networks,” IEEE Trans. Wireless +Commun., vol. 20, no. 4, pp. 2634–2651, 2020. +[18] H. Xiao, K. Rasul, and R. Vollgraf, “Fashion-mnist: a novel image +dataset for benchmarking machine learning algorithms,” arXiv preprint +arXiv:1708.07747, 2017. + diff --git a/ENE5T4oBgHgl3EQfUg_2/content/tmp_files/load_file.txt b/ENE5T4oBgHgl3EQfUg_2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e0c1419682bd61ac56ba488785e024b0e0a543f9 --- /dev/null +++ b/ENE5T4oBgHgl3EQfUg_2/content/tmp_files/load_file.txt @@ -0,0 +1,470 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf,len=469 +page_content='STAR-RIS Assisted Over-the-Air Vertical Federated Learning in Multi-Cell Wireless Networks Xiangyu Zeng∗†‡, Yijie Mao∗, and Yuanming Shi∗ ∗School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China †Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, China ‡University of Chinese Academy of Sciences, Beijing 100049, China E-mail: {zengxy, maoyj, shiym}@shanghaitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='cn Abstract—Vertical federated learning (FL) is a critical enabler for distributed artificial intelligence services in the emerging 6G era, as it allows for secure and efficient collaboration of machine learning among a wide range of Internet of Things devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' However, current studies of wireless FL typically con- sider a single task in a single-cell wireless network, ignoring the impact of inter-cell interference on learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' In this paper, we investigate a simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) assisted over-the-air computation based vertical FL system in multi-cell networks, in which a STAR-RIS is deployed at the cell edge to facilitate the completion of different FL tasks in different cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' We establish the convergence of the proposed system through theoretical analysis and introduce the Pareto boundary of the optimality gaps to characterize the trade-off among cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Based on the analysis, we then jointly design the transmit and receive beamforming as well as the STAR-RIS transmission and reflection coefficient matrices to minimize the sum of the gaps of all cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' To solve the non-convex resource allocation problem, we introduce a successive convex approximation based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Numerical experiments demonstrate that compared with con- ventional approaches, the proposed STAR-RIS assisted vertical FL model and the cooperative resource allocation algorithm achieve much lower mean-squared error for both uplink and downlink transmission in multi-cell wireless networks, resulting in improved learning performance for vertical FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' INTRODUCTION Federated learning (FL) is a machine learning (ML) ap- proach that enables multiple parties to collaboratively train a learning model without revealing their individual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' This is beneficial in a variety of fields where data privacy is a concern, as FL allows parties to maintain control over their own data while still benefiting from the combined knowledge of all parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' In modern wireless Internet of Things (IoT) networks, data is often collected from various types of devices [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' To facilitate data analysis in such settings, vertical FL, a variation of FL that is designed to address the challenges of training machine learning models on vertically partitioned data silos, is commonly adopted [2]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' One major issue that prevents the implementation of (ver- tical) FL in real-world application is the communication latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' To address this issue, over-the-air computation (Air- Comp) has been proposed to facilitate fast wireless data aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' By utilizing the superposition property of wire- less multiple access channels (MAC) to concurrently transmit and aggregate local updates, AirComp significantly reduces communication latency compared to orthogonal transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Previous research has explored the use of AirComp in FL, such as the joint design of device selection and beamforming for fast global model aggregation in [5], and the development of a broadband analog aggregation scheme for low latency FL with linear growth of latency reduction ratio in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' On the other hand, the coexistence of multiple FL tasks in multi-cell networks has yet to be fully explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Though the authors in [8] have studied the bandwidth allocation for multiple FL tasks, the system model is limited to a single-cell network and the impact of inter-cell interference on FL performance remains unplumbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' It has been well investigated that reconfigurable intelligent surface (RIS), a metasurface composed of reconfigurable passive elements, can modify the propagation environment of wireless signal and reduce multi-cell interference [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' However, conventional RISs are reflecting only with limited wireless coverage [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The recently introduced simultaneous transmitting and reflect- ing RIS (STAR-RIS), which allows the source and destination to be located at either side of the metasurface, has been recognized as a promising strategy to enhance the coverage of each cell and further reduce inter-cell interference [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' STAR-RIS is therefore a promising technique to facilitate FL in multi-cell networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' To the best of our knowledge, STAR- RIS assisted vertical FL has not been studied yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' In this paper, inspired by the benefits of AirComp for global aggregation [12] and the merits of STAR-RIS in multi-cell networks, we fill the research gap and propose a STAR-RIS assisted AirComp-based vertical FL in multi-cell networks, where a STAR-RIS is deployed at the cell edge to assist each cell in completing different FL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Through theoretical analysis, we demonstrate the convergence of our proposed vertical FL process and introduce the Pareto boundary of the gap region to characterize the trade-off performance among multiple cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' This allows us to formulate an optimization problem with the aim of minimizing the sum of error-induced gaps for all cells using the proposed algorithm based on suc- cessive convex approximation (SCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Numerical experiments confirm the validity of our theoretical analysis and show the superiority of our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' SYSTEM MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Learning Framework Consider a STAR-RIS assisted multi-cell wireless net- work consisting of M base stations (BS) with N an- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='05545v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='IT] 13 Jan 2023 tennas, where BS m ∈ M = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' , M} aims to train an ML model by coordinating Km single- antenna devices located in cell m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Specifically, device k ∈ Km = ��m−1 l=1 Kl + 1, �m−1 l=1 Kl + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' , �m−1 l=1 Kl+ Km} is associated with BS m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' And there is one STAR-RIS equipped with Q passive reflecting/transmitting elements, de- ployed at the cell-edge of all cells to boost the signal strength of edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Each cell is equipped with a vertically partitioned dataset, where different devices hold different features of the same samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' For simplicity, we assume that each cell has the same number of samples and that devices within each cell contain the same number of non-overlapping features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Let Dm = {(xi m,1, · · · , xi m,Km), yi m}Lm i=1 denote the whole training dataset of Lm samples in cell m, where xi m,k denotes the partial features of sample i located at device k in cell m, and yi m denotes the corresponding label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' In vertical FL, it is assumed that the BS holds all labels ym = {yi m}Lm i=1, and device k is only available to its own local feature set Dm,k = {xi m,k}Lm i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' And xi m = [(xi m,1)T, · · · , (xi m,Km)T]T denotes the overall feature vector of sample i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The goal of vertical FL in cell m is to collaboratively learn a global model wm (concatenated vector of wk for k ∈ Km) that maps an input to the corresponding prediction through a continuously differentiable function σ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Since features of one sample are distributed at different devices, we assume that device k maps the local feature xk to local prediction result gk(wk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' This paper considers a linear form for the local prediction function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=', gk(wk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' xk) = wT k xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' By ag- gregating local prediction results, the final prediction in cell m can be obtained by σ(wm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' xm) = σ(� k∈Km gk(wk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' xk)) = σ(wT mxm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' In order to learn the global model wm in cell m, we propose to minimize the loss function as min wm F(wm) = 1 Lm Lm � i=1 f � σ(wT mxi m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' yi m � , (1) where f(·) is the sample-wise loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' In our multi-cell system, each cell performs a unique FL task using the full batch gradient descent (GD) approach, which is described in the following subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' We assume universal frequency reuse, meaning that all cells share the same frequency channel, leading to inter-cell interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' GD Algorithm for Vertical FL In this subsection, we introduce the framework of GD algorithm for vertical FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' For brevity, the subscript of cell m is omitted for Lm, wm, xm, ym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The GD algorithm specified in this subsection is applied for all cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Let ∇F(w) denote the gradient of F respect to w, which is calculated as ∇F(w) = 1 L L � i=1 ∇f(σ(wTxi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' yi), (2) where ∇f(σ(wTxi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' yi) denote the gradient of f(σ(wTxi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' yi) respect to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Based on the chain rule, the gradient of f is rewritten as ∇f(σ(wTxi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' yi) = G(wTxi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' yi)xi, (3) where G(wTxi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' yi) = ∂f(σ(wTxi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' yi)/∂wTxi is an auxil- iary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Hence, ∇F(w) can be rewritten as ∇F(w) = 1 L L � i=1 G(wTxi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' yi)xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' (4) Recall that the BS holds all labels y, so G(wTxi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' yi) can be calculated at the BS only if the BS can access the aggregation of local predictions {wTxi}L i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Specifically, at the t-th communication round, the BS and the edge devices in each cell perform the following three procedures: Broadcasting: The BS computes {G((w(t))Txi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' yi)}L i=1 and broadcasts the result back to its corresponding devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Local model update: After broadcasting, device k com- putes the partial gradient ∇kF(wk) with local data Dk, given as ∇kF(wk) = 1 L L � i=1 G(wTxi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' yi)xi k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' (5) Each device can thus update its local model by taking a step of GD with learning rate µ(t) as w(t+1) k = w(t) k − µ(t)∇kF(w(t) k ), (6) where w(t) k is the local model of device k at the t-th round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Local prediction and global aggregation: device k com- putes the local prediction results {(w(t+1) k )Txi k}L i=1 and sends to the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' And BS aggregates them to get final prediction result {(w(t+1))Txi}L i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Since the BS only needs the aggregation of local prediction results, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=', neither local features nor local models need be uploaded to the BS, which significantly enhances privacy pro- tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' In addition, the communication efficiency is improved since the local prediction result is usually low-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Communication Model In this subsection, the proposed communication model is delineated with a special focus on the STAR-RIS assisted uplink and downlink transmission models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' 1) STAR-RIS: The STAR-RIS is a type of RIS that can produce omnidirectional radiation by implementing equiva- lent electric and magnetic currents in its hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' It has three protocols for use in wireless networks: energy splitting, mode switching, and time switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' In this article, we focus on the mode-switching protocol, in which each element of the STAR-RIS can operate in either the reflection mode (R mode) or the transmission mode (T mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Such on-off type of operating protocol is simpler to implement compared to the energy splitting protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Specifically, one group consists of Qt elements operating in the T mode, while the other group contains Qr elements operating in the R mode, where Qt + Qr = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Accordingly, the STAR-RIS transmission- coefficient and reflection-coefficient matrices are given by Θt = diag �� βt 1ejθt 1, � βt 2ejθt 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' , � βt Qejθt Q � and Θr = diag �� βr 1ejθr 1, � βr 2ejθr 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' , � βr Qejθr Q � , respec- tively, where βt q, βr q ∈ {0, 1}, βt q + βr q = 1, and θt q, θr q ∈ [0, 2π), ∀q ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' , Q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The M cells can be divided into two groups Mr and Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Specifically, cell m is in the reflection dimension with m ∈ Mr and in the transmission dimension with m ∈ Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Let hm,k ∈ CN, hr k ∈ CQ and Gm ∈ CQ×N denote the equivalent channels from edge device k to BS m, from edge device k to the STAR-RIS, and from the STAR-RIS to BS m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The combined channel from the k-th edge device to the BS m via the STAR-RIS can be written as ¯hm,k = � hm,k + GH mΘthr k, ∀m ∈ Mt, hm,k + GH mΘrhr k, ∀m ∈ Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Note that the uplink and downlink STAR-RIS matrices can be separatively designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' For simplify, we write Θt and Θr for uplink and downlink transmission in terms of Θul and Θdl, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' 2) Uplink transmission: In the uplink transmission, we assume the devices communicate with the BS via AirComp, which has a wide range of FL applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Specifically, we denote sk = [s1 k, s2 k, · · · , sL k ]T ∈ CL as the local prediction results at device k, where the local prediction result of the i-th sample si k = wT k xi k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' At each time slot i ∈ {1, 2, · · · , Lm}, each device in cell m sends the corresponding prediction result of the i-th sample to BS m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' And we assume that sk is normalized with zero mean and unit variance [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' We denote gm(i) = � k∈Km si k as the target function to be estimated through AirComp at the i-th time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' To simplify the notation, we omit the time index by writing g(i) and si k as g and sul k , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' And we assume that the signals transmitted by all devices are synchronized at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Then the received signal at BS m is given by yul m = � k ¯hm,kbksul k + nul m, (7) where bk ∈ C is the transmit scalar at device k, and nul m is the additive white Gaussian noise with zero mean and variance (σul)2 at BS m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The transmit power constraint at device k is E(|bksul k |2) = |bk|2 ≤ P ul, where P ul > 0 is the maximum transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The scaled signal received at BS m is ¯gm = 1 √ηm rH myul m = 1 √ηm rH m � k∈K ¯hm,kbksul k + rH mnul m √ηm , (8) where rm ∈ CN is the receive beamforming vector and ηm is a normalizing factor for cell m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' To compensate for the phase distortion introduced by complex channel responses, the transmit scalar at device k in cell m is set to bk = √ηm (rH m¯hm,k)H |rHm¯hm,k|2 , ∀k ∈ Km, and ηm can be expressed as ηm = P ul mink∈Km |rH m¯hm,k|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Then the estimated function at BS for cell m is given as ˆgm = ℜ{¯gm} = ℜ{gm + 1 √ηm rH m � l̸=m � j∈Kl ¯hm,jbjsul j + rH mnul m √ηm � �� � eulm } = gm + ℜ{eul m} (9) 3) Downlink transmission: After obtaining the estimate ˆgm in the cell m, BS m computes G(ˆgm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' y) with noisy aggregation ˆgm, and then broadcasts the result to the associ- ated devices in Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' And we write G(ˆgm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' y) in terms of Gm for simplify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Without loss of generality, we assume that the transmitted signal follows the standard Gaussian distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=', Gm ∼ CN(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The received signal at device k is ydl k = � m ¯hH m,ktmGm + ndl k , (10) where tm denotes the transmit beamforming vector at BS m, and ndl k ∼ CN � 0, (σdl)2� is the additive white Gaus- sian noise with zero mean and variance (σdl)2 at device k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The maximum transmit power at BS m is P dl, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=', E(∥Gmtm∥2) = ∥tm∥2 ≤ P dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' To compensate for the phase distortion introduced by complex channel responses, the receive scalar at device k in cell m is set to rk = (¯hH m,ktm)H |¯hH m,ktm| 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The estimated Gm at device k is given as ˆGm,k = ℜ{rkydl k } = ℜ{Gm + (¯hH m,ktm)H ���¯hH m,ktm ��� 2 � �� l̸=m ¯hH l,ktlGl + ndl k � � � �� � ¯edl k } = Gm + ℜ{¯edl k }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' (11) Note that the uplink noise is embedded in function Gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' In order to directly describe the effective noise, we expand Gm to its first-order Taylor expansion as follows ˆGm,k = Gm + ℜ{¯edl k } = G(gm + ℜ{eul m};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' y) + ℜ{¯edl k } = G(gm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' y) + G ′(gm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' y)ℜ{eul m} + O(|ℜ{eul m}|2) + ℜ{¯edl k } ≈ G(gm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' y) + G ′(gm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' y)ℜ{eul m} + ℜ{¯edl k } � �� � edl k , (12) where G′(·) is the first derivative of G(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Assume that the noise amplitude is small, the term O(|ℜ{eul m}|2) is neglected, which implies the last approximation in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' CONVERGENCE ANALYSIS AND PROBLEM FORMULATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Convergence Analysis In previous work [12], [14], [15], the convergence analysis of the AirComp-based vertical FL process in each cell has been established under the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Assumption 1 (α-strongly convexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The function F(·) is assumed to be α-strongly convex on Rd with constant α, namely, for all x, y ∈ Rd, we have F(y) ≥ F(x) + ∇F(x)T(y − x) + α 2 ∥y − x∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Assumption 2 (β-smoothness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The function F(·) is assumed to be β-smooth on Rd with constant β, namely, for all x, y ∈ Rd, we have F(y) ≤ F(x) + ∇F(x)T(y − x) + β 2 ∥y − x∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Theorem 1 (Convergence of vertical FL process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Suppose that Assumption 1 and 2 hold, setting the learning rate to be 0 < µ(t) ≤ 1 β , then the expected optimality gap after T communication rounds is upper bounded by E � F(w(T ) m ) − F(w∗ m) � ≤ ρT E � F(w(0) m ) − F(w∗ m) � + 1 2βL2 T −1 � t=0 ρT −t−1 � k∈Km � Φ1,kE[|ℜ{eul m}|2] + Φ2,kE[|ℜ{¯edl k }|2] � , (13) where ρ = 1−α/β, Φ1,k = �L i=1 ∥(Gi m,k) ′xi k∥2 2 and Φ2,k = �L i=1 ∥xi k∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Please refer to previous work [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Problem Formulation According to Theorem 1, the convergence optimality gap is largely determined by the mean-squared-error (MSE) of both gm and Gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' However, solely optimizing MSE for each cell through AirComp may result in significant inter-cell interference in the considered multi-cell wireless networks, which can negatively impact the learning performance of other cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' As such, it is necessary to carefully balance the learning performance among various FL tasks in multiple cells through a cooperative design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' We begin by identifying the gap region G, to be the set of tuples (∆1, ∆2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' , ∆M), which represents the instantaneous errors that cause gaps in all cells, and can be achieved simul- taneously under specific downlink and uplink transmission power constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The gap region G can be represented as G = � {(∆1, ∆2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' , ∆M)|∆m ≥ Gapm, ∀m ∈ M}, (14) where Gapm = � k∈Km � Φ1,kE[|ℜ{eul m}|2] + Φ2,kE[|ℜ{¯edl k }|2] � , (15) E[|ℜ{eul m}|2] = � l̸=m,j∈Kl ηl|rH m¯hm,j|2 ηm|rH l ¯hl,j|2 + ∥rm∥2σ2 ul ηm , E[|ℜ{¯edl k }|2] = � l̸=m |¯hH l,ktl|2 + (σdl)2 ���¯hH m,ktm ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' (16) As previously stated, in order to decrease the error-induced gap in one cell, the gaps of other cells maybe increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' In light of this, our objective is to find a suitable solution that allows us to achieve the Pareto boundary of the gap region G, so as to balance the performance of learning among multiple cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' In this context, the Pareto optimality of a tuple is described as follows [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Here, we leverage the profiling technique [17] to char- acterize the Pareto boundary by coordinating all BSs to minimize the sum of Gap of all cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Specifically, let κ = [κ1, κ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' , κM] denote a given profiling vector, which satisfies κm ≥ 0, ∀m ∈ M, and � m∈M κm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The gap tuple on Pareto boundary can be obtained by solving the following problem minimize ζ,{rm},{tm},Θt,Θr ζ (17a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Gapm ≤ κmζ, ∀m ∈ M (17b) ζ ≥ 0, (17c) where ζ denotes the sum of the gaps of all cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Thus, the gap tuple can be represented as (∆1, ∆2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' , ∆M) = (κ1ζ, κ2ζ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' , κMζ), where a smaller value of κm implies a more stringent requirement for the gap of cell m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Denote ζ = ζul +ζdl, where ζul and ζdl are used to quan- tify the sum of instantaneous error-induced gaps generated by uplink and downlink transmissions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Hence, we rewrite problem (17) as minimize ζul,ζdl,{rm},{tm},Θt,Θr ζul + ζdl (18a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Gapul m ≤ κmζul, ∀m ∈ M (18b) Gapdl m ≤ κmζdl, ∀m ∈ M (18c) ζul ≥ 0 (18d) ζdl ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' (18e) The downlink and uplink transmissions can be decoupled in problem (18), which allows us to separately optimize the downlink and uplink transmission resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' OPTIMIZATION FRAMEWORK In this section, we specify the optimization framework for solving the uplink and downlink optimization problems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Uplink Optimization For the uplink aggregation, the optimization problem is minimize ζul,{rm},Θul ζul (19a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' � l̸=m � j∈Kl ηl|rH m¯hm,j|2 ηm|rH l ¯hl,j|2 (19b) + ∥rm∥2(σul)2 ηm ≤ κmζul, ∀m ∈ M (19c) ζul ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' (19d) By setting optimzing varibales qi = ri/√ηi, ∀i ∈ M, the problem can be converted to minimize ζul,{qm},Θul ζul (20a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' � l̸=m � j∈Kl |qH m¯hm,j|2 |qH l ¯hl,j|2 + (σul)2∥qH m∥2 ≤ κmζul, ∀m ∈ M (20b) |qH m¯hm,k|2 ≥ 1 Pul , ∀m, ∀k ∈ Km (20c) (19d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Then we let |qH m¯hm,j|2 |qH l ¯hl,j|2 ≤ bl,j, the optimization problem relaxes to minimize ζul,{qm,b},Θul ζul (21a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' � l̸=m � j∈Kl bl,j + (σul)2∥qH m∥2 ≤ κmζul, ∀m (21b) |qH m¯hm,j|2 |qH l ¯hl,j|2 ≤ bl,j, ∀l, j (21c) (19d), (20c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' However, constraint (21c) is still non-convex, then we use the SCA method to transform (21c) into a linear con- straint which satisfies the property of convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Let al,j = [ℜ(qH l ¯hl,j), ℑ(qH l ¯hl,j)], the corresponding approximated lin- ear constraint is |qH m¯hm,j|2 bl,j ≤ ∥al,j∥2 ≤ ∥a(t) l,j ∥2 + 2(a(t) l,j )T(al,j − a(t) l,j ) (22) and ∥a(t) m,k∥2 + 2(a(t) m,k)T(am,k − a(t) m,k) ≥ 1 Pul .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' (23) The origin problem (21) is then approximated as minimize ζul,{qm,b,a},Θul ζul s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' al,j = [ℜ(qH l ¯hl,j), ℑ(qH l ¯hl,j)], ∀l, j (19d), (21b), (22), (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' (24) And we can observe that the above problem turns out to be highly intractable due to the non-convexity of multiplication between variables q and Θul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Hence, a classical alternative optimization algorithm can be used to solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Downlink Optimization For the downlink dissemination, the optimization problem can be written as minimize ζdl,{tm},Θdl ζdl (25a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' � k∈Km � l̸=m |¯hH l,ktl|2 + (σdl)2 ���¯hH m,ktm ��� 2 ≤ κmζdl, ∀m (25b) ∥tm∥2 ≤ P dl, (25c) ζdl ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' (25d) By letting � l̸=m |¯hH l,ktl|2+(σdl)2 |¯hH m,ktm| 2 ≤ dk, the optimization prob- lem is relaxed to minimize ζdl,{tm},Θdl ζdl (26a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' � k∈Km dk ≤ κmζdl, ∀m ∈ M (26b) � l̸=m |¯hH l,ktl|2 + (σdl)2 ���¯hH m,ktm ��� 2 ≤ dk, (26c) (25c), (25d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Similar as the uplink optimization, we can still convert (26c) to linear constraints using the SCA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' By setting cm,k = [ℜ(¯hH m,ktm), ℑ(¯hH m,ktm)], the relaxed problem is given as minimize ζdl,{tm,c},Θdl ζdl (27a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' � l̸=m |¯hH l,ktl|2 + (σdl)2 dk ≤ ∥c(t) m,k∥2 + 2(c(t) m,k)T(cm,k − c(t) m,k), ∀m, k (27b) cm,k = [ℜ(¯hH m,ktm), ℑ(¯hH m,ktm)], ∀m, k (27c) (25c), (25d), (26b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The above problem (27) can be solved in the same way as uplink optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' SIMULATION RESULTS In this section, we conduct extensive numerical experi- ments to evaluate the performance of the proposed SCA algorithm for the STAR-RIS assisted AirComp-based vertical FL system in multi-cell wireless network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' We consider a STAR-RIS assisted two-cell wireless vertical FL network in a two-dimensional space, where the coordi- nates of the BSs are (0m, 0m) and (40m, 0m), the STAR- RIS is deployed at the edge of two cells, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=', (20m, 0m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' And the devices in each cell are uniformly located within a circular region centered at their corresponding BS with radius 20 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' All channel coefficients are modeled as h = ρ−α/2 �� β 1 + β hLoS + � 1 1 + β hNLoS � (28) and vary independently over different rounds, where ρ denotes the distance between the transmitter and the receiver, α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='5 denotes the pathloss exponent, β = 5 dB represents the Ri- cian factor, hLoS denotes the line-of-sight (LoS) component, and hNLoS denotes the non-line-of-sight (NLoS) exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' In addition, the noise power are set to � σul�2 = � σdl�2 = −10dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' All simulation results in the following are obtained by averaging over 100 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' We first evaluate the performance of uplink aggregation using AirComp and downlink dissemination error by consid- ering the MSE as the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' 1, the MSE 5 10 15 20 25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='12 (a) MSE of AirComp versus the num- ber of elements at STAR-RIS when N = 8 and Km = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='045 (b) Downlink MSE versus the num- ber of elements at STAR-RIS when N = 8 and Km = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Performance of uplink aggregation via AirComp under different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' 10 20 30 40 50 60 70 80 90 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='7 (a) Training loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Round 10 20 30 40 50 60 70 80 90 100 40 50 60 70 80 90 100 Average Testing Accuracy (%) (b) Testing accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Round Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Performance of AirComp assisted Vertical FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' decreases as the number of STAR-RIS elements increases for both downlink and uplink transmission, indicating that STAR- RIS can effectively enhance the signal transmission quality, particularly when it has a large number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' We further evaluate the performance of our proposed STAR-RIS assisted vertical FL system, where Km = 4 devices in each cell cooperatively train a regularized logistic regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The number of antennas at each BS is N = 8, and the number of elements at STAR-RIS is Q = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' We simulate the image classification task on Fashion-MNIST dataset [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' And we assume that each cell perform a different binary classification task for simplify (0-1 in cell 1, 2-3 in cell 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The traditional binary cross-entropy loss function is given as F(w) = − 1 L L � i=1 � yi � wxi� − ln � 1 + exp(wxi) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The learning rate µ(t) is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' We consider the noiseless case as the performance upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' 2 shows that our proposed STAR-RIS assisted system converges quickly and achieves 96% testing accuracy in inference, which is far ahead compared with the other two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' And it is even close to the performance upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' CONCLUSION In this paper, we proposed a STAR-RIS assisted AirComp- based vertical FL system in multi-cell networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' To be specific, a STAR-RIS is deployed at the cell edge to facilitate the completion of different FL tasks by each cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' The Pareto boundary of the gap region is introduced to characterize the trade-off of learning performance among cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' We then formulate an optimization problem to minimize the sum of error-induced gaps across all cells, which is then solved by SCA-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'} +page_content=' Our simulation results demonstrate that the proposed STAR-RIS assisted system can significantly improve the learning performance in both training and infer- ence phases thanks to its powerful capability of reducing 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SATTAROV, E.T. ALIEV +Abstract. We consider (1, 2)-rational functions given on the field of p-adic numbers Qp. +In general, such a function has four parameters. We study the case when such a function +has two fixed points and show that when there are two fixed points then (1, 2)-rational +function is conjugate to a two-parametric (1, 2)-rational function. +Depending on these +two parameters we determine type of the fixed points, find Siegel disks and the basin of +attraction of the fixed points. Moreover, we classify invariant sets and study ergodicity +properties of the function on each invariant set. We describe 2- and 3-periodic orbits of +the p-adic dynamical systems generated by the two-parametric (1, 2)-rational functions. +1. Introduction and preliminaries +A function is called a (n, m)-rational function if and only if it can be written in the +form f(x) = Pn(x) +Qm(x), where Pn(x) and Qm(x) are polynomial functions with degree n and m +respectively, Qm(x) is non zero polynomial. +In this paper we study dynamical systems generated by a (1.2-)rational function. Our +investigations based on methods of [1], [3], [13]-[17]. For motivations of the study see [2], +[4]-[6], [10]-[12] and the references therein. +Let us give main definitions. Let Q be the field of rational numbers. The greatest common +divisor of the positive integers n and m is denotes by (n, m). Every rational number x ̸= 0 +can be represented in the form x = pr n +m, where r, n ∈ Z, m is a positive integer, (p, n) = 1, +(p, m) = 1 and p is a fixed prime number. +The p-adic norm of x ∈ Q is given by +|x|p = +� +p−r, +for x ̸= 0, +0, +for x = 0. +It has the following properties: +1) |x|p ≥ 0 and |x|p = 0 if and only if x = 0, +2) |xy|p = |x|p|y|p, +3) the strong triangle inequality +|x + y|p ≤ max{|x|p, |y|p}, +3.1) if |x|p ̸= |y|p then |x + y|p = max{|x|p, |y|p}, +2010 Mathematics Subject Classification. 46S10, 12J12, 11S99, 30D05, 54H20. +Key words and phrases. Rational dynamical systems; fixed point; invariant set; Siegel disk; complex +p-adic field. +1 + +2 +I.A. SATTAROV, E.T. ALIEV +3.2) if |x|p = |y|p then |x + y|p ≤ |x|p. +The completion of Q with respect to p-adic norm defines the p-adic field which is denoted +by Qp (see [8]). +For any a ∈ Qp and r > 0 denote +Ur(a) = {x ∈ Qp : |x − a|p < r}, +Vr(a) = {x ∈ Qp : |x − a|p ≤ r}, +Sr(a) = {x ∈ Qp : |x − a|p = r}. +A function f : Ur(a) → Qp is said to be analytic if it can be represented by +f(x) = +∞ +� +n=0 +fn(x − a)n, +fn ∈ Qp, +which converges uniformly on the ball Ur(a). +Now let f : U → U be an analytic function. Denote f n(x) = f ◦ · · · ◦ f +� +�� +� +n +(x). +If f(x0) = x0 then x0 is called a fixed point. The set of all fixed points of f is denoted +by Fix(f). A fixed point x0 is called an attractor if there exists a neighborhood U(x0) of +x0 such that for all points x ∈ U(x0) it holds lim +n→∞ f n(x) = x0. If x0 is an attractor then its +basin of attraction is +A(x0) = {x ∈ Qp : f n(x) → x0, n → ∞}. +A fixed point x0 is called repeller if there exists a neighborhood U(x0) of x0 such that +|f(x) − x0|p > |x − x0|p for x ∈ U(x0), x ̸= x0. +Let x0 be a fixed point of a function f(x). Put λ = f ′(x0). The point x0 is attractive if +0 < |λ|p < 1, indifferent if |λ|p = 1, and repelling if |λ|p > 1. +The ball Ur(x0) is said to be a Siegel disk if each sphere Sρ(x0), ρ < r is an invariant +sphere of f(x), i.e. if x ∈ Sρ(x0) then all iterated points f n(x) ∈ Sρ(x0) for all n = 1, 2 . . . . +The union of all Siegel desks with the center at x0 is said to a maximum Siegel disk and is +denoted by SI(x0). +Let f : A → A and g : B → B be two maps. f and g are said to be topologically conjugate +if there exists a homeomorphism h : A → B such that, h◦f = g ◦h. The homeomorphism h +is called a topological conjugacy. Mappings which are topologically conjugate are completely +equivalent in terms of their dynamics. +In this paper we consider (1, 2)-rational function f : Qp → Qp defined by +f(x) = +ax + b +x2 + cx + d, +x ̸= ˆx1,2 = −c ± +√ +c2 − 4d +2 +(1.1) +where the parameters of the function satisfy the following conditions +a ̸= 0, +a, b, c, d, +� +c2 − 4d ∈ Qp. +We study p-adic dynamical systems generated by the rational function (1.1). The equa- +tion f(x) = x for fixed points of the function (1.1) is equivalent to the equation +x3 + cx2 + (d − a)x − b = 0. +(1.2) +The equation (1.2) may have three solutions with one of the following relations: + +ERGODICITY AND PERIODIC ORBITS +3 +(i) one solution having multiplicity three; +(ii) two solutions, one of which has multiplicity two; +(iii) three distinct solutions. +Remark 1. Since the behavior of dynamical system depends on the set of fixed points, each +of the above mentioned case (i)-(iii) has its own character of dynamics. In [15] the case (i) +was considered. In this paper we consider the case (ii), i.e., we investigate the behavior of +the trajectories of an arbitrary (1, 2)-rational dynamical system in Qp when there are two +fixed points for f. The case (iii) will be considered in a separate paper. +The paper is organized as follows. In Section 2 under some assumptions we show that +four-parametric function (1.1) is conjugate to a two-parametric (1,2)-rational function. In +Section 3 we study the p-adic dynamics generated by the two-parametric function and give +Siegel disks, the basin of attractions and classification of all invariant sets. In Section 4 we +investigate ergodicity of this dynamical systems on invariant sets. In Section 5 we describe +2- and 3-periodic orbits. +2. A function conjugate to (1.1) +Denote by x1 and x2 the two solutions of the equation (1.2), where x2 has multiplicity +two. Then we have x3 + cx2 + (d − a)x − b = (x − x1)(x − x2)2 and + + + + + +x1 + 2x2 = −c +x2 +2 + 2x1x2 = d − a +x1x2 +2 = b. +(2.1) +Let homeomorphism h : Qp → Qp be defined by h(t) = t+x2. We note that, the function +f is topologically conjugate to function h−1 ◦ f ◦ h. We have +(h−1 ◦ f ◦ h)(t) = −x2t2 + Bt +t2 + Dt + B , +(2.2) +where B = x2 +2 + cx2 + d and D = 2x2 + c. +In [13] the case x2 ̸= 0 is studied. +Thus in this paper we consider the case x2 = 0 in (2.2). If x2 = 0, then B = d = a and +D = c. Thus we have the following proposition +Proposition 1. Any (1,2)-rational function having two distinct fixed points is topologically +conjugate to one of the following functions +f(x) = +ax2 + bx +x2 + cx + b, +ab(a − c) ̸= 0, +a, b, c ∈ Qp, +and +f(x) = +ax +x2 + cx + a, +ac ̸= 0, +a, c, ∈ Qp. +(2.3) +where x ̸= ˆx1,2 = −c± +√ +c2−4a +2 +. +We study the dynamical system (Qp, f) with f given by (2.3). + +4 +I.A. SATTAROV, E.T. ALIEV +3. p-Adic dynamics of (2.3) +Note that, the function (2.3) has two fixed points x1 = 0 and x2 = −c. We have +f ′(x1) = 1 and f ′(x2) = 1 − c2 +a . +Thus, the point x1 is an indifferent point for (2.3), i.e., x1 is a center of some Siegel disk +SI(x1). In this section we determine the character of the fixed point x2 for each cases. +Then we find Siegel disk or basin of attraction of the fixed point x2, when x2 is indifferent +or attractive, respectively. In the case where x2 is repelling, we find open ball Ur(x2), such +that the inequality |f(x) − x2|p > |x − x2|p holds for all x ∈ Ur(x2). Moreover, we study a +relation between the sets SI(x1) and SI(x2) when x2 is an indifferent. +For any x ∈ Qp, x ̸= ˆx1,2, by simple calculations we get +|f(x)|p = |x|p · +|a|p +|x − ˆx1|p|x − ˆx2|p +. +(3.1) +Denote +P = {x ∈ Qp : ∃n ∈ N ∪ {0}, f n(x) ∈ {ˆx1, ˆx2}}, +α = min{|ˆx1|p, |ˆx2|p} and β = max{|ˆx1|p, |ˆx2|p}. +(3.2) +Since ˆx1 + ˆx2 = −c, we have |c|p ≤ α for α = β and |c|p = β for α < β. Also, since +ˆx1ˆx2 = a, we have |a|p = αβ. +Theorem 1. The p-adic dynamical system generated by the function (2.3) has the following +properties: +1. SI(x1) = Uα(0). +2. If |c|p < α = β, then x2 is indifferent fixed point for (2.3) and +SI(x2) = SI(x1). +3. If |c|p = α = β and |a − c2|p = α2, then x2 is indifferent fixed point for (2.3) and +SI(x2) = Uα(x2), +SI(x2) ∩ SI(x1) = ∅. +4. If |c|p = α = β and |a − c2|p < α2, then x2 is attractive fixed point for (2.3) and +A(x2) = Uα(x2) ⊂ Sα(0). +5. If α < β, then x2 ∈ Sβ(0) is repelling fixed point for (2.3) and the inequality +|f(x) − x2|p > |x − x2|p holds for all x ∈ Uβ(x2), x ̸= x2. +Proof. 1. Let x ∈ Sr(x1), i.e., |x|p = r. Then, from the equalities (3.1), (3.2) and the +properties of the p-adic norm, we have the following +|f(x)|p = + + + + + + + +r, +if r < α, +≥ α, +if α ≤ r ≤ β, +|a|p +r , +if r > β. +From this equality, f(Sr(x1)) ⊂ Sr(x1) for arbitrary r < α, i.e. we have SI(x1) = Uα(0). + +ERGODICITY AND PERIODIC ORBITS +5 +2. Note that |a|p = αβ. If |c|p < α = β, then |f ′(x2)|p = +���1 − c2 +a +��� +p = 1. From this x2 is +indifferent fixed point. Let x ∈ Sr(x2), i.e., |x − x2|p = r. Then from the equality +|f(x) − x2|p = |x − x2|p · +|x2(x − x2) + (x2 +2 − a)|p +|(x − x2) + ˆx1|p|(x − x2) + ˆx2|p +(3.3) +we have |f(x) − x2|p = r for all r < α and |f(x) − x2|p ≥ r for r = α. Thus, f(Sr(x2)) ⊂ +Sr(x2) for arbitrary r < α, i.e. we have SI(x2) = Uα(x2). In this case, we have |x2|p = +|c|p < α, so x2 ∈ Uα(0) = SI(x1). Since these two Siegel disks have the same radii and +share a common point, they are the same, i.e., SI(x2) = SI(x1). +3. If |c|p = α = β and |a − c2|p = α2, then |f ′(x2)|p = +��� a−c2 +a +��� +p = 1. From this x2 is +indifferent fixed point. As above, from equation (3.3) we get SI(x2) = Uα(x2). However, +in this case x2 ∈ Sα(0), so SI(x2) ∩ SI(x1) = ∅. +4. If |c|p = α = β and |a − c2|p < α2, then |f ′(x2)|p = +��� a−c2 +a +��� +p < 1. From this x2 is +attractive fixed point. Note that |x2|p = α. Let x ∈ Uα(x2) ⊂ Sα(0). Then from equality +(3.3) and using the strong triangle inequality of the p-adic norm we derive the relation +|f(x) − x2|p < |x − x2|p for all x ∈ Uα(x2). Similarly, if x /∈ Uα(x2), then we have the +relation |f(x) − x2|p ≥ α. +Note that, the set of valuations of p-adic norm is {pm| m ∈ Z}. +Thus, the relation +|f(x) − x2|p < |x − x2|p is equivalent to the relation |f(x) − x2|p ≤ 1 +p|x − x2|p. This means +that the map f : Uα(x2) → Uα(x2) is a contraction map. According to the properties of +contraction map, we have the equality A(x2) = Uα(x2). +5. If α < β, then we have |x2|p = β, i.e., x2 ∈ Sβ(0). Also, |f ′(x2)|p = +���1 − c2 +a +��� +p = β +α > 1. +Let x ∈ Sr(x2), i.e., |x − x2|p = r. Then from the equality (3.3) we get +|f(x) − x2|p = + + + + + + + + + + + + + + + + + +β +α|x − x2|p, +if r < α, +≥ β, +if r = α, +β, +if α < r < β, +≤ β, +if r = β, +β, +if r > β. +From this we conclude that the inequality |f(x)−x2|p > |x−x2|p is holds for all x ∈ Uβ(x2), +x ̸= x2. +□ +Corollary 1. • The spheres Sr(x1) is invariant for f if and only if r < α. +• The spheres Sr(x2) is invariant for f if and only if one of the statements holds +a) |c|p < α = β and r < α; +b) |c|p = α = β, |a − c2|p = α2 and r < α. + +6 +I.A. SATTAROV, E.T. ALIEV +4. Ergodicity of the dynamical systems on invariant spheres +Recall that an invariant measure is a measure that is preserved by some function. In +ergodic theory of dynamical systems an invariant measure is very important . +Let G be a topological group. If G is abelian and locally compact, then it is well known +[7] that it has a nonzero translation-invariant measure µ, which is unique up to scalar. This +is called the Haar measure. +In the field of p-adic numbers let Σ be the minimal σ-algebra containing all open and +closed (clopen) subsets. +A measure µ(Vρ) = ρ, Vρ ∈ Σ is usually called a Haar measure, where Vρ is a ball with +radius ρ. +However, in some cases, the problem of studying the dynamical system of a function that +mapping a compact subset of Qp to itself arises. At this time, is needed a measure defined +on σ-algebra with the unit a compact set. If this compact set has some algebraic structure, +then can we look at the natural Haar measure? If the considered compact set is a ball or a +sphere, the answer to this question is positive, which is given as follows in [16]. +Let Vr(a) be the ball (Sr(a) be the sphere) with the center at the point a ∈ Qp and B is +the algebra generated by clopen subsets of Vr(a) (Sr(a)). It is known that every element of +B is a union of some balls Vρ(s) ⊂ Vr(a), s ∈ Vr(a) (Vρ(s) ⊂ Sr(a), s ∈ Sr(a)). +Theorem 2. [16] A measure ¯µ : B → pZ is a Haar measure if it is defined by ¯µ(Vρ(s)) = ρ +for all Vρ(s) ∈ B. +Also, ergodic theory often deals with ergodic transformations. Here is the definition: +Definition 1. [18] Let T : X → X be a measure-preserving transformation on a measure +space (X, Σ, µ), with µ(X) = 1. Then T is ergodic if for every E in Σ with T −1(E) = E, +either µ(E) = 0 or µ(E) = 1. +In this section we are interested in ergodicity (with respect to Haar measure) of the +dynamical systems on invariant spheres with the center at the fixed point.. +Remark 2. Corollary 1 in the previous section gives a classification of invariant spheres +centered at a fixed point. Also, in part 2 of Theorem 1, it is proved that maximal Siegel discs +consisting of union of invariant spheres fall on top of each other. Therefore, the center of +invariant spheres is not significant when |c|p < α = β. However, when |c|p = α = β, it +is necessary to consider separately the ergodicity of dynamical systems in invariant spheres +with centers x1 and x2. +For each invariant sphere we consider a measurable space (Sr(xi), B), here B is the algebra +generated by closed subsets of Sr(xi), i = 1, 2. Every element of B is a union of some balls +Vρ(s) ⊂ Sr(xi). +A measure ¯µ : B → R is a Haar measure if it is defined by ¯µ(Vρ(s)) = ρ for all s ∈ Sr(xi) +and ρ ∈ pZ such that Vρ(s) ⊂ Sr(xi). +Note that Sr(xi) = Vr(xi) \ V r +p (xi). So, we have ¯µ(Sr(xi)) = r(1 − 1 +p). + +ERGODICITY AND PERIODIC ORBITS +7 +We consider normalized (probability) Haar measure: +µ(Vρ(s)) = ¯µ(Vρ(s)) +¯µ(Sr(xi)) = +pρ +(p − 1)r. +Theorem 3. Let Sr(xi), i = 1, 2 be invariant sphere for the function f given by (2.3). +Then the function f : Sr(xi) → Sr(xi) is an isometry. +Proof. By the Corollary 1, if the sphere Sr(xi), i = 1, 2 is invariant for (2.3), then r < α. +Let i = 1. From relation x, y ∈ Sr(x1) we have |x|p = |y|p = r. Then, we get the following +|f(x) − f(y)|p = |x − y|p · +|a|p|a − xy|p +|(x − ˆx1)(x − ˆx2)(y − ˆx1)(y − ˆx2)|p +. +(4.1) +Note that |a|p = αβ and |x|p = |y|p = r < α ≤ β. Then, +|f(x) − f(y)|p = |x − y|p · α2β2 +α2β2 = |x − y|p. +Consequently, the function f : Sr(x1) → Sr(x1) is an isometry. +Let i = 2. Then by Corollary 1 we have two cases. If |c|p < α = β , then by Remark +2, this case overlaps with case i = 1. If |c|p = α = β and |a − c2|p = α2, then by part 3 +of Theorem 1, we have the relation Sr(x2) ⊂ Sα(0) for all invariant sphere. So, we have +|x − x2|p = r < α and |x|p = α for all x ∈ Sr(x2). +Let x, y ∈ Sr(x2). Then +|f(x) − f(y)|p = |x − y|p · +|a|p|(a − x2 +2) + x2(x2 − y) + y(x2 − x)|p +|[(x − x2) + ˆx1][(x − x2) + ˆx2][(y − x2) + ˆx1][(y − x2) + ˆx2]|p +. +Note that |a|p = α2, |x − x2|p = |y − x2|p = r < α and |a − x2 +2|p = |a − c2|p = α2. Then, +|f(x) − f(y)|p = |x − y|p · α4 +α4 = |x − y|p. +Consequently, the function f : Sr(x2) → Sr(x2) is an isometry. +□ +Corollary 2. Let the conditions of the above theorem be satisfied. Then f : Sr(xi) → Sr(xi), +i = 1, 2 is a measure-preserving transformation on a measure space (Sr(xi), B, µ), where µ +is a normalized Haar measure. +In [16], given an important results about the dynamics of isometric maps, and since the +function (2.3) we are considering is also an isometry, the results obtained in [16] are also +relevant for the dynamics of the function (2.3), i.e., if Sr(xi), i = 1, 2 is invariant sphere for +the function f given by (2.3), then we have the following: +• The function f : Sr(xi) → Sr(xi), i = 1, 2 is bijection. +• For any initial point x ∈ Sr(xi), i = 1, 2 (except fixed point) the orbit {f n(x)| n ∈ N} +isn’t convergent. +The result of the following Lemma is given as a condition in [16]. Let Sr(xi), i = 1, 2 be +invariant sphere for the function f given by (2.3), then we denote ρ(r, x) = |f(x) − x|p for +x ∈ Sr(xi). + +8 +I.A. SATTAROV, E.T. ALIEV +Lemma 1. If r ̸= |c|p, then for the function f given by (2.3) the value ρ(r, x) does not +depend to x. +Proof. We consider all cases in Corollary 1. Let i = 1. Then r < α. By simple calculation +we get +ρ(r, x) = +���� +ax +x2 + cx + a − x +���� +p += |x|2 +p · +|x + c|p +|x − ˆx1|p|x − ˆx2|p += + + + +r2|c|p +αβ , +if r < |c|p, +r3 +αβ, +if r > |c|p. +Let i = 2. In this case, according to Remark 2, it is sufficient to prove the Lemma when +|c|p = α = β. So, we have r = |x − x2|p = |x + c|p < α and +ρ(r, x) = +���� +ax +x2 + cx + a − x +���� +p += |x + c|p · +|(x + c) − c|2 +p +|(x + c) + ˆx1|p|(x + c) + ˆx2|p += r. +□ +So, we denote ρ(r) = |f(x) − x|p for all x ∈ Sr(xi), i = 1, 2, r ̸= |c|p. In that case, we +have the following assertions from [16]: +• The ball with radius ρ(r) is minimal invariant ball for f : Sr(xi) → Sr(xi), i = 1, 2, +r ̸= |c|p. +• Let µ be normalized Haar measure on Sr(xi). Then +a) the dynamical system (Sr(xi), f, µ) is not ergodic for all p ≥ 3; +b) the dynamical system (Sr(xi), f, µ) may be ergodic if and only if r = 2ρ(r) for +p = 2. +Let p = 2. +Then according to the above the dynamical system (Sr(x2), f, µ) is not +ergodic, because r = ρ(r) for i = 2. +If i = 1, then x1 = 0 and we consider the dynamical system (Sr(0), f, µ). +Recall Z2 = {x ∈ Q2 : |x|2 ≤ 1}. So we have 1 + 2Z2 = S1(0). The following theorem +gives a criterion of ergodicity for the rational functions mapping S1(0) to itself: +Theorem 4. [9] Let f, g : 1 + 2Z2 → 1 + 2Z2 be polynomials whose coefficients are 2-adic +integers. +Set f(x) = � +i aixi, g(x) = � +i bixi, and +A1 = +� +i odd +ai, +A2 = +� +i even +ai, +B1 = +� +i odd +bi, +B2 = +� +i even +bi. +The rational function R = +f +g is ergodic if and only if one of the following situations +occurs: +(1) A1 = 1(mod 4), A2 = 2(mod 4), B1 = 0(mod 4) and B2 = 1(mod 4). +(2) A1 = 3(mod4), A2 = 2(mod 4), B1 = 0(mod 4) and B2 = 3(mod 4). +(3) A1 = 1(mod 4), A2 = 0(mod 4), B1 = 2(mod 4) and B2 = 1(mod 4). +(4) A1 = 3(mod 4), A2 = 0(mod 4), B1 = 2(mod 4) and B2 = 3(mod 4). +(5) One of the previous cases with f and g interchanged. + +ERGODICITY AND PERIODIC ORBITS +9 +Consider x = g(t) = r−1t for t ∈ S1(0), then g−1 ◦ f ◦ g : S1(0) → S1(0). Let B (resp. +B1) be the algebra generated by closed subsets of Sr(0) (resp. S1(0)), and µ (resp. µ1) be +normalized Haar measure on B (resp. B1). +Theorem 5. [14] The dynamical system (Sr(0), f, µ) is ergodic if and only if +(S1(0), g−1 ◦ f ◦ g, µ1) is ergodic. +Now using the above mentioned results for (2.3) when p = 2 and we prove the following +Theorem 6. Let p = 2. Then the dynamical system (Sr(0), f, µ) is ergodic if and only if +|c|2 = β and r = α +2 . +Proof. Let r = 2l, α = 2m, β = 2k and |c|2 = 2q. Since α ≤ β we have m ≤ k. Also, since +c = −ˆx1 − ˆx2 and a = ˆx1ˆx2 we have q ≤ k and |a|2 = 2m+k. +Note that the sphere S2l(0) is invariant for f iff l < m. +We consider the function +g : S1(0) → Sr(0) defined by x = g(t) = 2−lt. Note that the function +g−1(f(g(t))) : S1(0) → S1(0) has the following form +g−1(f(g(t))) = +t +2−2l +a t2 + 2−lc +a t + 1 +, +(4.2) +for the function f given by (2.3). Note that k, l, m, q ∈ Z, l < m ≤ k and q ≤ k. So we +have the inequalities l − m ≤ −1 and l − k ≤ −1. In (4.2) we can easily see the following +���� +2−2l +a t2 +���� +2 += 22l−(m+k) ≤ 2−2, +���� +2−lc +a t +���� +2 += 2l+q−(m+k) ≤ 2−1. +Consequently, +t =: γ1(t), +is such that γ1 : 1 + 2Z2 → 1 + 2Z2 +and +2−2l +a t2 + 2−lc +a t + 1 =: γ2(t) is such that γ2 : 1 + 2Z2 → 1 + 2Z2. +Hence the function (4.2) satisfies all condition of Theorem 4, therefore using this theorem, +we get +A1 = 1, +A2 = 0, +B1 = 2−lc +a +and B2 = 2−2l +a ++ 1. +Moreover, +A1 = 1(mod 4), +A2 = 0(mod 4), +B1 ∈ 2m+k−(l+q)(1 + 2Z2) and B2 = 1(mod 4). +By these relations and Theorem 4 we get m+k−(l+q) = (m−l)+(k−q) = 1. Note that +l < m and q ≤ k. Therefore we conclude that the dynamical system (S1(0), g−1 ◦ f ◦ g, µ1) +is ergodic if and only if q = k and l = m − 1, i.e., |c|2 = β and r = α +2 . Consequently, by +Theorem 5, (Sr(0), f, µ) is ergodic if and only if |c|2 = β and r = α +2 . +□ + +10 +I.A. SATTAROV, E.T. ALIEV +5. Periodic orbits +In this section we are interested in periodic trajectories and their characteristics. Since +our function is an isometry on an invariant sphere, we get the following result about periodic +trajectories from [16]: +Theorem 7. If the dynamical system (Sr(xi), f), i = 1, 2 has n-periodic orbit +y0 → y1 → ... → yn → y0, +then the following statements hold: +1. yk ∈ Vρ(r)(y0) for all k ∈ {1, 2, ..., n}; +2. Character of periodic points is indifferent; +3. If ρ ≤ ρ(r), then we have f(Sρ(yk)) ⊂ Sρ(yk+1) for any k ∈ {0, 1, ...n − 1} and +f(Sρ(yn)) ⊂ Sρ(y0). +Now we prove the following theorems about the existence of 2-periodic and 3-periodic +trajectories: +Theorem 8. If +√ +c2 − 2a ∈ Qp, then the function (2.3) has unique 2-periodic orbit {t1, t2}, +where t1,2 = −c ± +√ +c2 − 2a. +Proof. We consider the equation +f 2(x) − x +f(x) − x = 0. +Then we obtain the following +(x2 + 2cx + 2a)(x2 + cx + a) = 0. +Since x2 + cx + a ̸= 0, we get x2 + 2cx + 2a = 0, and t1,2 = −c ± +√ +c2 − 2a. +□ +Theorem 9. Let Sr(xi), i = 1, 2 be invariant sphere for (2.3) and assume that the param- +eter a ∈ Sr(xi). Then the function (2.3) has 3-periodic orbit +� +a, f(a), f 2(a) +� +if and only +if +(a, c) ∈ +� +(h(q), qh(q) − 1) : q ∈ Qp \ +� +0, −1, −2 +3 +� +, |h(q)|p = r +� +, +for i = 1, +(5.1) +(a, c) ∈ +� +(h(q), qh(q) − 1) : q ∈ Qp \ +� +0, −1, −2 +3 +� +, |h(q)(q + 1) − 1|p = r +� +, +for i = 2, +(5.2) +where h(q) = +3q2+2q +6q3+11q2+6q+1. +Proof. We consider the equation +f 3(x) − x +f(x) − x = 0. +By simplifying this equation, we get the following equation +P(x) = x6 + 6cx5 + (11c2 + 6a)x4 + (6c3 + 20ac)x3 + (15ac2 + 9a2)x2 + 12a2cx + 3a3 = 0. + +ERGODICITY AND PERIODIC ORBITS +11 +Necessity. Let a ∈ Sr(xi) be a 3-periodic point. Then P(a) = 0 and from this we have the +equality +a3 + 6(c + 1)a2 + (11c + 9)(c + 1)a + 3(2c + 1)(c + 1)2 = 0. +(5.3) +According to equality (5.3), since a ̸= 0, we have c ̸= −1. Denote +q = c + 1 +a +. +Then by (5.3) we get (6q3 + 11q2 + 6q + 1)a − (3q2 + 2q) = 0. +If we denote +a := h(q) = +3q2 + 2q +6q3 + 11q2 + 6q + 1, +then c = qh(q) − 1. Notice that h(q) is undefined at q = −1. Applying the conditions that +a(c + 1) ̸= 0 we see that q ̸= 0 and q ̸= − 2 +3. +For i = 1, we have |a|p = |h(q)|p = r, analogically for i = 2 we have +|a + c|p = |h(q)(q + 1) − 1|p = r. Summarizing the above, we get (5.1) and (5.2). +Sufficiency. +Let conditions (5.1) and (5.2) be satisfied. +Then it is easy to see that +P(a) = 0. Hence, a ∈ Sr(xi) is 3-periodic point for f given by (2.3). +□ +6. Availability of data +The datasets supporting the conclusions of this article are included in the article. +Acknowledgements +We thank our supervisor U.A. Rozikov for the useful discussions. +References +[1] S. 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Chaos, Solitons and Fractals, 105 (2017), 260–270. +[15] U.A. Rozikov, I.A. Sattarov. S. Yam. p-adic dynamical systems of the function +ax +x2 + a. p-Adic Numbers, +Ultrametric Analysis and Applications, 11(1) (2019), 77–87. +[16] I.A. Sattarov. Group structure of the p-adic ball and dynamical system of isometry on a sphere. +arXiv:2208.03513, doi.org/10.48550/arXiv.2208.03513 +[17] I.A. Sattarov. p-adic (3, 2)-rational dynamical systems. p-Adic Numbers, Ultrametric Analysis and +Applications, 7(1) (2015), 39–55. +[18] P.Walters, An introduction to ergodic theory. Springer, Berlin-Heidelberg-New York, (1982). +I. A. Sattarov, Namangan Satate University, 316, Uychi str., 160100, Namangan, Uzbekistan. +Email address: sattarovi-a@yandex.ru +E. T. Aliev, Namangan Institute of Engineering Technology, 7, Kosonsoy str., 160115, +Namangan, Uzbekistan. +Email address: aliev-erkinjon@mail.ru + diff --git a/EdE1T4oBgHgl3EQfEgNA/content/tmp_files/load_file.txt b/EdE1T4oBgHgl3EQfEgNA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..851b93ed8d928dbff32465c37f3426ca3d489d9c --- /dev/null +++ b/EdE1T4oBgHgl3EQfEgNA/content/tmp_files/load_file.txt @@ -0,0 +1,552 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf,len=551 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='02890v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='DS] 7 Jan 2023 ERGODICITY AND PERIODIC ORBITS OF p-ADIC (1, 2)-RATIONAL DYNAMICAL SYSTEMS WITH TWO FIXED POINTS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' SATTAROV, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ALIEV Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We consider (1, 2)-rational functions given on the field of p-adic numbers Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In general, such a function has four parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We study the case when such a function has two fixed points and show that when there are two fixed points then (1, 2)-rational function is conjugate to a two-parametric (1, 2)-rational function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Depending on these two parameters we determine type of the fixed points, find Siegel disks and the basin of attraction of the fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Moreover, we classify invariant sets and study ergodicity properties of the function on each invariant set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We describe 2- and 3-periodic orbits of the p-adic dynamical systems generated by the two-parametric (1, 2)-rational functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Introduction and preliminaries A function is called a (n, m)-rational function if and only if it can be written in the form f(x) = Pn(x) Qm(x), where Pn(x) and Qm(x) are polynomial functions with degree n and m respectively, Qm(x) is non zero polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In this paper we study dynamical systems generated by a (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2-)rational function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Our investigations based on methods of [1], [3], [13]-[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' For motivations of the study see [2], [4]-[6], [10]-[12] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let us give main definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let Q be the field of rational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The greatest common divisor of the positive integers n and m is denotes by (n, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Every rational number x ̸= 0 can be represented in the form x = pr n m, where r, n ∈ Z, m is a positive integer, (p, n) = 1, (p, m) = 1 and p is a fixed prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The p-adic norm of x ∈ Q is given by |x|p = � p−r, for x ̸= 0, 0, for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' It has the following properties: 1) |x|p ≥ 0 and |x|p = 0 if and only if x = 0, 2) |xy|p = |x|p|y|p, 3) the strong triangle inequality |x + y|p ≤ max{|x|p, |y|p}, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) if |x|p ̸= |y|p then |x + y|p = max{|x|p, |y|p}, 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 46S10, 12J12, 11S99, 30D05, 54H20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Rational dynamical systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' fixed point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' invariant set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Siegel disk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' complex p-adic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 1 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' SATTAROV, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ALIEV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) if |x|p = |y|p then |x + y|p ≤ |x|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The completion of Q with respect to p-adic norm defines the p-adic field which is denoted by Qp (see [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' For any a ∈ Qp and r > 0 denote Ur(a) = {x ∈ Qp : |x − a|p < r}, Vr(a) = {x ∈ Qp : |x − a|p ≤ r}, Sr(a) = {x ∈ Qp : |x − a|p = r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' A function f : Ur(a) → Qp is said to be analytic if it can be represented by f(x) = ∞ � n=0 fn(x − a)n, fn ∈ Qp, which converges uniformly on the ball Ur(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Now let f : U → U be an analytic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Denote f n(x) = f ◦ · · · ◦ f � �� � n (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If f(x0) = x0 then x0 is called a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The set of all fixed points of f is denoted by Fix(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' A fixed point x0 is called an attractor if there exists a neighborhood U(x0) of x0 such that for all points x ∈ U(x0) it holds lim n→∞ f n(x) = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If x0 is an attractor then its basin of attraction is A(x0) = {x ∈ Qp : f n(x) → x0, n → ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' A fixed point x0 is called repeller if there exists a neighborhood U(x0) of x0 such that |f(x) − x0|p > |x − x0|p for x ∈ U(x0), x ̸= x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let x0 be a fixed point of a function f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Put λ = f ′(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The point x0 is attractive if 0 < |λ|p < 1, indifferent if |λ|p = 1, and repelling if |λ|p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The ball Ur(x0) is said to be a Siegel disk if each sphere Sρ(x0), ρ < r is an invariant sphere of f(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' if x ∈ Sρ(x0) then all iterated points f n(x) ∈ Sρ(x0) for all n = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The union of all Siegel desks with the center at x0 is said to a maximum Siegel disk and is denoted by SI(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let f : A → A and g : B → B be two maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' f and g are said to be topologically conjugate if there exists a homeomorphism h : A → B such that, h◦f = g ◦h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The homeomorphism h is called a topological conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Mappings which are topologically conjugate are completely equivalent in terms of their dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In this paper we consider (1, 2)-rational function f : Qp → Qp defined by f(x) = ax + b x2 + cx + d, x ̸= ˆx1,2 = −c ± √ c2 − 4d 2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) where the parameters of the function satisfy the following conditions a ̸= 0, a, b, c, d, � c2 − 4d ∈ Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We study p-adic dynamical systems generated by the rational function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The equa- tion f(x) = x for fixed points of the function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) is equivalent to the equation x3 + cx2 + (d − a)x − b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) The equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) may have three solutions with one of the following relations: ERGODICITY AND PERIODIC ORBITS 3 (i) one solution having multiplicity three;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (ii) two solutions, one of which has multiplicity two;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (iii) three distinct solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Since the behavior of dynamical system depends on the set of fixed points, each of the above mentioned case (i)-(iii) has its own character of dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In [15] the case (i) was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In this paper we consider the case (ii), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', we investigate the behavior of the trajectories of an arbitrary (1, 2)-rational dynamical system in Qp when there are two fixed points for f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The case (iii) will be considered in a separate paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In Section 2 under some assumptions we show that four-parametric function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) is conjugate to a two-parametric (1,2)-rational function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In Section 3 we study the p-adic dynamics generated by the two-parametric function and give Siegel disks, the basin of attractions and classification of all invariant sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In Section 4 we investigate ergodicity of this dynamical systems on invariant sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In Section 5 we describe 2- and 3-periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' A function conjugate to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) Denote by x1 and x2 the two solutions of the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2), where x2 has multiplicity two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then we have x3 + cx2 + (d − a)x − b = (x − x1)(x − x2)2 and \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 x1 + 2x2 = −c x2 2 + 2x1x2 = d − a x1x2 2 = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) Let homeomorphism h : Qp → Qp be defined by h(t) = t+x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We note that, the function f is topologically conjugate to function h−1 ◦ f ◦ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We have (h−1 ◦ f ◦ h)(t) = −x2t2 + Bt t2 + Dt + B , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) where B = x2 2 + cx2 + d and D = 2x2 + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In [13] the case x2 ̸= 0 is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Thus in this paper we consider the case x2 = 0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If x2 = 0, then B = d = a and D = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Thus we have the following proposition Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Any (1,2)-rational function having two distinct fixed points is topologically conjugate to one of the following functions f(x) = ax2 + bx x2 + cx + b, ab(a − c) ̸= 0, a, b, c ∈ Qp, and f(x) = ax x2 + cx + a, ac ̸= 0, a, c, ∈ Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) where x ̸= ˆx1,2 = −c± √ c2−4a 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We study the dynamical system (Qp, f) with f given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 4 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' SATTAROV, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ALIEV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' p-Adic dynamics of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) Note that, the function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) has two fixed points x1 = 0 and x2 = −c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We have f ′(x1) = 1 and f ′(x2) = 1 − c2 a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Thus, the point x1 is an indifferent point for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', x1 is a center of some Siegel disk SI(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In this section we determine the character of the fixed point x2 for each cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then we find Siegel disk or basin of attraction of the fixed point x2, when x2 is indifferent or attractive, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In the case where x2 is repelling, we find open ball Ur(x2), such that the inequality |f(x) − x2|p > |x − x2|p holds for all x ∈ Ur(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Moreover, we study a relation between the sets SI(x1) and SI(x2) when x2 is an indifferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' For any x ∈ Qp, x ̸= ˆx1,2, by simple calculations we get |f(x)|p = |x|p · |a|p |x − ˆx1|p|x − ˆx2|p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) Denote P = {x ∈ Qp : ∃n ∈ N ∪ {0}, f n(x) ∈ {ˆx1, ˆx2}}, α = min{|ˆx1|p, |ˆx2|p} and β = max{|ˆx1|p, |ˆx2|p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) Since ˆx1 + ˆx2 = −c, we have |c|p ≤ α for α = β and |c|p = β for α < β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Also, since ˆx1ˆx2 = a, we have |a|p = αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The p-adic dynamical system generated by the function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) has the following properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' SI(x1) = Uα(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If |c|p < α = β, then x2 is indifferent fixed point for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) and SI(x2) = SI(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If |c|p = α = β and |a − c2|p = α2, then x2 is indifferent fixed point for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) and SI(x2) = Uα(x2), SI(x2) ∩ SI(x1) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If |c|p = α = β and |a − c2|p < α2, then x2 is attractive fixed point for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) and A(x2) = Uα(x2) ⊂ Sα(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If α < β, then x2 ∈ Sβ(0) is repelling fixed point for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) and the inequality |f(x) − x2|p > |x − x2|p holds for all x ∈ Uβ(x2), x ̸= x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let x ∈ Sr(x1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', |x|p = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then, from the equalities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) and the properties of the p-adic norm, we have the following |f(x)|p = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 r, if r < α, ≥ α, if α ≤ r ≤ β, |a|p r , if r > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' From this equality, f(Sr(x1)) ⊂ Sr(x1) for arbitrary r < α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' we have SI(x1) = Uα(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ERGODICITY AND PERIODIC ORBITS 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that |a|p = αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If |c|p < α = β, then |f ′(x2)|p = ���1 − c2 a ��� p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' From this x2 is indifferent fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let x ∈ Sr(x2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', |x − x2|p = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then from the equality |f(x) − x2|p = |x − x2|p · |x2(x − x2) + (x2 2 − a)|p |(x − x2) + ˆx1|p|(x − x2) + ˆx2|p (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) we have |f(x) − x2|p = r for all r < α and |f(x) − x2|p ≥ r for r = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Thus, f(Sr(x2)) ⊂ Sr(x2) for arbitrary r < α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' we have SI(x2) = Uα(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In this case, we have |x2|p = |c|p < α, so x2 ∈ Uα(0) = SI(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Since these two Siegel disks have the same radii and share a common point, they are the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', SI(x2) = SI(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If |c|p = α = β and |a − c2|p = α2, then |f ′(x2)|p = ��� a−c2 a ��� p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' From this x2 is indifferent fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' As above, from equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) we get SI(x2) = Uα(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' However, in this case x2 ∈ Sα(0), so SI(x2) ∩ SI(x1) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If |c|p = α = β and |a − c2|p < α2, then |f ′(x2)|p = ��� a−c2 a ��� p < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' From this x2 is attractive fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that |x2|p = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let x ∈ Uα(x2) ⊂ Sα(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then from equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) and using the strong triangle inequality of the p-adic norm we derive the relation |f(x) − x2|p < |x − x2|p for all x ∈ Uα(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Similarly, if x /∈ Uα(x2), then we have the relation |f(x) − x2|p ≥ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that, the set of valuations of p-adic norm is {pm| m ∈ Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Thus, the relation |f(x) − x2|p < |x − x2|p is equivalent to the relation |f(x) − x2|p ≤ 1 p|x − x2|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' This means that the map f : Uα(x2) → Uα(x2) is a contraction map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' According to the properties of contraction map, we have the equality A(x2) = Uα(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If α < β, then we have |x2|p = β, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', x2 ∈ Sβ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Also, |f ′(x2)|p = ���1 − c2 a ��� p = β α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let x ∈ Sr(x2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', |x − x2|p = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then from the equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) we get |f(x) − x2|p = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 β α|x − x2|p, if r < α, ≥ β, if r = α, β, if α < r < β, ≤ β, if r = β, β, if r > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' From this we conclude that the inequality |f(x)−x2|p > |x−x2|p is holds for all x ∈ Uβ(x2), x ̸= x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' □ Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' • The spheres Sr(x1) is invariant for f if and only if r < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The spheres Sr(x2) is invariant for f if and only if one of the statements holds a) |c|p < α = β and r < α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' b) |c|p = α = β, |a − c2|p = α2 and r < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 6 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' SATTAROV, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ALIEV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Ergodicity of the dynamical systems on invariant spheres Recall that an invariant measure is a measure that is preserved by some function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In ergodic theory of dynamical systems an invariant measure is very important .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let G be a topological group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If G is abelian and locally compact, then it is well known [7] that it has a nonzero translation-invariant measure µ, which is unique up to scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' This is called the Haar measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In the field of p-adic numbers let Σ be the minimal σ-algebra containing all open and closed (clopen) subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' A measure µ(Vρ) = ρ, Vρ ∈ Σ is usually called a Haar measure, where Vρ is a ball with radius ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' However, in some cases, the problem of studying the dynamical system of a function that mapping a compact subset of Qp to itself arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' At this time, is needed a measure defined on σ-algebra with the unit a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If this compact set has some algebraic structure, then can we look at the natural Haar measure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If the considered compact set is a ball or a sphere, the answer to this question is positive, which is given as follows in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let Vr(a) be the ball (Sr(a) be the sphere) with the center at the point a ∈ Qp and B is the algebra generated by clopen subsets of Vr(a) (Sr(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' It is known that every element of B is a union of some balls Vρ(s) ⊂ Vr(a), s ∈ Vr(a) (Vρ(s) ⊂ Sr(a), s ∈ Sr(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' [16] A measure ¯µ : B → pZ is a Haar measure if it is defined by ¯µ(Vρ(s)) = ρ for all Vρ(s) ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Also, ergodic theory often deals with ergodic transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Here is the definition: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' [18] Let T : X → X be a measure-preserving transformation on a measure space (X, Σ, µ), with µ(X) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then T is ergodic if for every E in Σ with T −1(E) = E, either µ(E) = 0 or µ(E) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In this section we are interested in ergodicity (with respect to Haar measure) of the dynamical systems on invariant spheres with the center at the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='. Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Corollary 1 in the previous section gives a classification of invariant spheres centered at a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Also, in part 2 of Theorem 1, it is proved that maximal Siegel discs consisting of union of invariant spheres fall on top of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Therefore, the center of invariant spheres is not significant when |c|p < α = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' However, when |c|p = α = β, it is necessary to consider separately the ergodicity of dynamical systems in invariant spheres with centers x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' For each invariant sphere we consider a measurable space (Sr(xi), B), here B is the algebra generated by closed subsets of Sr(xi), i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Every element of B is a union of some balls Vρ(s) ⊂ Sr(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' A measure ¯µ : B → R is a Haar measure if it is defined by ¯µ(Vρ(s)) = ρ for all s ∈ Sr(xi) and ρ ∈ pZ such that Vρ(s) ⊂ Sr(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that Sr(xi) = Vr(xi) \\ V r p (xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' So, we have ¯µ(Sr(xi)) = r(1 − 1 p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ERGODICITY AND PERIODIC ORBITS 7 We consider normalized (probability) Haar measure: µ(Vρ(s)) = ¯µ(Vρ(s)) ¯µ(Sr(xi)) = pρ (p − 1)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let Sr(xi), i = 1, 2 be invariant sphere for the function f given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then the function f : Sr(xi) → Sr(xi) is an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' By the Corollary 1, if the sphere Sr(xi), i = 1, 2 is invariant for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3), then r < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' From relation x, y ∈ Sr(x1) we have |x|p = |y|p = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then, we get the following |f(x) − f(y)|p = |x − y|p · |a|p|a − xy|p |(x − ˆx1)(x − ˆx2)(y − ˆx1)(y − ˆx2)|p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) Note that |a|p = αβ and |x|p = |y|p = r < α ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then, |f(x) − f(y)|p = |x − y|p · α2β2 α2β2 = |x − y|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Consequently, the function f : Sr(x1) → Sr(x1) is an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then by Corollary 1 we have two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If |c|p < α = β , then by Remark 2, this case overlaps with case i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If |c|p = α = β and |a − c2|p = α2, then by part 3 of Theorem 1, we have the relation Sr(x2) ⊂ Sα(0) for all invariant sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' So, we have |x − x2|p = r < α and |x|p = α for all x ∈ Sr(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let x, y ∈ Sr(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then |f(x) − f(y)|p = |x − y|p · |a|p|(a − x2 2) + x2(x2 − y) + y(x2 − x)|p |[(x − x2) + ˆx1][(x − x2) + ˆx2][(y − x2) + ˆx1][(y − x2) + ˆx2]|p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that |a|p = α2, |x − x2|p = |y − x2|p = r < α and |a − x2 2|p = |a − c2|p = α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then, |f(x) − f(y)|p = |x − y|p · α4 α4 = |x − y|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Consequently, the function f : Sr(x2) → Sr(x2) is an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let the conditions of the above theorem be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then f : Sr(xi) → Sr(xi), i = 1, 2 is a measure-preserving transformation on a measure space (Sr(xi), B, µ), where µ is a normalized Haar measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In [16], given an important results about the dynamics of isometric maps, and since the function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) we are considering is also an isometry, the results obtained in [16] are also relevant for the dynamics of the function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', if Sr(xi), i = 1, 2 is invariant sphere for the function f given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3), then we have the following: The function f : Sr(xi) → Sr(xi), i = 1, 2 is bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' For any initial point x ∈ Sr(xi), i = 1, 2 (except fixed point) the orbit {f n(x)| n ∈ N} isn’t convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The result of the following Lemma is given as a condition in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let Sr(xi), i = 1, 2 be invariant sphere for the function f given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3), then we denote ρ(r, x) = |f(x) − x|p for x ∈ Sr(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 8 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' SATTAROV, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ALIEV Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If r ̸= |c|p, then for the function f given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) the value ρ(r, x) does not depend to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We consider all cases in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then r < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' By simple calculation we get ρ(r, x) = ���� ax x2 + cx + a − x ���� p = |x|2 p · |x + c|p |x − ˆx1|p|x − ˆx2|p = \uf8f1 \uf8f2 \uf8f3 r2|c|p αβ , if r < |c|p, r3 αβ, if r > |c|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In this case, according to Remark 2, it is sufficient to prove the Lemma when |c|p = α = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' So, we have r = |x − x2|p = |x + c|p < α and ρ(r, x) = ���� ax x2 + cx + a − x ���� p = |x + c|p · |(x + c) − c|2 p |(x + c) + ˆx1|p|(x + c) + ˆx2|p = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' □ So, we denote ρ(r) = |f(x) − x|p for all x ∈ Sr(xi), i = 1, 2, r ̸= |c|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In that case, we have the following assertions from [16]: The ball with radius ρ(r) is minimal invariant ball for f : Sr(xi) → Sr(xi), i = 1, 2, r ̸= |c|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let µ be normalized Haar measure on Sr(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then a) the dynamical system (Sr(xi), f, µ) is not ergodic for all p ≥ 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' b) the dynamical system (Sr(xi), f, µ) may be ergodic if and only if r = 2ρ(r) for p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then according to the above the dynamical system (Sr(x2), f, µ) is not ergodic, because r = ρ(r) for i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If i = 1, then x1 = 0 and we consider the dynamical system (Sr(0), f, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Recall Z2 = {x ∈ Q2 : |x|2 ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' So we have 1 + 2Z2 = S1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The following theorem gives a criterion of ergodicity for the rational functions mapping S1(0) to itself: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' [9] Let f, g : 1 + 2Z2 → 1 + 2Z2 be polynomials whose coefficients are 2-adic integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Set f(x) = � i aixi, g(x) = � i bixi, and A1 = � i odd ai, A2 = � i even ai, B1 = � i odd bi, B2 = � i even bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The rational function R = f g is ergodic if and only if one of the following situations occurs: (1) A1 = 1(mod 4), A2 = 2(mod 4), B1 = 0(mod 4) and B2 = 1(mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (2) A1 = 3(mod4), A2 = 2(mod 4), B1 = 0(mod 4) and B2 = 3(mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (3) A1 = 1(mod 4), A2 = 0(mod 4), B1 = 2(mod 4) and B2 = 1(mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (4) A1 = 3(mod 4), A2 = 0(mod 4), B1 = 2(mod 4) and B2 = 3(mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (5) One of the previous cases with f and g interchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ERGODICITY AND PERIODIC ORBITS 9 Consider x = g(t) = r−1t for t ∈ S1(0), then g−1 ◦ f ◦ g : S1(0) → S1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' B1) be the algebra generated by closed subsets of Sr(0) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' S1(0)), and µ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' µ1) be normalized Haar measure on B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' [14] The dynamical system (Sr(0), f, µ) is ergodic if and only if (S1(0), g−1 ◦ f ◦ g, µ1) is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Now using the above mentioned results for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) when p = 2 and we prove the following Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then the dynamical system (Sr(0), f, µ) is ergodic if and only if |c|2 = β and r = α 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let r = 2l, α = 2m, β = 2k and |c|2 = 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Since α ≤ β we have m ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Also, since c = −ˆx1 − ˆx2 and a = ˆx1ˆx2 we have q ≤ k and |a|2 = 2m+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that the sphere S2l(0) is invariant for f iff l < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We consider the function g : S1(0) → Sr(0) defined by x = g(t) = 2−lt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that the function g−1(f(g(t))) : S1(0) → S1(0) has the following form g−1(f(g(t))) = t 2−2l a t2 + 2−lc a t + 1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) for the function f given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that k, l, m, q ∈ Z, l < m ≤ k and q ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' So we have the inequalities l − m ≤ −1 and l − k ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) we can easily see the following ���� 2−2l a t2 ���� 2 = 22l−(m+k) ≤ 2−2, ���� 2−lc a t ���� 2 = 2l+q−(m+k) ≤ 2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Consequently, t =: γ1(t), is such that γ1 : 1 + 2Z2 → 1 + 2Z2 and 2−2l a t2 + 2−lc a t + 1 =: γ2(t) is such that γ2 : 1 + 2Z2 → 1 + 2Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Hence the function (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) satisfies all condition of Theorem 4, therefore using this theorem, we get A1 = 1, A2 = 0, B1 = 2−lc a and B2 = 2−2l a + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Moreover, A1 = 1(mod 4), A2 = 0(mod 4), B1 ∈ 2m+k−(l+q)(1 + 2Z2) and B2 = 1(mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' By these relations and Theorem 4 we get m+k−(l+q) = (m−l)+(k−q) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that l < m and q ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Therefore we conclude that the dynamical system (S1(0), g−1 ◦ f ◦ g, µ1) is ergodic if and only if q = k and l = m − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', |c|2 = β and r = α 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Consequently, by Theorem 5, (Sr(0), f, µ) is ergodic if and only if |c|2 = β and r = α 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' □ 10 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' SATTAROV, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ALIEV 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Periodic orbits In this section we are interested in periodic trajectories and their characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Since our function is an isometry on an invariant sphere, we get the following result about periodic trajectories from [16]: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If the dynamical system (Sr(xi), f), i = 1, 2 has n-periodic orbit y0 → y1 → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' → yn → y0, then the following statements hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' yk ∈ Vρ(r)(y0) for all k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Character of periodic points is indifferent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If ρ ≤ ρ(r), then we have f(Sρ(yk)) ⊂ Sρ(yk+1) for any k ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='n − 1} and f(Sρ(yn)) ⊂ Sρ(y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Now we prove the following theorems about the existence of 2-periodic and 3-periodic trajectories: Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If √ c2 − 2a ∈ Qp, then the function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) has unique 2-periodic orbit {t1, t2}, where t1,2 = −c ± √ c2 − 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We consider the equation f 2(x) − x f(x) − x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then we obtain the following (x2 + 2cx + 2a)(x2 + cx + a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Since x2 + cx + a ̸= 0, we get x2 + 2cx + 2a = 0, and t1,2 = −c ± √ c2 − 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' □ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let Sr(xi), i = 1, 2 be invariant sphere for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) and assume that the param- eter a ∈ Sr(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then the function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) has 3-periodic orbit � a, f(a), f 2(a) � if and only if (a, c) ∈ � (h(q), qh(q) − 1) : q ∈ Qp \\ � 0, −1, −2 3 � , |h(q)|p = r � , for i = 1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) (a, c) ∈ � (h(q), qh(q) − 1) : q ∈ Qp \\ � 0, −1, −2 3 � , |h(q)(q + 1) − 1|p = r � , for i = 2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) where h(q) = 3q2+2q 6q3+11q2+6q+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We consider the equation f 3(x) − x f(x) − x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' By simplifying this equation, we get the following equation P(x) = x6 + 6cx5 + (11c2 + 6a)x4 + (6c3 + 20ac)x3 + (15ac2 + 9a2)x2 + 12a2cx + 3a3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ERGODICITY AND PERIODIC ORBITS 11 Necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let a ∈ Sr(xi) be a 3-periodic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then P(a) = 0 and from this we have the equality a3 + 6(c + 1)a2 + (11c + 9)(c + 1)a + 3(2c + 1)(c + 1)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) According to equality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3), since a ̸= 0, we have c ̸= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Denote q = c + 1 a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) we get (6q3 + 11q2 + 6q + 1)a − (3q2 + 2q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If we denote a := h(q) = 3q2 + 2q 6q3 + 11q2 + 6q + 1, then c = qh(q) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Notice that h(q) is undefined at q = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Applying the conditions that a(c + 1) ̸= 0 we see that q ̸= 0 and q ̸= − 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' For i = 1, we have |a|p = |h(q)|p = r, analogically for i = 2 we have |a + c|p = |h(q)(q + 1) − 1|p = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Summarizing the above, we get (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let conditions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then it is easy to see that P(a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Hence, a ∈ Sr(xi) is 3-periodic point for f given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Availability of data The datasets supporting the conclusions of this article are included in the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Acknowledgements We thank our supervisor U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Rozikov for the useful discussions.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', 160100, Namangan, Uzbekistan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Email address: sattarovi-a@yandex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='ru E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Aliev, Namangan Institute of Engineering Technology, 7, Kosonsoy str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', 160115, Namangan, Uzbekistan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Email address: aliev-erkinjon@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='ru' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} diff --git a/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf b/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6050b7e202e31edce55cccf160198db49f9c2654 --- /dev/null +++ b/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:96b5c80450a1e336a9a816f0afe2a072b72db8995a3aae59fa219a264ffeea1b +size 1804511 diff --git a/EtFJT4oBgHgl3EQfCSzh/vector_store/index.faiss b/EtFJT4oBgHgl3EQfCSzh/vector_store/index.faiss new file mode 100644 index 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b/FdAzT4oBgHgl3EQfxP4S/content/tmp_files/2301.01733v1.pdf.txt @@ -0,0 +1,1643 @@ +Enhancing the Accuracy of Density Functional Tight Binding Models Through +ChIMES Many-body Interaction Potentials +Nir Goldman,1, 2 Laurence E. Fried,1 Rebecca K. Lindsey,3 C. Huy Pham,1 and R. +Dettori1 +1)Physical and Life Sciences Directorate, Lawrence Livermore National Laboratory, +Livermore, CA 94550 USAa) +2)Department of Chemical Engineering, University of California, Davis, California 95616, +United States +3)Department of Chemical Engineering, University of Michigan, Ann Arbor, +Michigan 48109, United States +(Dated: 5 January 2023) +Semi-empirical quantum models such as Density Functional Tight Binding (DFTB) are +attractive methods for obtaining quantum simulation data at longer time and length scale +than possible with standard approaches. However, application of these models can require +lengthy effort due to the lack of a systematic approach for their development. In this work, +we discuss use of the Chebyshev Interaction Model for Efficient Simulation (ChIMES) to +create rapidly parameterized DFTB models which exhibit strong transferability due to the +inclusion of many-body interactions that might otherwise be underestimated. We apply +our modeling approach to silicon polymorphs and review previous work on titanium hy- +dride. We also review creation of a general purpose DFTB/ChIMES model for organic +molecules and compounds that approaches hybrid functional and coupled cluster accuracy +with two orders of magnitude fewer parameters than similar neural network approaches. +In all cases, DFTB/ChIMES yields similar accuracy to the underlying quantum method +with orders of magnitude improvement in computational cost. Our developments provide +a way to create computationally efficient and highly accurate semi-empirical models for +studies where physical and chemical properties can be difficult to interrogate directly and +there is historically a significant reliance on theoretical approaches for interpretation and +validation of experimental results. +a)Electronic mail: ngoldman@llnl.gov +1 +arXiv:2301.01733v1 [cond-mat.mtrl-sci] 4 Jan 2023 + +I. +INTRODUCTION +Atomistic calculation approaches for materials modeling can be used as an independent route +to aid in new materials synthesis1, characterizing mixtures for use as fuel2,3, or quantifying rates +for chemical decomposition of organic materials4. These types of studies generally rely on quan- +tum mechanical approaches such as Kohn-Sham Density Functional Theory (DFT) in order to aid +in experimental interpretation and/or new materials design. In particular, DFT has been shown ex- +tensively to yield accurate descriptions of condensed phase physical and chemical data, such as the +material equation of state under compressive or tensile loads5, heats of formation/mixture of new +phases6,7, and the energetics of chemical bond breaking and forming under reactive conditions8. +However, standard DFT is also renown for its significant computational expense and poor com- +putational scaling (generally O(N 3)) resulting from solving for the Kohn-Sham eigenstates. As a +result, DFT molecular dynamics (MD) simulations can be limited to system sizes of hundreds of +atoms for timescales of tens of picoseconds or smaller for many systems9. In contrast, many pro- +cesses of interest have properties that can span orders of magnitude larger scales, including large- +scale carbon heterocycle synthesis10, the rational design of 3D materials11, and defect formation +and grain boundary interactions in crystalline systems12. Thus, the need for alternate simulation +approaches remains a highly active research area where the goal is to develop methods that can +harness the accuracy of DFT while yielding vastly improved computational efficiency and scaling. +In this regard, machine learning approaches for the development of interatomic atomic po- +tentials have been an effective route for modeling materials under reactive and nonreactive +conditions13,14. For example, neural networks have been used successfully to model structural +properties of catalytic materials15 as well as the phase stability of high-entropy ceramics16. Gaus- +sian Process Regression in the form of the Gaussian Approximation Potential (GAP) has been +used for a number of materials, including silicon based materials17. Regardless, the development +of these potentials tends to remain a highly labor-intensive task, where frequently a high-degree +of expertise and months to years of human effort are required for a single application area. As +a result, it can be difficult for these efforts to keep up with experimental needs particularly in +the area of materials synthesis, where the number of permutations of different starting materials, +thermodynamic conditions, and catalysts can be combinatorially large. +Semi-empirical quantum mechanical approaches hold promise as a middle ground for acceler- +ated simulations with a high degree of accuracy. These methods combine approximate quantum +2 + +mechanics with empirical functions to yield approaches that can achieve several orders of magni- +tude longer time scales in quantum MD simulations.18,19 In addition, semi-empirical approaches +tend to utilize significantly fewer computational resources, allowing for ensembles of statistically +independent trajectories and improved statistical sampling of desired properties.20 These methods +also tend to show much stronger transferability to systems and conditions outside of their training +set compared to interatomic potentials, in part due to the accuracy of the approximate quantum +mechanics and subsequent reduced reliance on empirical functions.21 +Density Functional Tight Binding (DFTB) is one such semi-empirical quantum mechanical +method22,23 that has had widespread success in modeling both gas-phase molecules24 as well +as condensed matter under inert and reactive conditions25–27, including extreme pressures and +temperatures28,29. The DFTB total energy is derived from an expansion of the Kohn-Sham en- +ergy to either second or third-order in charge fluctuations, resulting in the following expression: +EDFTB = EBS + ECoul + Erep. +(1) +Here, EBS corresponds to the band structure energy, ECoul is the charge fluctuation term, and +Erep is the repulsive energy. EBS is calculated as a sum over occupied electronic states from the +DFTB Hamiltonian. The DFTB Hamiltonian matrix elements are determined from pre-tabulated +Slater-Koster tables derived from reference calculations with a minimal basis set. The onsite +matrix elements are the free-atom orbital energies and the off-site terms are computed with a two- +center approximation where both wavefunctions and electron density are subjected to confining +potentials. Erep corresponds to ion-ion repulsions, as well as Hartree and exchange-correlation +double counting terms. This term can be expressed as an empirical function where parameters +are fit to reproduce high-level quantum or experimental reference data. In practice, an additional +dispersion correction can be included, including those in standard use for DFT calculations30,31. +DFTB is approximately three orders of magnitude more efficient than DFT calculations though +it also tends to exhibit O(N 3) scaling due to the need to solve for the band structure eigenstates. +DFTB has been shown to exhibit transferability across element types and diverse conditions32–34 +and has been applied to a broad range of materials35–39. +However, DFTB model development can be challenging in terms of optimizing the hyperpa- +rameters needed for the approximate quantum mechanical parts of the calculations. These include +the separate confining potentials for the wavefunctions and electron density (which can be differ- +ent for each angular momentum channel of an element),38 choice of second-order vs. third-order +3 + +charge fluctuations for the energy expression40, and whether to use density or potential superposi- +tion when computing the Slater-Koster tables.36,41 The DFTB Hamiltonian tends to be highly sen- +sitive to these options42, and in general there does not exist a predefined recipe for how to choose +these parameters nor how to explore that specific phase space. Prediction of physical and/or chem- +ical properties are in turn are closely coupled to the empirical repulsive energy, which itself has a +wide variety of options in terms of functional form and data to be fit35. ERep is usually taken to be +strictly pairwise (two-center), though a number of systems can require many-body terms as well +for accurate predictions28. Novel approaches for determination of ERep include constrained spline +optimization34, neural networks43,44, and Gaussian Process Regression45,46. Machine learning ap- +proaches though tend to be highly data intensive14 and prone to overfitting21, which can pose dif- +ficulties for any method that leverages these techniques. Thus, DFTB method development would +be holistically improved through a more automatic method for parameterization, where candidate +models could be screened rapidly and efficiently, thereby allowing the user to quickly determine +an optimal model for their specific needs. +In this work, we discuss our recent efforts to overcome these issues through use of the Cheby- +shev Interaction Model for Efficient simulation (ChIMES),47,48 which can be used to determine +ERep for molecular and condensed phase systems relatively quickly and with comparatively lower +data requirements. ChIMES is a many-body reactive force field based on linear combinations +of Chebyshev polynomials. It was initially developed for pure MD simulation (i.e., where all +aspects of a quantum mechanical calculation have been mapped onto the ChIMES functional +form). This has included both non-reactive and reactive materials, such as water under ambi- +ent and high pressure-temperature conditions49,50, high pressure C/O systems51,52, and detonating +energetic materials53. DFTB/ChIMES models have been created for a wide variety of materials, +including actinides and their oxides54,55, titanium-based systems36, and silicon (discussed below). +Additionally, ChIMES has been used to improve the accuracy of DFTB by including many-body +energies and forces through ∆-learning, where ChIMES augments a pre-existing DFTB param- +eterization for organic materials under ambient56 and reactive conditions39. We note that similar +to other machine-learning methods21, ChIMES can be used within any semi-empirical quantum +mechanical approach. However, we choose to focus on DFTB due to its close resemblance to +Kohn-Sham DFT as well as its proven accuracy for a variety of materials and conditions. +We begin with a brief discussion of the ChIMES formalism, including discussion of its func- +tional form and methods for optimization. Next we present some recent results on a general pur- +4 + +pose DFTB/ChIMES model for silicon polymorphs, which has remained an outstanding issue in +DFTB model development. We note that all DFTB calculations discussed within this work were +performed with the DFTB+ code57,58. We then summarize previous work on a semi-automated +workflow for screening DFTB hyperparameters and ERep determination in creating a models for +TiH2, a candidate hydrogen storage material with several potential uses. Finally, we review our +recent results in using ChIMES to create DFTB models that approach hybrid-functional and cou- +pled cluster accuracy for organic compounds and molecular solids. In all cases, the advantages to +use of DFTB/ChIMES lies in its rapid parameterization time, small data requirements relative to +other machine-learned approaches, and the relative ease with which overfitting can be prevented +due to regularization within linear optimization approaches as well as the orthogonal nature of the +underlying basis set. +II. +METHODS +A. +ChIMES Formalism +The design philosophy behind ChIMES is based on a many-body expansion of the DFT total +energy. Briefly, the DFT total energy can be thought of as a sum of contributions of clusters +containing different numbers of atoms: +EDFT = +na +� +i1 +1Ei1+ +na +� +i1>i2 +2Ei1i2+ +na +� +i1>i2>i3 +3Ei1i2i3+ +na +� +i1>i2>i3>i4 +4Ei1i2i3i4+· · ·+ +na +� +i1>i2... inB−1>inB +nBEi1i2...inB. +(2) +Here, the one-body energies, 1Ei1, correspond to the atomic energy constants, the two-body ener- +gies, 2Ei1i2, to all pair-wise energies with indices {i1, i2}, the three-body energies, 3Ei1i2i3, to all +triplet energies with indicies {i1, i2, i3}, etc., all the way up to some predeterimed maximum bod- +iedness, nB. These terms are summed over all cluster combinations within the system containing +na total number of atoms. +In the ChIMES formalism, we represent each of the terms in our n-body expansion as a lin- +ear combination of Chebyshev polynomials. Chebyshev polynomials of the first kind of order m +are defined by the expression Tm (cos θ) = cos (mθ), more commonly written as Tm(x), where +x = cos θ and thus exists over the range [−1, 1]. Chebyshev polynomials offer a number of dis- +tinct advantages for interpolation that bear mentioning. Chebyshev polynomials of the first kind +5 + +are orthogonal with respect to the weighting function 1/ +√ +1 − x2. They can be computed with +a recurrence relationship and define a complete basis set, allowing for arbitrary complexity in a +potential energy surface. Their orthogonality allows for simple regularization where higher-order +polynomial coefficients can be set to zero without necessarily adversely affecting the quality of +the optimization. Polynomial expansions with Chebyshev polynomials of the first kind will have +exponentially decreasing coefficients for higher-order terms due to their monic form, helping to +prevent overfitting. In addition, they yield a “nearly optimal” error function, where the error in +an expansion will closely resemble a minimax polynomial. The derivaties of Chebyshev polyno- +mials of the first kind are related to Chebyshev polynomials of the kind Um(x) by the expression +dTm/dx = mUm−1, where Um (cos θ) = sin [(n + 1) θ] /sin θ. Chebyshev polynomials of the +second kind also form an orthogonal basis set (with respect to the weighting function +√ +1 − x2) +and can also be generated via a recurrence relation. This can allow for arbitrary complexity for +structural optimization or molecular dynamics calculations, where atomic forces are needed. +As a result, we can now write the two-body (2B) energy term in Equation 2 as the following +expression: +2Ei1i2 = fp (ri1i2) + f +ei1ei2 +c +(ri1i2) +O2 +� +m=1 +C +ei1ei2 +m +Tm(s +ei1ei2 +i1i2 ) +(3) +In this case, C +ei1ei2 +m +is the corresponding permutationally invariant coefficient for the interaction +between atom types ei1 and ei2, taken from the set of all possible element types, {e}. Tm +� +s +ei1ei2 +i1i2 +� +represents a Chebyshev polynomial of order m, and s +ei1ei2 +i1i2 +is the pair distance transformed to +occur over the interval [−1, 1] using a Morse-like function59,60. For that coordinate transform, +s +ei1ei2 +i1i2 +∝ exp (−ri1i2/λe1e2) and λe1e2 is an element-pair distance scaling constant, usually taken +to be the peak position of the first coordination shell. Further details are discussed in Ref. 47. The +term f +ei1ei2 +c +(ri1i2) is a Tersoff cutoff function61 which is set to zero beyond a maximum distance +defined for a given {e1, e2} pair set. In order to prevent sampling of ri1i2 distances below what is +sampled in our DFT training set, we introduce use of a smooth penalty function fp(ri1i2). +We can create greater than two-body orthogonal polynomials by defining a cluster of size n and +taking the product of the Chebyshev polynomials derived from the constituent +�n +2 +� +unique pairs. +For example, the three-body polynomials will be products of +�3 +2 +� += 3 two-body polynomials. We +thus write the ChIMES three-body (3B) energy as the following: +6 + +3Ei1i2i3 = f +ei1ei2 +c +(ri1i2) f +ei1ei3 +c +(ri1i3) f +ei2ei3 +c +(ri2i3) +O3 +� +m=0 +O3 +� +p=0 +O3 +� +q=0 +′ +C +ei1ei2ei3 +mpq +Tm +� +s +ei1ei2 +i1i2 +� +Tp +� +s +ei1ei3 +i1i3 +� +Tq +� +s +ei2ei3 +i2i3 +� +. +(4) +We take a triple sum for the i1i2, i1i3, and i2i3 polynomials over the hypercube up to O3, and +include a single permutationally invariant coefficient for each set of powers and atom types, +C +ei1ei2ei3 +mpq +. We use the primed sum to denote that only terms for which two or more of the m, p, q +polynomial powers are greater than zero are included in order to guarantee that three distinct +atom-centers are evaluated. The expression for 3Ei1i2i3 also contains the fc smoothly varying cut- +off functions for each constituent pair distance. Penalty functions are not included in this case and +instead are handled entirely by the two-body interaction. +Higher bodied terms are included in ChIMES in a similar fashion. For example, four-body (4B) +terms are regularly included in ChIMES optimizations53, where 4Ei1i2i3i4 is now determined from +the sum over the product of the +�4 +2 +� += 6 constituent pair-wise polynomials multiplied by a single +permutationally invariant coefficient. In practice, even higher bodied terms could be included in +ChIMES, though this can lead to a combinatorially large polynomial space and hence parameter +explosion that can lead to overfitting and excessive computational expense. Hence, the norm +with ChIMES optimization is generally to include up to four-body terms, though DFTB/ChIMES +models tend to be converged with up to three-body terms, only.36,39,54–56 +Optimal ChIMES parameters (the coefficients of linear combination) can then readily be deter- +mined through the overdetermined matrix equation wAC = wBrep. The matrix A corresponds +to the derivatives of the ChIMES energy or force expression with respect to the fitting coefficients. +The column vectors C and Brep correspond to the linear ChIMES coefficients for which we are +solving and the numerical values for the training data, respectively. The symbol w corresponds +to a diagonal matrix of weights to be applied to the elements of Brep and rows of A. This linear +least-squares optimization problem can be solved for with any number of established algorithms, +discussed below. +B. +ChIMES optimization for ERep or ∆-learning +The ChIMES training set for determination of ERep or ∆-learning proceed in a similar fashion. +ERep training is computed by calculating DFTB forces (F), stress tensor components (σ), and +7 + +possibly system energies Etot for each configuration in the training set with the chosen set of +Hamiltonian parameters (i.e., {Rψ}, {Rn}, density or potential superposition, second or third- +order DFTB) with zero values for those components from ERep. These “repulsive energy free” +results are then subtracted from the DFT values for those quantities, i.e., +Eτ∗ +Rep = Eτ +DFTi − Eτ +QM,DFTBi +F τ∗ +Repαi = F τ +DFTαi − F τ +QM,DFTBαi +στ∗ +Repαβ = στ +DFTαβ − στ +QM,DFTBαβ +(5) +Here, τ corresponds to a specific MD configuration, α and β to the cartesian directions, and i is +the atomic index. In practice, we have used the diagonal components of the stress tensor, only +(i.e., α = β in Equation 5). The ‘*’ is used to denote that the quantities being computed are part of +the training set, and ‘QM,DFTB’ refers to the quantum components of the DFTB calculation, i.e., +only forces and stresses from EBS and ECoul. Calculation of a ∆-learning training set is identical +with the exception that the quantities in Equation 5 are no longer repulsive energy free but instead +contain terms from the DFTB repulsive energy model of choice. This results in the following +objective function: +Fobj = +� +� +� +� 1 +Nd +× +� M +� +τ=1 +N +� +i=1 +3 +� +α=1 +w2 +Fαi (∆Fαi)2 + +M +� +τ=1 +3 +� +α=1 +w2σαα (∆σαα)2 + +M +� +τ=1 +w2 +Ei (∆Ei)2 +� +, +(6) +where M is the total number of configurations in the training set and Nd is the total number of data +entries (3MN force components plus 3M stress tensor components plus M energy components). +In addition, ∆Fαi = F τ +ChIMESαi − F τ∗ +Repαi, ∆σαβ = στ +ChIMESαβ − στ∗ +Repαβ, and ∆Ei = Eτ +ChIMESi − +Eτ∗ +Repi. +ChIMES bears some resemblance to the Atomic Cluster Expansion approach (ACE)62,63, where +many-body interactions are represented by a product of Chebyshev polynomials and real spherical +harmonics. These models also differ from ChIMES in that the underlying polynomial basis set is +atom-centered (similar in spirit to an embedded atom model64) rather than using a cluster approach +as we adopt here. Similarly, the spectral neighbor analysis potential (SNAP) uses bispectrum +components to compute the total energy of a system as a sum over atom energies, which are +expressed as a weighted sum over bispectrum components65. +8 + +C. +Linear least-squares approaches for ChIMES optimization +The ChIMES potential is linear with respect to the fitting coefficients, which allows for use +of powerful global optimization tools that are unavailable to non-linear machine-learned models. +In our efforts, we have focussed on the Singular Value Decomposition (SVD) and Least-Angle +Regression (LARS) with Least Absolute Selection and Shrinkage Operator (LASSO) regulariza- +tion methods. We now offer a brief discussion of each method and leave details to the pertinent +references. +SVD66 solves for optimal fitting coefficients directly by performing an eigendecomposition +of the generally rectangular A matrix and computing its pseudo-inverse. Regularization can be +performed by setting singular values (eigenvalues of the square matrix in the SVD decomposition) +with an absolute value below a given threshold to zero. In our work, we take this parameter to be +Dmaxϵ, where Dmax is the maximum singular value of A and ϵ is a factor below a value of one. +LARS is a type of forward step-wise or iterative regression67,68. Here, all model coefficients are +initialized to zero and the covariate (i.e., polynomial values) most correlated to the error residual is +determined (i.e., those having the most significant impact on the fit). The corresponding ChIMES +parameter is modified incrementally to minimize the error residual until a second covariate yields +an equal correlation. At this point, it is included in the active parameter set and both coefficients are +modified simultaneously. The process continues until all coefficients are included in the solution, +at which point a result equivalent to ordinary least squares fitting is obtained. In practice, LARS +optimization can be performed using only a subset of all possible parameters. +LASSO69 is an L1-norm regularization method whereby regularization is based on the sum +of the absolute values of the fitting coefficients, which has the effect of shrinking a subset of +parameters to zero. In this case, the objective function Fobj (Equation 6) is minimized with the +following additional constraint: +F LASSO +obj += Fobj + 2α +ni +� +i=1 +|ci| . +(7) +Here, ni is the total number of unique fitting parameters, ci. The parameter α regularizes the +magnitude of the fitting coefficients, which reduces possible overfitting. The LASSO method +can be implemented as a variant of LARS where parameters are either added or removed at each +solution stage. We find the LASSO variant of LARS to be numerically stable for ill-conditioned +A matrices, which are often found in force matching. +9 + +III. +RESULTS +A. +DFTB/ChIMES Models for Silicon Polymorphs +Silicon has proven to be a significant challenge for DFTB model parameterization likely due to +the fact that its different polymorphs can have different coordination numbers and nearest neigh- +bor distances. This yields a variety of bond lengths and energies that need to be accounted for in +order to obtain a single, transferable DFTB model that does not have to be specific for a given solid +phase. Previous work has shown that standard two-body repulsive energies do not exhibit sufficient +complexity to accurately account for several Si phases with different bonding environments,34 in +contrast to carbon, where multiple phases can be represented by a single two-body polynomial +expansion70. Neural network (NN) approaches have been used for the repulsive energy in order +to account for many-body interactions in ERep,44 and the results are promising. NN approaches +though generally require large amounts of data and can frequently optimize to local minima, po- +tentially complicating their use. Here, we attempt to overcome this issue by creating a many-body +ChIMES ERep for silicon that is transferable to a number of different Si polymorphs as well as +prediction of vibrational spectra and calculation of defect formation energies. +In our work, we target two previous Si DFTB parameterizations, pbc-0-371 and siband-1-1,41 +which have different strengths and weaknesses. The pbc-0-3 parameter was creating using density +superposition (i.e., the quantum mechanical potential VQM (ρ) was expressed as V (ρA + ρB) for +atoms A and B) , which tends to be preferred due to its improved representation of chemical +bonding and vibrations36. However, d-orbital interactions were not tabulated aside from the d- +orbital onsite energy, which could have ramifications for some material properties. In contrast, +the siband-1-1 parameter set was specifically created with d-orbital interactions but with potential +superposition (i.e., VQM (ρ) = V (ρA) + V (ρB)) in order to yield accurate prediction of electronic +properties, including the electronic band structure of Si-containing solids. In addition, the siband- +1-1 parameter set does not contain a repulsive energy of any sort, precluding its use in structural +relaxation or MD simulation which severely limits its usefulness overall. +Our goal is to thus to create new ChIMES ERep potentials for each set of Slater-Koster interac- +tion parameters using identical DFT training data and ChIMES hyperparameters in order to com- +pare and contrast the effectiveness of each as a possible one-fits-all model. Calculations for our sil- +icon DFT dataset were performed using the Vienna ab initio Simulation Package (VASP)72–74, with +10 + +projector-augmented wave function (PAW) pseudopotentials75,76 and the Perdew-Burke-Ernzerhof +exchange-correlation functional (PBE)77. We found our results to be converged with a planewave +cutoff of 500 eV, which was used in all of the calculations discussed here. We have used an +electron density convergence criteria of 10−6 eV, with a force convergence of 10−2 eV/ ˚A for all +geometry/cell optimizations. The Mermin functional78 smearing was set to 0.03 eV for all calcu- +lations performed in this work. The system energy and pressure was found to be converged with +sampling of the Brillouin Zone with a 2 × 2 × 2 Monkhorst-Pack mesh79 for all supercells. We +then generated cold curves for each phase by isotropically expanding and contracting the simula- +tion cell lattice. Here, we used a diamond structure supercell of 64 atoms, a bcc structure of 54 +atoms, a simple cubic structure of 64 atoms, and a graphene sheet of 32 atoms. This yielded an +initial set of 463 configurations for our ChIMES ERep optimization. +In order to sample forces from a variety different configurations, we have also included MD +data for the diamond and graphene phases, using the same number of atoms in each supercell as +before. These supercells were isotropically expanded and contracted between 90% to 110% of +the ground-state density. Each MD simulation was run for ∼5 picoseconds at 600 K, from which +we took snapshots at fixed intervals of ∼200 femtoseconds for our training set. This yielded +an additional 405 configurations for our ChIMES ERep determination. In all, our final training set +contained a total of 838 configurations of different silicon phases. ChIMES ERep optimization was +then performed using values of rmin = 2.0 ˚A and rmax = 4.0 ˚A. The value of rmax was informed +in part from previous development of a neural network repulsive energy,34 which resulted in a +minimization of the root mean square (RMS) error in our fit. In addition, we found that a value of +rmax = 4.0 ˚A yielded an improved description of the expanded states in our training set, where the +bonded interactions between Si atoms is longer than the ground-state. +We now refer to our ChIMES model based on pbc-0-3 as pbc/ChIMES and our model based +on siband-1-1 as siband/ChIMES. Both pbc/ChIMES and siband/ChIMES were created with a 2B +order of 12, 3B order of 8, and a LASSO regularization parameter (α) value of 10−3, similar to +previous efforts36. We have used the Morse coordinate transform with a value of λ = 2.4 ˚A, which +corresponds to the first peak in the diamond phase radial distribution function. For pbc/ChIMES, +this yielded an overall RMS error of 1.44 eV/ ˚A in the forces, 0.43 GPA in the pressure, and +0.038 eV/atom in energy. The RMS errors for siband/ChIMES were slightly higher, with values +of 2.22 eV/ ˚A for the forces, 0.55 GPa for the pressure, and 0.16 eV/atom for the energy. Use +of a Chebyshev basis set 2B order of 16, 3B order of 12, and 4B order of 4 yielded reduction in +11 + +the RMS errors of < 1% with similarly marginal improvement in validation quantities such as the +computed defect energies. Use of a value of λ = 3.0 ˚A also had only a small effect on the resulting +model. All ChIMES/DFTB calculations were performed with self-consistent charges using similar +parameters to our DFT calculations. This included charge convergence criteria of 2.72 × 10−5 eV +(10−6 au), a force convergence of 10−2 eV/ ˚A for all geometry optimizations, and 2×2×2 k-point +mesh for all calculations. +TABLE I: Ground state energies relative to diamond (∆Ediam) in eV/atom and nearest neighbor +distances (NN) in ˚A for the Si polymorphs considered in this work. +diamond +bcc +simple cubic +graphene +bc8 +NN ∆Ediam NN ∆Ediam NN ∆Ediam NN ∆Ediam NN ∆Ediam +pbc/ChIMES +2.37 +0.00 +2.67 +0.55 +2.53 +0.30 +2.23 +0.70 +2.37 +0.14 +siband/ChIMES 2.36 +0.00 +2.65 +0.53 +2.54 +0.31 +2.26 +0.59 +2.39 +0.15 +DFT +2.37 +0.00 +2.68 +0.54 +2.53 +0.30 +2.25 +0.65 +2.39 +0.16 +In order to test the applicability of our ChIMES/DFTB models to different of Si phases, we have +computed the relative energies and nearest neighbor distances for several polymorphs, including +those in our training set as well as the bc8 phase (Table I). Our results indicate strong agreement +with DFT for both models. We observe close agreement for all properties for both pbc/ChIMES +and siband/ChIMES, where the energy of each phase relative to the diamond ground-state tends +to agree with DFT within 0.01 eV, and the subsequent nearest neighbor distances agree within +0.01 − 0.02 ˚A. The graphene phase is a small exception, where pbc/ChIMES yielded a relative +energy of 0.70 eV/atom and siband/ChIMES a relative energy of 0.59 eV, compared to a value of +0.65 eV for DFT. However, both models still yield the correct energetic ordering of the phases. +Similar to previous efforts34,44, we have determined cold energy curves under isotropic com- +pression and expansion for all phases in this study (Fig. 1). +Overall, both pbc/ChIMES and +siband/ChIMES yield close agreement with DFT. Both models have particularly close agreement +for the diamond and simple cubic phases. The siband/ChIMES model exhibited a small oscil- +lation in the bcc cold curve at a nearest neighbor distance of 2.7 ˚A which is not present in the +pbc/ChIMES result. However, the agreement with DFT is reasonable for both models. The largest +disagreement for pbc/ChIMES is with graphene, where it yields a more positive curvature at ex- +panded densities, whereas siband/ChIMES yields closer agreement to DFT overall. Both models +12 + +−5.6 +−5.4 +−5.2 +−5 +−4.8 +−4.6 +−4.4 +−4.2 +−4 + 2.2 + 2.3 + 2.4 + 2.5 + 2.6 + 2.7 + 2.8 + 2.9 + 3 +Energy/atom (eV) +NN (Å) +(a) pbc/ChIMES +−5.6 +−5.4 +−5.2 +−5 +−4.8 +−4.6 +−4.4 +−4.2 +−4 + 2.2 + 2.3 + 2.4 + 2.5 + 2.6 + 2.7 + 2.8 + 2.9 + 3 +Energy/atom (eV) +NN (Å) +(b) siband/ChIMES +FIG. 1: Cold curves for several silicon polymorphs from pbc/ChIMES and siband/ChIMES +DFTB models (points) compared to results from DFT (solid lines). The black curves correspond +to the diamond phase, blue to bcc, red to simple cubic, and the green to graphene. The orange +marks correspond to the bc8 phase and were not a part of the training set. +predict very similar agreement for the bc8 phase, where each yielded a small oscillation in the +cold curve around 2.5 ˚A. This is likely due to insufficient sampling of these Si-Si distances and +bonding environments in our training set. Regardless, these results indicate strong agreement for +energy vs. volume relationships, which could indicate accurate force prediction from each model. +We now assess the force output from each model through comparison of the resulting vibra- +tional density of states (VDOS) for the diamond phase to results from DFT (Fig. 2). These were +computed from Fourier Transform of the velocity autocorrelation function which was determined +from MD simulations at constant volume-temperature (NVT), conducted at 600 K, using a Nos´e- +Hoover thermostatted chain80–82 and run for 15–20 ps using a timestep of 1 ps. Our results for +pbc/ChIMES indicate fairly close agreement with DFT. Prediction of the lowest lying vibrational +peak is off by only ∼7 cm−1, with a value of 134 cm−1 compared to a value of 127 cm−1 from +DFT. DFT yields a small peak at 231 cm−1 which appears as a broad, higher intensity shoulder +at 224 cm−1 in the pbc/ChIMES spectrum. The remaining peaks in the spectrum show similarly +strong agreement with some variation in the intensity of the peaks, including accurate prediction +from pbc/ChIMES of the vibron peak at 450 cm−1 compared to a frequency of 453 cm−1 from +DFT. +13 + +−20 + 0 + 20 + 40 + 60 + 80 + 100 + 120 + 140 + 160 + 180 + 200 + 100 + 200 + 300 + 400 + 500 +Intensity +Frequency (cm−1) +FIG. 2: Vibrational density of states for the Si diamond phase, computed at 600 K. The red line +corresponds to pbc/ChIMES. the blue line to siband/ChIMES, and the black dashed line to DFT. +In contrast, siband/ChIMES shows slightly less accurate agreement with DFT overall. The +agreement for the lowest vibrational peak is fairly close, with a frequency of 120 cm−1. The +remainder of the siband/ChIMES spectrum yields an accurate overall shape of the VDOS, though +with some errors in peak positions and intensities. There is some deviation in the siband/ChIMES +spectrum for next two vibrational peaks, where we observe a frequency of 173 cm−1 for the second +lowest frequency peak compared to a value of 188 cm−1 from DFT and a frequency of 217 cm−1 +for the low intensity peak after that compared to the previously mentioned DFT peak at 231 cm−1. +The siband/ChIMES spectrum yields a close match in intensity and frequency with DFT for the +VDOS peak at 344 cm−1. However, the subsequent two peaks are red shifted in frequency and +lower in intensity, with values of peak positions of 413 and 472 cm−1, compared to values of 396 +and 453 cm−1 from DFT. The improved VDOS determination from pbc/ChIMES could be due +in part to its parameterization with density superposition, which has been shown to yield more +accurate predictions over potential superposition36. We note that these peak position differences +discussed here correspond to small changes in energy, where 20 cm−1 corresponds to ∼ 2.5 × +10−3 eV. Hence, it is possible that siband/ChIMES will still yield sufficiently accurate forces for +some applications. +14 + +(a) Vacancy +(b) Tetrahedral +(c) Hexagonal +FIG. 3: Images of the diamond phase point defects investigated in this study. All defects are +shown as a red sphere for the sake of clarity. +TABLE II: Defect formation energies for the Si diamond phase. All energies are in eV. +Defect +pbc/ChIMES siband/ChIMES DFT (PBE) +vacancy +3.45 +4.60 +3.84 +tetrahedral +5.11 +4.88 +3.84 +hexagonal +5.87 +4.79 +3.61 +Finally, we have computed defect formation energies from our DFTB/ChIMES models (Fig. 3). +Here, we have investigated a single Si atom vacancy as well as an interstitial atom in either a +hexagonal or tetrahedral site, which were determined from use of the pymatgen software suite83. +The tetrahedral interstitial site occurs where an additional Si atom is coordinated by four atoms +from the lattice, whereas the hexagonal interstitial site occurs when the additional Si atom re- +sides in a hexagonal opening within the lattice. The defect formation energy Eform is computed +as Eform = Edef − NdefEdiam, where Edef is the total energy of the defect containing system, +Ndef is the number of Si atoms in that configuration, and Ediam is the energy per atom of the +perfect diamond phase. Similar to previous Si DFTB efforts44, calculations were initialized from +an optimized 216 atom supercell where we retained a Monkhorst-Pack mesh of 2 × 2 × 2, after +which we created the point defect and optimized the ionic positions of each configuration using +the same k-point mesh. Our results indicate some agreement with previous PBE-DFT calculations +from Ref. 44. The pbc/ChIMES model agrees with the DFT vacancy energy within 0.4 eV, but +yields results that are 1–2 eV too high for both interstitial energies. In particular, the three defect +energies from pbc/CHIMES differ over a range of over 2.4 eV, with the both interstitial energies +15 + +yielding larger results than that of the vacancy. In contrast, the result from DFT all lie relatively +close together (within a range of 0.23 eV) and DFT exhibits equal formation energy values for +the vacancy and tetrahedral interstitial. It is likely that the interstitial energies would be decreased +with full accounting of d-orbital off-site interactions, which are absent in the original pbc-0-3 +parameter set. The siband/ChIMES model yields defect formation energies that are consistently +∼1 eV too high relative to DFT. However, the siband/ChIMES results differ over an energy range +of 0.28 eV, yielding improved agreement with DFT in this respect. It is likely that there would +be some variation in DFT results depending on the choice of exchange-correlation function and +possible inclusion of a dispersion energy correction. +Overall, our we able to create two new DFTB/ChIMES models that more closely approach a +single-purpose approach for silicon phases under different conditions. The pbc/ChIMES model ap- +pears to yield a somewhat improved description of atomic forces, whereas as the siband/ChIMES +model yields more systematically consistent defect formation energies that could make it prefer- +able for some calculations. As mentioned, some of the limitations of the pbc/ChIMES model could +possibly be overcome through inclusion of d-orbital two-center interactions in the corresponding +Slater-Koster file. Regardless, we now provide a repulsive energy for the siband-1-1 parameter +set, which will allow its use for structural relaxations and/or dynamics calculations in addition to +its accuracy for electronic properties. It is possible that the slightly longer cutoff radius for our +ChIMES ERep could be mitigated through optimization of the choice of DFTB confining radii +(discussed in the next section). Future improvement of these models could also involve inclusion +of data from MD simulations of amorphous or defect containing systems at different temperatures +and pressures. +B. +Semi-automated Workflow for DFTB/ChIMES Model Creation +In this subsection, we summarize previous work on TiH236 which indicates the utility in using a +ChIMES ERep in a semi-automated fashion to screen for optimal confining radii in a Slater-Koster +file parameterization. TiH2 has a number of industrial uses as a functional material, including +in hydrogen storage alloys84, superconductors85, and as a blending agent for porous foams86. Its +ground-state structure exhibits face-centered-cubic (fcc) symmetry, with the (111) facet computed +to have the lowest surface energy (Fig. 4). Several adsorption sites are illustrated, including Top +(directly above a Ti atom), Hollow (in an interstitial cavity), and several Bridge sites (existing in +16 + +between Ti-Ti and H-H nearest neighbors) sites. TiH2 is a somewhat ideal system for DFTB model +development in that DFT calculations on small supercells are relatively tractable, which allows for +straightforward validation. DFT calculations though are generally too computationally inefficient +for the larger supercells needed to model grain boundaries and crystalline defects at sufficiently +low concentration, allowing for several applications of a new TiH2 DFTB model in future studies. +Ti +H +x +o +o +o +o +o +o +o +x2 +x1 +(111) surface +(011) surface +Bulk +FIG. 4: Pictures of TiH2 bulk and surfaces. The left panel shows the bulk fcc lattice. The middle +panel shows the (111) crystalline surface the Top (marked with an ‘O’) and Hollow (‘X’) +adsorption sites indicated. The right panel shows the (011) crystalline surface with the Top (‘O’), +Bridge-1 (‘X1’) and Bridge-2 (‘X2’) sites indicated. Reprinted with permission from Journal of +Chemical Theory and Computation 2021 17 (7), 4435-4448. Copyright 2021, American +Chemical Society. +Here, we have leveraged rapid ChIMES ERep optimzation by creating a workflow that allowed +us to perform a semi-exhaustive search of all DFTB and ChIMES hyperparameters (Fig. 5). We +first compute a matrix of thirty Slater-Koster files from titanium wavefunction confining radii +(RTi +ψ ) and density confining radii (RTi +n ) sampled over a range of 3.2 ≤ RTi +ψ +≤ 5.0 au and +6.0 ≤ RTi +n +≤ 17.0 au. Hydrogen interaction parameters were taken from the miomod-hh-0-1 +parameter set. Model down selection could then be performed over the entire grid Slater-Koster +tables through comparison to our selected validation data, which allowed us to determine optimal +ChIMES polynomial orders and the LASSO regularization parameter. +For this work, our DFT training set consisted of molecular dynamics simulations of unit cell +configurations (12 atoms total), run for 5 ps at 400 K with simulation cells initially optimized to +pressures in a range from −8 to 100 GPa. All MD calculations were run in the constant temperature +and volume (NV T) ensemble with Nos´e-Hoover thermostat chains80–82 and a timestep of 0.2 fs. +The slightly elevated temperature and wide pressure range including negative pressure were chosen +in order to yield a broad sampling of the underlying potential energy surface. Atomic forces and +the diagonal of the stress tensor were then sampled from MD configurations at fixed time intervals +17 + +bbSelect !", $!, $"; +Create SKF files. +Compute DFTB training set: +⃗&DFT − ⃗&DFTB (no ERep) +(##,DFT −(##,DFTB (no ERep) +Desired accuracy +achieved? +Validation set: +• +Bulk: lattice const., VDOS, 1H and 2H +vacancies. +• +(001), (011), and (111) surface energies +• +Eads on (011) and (111) surfaces (5 total) +• +(011) and (111) surface and subsurface +H vacancy energies (8 total) +Choose ChIMES 2B, 3B, +4B orders, cutoff radii +and determine )$%&. +Yes +No +Complete +Compute +DFT-MD +data +FIG. 5: Flowchart for creation of DFTB Erep models through ChIMES force field +parameterization. Reprinted with permission from Journal of Chemical Theory and Computation +2021 17 (7), 4435-4448. Copyright 2021, American Chemical Society. +of ∼ 160 fs in order ensure configurations were as statistically uncorrelated as possible. This +yielded up to 30 MD snapshots for each pressure. Inclusion of system energies in our training data +did not appear to improve the quality of our optimization and hence were omitted. In addition, in +order to sample hyper- and hypo-coordinated configurations in the system, we included MD data +for a unit cell with a single hydrogen interstitial or single vacancy site, each run for 5 ps. This +yielded a total of 153 unit cell-sized configurations for our training set. Validation calculations +for all of our DFTB/ChIMES models were performed on the bulk lattice constant, single and +double hydrogen vacancy energies, and the vibrational density of states. We also validated our +models against a number of surface properties, including the surface energies of the (001), (011) +and (111) facets, five different hydrogen adsorption energies on the (011) and (111) surfaces, and +surface and subsurface hydrogen vacancy energies on the same two facets. Validation data for +hydrogen interactions with the (001) surface were omitted from our study due to the presence of a +significant surface dipole on this facet. +Once again, all DFT calculations were performed with VASP using PAW pseudopotentials and +PBE. We found our results to be converged with a planewave cutoff of 400 eV and an energy +18 + +convergence criteria of 10−6 eV, both of which were used for the results reported here. Fourth +order Methfessel-Paxton smearing87 was used with a value of 0.13 eV for all geometry and cell +lattice optimizations in order to ensure energy convergence without dependence on the electronic +smearing temperature. The Mermin functional78 with the same electronic temperature was used for +all MD calculations in order to avoid spurious forces due to possible negative occupation numbers +from the Methfessel-Paxton approach. Brillouin Zone sampling for all TiH2 unit cell calculations +was performed with a 10 × 10 × 10 k-point mesh, whereas we used a mesh of 5 × 5 × 5 for +32 formula unit (96 atom) bulk calculations. We used system sizes of 168 atoms/7 layers for +the (001) surface, 144 atoms/6 layers for the (011) surface, and 192 atoms/8 layers for the (111) +surface, each with a vacuum of 20 ˚A and a k-point mesh of 5×5×1 in the direction of the surface. +DFTB+ calculations were performed using self-consistent charges (SCC)22 and charge conver- +gence criteria of 2.72 × 10−5 eV (10−6 au). Inclusion of an external van der Waals correction31,88 +is beyond the scope of our present study. We have performed “shell-resolved” SCC calculations, +where separate Hubbard U parameters were determined for each orbital angular momentum shell. +All minimum and cutoff radii for the ChIMES ERep were set to include the first coordination shell +sampled in our training set, only: 2.5 ≤ rTiTi ≤ 3.5 ˚A and 1.5 ≤ rHTi ≤ 2.5 ˚A . We use values +of λTiTi = 3.0 ˚A and λHTi = 2.0 ˚A for the Morse-like coordinate transforms. H-H repulsive +interaction were not sampled in our training set and were thus also taken from the miomod-hh-0-1 +parameter set. +Our results for a subset of our validation data (Fig. 6) allow us to describe general trends +regarding the confining radii. We observe an approximate linear relationship between RTi +ψ and +RTi +n in terms of the accuracy of the E111 energy, where the most accurate surface energy results +from either small or large choice for both radii. All of the DFTB/ChIMES models created in this +iteration tend to under-predict the (E001/E111) ratio (i.e., the ratio of highest to lowest surface +energies in our study) relative to our DFT calculations, where we observe values of 1.35–1.44 +compared to the DFT ratio of 1.70. We note that is is likely in part due to the surface dipole +moment present in our construction of the (001) facet. In addition, our results indicate a much +smaller dependence on choice of RTi +n for a given RTi +ψ . We note that there can be strong dependence +of the surface energies on choice of DFT functional (e.g., Ref. 89), although the relative energetic +ordering tends to be consistent. +Our final set of hyper-parameter values includes {RTi +ψ = 3.6 au, RTi +n = 6.0 au} and {O2B = 8, +O3B = 4}, optimized with LASSO/LARS and regularization of α = 10−3. This model yields +19 + + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 3 + 3.5 + 4 + 4.5 + 5 +RTi +n (au) +RTi +ψ (au) +−0.2 +−0.15 +−0.1 +−0.05 + 0 + 0.05 + 0.1 + 0.15 +Fractional Deviation of E111 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 3 + 3.5 + 4 + 4.5 + 5 +RTi +n (au) +RTi +ψ (au) +−0.35 +−0.34 +−0.33 +−0.32 +−0.31 +−0.3 +−0.29 +−0.28 +−0.27 +−0.26 +∆(E001/E111) +FIG. 6: Results for sweep of values of RTi +ψ and RTi +n , where the ChIMES ERep was determined +with a 2B order of 12 and 3B order of 8. The top panel corresponds to the fractional deviation of +the surface energy, +� +EDFTB +111 +− EDFT +111 +� +/EDFT +111 , and the middle panel to the deviation of +(E001/E111) relative to DFT. Reprinted with permission from Journal of Chemical Theory and +Computation 2021 17 (7), 4435-4448. Copyright 2021, American Chemical Society. +RMS errors of 0.076 eV/ ˚A for hydrogen forces, 0.056 eV/ ˚A for titanium forces, and 0.35 GPa for +the stress tensor diagonal. Results for bulk properties indicate that DFTB/ChIMES yields a lattice +constant with errors of only ∼0.4% and 1.0% from DFT and experiment90, respectively. However, +our model yields a hydrogen bulk vacancy energy (Evac) that is ∼0.5 eV too small. We found +that a systematic ∼0.5 eV underestimation of vacancy energies in a variety of environments and +concentrations was typical for all ChIMES parameterizations created in this work, which could be +rectified with improved training data or adaptations to DFTB such as the inclusion of multi-center +terms in the Hamiltonian.91. +Overall, our final model yields accurate surface energies for all three low-index facets investi- +gated in this study (Table III). In particular, the E011 and E111 values are nearly identical to those +from DFT. The E001 value from DFTB/ChIMES is around 17% lower than than that for our DFT +calculations (0.114 vs. 0.136 eV/ ˚A2). This could be due in part to the internal electric field on the +(001) surface configuration studied here, as mentioned. DFTB generally can underestimate surface +electrostatic interactions due to its determination of atom-centered point charges only in Coulom- +bic interactions92. Our DFTB/ChIMES results show similarly strong agreement with hydrogen +surface adsorption energies (Table IV). We compute the correct energetic ordering of adsorption +on the (111) Top and Hollow sites, though the Hollow site energy is 0.35 eV smaller than that from +20 + +DFT. We see similar agreement with DFT for the (011) surface. Here, DFTB/ChIMES show close +agreement for Top site adsorption with a difference of only 0.05 eV from DFT. Our model yields +Bridge-1 and Bridge-2 adsorption energies that differ from DFT by 0.29 eV and 0.21 eV, respec- +tively, and incorrectly predicts that the Top site is the lowest energetically of the three. Regardless, +these values are similar in energy for all surface sites and we have overall favorable agreement. +TABLE III: TiH2 surface energies (in eV/ ˚A2). Reprinted with permission from Journal of +Chemical Theory and Computation 2021 17 (7), 4435-4448. Copyright 2021, American +Chemical Society. +Surface DFTB/ChIMES DFT +111 +0.080 +0.080 +011 +0.105 +0.101 +001 +0.114 +0.136 +TABLE IV: Surface hydrogen adsorption energies on TiH2 surface sites (in eV). Reprinted with +permission from Journal of Chemical Theory and Computation 2021 17 (7), 4435-4448. +Copyright 2021, American Chemical Society. +Surface +Site +DFTB/ChIMES DFT +111 +Top +-1.888 +-1.760 +Hollow +-2.081 +-2.440 +011 +Top +-2.383 +-2.332 +Bridge-1 +-2.154 +-2.442 +Bridge-2 +-2.132 +-2.342 +Our results indicate DFTB/ChIMES models can be accurately determined based on relatively +small training data (unit cell MD calculations in this work), even for physically complex sys- +tems such as those containing surface chemistry. Further refinement of our TiH2 model could +involve inclusion of training data from additional phases and thermodynamic state points. Re- +gardless, our current effort yields accurate results for bulk and surface TiH2 properties, and our +model shows strong transferability to bulk α-Ti and gas phase TiH4 (not shown here). The small +training set could yield significant advantages for computationally challenging systems such as +21 + +magnetic materials and their interfaces, where DFT data is limited and difficult to generate. Over- +all, our DFTB/ChIMES approach can have particular impact on myriad of research areas, such as +interpretation of imaging and spectroscopy studies on bulk and interfacial systems, where there is +traditionally a strong coupling with atomistic simulation approaches. +C. +∆-learning to Enhance the Accuracy of DFTB for Organic Materials +In this subsection we review our recent efforts to leverage a high-level quantum chemical +database to create an “out-of-the-box” model with accuracy beyond standard DFT approaches +(e.g., PBE) that is generally applicable to many organic molecular systems56. In this work, we have +used the ANI-1x quantum chemical data set93,94 to create a DFTB/ChIMES model that approaches +hybrid-functional and/or coupled cluster accuracy. Here, ChIMES is used as a ∆-learning po- +tential where we have included it as an extra energy term to the 3ob-3-1 parameterization40,95, +which includes third-order charge fluctuation terms in the DFTB energy. This parameterization is +known to yield reliable accuracy for many organic molecules and thus was a reasonable starting +point for our efforts. We have found that the advantage of ChIMES over a neural network ap- +proach is two-fold: (1) the training set requirements of ChIMES is significantly lower, where only +a small fraction of the ANI-1x dataset was required to achieve a high degree of accuracy, and (2) +our ChIMES potential required two-order of magnitude fewer parameters than several recent NN- +based semi-empirical approaches. These effects allow for a much easier to parameterize model +that is less likely to be hampered by overfitting. +The original ANI-1x database was developed for the creation of ML-based general-purpose +organic potentials where the data set was determined through an active learning process94, result- +ing in approximately 5 million molecular equilibrium and non-equilibrium configurations. Our +∆-learning optimization used an iterative approach by first creating a subset of ANI-1x called +“sub ANI-1x” that only contained results computed from CCSD(T) (coupled-cluster consider- +ing single, double, and perturbative triple excitations) and using a well-known hybrid functional, +wB97X96. This corresponded to 459,464 molecular confirmations from computed from 1895 +unique molecules, or ∼10% of the original ANI-1x database. We note that there are no atomic +force data from CCSD(T)/CBS calculations. Hence, we used wB97X results computed with a +large basis set (def2-TZVPP) data for fitting purposes, with the remainder of the data set available +for validation. +22 + +We then used an iterative approach to ChIMES optimization (Fig. 7) where we first randomly +selected only 1% of sub ANI-1x and performed an initial ChIMES optimization. Validation cal- +culations agains the remainder of sub ANI-1x resulted in some large deviations in the computed +energies and forces. We then selected an additional equivalent of 1% of the data set from con- +figurations with the highest force deviations and added them to our training set and repeated the +process, where each increment of the training process would include the equivalent of an additional +1% of sub ANI-1x. Our DFTB/ChIMES ∆-learning was converged after three iterations of our +optimization scheme, using only 3% of sub ANI-1x or 0.3% of the original ANI-1x database. Our +model was ultimately validated against the entire sub ANI-1x data set, though its size is somewhat +arbitrary and it is possible that a smaller subset of ANI-1x could have been used for our purposes. +-2000 -1000 +0 +1000 +2000 +FDFT(kcal/mol/Å) +-2000 +-1000 +0 +1000 +2000 +FDFTB/ChIMES(kcal/mol/Å) +-2000 -1000 +0 +1000 +2000 +FDFT(kcal/mol/Å) +-2000 +-1000 +0 +1000 +2000 +-2000 -1000 +0 +1000 +2000 +FDFT(kcal/mol/Å) +-2000 +-1000 +0 +1000 +2000 +-20 +0 +20 +40 +60 +EDFT(kcal/mol/atom) +-20 +0 +20 +40 +60 +EDFTB/ChIMES(kcal/mol/atom) +circle 0 +1% train and 99% validation +-20 +0 +20 +40 +60 +EDFT(kcal/mol/atom) +-20 +0 +20 +40 +60 +circle 1 +2% train and 98% validation +-20 +0 +20 +40 +60 +EDFT(kcal/mol/atom) +-20 +0 +20 +40 +60 +circle 2 +3% train and 97% validation +a) +b) +c) +d) +e) +f) +FIG. 7: Comparison of energies per atom (top panels) and forces (bottom panels) predicted by +DFT (wB97X) and DFTB/ChIMES for all configurations in the validation set. The dataset used +here is ‘sub ANI-1x’, ∼10% of the full ANI-1x. Reprinted with permission from J. Phys. Chem. +Lett. 2022 13 (13), 2934-2942. Copyright 2022, American Chemical Society. +Our final model used ChIMES polynomial orders of {2B = 24, 3B = 10, 4B = 0} with a +somewhat long radial cutoff of 4.0 ˚A used for all atom pairs. This longer cutoff helped account +for some dispersion interactions that would otherwise be absent from standard DFTB calculations, +though future efforts will involve shorter cutoffs combined with a dispersion interaction model. +23 + +Further details about our ChIMES model for organics can be found in the Supporting Information +in Ref. 56. Ultimately, our DFTB/ChIMES model resulted in 5546 parameters and was trained to +∼372k data points. This is in contrast to the recently developed AIQM1 semi-empirical quantum +model, which utilizes an NN trained to the entire ANI-1x data set, resulting in 322,660 parameters. +Similarly, a recent DFTB-NN approach using deep-tensor neural networks used a training set of +∼800k data points, resulting in 228,865 parameters. +TABLE V: Performance of DFTB and DFTB/ChIMES in predicting reference energies and/or +atomic forces in the GDB-10to13, ISO34, and GDML data set. The MAE and RMSE for the +energies and forces (labeled with subscripts ‘E’ and ‘F’) are in kcal/mol and kcal/mol- ˚A, +respectively. Reference molecular energies and atomic forces in the GDB-10to13 data set are at +the wB97X/6-31G* level of theory. Isomerization energies in the ISO34 data set are a mixture of +experimental- and CCSD(T) extrapolation energies. The CCSD(T)/cc-pVTZ atomic forces of +2000 configurations of ethanol in the GDML data set are used for comparison. Reprinted with +permission from J. Phys. Chem. Lett. 2022 13 (13), 2934-2942. Copyright 2022, American +Chemical Society. +GDB-10to13 +ISO34 +GDML +method +MAEE/RMSEE MAEF/RMSEF MAEE/RMSEE MAEF/RMSEF +DFTB +9.10/11.70 +6.34/9.85 +3.69/4.96 +4.52/6.12 +DFTB/ChIMES +3.57/4.72 +3.62/5.33 +2.06/2.56 +2.72/3.61 +ANI-197 +3.12/4.74 +3.96/7.09 +- +- +ANI-1x97 +2.30/3.21 +3.67/6.01 +- +- +DFTB-NNrep98 +- +- +2.21/3.30 +- +PBE098 +- +- +1.82/2.48 +- +We then tested the transferability of our DFTB/ChIMES model through comparison to different +quantum chemical data that were computed at the wB97X or CCSD(T) level but were not a part +of ANI-1x (Table V). For example, the GDB-10to13 data set97 consists of the molecular energies +and forces at the wB97X level of nearly 3000 molecules containing 10-13 C, N, or O atoms for a +total of 47,670 configurations. Our DFTB/ChIMES model exhibits a 60% reduction in the mean +average error (MAE) and RMSE error in the energies and a 45 % decrease in the forces over +24 + +standard DFTB. The accuracy of DFTB/ChIMES is similar to values from the ANI-1 and ANI- +1x neural network interatomic potentials97 (i.e., stand-alone potentials without explicit quantum +mechanical elements), and are smaller than the variations between wB97X itself and higher levels +of theory such as CCSD(T) and MP2 (4.9/5.9 kcal/mol for energies and 4.6/5.9 kcal/mol- ˚A for +forces)93. +Our DFTB/ChIMES model is validated against additional CCSD(T) reference data from the +ISO34 data set99, which consists of energies of 34 isomers containing the elements C, H, N, +and O. We observe that the accuracy of DFTB/ChIMES is much better than that for standard +DFTB, is slightly improved over that from DFTB-NNrep, and approaches the PBE0 data given +in Ref. 98. To test the performance of our model on high accuracy force data specifically, we +compare DFTB/ChIMES with the CCSD(T)/cc-pVTZ data for 2000 configurations of ethanol in +the GDML data set100 (54,000 data points total). Again our DFTB/ChIMES gives an improvement +over standard DFTB as MAE and RMSE are both reduced by ∼40%. A direct force comparison to +DFTB-NNrep or the ISO34 reference was unavailable. Additional validation of our model included +calculation of the n-butane dihedral potential and correct prediction of the energetic ordering of +coumarin molecular crystals. +We have also validated DFTB/ChIMES against vibrational frequencies of 342 gas-phase +molecules from the Computational Chemistry Comparison and Benchmark Database or CC- +CBDB (https://cccbdb.nist.gov/), computed with MP2/cc-pVTZ and wB97XD (with dispersion +correction), amongst other methods (Fig. 8). Here, DFTB/ChIMES yields errors in the frequency +prediction of MAE/RMSE = 36/61 cm−1, indicating improved accuracy over PBE and with similar +accuracy to accuracy to wB97XD. In all of our validation tests, DFTB/ChIMES shows marked +improvement over standard DFTB and PBE, and shows similar accuracy to results from wB97X or +other higher-levels of theory. Further details of all validation calculations are provided in Ref. 56. +Lastly, though the DFTB/ChIMES model developed here is trained on gas phase molecular +data, we have also tested its performance in reproducing the structural properties of bulk graphite +and diamond. We compare predicted density and lattice parameters from different methods in +Table VI. +For graphite, all computational models considered here give an accurate descrip- +tion of the in-plane lattice parameters. DFTB and PBE overestimate the interlayer separation +(c/2) by over 25% and 30%, respectively, due to their under-prediction of dispersion interactions. +DFTB/ChIMES predicts the lattice parameters and density in excellent agreement with the exper- +imental value, with a deviation of less than 1%. For diamond, the computed values using DFTB, +25 + +0 +1000 +2000 +3000 +4000 +Frequency (cm +-1) +Distribution +MP2 +ωB97XD +DFTB/ChIMES +PBE +DFTB +FIG. 8: The distribution of the calculated frequency values using DFTB and DFTB/ChIMES for +342 neutral molecules taken from the CCCBDB database. The MP2 and DFT (PBE and +wB97XD) calculations using cc-pVTZ basis set in the CCCBDB are selected for comparison. +Reprinted with permission from J. Phys. Chem. Lett. 2022 13 (13), 2934-2942. Copyright 2022, +American Chemical Society. +DFTB/ChIMES, and PBE-DFT differ by ∼1% from experimental values for lattice parameters +and ∼3% for the density. +Ultimately, we have shown that ChIMES can be used to extend DFTB to hybrid functional +accuracy or greater. DFTB/ChIMES has the capability of reproducing vast quantities of high-level +reference data while requiring only a small fraction of it for training. On the basis of the results +presented here, DFTB/ChIMES represents a promising direction for developing general purpose +quantum models that are applicable to a wide range of materials and conditions. The small training +set required as well as the small number of potential parameters relative to neural network methods +could yield significant advantages for future development of computational efficient models with +up to coupled cluster accuracy. The ease of parameterization and transferability of DFTB/ChIMES +26 + +TABLE VI: Comparison of predicted density and lattice parameters of graphite and diamond for +DFTB, DFTB/ChIMES, PBE-DFT with experimental data. Reprinted with permission from J. +Phys. Chem. Lett. 2022 13 (13), 2934-2942. Copyright 2022, American Chemical Society. +phase +method +density (g/cm3) a( ˚A) c/2( ˚A) +graphite Expt.101 +2.26 +2.462 3.356 +PBE-DFT102 +1.71 +2.470 4.420 +DFTB/ChIMES +2.25 +2.461 3.379 +DFTB +1.77 +2.474 4.248 +diamond Expt.101 +3.51 +3.567 +PBE-DFT70 +3.48 +3.580 +DFTB/ChIMES +3.42 +3.600 +DFTB +3.42 +3.600 +allows for high-level quantum theory accuracy in systems where traditional wavefunction or hybrid +functional methods are far too computationally intensive for intensive use. +IV. +DISCUSSION AND FUTURE WORK +ChIMES was initially developed as a method for creating many-body force fields for molecular +dynamics simulations. However, it has also proven robust as a repulsive energy for DFTB models, +where the standard two-center approach for both quantum mechanical and repulsive terms can be +insufficient for many systems. The strength in ChIMES as an element of a semi-empirical quantum +model or MD model lies in its use of linear combinations of many-body Chebyshev polynomials, +where the nearly optimal nature of the polynomials as well as the linear least-squares fitting allow +for rapid optimizations that require far fewer parameters and significantly smaller data sets than the +neural network models reviewed here. In addition, ChIMES adds very little extra computational +time to DFTB calculations, where the matrix diagonalization and SCC convergence use the vast +majority of the CPU effort. +Future work will involve extending ChIMES to systems with four or more elements, where de- +velopment of training sets and proper validation approaches remains an open question. It is likely +that these ChIMES models will require larger data sets and the potentials themselves will have +27 + +significantly more parameters than those presented in this work due to the combinatorial effect +of forming many-body clusters with different possible combinations of elements. Determination +of ERep for these systems will likely yield significant advantages over pure interatomic potentials +due to the short-ranged nature of the repulsive energy as well as the general accuracy of the quan- +tum mechanical parts of DFTB. Both of these considerations make creation of DFTB/ChIMES +model in general more tractable than optimizing ChIMES on its own as an atomistic force field. +DFTB/ChIMES can serve as either a stand-alone model for running dynamics and determining +physical and chemical properties of a system, or as a surrogate for DFT in a “boot-strapping” op- +timization, where it can serve to generate reasonably high fidelity training data for pure ChIMES +MD models. Overall, our approach can be used to enhance the speed of quantum accurate pre- +dictions for both molecular and condensed matter systems, where there is a historic reliance on +computationally intensive quantum simulations for predictions of chemical and physical properties +related to experiments. +ACKNOWLEDGMENTS +This work performed under the auspices of the U.S. Department of Energy by Lawrence Liv- +ermore National Laboratory under Contract DE-AC52-07NA27344. The assigned release number +is LLNL-JRNL-XXXXXX. +28 + +REFERENCES +1K. R. S. Chandrakumar, A. J. Page, S. Irle, +and K. Morokuma, “Carbon coating precedes +SWCNT nucleation on silicon nanoparticles: Insights from QM/MD simulations,” J. Phys. +Chem. C 117, 4238–4244 (2013). +2A. Sharma, G. D. Cody, and R. J. Hemley, “In situ diamond-anvil cell observations of methano- +genesis at high pressures and temperatures,” Energy & Fuels 23, 5571 (2009). +3W. Peiman, I. Pioro, K. Gabriel, and M. 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A 114, 11814–11824 (2010). +37 + diff --git a/FdAzT4oBgHgl3EQfxP4S/content/tmp_files/load_file.txt b/FdAzT4oBgHgl3EQfxP4S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6707d0a77cc26cf6f7d810b6c33be9bd9f3aeaaf --- /dev/null +++ b/FdAzT4oBgHgl3EQfxP4S/content/tmp_files/load_file.txt @@ -0,0 +1,1765 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf,len=1764 +page_content='Enhancing the Accuracy of Density Functional Tight Binding Models Through ChIMES Many-body Interaction Potentials Nir Goldman,1, 2 Laurence E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Fried,1 Rebecca K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Lindsey,3 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Huy Pham,1 and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Dettori1 1)Physical and Life Sciences Directorate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Lawrence Livermore National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Livermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' CA 94550 USAa) 2)Department of Chemical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Davis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' California 95616,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' United States 3)Department of Chemical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Michigan 48109,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' United States (Dated: 5 January 2023) Semi-empirical quantum models such as Density Functional Tight Binding (DFTB) are attractive methods for obtaining quantum simulation data at longer time and length scale than possible with standard approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' However, application of these models can require lengthy effort due to the lack of a systematic approach for their development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In this work, we discuss use of the Chebyshev Interaction Model for Efficient Simulation (ChIMES) to create rapidly parameterized DFTB models which exhibit strong transferability due to the inclusion of many-body interactions that might otherwise be underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We apply our modeling approach to silicon polymorphs and review previous work on titanium hy- dride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We also review creation of a general purpose DFTB/ChIMES model for organic molecules and compounds that approaches hybrid functional and coupled cluster accuracy with two orders of magnitude fewer parameters than similar neural network approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In all cases, DFTB/ChIMES yields similar accuracy to the underlying quantum method with orders of magnitude improvement in computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Our developments provide a way to create computationally efficient and highly accurate semi-empirical models for studies where physical and chemical properties can be difficult to interrogate directly and there is historically a significant reliance on theoretical approaches for interpretation and validation of experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' a)Electronic mail: ngoldman@llnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='gov 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='01733v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='mtrl-sci] 4 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' INTRODUCTION Atomistic calculation approaches for materials modeling can be used as an independent route to aid in new materials synthesis1, characterizing mixtures for use as fuel2,3, or quantifying rates for chemical decomposition of organic materials4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' These types of studies generally rely on quan- tum mechanical approaches such as Kohn-Sham Density Functional Theory (DFT) in order to aid in experimental interpretation and/or new materials design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In particular, DFT has been shown ex- tensively to yield accurate descriptions of condensed phase physical and chemical data, such as the material equation of state under compressive or tensile loads5, heats of formation/mixture of new phases6,7, and the energetics of chemical bond breaking and forming under reactive conditions8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' However, standard DFT is also renown for its significant computational expense and poor com- putational scaling (generally O(N 3)) resulting from solving for the Kohn-Sham eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' As a result, DFT molecular dynamics (MD) simulations can be limited to system sizes of hundreds of atoms for timescales of tens of picoseconds or smaller for many systems9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In contrast, many pro- cesses of interest have properties that can span orders of magnitude larger scales, including large- scale carbon heterocycle synthesis10, the rational design of 3D materials11, and defect formation and grain boundary interactions in crystalline systems12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Thus, the need for alternate simulation approaches remains a highly active research area where the goal is to develop methods that can harness the accuracy of DFT while yielding vastly improved computational efficiency and scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In this regard, machine learning approaches for the development of interatomic atomic po- tentials have been an effective route for modeling materials under reactive and nonreactive conditions13,14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' For example, neural networks have been used successfully to model structural properties of catalytic materials15 as well as the phase stability of high-entropy ceramics16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Gaus- sian Process Regression in the form of the Gaussian Approximation Potential (GAP) has been used for a number of materials, including silicon based materials17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Regardless, the development of these potentials tends to remain a highly labor-intensive task, where frequently a high-degree of expertise and months to years of human effort are required for a single application area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' As a result, it can be difficult for these efforts to keep up with experimental needs particularly in the area of materials synthesis, where the number of permutations of different starting materials, thermodynamic conditions, and catalysts can be combinatorially large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Semi-empirical quantum mechanical approaches hold promise as a middle ground for acceler- ated simulations with a high degree of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' These methods combine approximate quantum 2 mechanics with empirical functions to yield approaches that can achieve several orders of magni- tude longer time scales in quantum MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='18,19 In addition, semi-empirical approaches tend to utilize significantly fewer computational resources, allowing for ensembles of statistically independent trajectories and improved statistical sampling of desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='20 These methods also tend to show much stronger transferability to systems and conditions outside of their training set compared to interatomic potentials, in part due to the accuracy of the approximate quantum mechanics and subsequent reduced reliance on empirical functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='21 Density Functional Tight Binding (DFTB) is one such semi-empirical quantum mechanical method22,23 that has had widespread success in modeling both gas-phase molecules24 as well as condensed matter under inert and reactive conditions25–27, including extreme pressures and temperatures28,29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The DFTB total energy is derived from an expansion of the Kohn-Sham en- ergy to either second or third-order in charge fluctuations, resulting in the following expression: EDFTB = EBS + ECoul + Erep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' (1) Here, EBS corresponds to the band structure energy, ECoul is the charge fluctuation term, and Erep is the repulsive energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' EBS is calculated as a sum over occupied electronic states from the DFTB Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The DFTB Hamiltonian matrix elements are determined from pre-tabulated Slater-Koster tables derived from reference calculations with a minimal basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The onsite matrix elements are the free-atom orbital energies and the off-site terms are computed with a two- center approximation where both wavefunctions and electron density are subjected to confining potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Erep corresponds to ion-ion repulsions, as well as Hartree and exchange-correlation double counting terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This term can be expressed as an empirical function where parameters are fit to reproduce high-level quantum or experimental reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In practice, an additional dispersion correction can be included, including those in standard use for DFT calculations30,31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' DFTB is approximately three orders of magnitude more efficient than DFT calculations though it also tends to exhibit O(N 3) scaling due to the need to solve for the band structure eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' DFTB has been shown to exhibit transferability across element types and diverse conditions32–34 and has been applied to a broad range of materials35–39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' However, DFTB model development can be challenging in terms of optimizing the hyperpa- rameters needed for the approximate quantum mechanical parts of the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' These include the separate confining potentials for the wavefunctions and electron density (which can be differ- ent for each angular momentum channel of an element),38 choice of second-order vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' third-order 3 charge fluctuations for the energy expression40, and whether to use density or potential superposi- tion when computing the Slater-Koster tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='36,41 The DFTB Hamiltonian tends to be highly sen- sitive to these options42, and in general there does not exist a predefined recipe for how to choose these parameters nor how to explore that specific phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Prediction of physical and/or chem- ical properties are in turn are closely coupled to the empirical repulsive energy, which itself has a wide variety of options in terms of functional form and data to be fit35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' ERep is usually taken to be strictly pairwise (two-center), though a number of systems can require many-body terms as well for accurate predictions28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Novel approaches for determination of ERep include constrained spline optimization34, neural networks43,44, and Gaussian Process Regression45,46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Machine learning ap- proaches though tend to be highly data intensive14 and prone to overfitting21, which can pose dif- ficulties for any method that leverages these techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Thus, DFTB method development would be holistically improved through a more automatic method for parameterization, where candidate models could be screened rapidly and efficiently, thereby allowing the user to quickly determine an optimal model for their specific needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In this work, we discuss our recent efforts to overcome these issues through use of the Cheby- shev Interaction Model for Efficient simulation (ChIMES),47,48 which can be used to determine ERep for molecular and condensed phase systems relatively quickly and with comparatively lower data requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' ChIMES is a many-body reactive force field based on linear combinations of Chebyshev polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' It was initially developed for pure MD simulation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=', where all aspects of a quantum mechanical calculation have been mapped onto the ChIMES functional form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This has included both non-reactive and reactive materials, such as water under ambi- ent and high pressure-temperature conditions49,50, high pressure C/O systems51,52, and detonating energetic materials53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' DFTB/ChIMES models have been created for a wide variety of materials, including actinides and their oxides54,55, titanium-based systems36, and silicon (discussed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Additionally, ChIMES has been used to improve the accuracy of DFTB by including many-body energies and forces through ∆-learning, where ChIMES augments a pre-existing DFTB param- eterization for organic materials under ambient56 and reactive conditions39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We note that similar to other machine-learning methods21, ChIMES can be used within any semi-empirical quantum mechanical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' However, we choose to focus on DFTB due to its close resemblance to Kohn-Sham DFT as well as its proven accuracy for a variety of materials and conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We begin with a brief discussion of the ChIMES formalism, including discussion of its func- tional form and methods for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Next we present some recent results on a general pur- 4 pose DFTB/ChIMES model for silicon polymorphs, which has remained an outstanding issue in DFTB model development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We note that all DFTB calculations discussed within this work were performed with the DFTB+ code57,58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We then summarize previous work on a semi-automated workflow for screening DFTB hyperparameters and ERep determination in creating a models for TiH2, a candidate hydrogen storage material with several potential uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Finally, we review our recent results in using ChIMES to create DFTB models that approach hybrid-functional and cou- pled cluster accuracy for organic compounds and molecular solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In all cases, the advantages to use of DFTB/ChIMES lies in its rapid parameterization time, small data requirements relative to other machine-learned approaches, and the relative ease with which overfitting can be prevented due to regularization within linear optimization approaches as well as the orthogonal nature of the underlying basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' ChIMES Formalism The design philosophy behind ChIMES is based on a many-body expansion of the DFT total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Briefly, the DFT total energy can be thought of as a sum of contributions of clusters containing different numbers of atoms: EDFT = na � i1 1Ei1+ na � i1>i2 2Ei1i2+ na � i1>i2>i3 3Ei1i2i3+ na � i1>i2>i3>i4 4Ei1i2i3i4+· · ·+ na � i1>i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' inB−1>inB nBEi1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='inB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' (2) Here, the one-body energies, 1Ei1, correspond to the atomic energy constants, the two-body ener- gies, 2Ei1i2, to all pair-wise energies with indices {i1, i2}, the three-body energies, 3Ei1i2i3, to all triplet energies with indicies {i1, i2, i3}, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=', all the way up to some predeterimed maximum bod- iedness, nB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' These terms are summed over all cluster combinations within the system containing na total number of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In the ChIMES formalism, we represent each of the terms in our n-body expansion as a lin- ear combination of Chebyshev polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Chebyshev polynomials of the first kind of order m are defined by the expression Tm (cos θ) = cos (mθ), more commonly written as Tm(x), where x = cos θ and thus exists over the range [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Chebyshev polynomials offer a number of dis- tinct advantages for interpolation that bear mentioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Chebyshev polynomials of the first kind 5 are orthogonal with respect to the weighting function 1/ √ 1 − x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' They can be computed with a recurrence relationship and define a complete basis set, allowing for arbitrary complexity in a potential energy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Their orthogonality allows for simple regularization where higher-order polynomial coefficients can be set to zero without necessarily adversely affecting the quality of the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Polynomial expansions with Chebyshev polynomials of the first kind will have exponentially decreasing coefficients for higher-order terms due to their monic form, helping to prevent overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In addition, they yield a “nearly optimal” error function, where the error in an expansion will closely resemble a minimax polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The derivaties of Chebyshev polyno- mials of the first kind are related to Chebyshev polynomials of the kind Um(x) by the expression dTm/dx = mUm−1, where Um (cos θ) = sin [(n + 1) θ] /sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Chebyshev polynomials of the second kind also form an orthogonal basis set (with respect to the weighting function √ 1 − x2) and can also be generated via a recurrence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This can allow for arbitrary complexity for structural optimization or molecular dynamics calculations, where atomic forces are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' As a result, we can now write the two-body (2B) energy term in Equation 2 as the following expression: 2Ei1i2 = fp (ri1i2) + f ei1ei2 c (ri1i2) O2 � m=1 C ei1ei2 m Tm(s ei1ei2 i1i2 ) (3) In this case, C ei1ei2 m is the corresponding permutationally invariant coefficient for the interaction between atom types ei1 and ei2, taken from the set of all possible element types, {e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Tm � s ei1ei2 i1i2 � represents a Chebyshev polynomial of order m, and s ei1ei2 i1i2 is the pair distance transformed to occur over the interval [−1, 1] using a Morse-like function59,60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' For that coordinate transform, s ei1ei2 i1i2 ∝ exp (−ri1i2/λe1e2) and λe1e2 is an element-pair distance scaling constant, usually taken to be the peak position of the first coordination shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Further details are discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The term f ei1ei2 c (ri1i2) is a Tersoff cutoff function61 which is set to zero beyond a maximum distance defined for a given {e1, e2} pair set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In order to prevent sampling of ri1i2 distances below what is sampled in our DFT training set, we introduce use of a smooth penalty function fp(ri1i2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We can create greater than two-body orthogonal polynomials by defining a cluster of size n and taking the product of the Chebyshev polynomials derived from the constituent �n 2 � unique pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' For example, the three-body polynomials will be products of �3 2 � = 3 two-body polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We thus write the ChIMES three-body (3B) energy as the following: 6 3Ei1i2i3 = f ei1ei2 c (ri1i2) f ei1ei3 c (ri1i3) f ei2ei3 c (ri2i3) O3 � m=0 O3 � p=0 O3 � q=0 ′ C ei1ei2ei3 mpq Tm � s ei1ei2 i1i2 � Tp � s ei1ei3 i1i3 � Tq � s ei2ei3 i2i3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' (4) We take a triple sum for the i1i2, i1i3, and i2i3 polynomials over the hypercube up to O3, and include a single permutationally invariant coefficient for each set of powers and atom types, C ei1ei2ei3 mpq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We use the primed sum to denote that only terms for which two or more of the m, p, q polynomial powers are greater than zero are included in order to guarantee that three distinct atom-centers are evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The expression for 3Ei1i2i3 also contains the fc smoothly varying cut- off functions for each constituent pair distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Penalty functions are not included in this case and instead are handled entirely by the two-body interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Higher bodied terms are included in ChIMES in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' For example, four-body (4B) terms are regularly included in ChIMES optimizations53, where 4Ei1i2i3i4 is now determined from the sum over the product of the �4 2 � = 6 constituent pair-wise polynomials multiplied by a single permutationally invariant coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In practice, even higher bodied terms could be included in ChIMES, though this can lead to a combinatorially large polynomial space and hence parameter explosion that can lead to overfitting and excessive computational expense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Hence, the norm with ChIMES optimization is generally to include up to four-body terms, though DFTB/ChIMES models tend to be converged with up to three-body terms, only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='36,39,54–56 Optimal ChIMES parameters (the coefficients of linear combination) can then readily be deter- mined through the overdetermined matrix equation wAC = wBrep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The matrix A corresponds to the derivatives of the ChIMES energy or force expression with respect to the fitting coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The column vectors C and Brep correspond to the linear ChIMES coefficients for which we are solving and the numerical values for the training data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The symbol w corresponds to a diagonal matrix of weights to be applied to the elements of Brep and rows of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This linear least-squares optimization problem can be solved for with any number of established algorithms, discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' ChIMES optimization for ERep or ∆-learning The ChIMES training set for determination of ERep or ∆-learning proceed in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' ERep training is computed by calculating DFTB forces (F), stress tensor components (σ), and 7 possibly system energies Etot for each configuration in the training set with the chosen set of Hamiltonian parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=', {Rψ}, {Rn}, density or potential superposition, second or third- order DFTB) with zero values for those components from ERep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' These “repulsive energy free” results are then subtracted from the DFT values for those quantities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=', Eτ∗ Rep = Eτ DFTi − Eτ QM,DFTBi F τ∗ Repαi = F τ DFTαi − F τ QM,DFTBαi στ∗ Repαβ = στ DFTαβ − στ QM,DFTBαβ (5) Here, τ corresponds to a specific MD configuration, α and β to the cartesian directions, and i is the atomic index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In practice, we have used the diagonal components of the stress tensor, only (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=', α = β in Equation 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The ‘*’ is used to denote that the quantities being computed are part of the training set, and ‘QM,DFTB’ refers to the quantum components of the DFTB calculation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=', only forces and stresses from EBS and ECoul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Calculation of a ∆-learning training set is identical with the exception that the quantities in Equation 5 are no longer repulsive energy free but instead contain terms from the DFTB repulsive energy model of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This results in the following objective function: Fobj = � � � � 1 Nd × � M � τ=1 N � i=1 3 � α=1 w2 Fαi (∆Fαi)2 + M � τ=1 3 � α=1 w2σαα (∆σαα)2 + M � τ=1 w2 Ei (∆Ei)2 � , (6) where M is the total number of configurations in the training set and Nd is the total number of data entries (3MN force components plus 3M stress tensor components plus M energy components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In addition, ∆Fαi = F τ ChIMESαi − F τ∗ Repαi, ∆σαβ = στ ChIMESαβ − στ∗ Repαβ, and ∆Ei = Eτ ChIMESi − Eτ∗ Repi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' ChIMES bears some resemblance to the Atomic Cluster Expansion approach (ACE)62,63, where many-body interactions are represented by a product of Chebyshev polynomials and real spherical harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' These models also differ from ChIMES in that the underlying polynomial basis set is atom-centered (similar in spirit to an embedded atom model64) rather than using a cluster approach as we adopt here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Similarly, the spectral neighbor analysis potential (SNAP) uses bispectrum components to compute the total energy of a system as a sum over atom energies, which are expressed as a weighted sum over bispectrum components65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Linear least-squares approaches for ChIMES optimization The ChIMES potential is linear with respect to the fitting coefficients, which allows for use of powerful global optimization tools that are unavailable to non-linear machine-learned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In our efforts, we have focussed on the Singular Value Decomposition (SVD) and Least-Angle Regression (LARS) with Least Absolute Selection and Shrinkage Operator (LASSO) regulariza- tion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We now offer a brief discussion of each method and leave details to the pertinent references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' SVD66 solves for optimal fitting coefficients directly by performing an eigendecomposition of the generally rectangular A matrix and computing its pseudo-inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Regularization can be performed by setting singular values (eigenvalues of the square matrix in the SVD decomposition) with an absolute value below a given threshold to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In our work, we take this parameter to be Dmaxϵ, where Dmax is the maximum singular value of A and ϵ is a factor below a value of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' LARS is a type of forward step-wise or iterative regression67,68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Here, all model coefficients are initialized to zero and the covariate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=', polynomial values) most correlated to the error residual is determined (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=', those having the most significant impact on the fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The corresponding ChIMES parameter is modified incrementally to minimize the error residual until a second covariate yields an equal correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' At this point, it is included in the active parameter set and both coefficients are modified simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The process continues until all coefficients are included in the solution, at which point a result equivalent to ordinary least squares fitting is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In practice, LARS optimization can be performed using only a subset of all possible parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' LASSO69 is an L1-norm regularization method whereby regularization is based on the sum of the absolute values of the fitting coefficients, which has the effect of shrinking a subset of parameters to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In this case, the objective function Fobj (Equation 6) is minimized with the following additional constraint: F LASSO obj = Fobj + 2α ni � i=1 |ci| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' (7) Here, ni is the total number of unique fitting parameters, ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The parameter α regularizes the magnitude of the fitting coefficients, which reduces possible overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The LASSO method can be implemented as a variant of LARS where parameters are either added or removed at each solution stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We find the LASSO variant of LARS to be numerically stable for ill-conditioned A matrices, which are often found in force matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 9 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' DFTB/ChIMES Models for Silicon Polymorphs Silicon has proven to be a significant challenge for DFTB model parameterization likely due to the fact that its different polymorphs can have different coordination numbers and nearest neigh- bor distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This yields a variety of bond lengths and energies that need to be accounted for in order to obtain a single, transferable DFTB model that does not have to be specific for a given solid phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Previous work has shown that standard two-body repulsive energies do not exhibit sufficient complexity to accurately account for several Si phases with different bonding environments,34 in contrast to carbon, where multiple phases can be represented by a single two-body polynomial expansion70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Neural network (NN) approaches have been used for the repulsive energy in order to account for many-body interactions in ERep,44 and the results are promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' NN approaches though generally require large amounts of data and can frequently optimize to local minima, po- tentially complicating their use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Here, we attempt to overcome this issue by creating a many-body ChIMES ERep for silicon that is transferable to a number of different Si polymorphs as well as prediction of vibrational spectra and calculation of defect formation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In our work, we target two previous Si DFTB parameterizations, pbc-0-371 and siband-1-1,41 which have different strengths and weaknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The pbc-0-3 parameter was creating using density superposition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=', the quantum mechanical potential VQM (ρ) was expressed as V (ρA + ρB) for atoms A and B) , which tends to be preferred due to its improved representation of chemical bonding and vibrations36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' However, d-orbital interactions were not tabulated aside from the d- orbital onsite energy, which could have ramifications for some material properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In contrast, the siband-1-1 parameter set was specifically created with d-orbital interactions but with potential superposition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=', VQM (ρ) = V (ρA) + V (ρB)) in order to yield accurate prediction of electronic properties, including the electronic band structure of Si-containing solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In addition, the siband- 1-1 parameter set does not contain a repulsive energy of any sort, precluding its use in structural relaxation or MD simulation which severely limits its usefulness overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Our goal is to thus to create new ChIMES ERep potentials for each set of Slater-Koster interac- tion parameters using identical DFT training data and ChIMES hyperparameters in order to com- pare and contrast the effectiveness of each as a possible one-fits-all model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Calculations for our sil- icon DFT dataset were performed using the Vienna ab initio Simulation Package (VASP)72–74, with 10 projector-augmented wave function (PAW) pseudopotentials75,76 and the Perdew-Burke-Ernzerhof exchange-correlation functional (PBE)77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We found our results to be converged with a planewave cutoff of 500 eV, which was used in all of the calculations discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We have used an electron density convergence criteria of 10−6 eV, with a force convergence of 10−2 eV/ ˚A for all geometry/cell optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The Mermin functional78 smearing was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='03 eV for all calcu- lations performed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The system energy and pressure was found to be converged with sampling of the Brillouin Zone with a 2 × 2 × 2 Monkhorst-Pack mesh79 for all supercells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We then generated cold curves for each phase by isotropically expanding and contracting the simula- tion cell lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Here, we used a diamond structure supercell of 64 atoms, a bcc structure of 54 atoms, a simple cubic structure of 64 atoms, and a graphene sheet of 32 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This yielded an initial set of 463 configurations for our ChIMES ERep optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In order to sample forces from a variety different configurations, we have also included MD data for the diamond and graphene phases, using the same number of atoms in each supercell as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' These supercells were isotropically expanded and contracted between 90% to 110% of the ground-state density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Each MD simulation was run for ∼5 picoseconds at 600 K, from which we took snapshots at fixed intervals of ∼200 femtoseconds for our training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This yielded an additional 405 configurations for our ChIMES ERep determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In all, our final training set contained a total of 838 configurations of different silicon phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' ChIMES ERep optimization was then performed using values of rmin = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='0 ˚A and rmax = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='0 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The value of rmax was informed in part from previous development of a neural network repulsive energy,34 which resulted in a minimization of the root mean square (RMS) error in our fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In addition, we found that a value of rmax = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='0 ˚A yielded an improved description of the expanded states in our training set, where the bonded interactions between Si atoms is longer than the ground-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We now refer to our ChIMES model based on pbc-0-3 as pbc/ChIMES and our model based on siband-1-1 as siband/ChIMES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Both pbc/ChIMES and siband/ChIMES were created with a 2B order of 12, 3B order of 8, and a LASSO regularization parameter (α) value of 10−3, similar to previous efforts36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We have used the Morse coordinate transform with a value of λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='4 ˚A, which corresponds to the first peak in the diamond phase radial distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' For pbc/ChIMES, this yielded an overall RMS error of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='44 eV/ ˚A in the forces, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='43 GPA in the pressure, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='038 eV/atom in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The RMS errors for siband/ChIMES were slightly higher, with values of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='22 eV/ ˚A for the forces, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='55 GPa for the pressure, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='16 eV/atom for the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Use of a Chebyshev basis set 2B order of 16, 3B order of 12, and 4B order of 4 yielded reduction in 11 the RMS errors of < 1% with similarly marginal improvement in validation quantities such as the computed defect energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Use of a value of λ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='0 ˚A also had only a small effect on the resulting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' All ChIMES/DFTB calculations were performed with self-consistent charges using similar parameters to our DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This included charge convergence criteria of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='72 × 10−5 eV (10−6 au), a force convergence of 10−2 eV/ ˚A for all geometry optimizations, and 2×2×2 k-point mesh for all calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' TABLE I: Ground state energies relative to diamond (∆Ediam) in eV/atom and nearest neighbor distances (NN) in ˚A for the Si polymorphs considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' diamond bcc simple cubic graphene bc8 NN ∆Ediam NN ∆Ediam NN ∆Ediam NN ∆Ediam NN ∆Ediam pbc/ChIMES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='14 siband/ChIMES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='59 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='15 DFT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='54 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='65 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='16 In order to test the applicability of our ChIMES/DFTB models to different of Si phases, we have computed the relative energies and nearest neighbor distances for several polymorphs, including those in our training set as well as the bc8 phase (Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Our results indicate strong agreement with DFT for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We observe close agreement for all properties for both pbc/ChIMES and siband/ChIMES, where the energy of each phase relative to the diamond ground-state tends to agree with DFT within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='01 eV, and the subsequent nearest neighbor distances agree within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='01 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='02 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The graphene phase is a small exception, where pbc/ChIMES yielded a relative energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='70 eV/atom and siband/ChIMES a relative energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='59 eV, compared to a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='65 eV for DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' However, both models still yield the correct energetic ordering of the phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Similar to previous efforts34,44, we have determined cold energy curves under isotropic com- pression and expansion for all phases in this study (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Overall, both pbc/ChIMES and siband/ChIMES yield close agreement with DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Both models have particularly close agreement for the diamond and simple cubic phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The siband/ChIMES model exhibited a small oscil- lation in the bcc cold curve at a nearest neighbor distance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='7 ˚A which is not present in the pbc/ChIMES result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' However, the agreement with DFT is reasonable for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The largest disagreement for pbc/ChIMES is with graphene, where it yields a more positive curvature at ex- panded densities, whereas siband/ChIMES yields closer agreement to DFT overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Both models 12 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='6 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='4 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='2 −5 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='8 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='6 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='4 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='2 −4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='9 3 Energy/atom (eV) NN (Å) (a) pbc/ChIMES −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='6 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='4 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='2 −5 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='8 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='6 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='4 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='2 −4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='9 3 Energy/atom (eV) NN (Å) (b) siband/ChIMES FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 1: Cold curves for several silicon polymorphs from pbc/ChIMES and siband/ChIMES DFTB models (points) compared to results from DFT (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The black curves correspond to the diamond phase, blue to bcc, red to simple cubic, and the green to graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The orange marks correspond to the bc8 phase and were not a part of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' predict very similar agreement for the bc8 phase, where each yielded a small oscillation in the cold curve around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='5 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This is likely due to insufficient sampling of these Si-Si distances and bonding environments in our training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Regardless, these results indicate strong agreement for energy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' volume relationships, which could indicate accurate force prediction from each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We now assess the force output from each model through comparison of the resulting vibra- tional density of states (VDOS) for the diamond phase to results from DFT (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' These were computed from Fourier Transform of the velocity autocorrelation function which was determined from MD simulations at constant volume-temperature (NVT), conducted at 600 K, using a Nos´e- Hoover thermostatted chain80–82 and run for 15–20 ps using a timestep of 1 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Our results for pbc/ChIMES indicate fairly close agreement with DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Prediction of the lowest lying vibrational peak is off by only ∼7 cm−1, with a value of 134 cm−1 compared to a value of 127 cm−1 from DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' DFT yields a small peak at 231 cm−1 which appears as a broad, higher intensity shoulder at 224 cm−1 in the pbc/ChIMES spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The remaining peaks in the spectrum show similarly strong agreement with some variation in the intensity of the peaks, including accurate prediction from pbc/ChIMES of the vibron peak at 450 cm−1 compared to a frequency of 453 cm−1 from DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 13 −20 0 20 40 60 80 100 120 140 160 180 200 100 200 300 400 500 Intensity Frequency (cm−1) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 2: Vibrational density of states for the Si diamond phase, computed at 600 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The red line corresponds to pbc/ChIMES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' the blue line to siband/ChIMES, and the black dashed line to DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In contrast, siband/ChIMES shows slightly less accurate agreement with DFT overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The agreement for the lowest vibrational peak is fairly close, with a frequency of 120 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The remainder of the siband/ChIMES spectrum yields an accurate overall shape of the VDOS, though with some errors in peak positions and intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' There is some deviation in the siband/ChIMES spectrum for next two vibrational peaks, where we observe a frequency of 173 cm−1 for the second lowest frequency peak compared to a value of 188 cm−1 from DFT and a frequency of 217 cm−1 for the low intensity peak after that compared to the previously mentioned DFT peak at 231 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The siband/ChIMES spectrum yields a close match in intensity and frequency with DFT for the VDOS peak at 344 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' However, the subsequent two peaks are red shifted in frequency and lower in intensity, with values of peak positions of 413 and 472 cm−1, compared to values of 396 and 453 cm−1 from DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The improved VDOS determination from pbc/ChIMES could be due in part to its parameterization with density superposition, which has been shown to yield more accurate predictions over potential superposition36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We note that these peak position differences discussed here correspond to small changes in energy, where 20 cm−1 corresponds to ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='5 × 10−3 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Hence, it is possible that siband/ChIMES will still yield sufficiently accurate forces for some applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 14 (a) Vacancy (b) Tetrahedral (c) Hexagonal FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 3: Images of the diamond phase point defects investigated in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' All defects are shown as a red sphere for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' TABLE II: Defect formation energies for the Si diamond phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' All energies are in eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Defect pbc/ChIMES siband/ChIMES DFT (PBE) vacancy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='84 tetrahedral 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='88 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='84 hexagonal 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='87 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='79 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='61 Finally, we have computed defect formation energies from our DFTB/ChIMES models (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Here, we have investigated a single Si atom vacancy as well as an interstitial atom in either a hexagonal or tetrahedral site, which were determined from use of the pymatgen software suite83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The tetrahedral interstitial site occurs where an additional Si atom is coordinated by four atoms from the lattice, whereas the hexagonal interstitial site occurs when the additional Si atom re- sides in a hexagonal opening within the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The defect formation energy Eform is computed as Eform = Edef − NdefEdiam, where Edef is the total energy of the defect containing system, Ndef is the number of Si atoms in that configuration, and Ediam is the energy per atom of the perfect diamond phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Similar to previous Si DFTB efforts44, calculations were initialized from an optimized 216 atom supercell where we retained a Monkhorst-Pack mesh of 2 × 2 × 2, after which we created the point defect and optimized the ionic positions of each configuration using the same k-point mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Our results indicate some agreement with previous PBE-DFT calculations from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The pbc/ChIMES model agrees with the DFT vacancy energy within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='4 eV, but yields results that are 1–2 eV too high for both interstitial energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In particular, the three defect energies from pbc/CHIMES differ over a range of over 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='4 eV, with the both interstitial energies 15 yielding larger results than that of the vacancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In contrast, the result from DFT all lie relatively close together (within a range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='23 eV) and DFT exhibits equal formation energy values for the vacancy and tetrahedral interstitial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' It is likely that the interstitial energies would be decreased with full accounting of d-orbital off-site interactions, which are absent in the original pbc-0-3 parameter set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The siband/ChIMES model yields defect formation energies that are consistently ∼1 eV too high relative to DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' However, the siband/ChIMES results differ over an energy range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='28 eV, yielding improved agreement with DFT in this respect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' It is likely that there would be some variation in DFT results depending on the choice of exchange-correlation function and possible inclusion of a dispersion energy correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Overall, our we able to create two new DFTB/ChIMES models that more closely approach a single-purpose approach for silicon phases under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The pbc/ChIMES model ap- pears to yield a somewhat improved description of atomic forces, whereas as the siband/ChIMES model yields more systematically consistent defect formation energies that could make it prefer- able for some calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' As mentioned, some of the limitations of the pbc/ChIMES model could possibly be overcome through inclusion of d-orbital two-center interactions in the corresponding Slater-Koster file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Regardless, we now provide a repulsive energy for the siband-1-1 parameter set, which will allow its use for structural relaxations and/or dynamics calculations in addition to its accuracy for electronic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' It is possible that the slightly longer cutoff radius for our ChIMES ERep could be mitigated through optimization of the choice of DFTB confining radii (discussed in the next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Future improvement of these models could also involve inclusion of data from MD simulations of amorphous or defect containing systems at different temperatures and pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Semi-automated Workflow for DFTB/ChIMES Model Creation In this subsection, we summarize previous work on TiH236 which indicates the utility in using a ChIMES ERep in a semi-automated fashion to screen for optimal confining radii in a Slater-Koster file parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' TiH2 has a number of industrial uses as a functional material, including in hydrogen storage alloys84, superconductors85, and as a blending agent for porous foams86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Its ground-state structure exhibits face-centered-cubic (fcc) symmetry, with the (111) facet computed to have the lowest surface energy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Several adsorption sites are illustrated, including Top (directly above a Ti atom), Hollow (in an interstitial cavity), and several Bridge sites (existing in 16 between Ti-Ti and H-H nearest neighbors) sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' TiH2 is a somewhat ideal system for DFTB model development in that DFT calculations on small supercells are relatively tractable, which allows for straightforward validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' DFT calculations though are generally too computationally inefficient for the larger supercells needed to model grain boundaries and crystalline defects at sufficiently low concentration, allowing for several applications of a new TiH2 DFTB model in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Ti H x o o o o o o o x2 x1 (111) surface (011) surface Bulk FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 4: Pictures of TiH2 bulk and surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The left panel shows the bulk fcc lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The middle panel shows the (111) crystalline surface the Top (marked with an ‘O’) and Hollow (‘X’) adsorption sites indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The right panel shows the (011) crystalline surface with the Top (‘O’), Bridge-1 (‘X1’) and Bridge-2 (‘X2’) sites indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Reprinted with permission from Journal of Chemical Theory and Computation 2021 17 (7), 4435-4448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Copyright 2021, American Chemical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Here, we have leveraged rapid ChIMES ERep optimzation by creating a workflow that allowed us to perform a semi-exhaustive search of all DFTB and ChIMES hyperparameters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We first compute a matrix of thirty Slater-Koster files from titanium wavefunction confining radii (RTi ψ ) and density confining radii (RTi n ) sampled over a range of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='2 ≤ RTi ψ ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='0 au and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='0 ≤ RTi n ≤ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='0 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Hydrogen interaction parameters were taken from the miomod-hh-0-1 parameter set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Model down selection could then be performed over the entire grid Slater-Koster tables through comparison to our selected validation data, which allowed us to determine optimal ChIMES polynomial orders and the LASSO regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' For this work, our DFT training set consisted of molecular dynamics simulations of unit cell configurations (12 atoms total), run for 5 ps at 400 K with simulation cells initially optimized to pressures in a range from −8 to 100 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' All MD calculations were run in the constant temperature and volume (NV T) ensemble with Nos´e-Hoover thermostat chains80–82 and a timestep of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='2 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The slightly elevated temperature and wide pressure range including negative pressure were chosen in order to yield a broad sampling of the underlying potential energy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Atomic forces and the diagonal of the stress tensor were then sampled from MD configurations at fixed time intervals 17 bbSelect !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' ", $!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=', $";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Create SKF files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Compute DFTB training set: ⃗&DFT − ⃗&DFTB (no ERep) (##,DFT −(##,DFTB (no ERep) Desired accuracy achieved?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Validation set: Bulk: lattice const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=', VDOS, 1H and 2H vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' (001), (011), and (111) surface energies Eads on (011) and (111) surfaces (5 total) (011) and (111) surface and subsurface H vacancy energies (8 total) Choose ChIMES 2B, 3B, 4B orders, cutoff radii and determine )$%&.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Yes No Complete Compute DFT-MD data FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 5: Flowchart for creation of DFTB Erep models through ChIMES force field parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Reprinted with permission from Journal of Chemical Theory and Computation 2021 17 (7), 4435-4448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Copyright 2021, American Chemical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' of ∼ 160 fs in order ensure configurations were as statistically uncorrelated as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This yielded up to 30 MD snapshots for each pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Inclusion of system energies in our training data did not appear to improve the quality of our optimization and hence were omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In addition, in order to sample hyper- and hypo-coordinated configurations in the system, we included MD data for a unit cell with a single hydrogen interstitial or single vacancy site, each run for 5 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This yielded a total of 153 unit cell-sized configurations for our training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Validation calculations for all of our DFTB/ChIMES models were performed on the bulk lattice constant, single and double hydrogen vacancy energies, and the vibrational density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We also validated our models against a number of surface properties, including the surface energies of the (001), (011) and (111) facets, five different hydrogen adsorption energies on the (011) and (111) surfaces, and surface and subsurface hydrogen vacancy energies on the same two facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Validation data for hydrogen interactions with the (001) surface were omitted from our study due to the presence of a significant surface dipole on this facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Once again, all DFT calculations were performed with VASP using PAW pseudopotentials and PBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We found our results to be converged with a planewave cutoff of 400 eV and an energy 18 convergence criteria of 10−6 eV, both of which were used for the results reported here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Fourth order Methfessel-Paxton smearing87 was used with a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='13 eV for all geometry and cell lattice optimizations in order to ensure energy convergence without dependence on the electronic smearing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The Mermin functional78 with the same electronic temperature was used for all MD calculations in order to avoid spurious forces due to possible negative occupation numbers from the Methfessel-Paxton approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Brillouin Zone sampling for all TiH2 unit cell calculations was performed with a 10 × 10 × 10 k-point mesh, whereas we used a mesh of 5 × 5 × 5 for 32 formula unit (96 atom) bulk calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We used system sizes of 168 atoms/7 layers for the (001) surface, 144 atoms/6 layers for the (011) surface, and 192 atoms/8 layers for the (111) surface, each with a vacuum of 20 ˚A and a k-point mesh of 5×5×1 in the direction of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' DFTB+ calculations were performed using self-consistent charges (SCC)22 and charge conver- gence criteria of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='72 × 10−5 eV (10−6 au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Inclusion of an external van der Waals correction31,88 is beyond the scope of our present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We have performed “shell-resolved” SCC calculations, where separate Hubbard U parameters were determined for each orbital angular momentum shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' All minimum and cutoff radii for the ChIMES ERep were set to include the first coordination shell sampled in our training set, only: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='5 ≤ rTiTi ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='5 ˚A and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='5 ≤ rHTi ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='5 ˚A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We use values of λTiTi = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='0 ˚A and λHTi = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='0 ˚A for the Morse-like coordinate transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' H-H repulsive interaction were not sampled in our training set and were thus also taken from the miomod-hh-0-1 parameter set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Our results for a subset of our validation data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 6) allow us to describe general trends regarding the confining radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We observe an approximate linear relationship between RTi ψ and RTi n in terms of the accuracy of the E111 energy, where the most accurate surface energy results from either small or large choice for both radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' All of the DFTB/ChIMES models created in this iteration tend to under-predict the (E001/E111) ratio (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=', the ratio of highest to lowest surface energies in our study) relative to our DFT calculations, where we observe values of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='35–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='44 compared to the DFT ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We note that is is likely in part due to the surface dipole moment present in our construction of the (001) facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In addition, our results indicate a much smaller dependence on choice of RTi n for a given RTi ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We note that there can be strong dependence of the surface energies on choice of DFT functional (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 89), although the relative energetic ordering tends to be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Our final set of hyper-parameter values includes {RTi ψ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='6 au, RTi n = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='0 au} and {O2B = 8, O3B = 4}, optimized with LASSO/LARS and regularization of α = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This model yields 19 6 8 10 12 14 16 18 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='5 5 RTi n (au) RTi ψ (au) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='15 Fractional Deviation of E111 6 8 10 12 14 16 18 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='5 5 RTi n (au) RTi ψ (au) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='35 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='27 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='26 ∆(E001/E111) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 6: Results for sweep of values of RTi ψ and RTi n , where the ChIMES ERep was determined with a 2B order of 12 and 3B order of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The top panel corresponds to the fractional deviation of the surface energy, � EDFTB 111 − EDFT 111 � /EDFT 111 , and the middle panel to the deviation of (E001/E111) relative to DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Reprinted with permission from Journal of Chemical Theory and Computation 2021 17 (7), 4435-4448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Copyright 2021, American Chemical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' RMS errors of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='076 eV/ ˚A for hydrogen forces, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='056 eV/ ˚A for titanium forces, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='35 GPa for the stress tensor diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Results for bulk properties indicate that DFTB/ChIMES yields a lattice constant with errors of only ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='4% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='0% from DFT and experiment90, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' However, our model yields a hydrogen bulk vacancy energy (Evac) that is ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='5 eV too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We found that a systematic ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='5 eV underestimation of vacancy energies in a variety of environments and concentrations was typical for all ChIMES parameterizations created in this work, which could be rectified with improved training data or adaptations to DFTB such as the inclusion of multi-center terms in the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Overall, our final model yields accurate surface energies for all three low-index facets investi- gated in this study (Table III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In particular, the E011 and E111 values are nearly identical to those from DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The E001 value from DFTB/ChIMES is around 17% lower than than that for our DFT calculations (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='114 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='136 eV/ ˚A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This could be due in part to the internal electric field on the (001) surface configuration studied here, as mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' DFTB generally can underestimate surface electrostatic interactions due to its determination of atom-centered point charges only in Coulom- bic interactions92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Our DFTB/ChIMES results show similarly strong agreement with hydrogen surface adsorption energies (Table IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We compute the correct energetic ordering of adsorption on the (111) Top and Hollow sites, though the Hollow site energy is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='35 eV smaller than that from 20 DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We see similar agreement with DFT for the (011) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Here, DFTB/ChIMES show close agreement for Top site adsorption with a difference of only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='05 eV from DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Our model yields Bridge-1 and Bridge-2 adsorption energies that differ from DFT by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='29 eV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='21 eV, respec- tively, and incorrectly predicts that the Top site is the lowest energetically of the three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Regardless, these values are similar in energy for all surface sites and we have overall favorable agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' TABLE III: TiH2 surface energies (in eV/ ˚A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Reprinted with permission from Journal of Chemical Theory and Computation 2021 17 (7), 4435-4448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Copyright 2021, American Chemical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Surface DFTB/ChIMES DFT 111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='080 011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='101 001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='136 TABLE IV: Surface hydrogen adsorption energies on TiH2 surface sites (in eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Reprinted with permission from Journal of Chemical Theory and Computation 2021 17 (7), 4435-4448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Copyright 2021, American Chemical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Surface Site DFTB/ChIMES DFT 111 Top 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='888 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='760 Hollow 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='081 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='440 011 Top 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='383 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='332 Bridge-1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='154 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='442 Bridge-2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='132 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='342 Our results indicate DFTB/ChIMES models can be accurately determined based on relatively small training data (unit cell MD calculations in this work), even for physically complex sys- tems such as those containing surface chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Further refinement of our TiH2 model could involve inclusion of training data from additional phases and thermodynamic state points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Re- gardless, our current effort yields accurate results for bulk and surface TiH2 properties, and our model shows strong transferability to bulk α-Ti and gas phase TiH4 (not shown here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The small training set could yield significant advantages for computationally challenging systems such as 21 magnetic materials and their interfaces, where DFT data is limited and difficult to generate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Over- all, our DFTB/ChIMES approach can have particular impact on myriad of research areas, such as interpretation of imaging and spectroscopy studies on bulk and interfacial systems, where there is traditionally a strong coupling with atomistic simulation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' ∆-learning to Enhance the Accuracy of DFTB for Organic Materials In this subsection we review our recent efforts to leverage a high-level quantum chemical database to create an “out-of-the-box” model with accuracy beyond standard DFT approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=', PBE) that is generally applicable to many organic molecular systems56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In this work, we have used the ANI-1x quantum chemical data set93,94 to create a DFTB/ChIMES model that approaches hybrid-functional and/or coupled cluster accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Here, ChIMES is used as a ∆-learning po- tential where we have included it as an extra energy term to the 3ob-3-1 parameterization40,95, which includes third-order charge fluctuation terms in the DFTB energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This parameterization is known to yield reliable accuracy for many organic molecules and thus was a reasonable starting point for our efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We have found that the advantage of ChIMES over a neural network ap- proach is two-fold: (1) the training set requirements of ChIMES is significantly lower, where only a small fraction of the ANI-1x dataset was required to achieve a high degree of accuracy, and (2) our ChIMES potential required two-order of magnitude fewer parameters than several recent NN- based semi-empirical approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' These effects allow for a much easier to parameterize model that is less likely to be hampered by overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The original ANI-1x database was developed for the creation of ML-based general-purpose organic potentials where the data set was determined through an active learning process94, result- ing in approximately 5 million molecular equilibrium and non-equilibrium configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Our ∆-learning optimization used an iterative approach by first creating a subset of ANI-1x called “sub ANI-1x” that only contained results computed from CCSD(T) (coupled-cluster consider- ing single, double, and perturbative triple excitations) and using a well-known hybrid functional, wB97X96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This corresponded to 459,464 molecular confirmations from computed from 1895 unique molecules, or ∼10% of the original ANI-1x database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We note that there are no atomic force data from CCSD(T)/CBS calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Hence, we used wB97X results computed with a large basis set (def2-TZVPP) data for fitting purposes, with the remainder of the data set available for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 22 We then used an iterative approach to ChIMES optimization (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 7) where we first randomly selected only 1% of sub ANI-1x and performed an initial ChIMES optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Validation cal- culations agains the remainder of sub ANI-1x resulted in some large deviations in the computed energies and forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We then selected an additional equivalent of 1% of the data set from con- figurations with the highest force deviations and added them to our training set and repeated the process, where each increment of the training process would include the equivalent of an additional 1% of sub ANI-1x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Our DFTB/ChIMES ∆-learning was converged after three iterations of our optimization scheme, using only 3% of sub ANI-1x or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='3% of the original ANI-1x database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Our model was ultimately validated against the entire sub ANI-1x data set, though its size is somewhat arbitrary and it is possible that a smaller subset of ANI-1x could have been used for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='2000 -1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 7: Comparison of energies per atom (top panels) and forces (bottom panels) predicted by DFT (wB97X) and DFTB/ChIMES for all configurations in the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The dataset used here is ‘sub ANI-1x’, ∼10% of the full ANI-1x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Reprinted with permission from J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 2022 13 (13), 2934-2942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Copyright 2022, American Chemical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Our final model used ChIMES polynomial orders of {2B = 24, 3B = 10, 4B = 0} with a somewhat long radial cutoff of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='0 ˚A used for all atom pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This longer cutoff helped account for some dispersion interactions that would otherwise be absent from standard DFTB calculations, though future efforts will involve shorter cutoffs combined with a dispersion interaction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 23 Further details about our ChIMES model for organics can be found in the Supporting Information in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Ultimately, our DFTB/ChIMES model resulted in 5546 parameters and was trained to ∼372k data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' This is in contrast to the recently developed AIQM1 semi-empirical quantum model, which utilizes an NN trained to the entire ANI-1x data set, resulting in 322,660 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Similarly, a recent DFTB-NN approach using deep-tensor neural networks used a training set of ∼800k data points, resulting in 228,865 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' TABLE V: Performance of DFTB and DFTB/ChIMES in predicting reference energies and/or atomic forces in the GDB-10to13, ISO34, and GDML data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The MAE and RMSE for the energies and forces (labeled with subscripts ‘E’ and ‘F’) are in kcal/mol and kcal/mol- ˚A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Reference molecular energies and atomic forces in the GDB-10to13 data set are at the wB97X/6-31G* level of theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Isomerization energies in the ISO34 data set are a mixture of experimental- and CCSD(T) extrapolation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The CCSD(T)/cc-pVTZ atomic forces of 2000 configurations of ethanol in the GDML data set are used for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Reprinted with permission from J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 2022 13 (13), 2934-2942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Copyright 2022, American Chemical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' GDB-10to13 ISO34 GDML method MAEE/RMSEE MAEF/RMSEF MAEE/RMSEE MAEF/RMSEF DFTB 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='10/11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='70 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='34/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='85 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='69/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='96 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='52/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='12 DFTB/ChIMES 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='57/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='72 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='62/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='06/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='56 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='72/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='61 ANI-197 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='12/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='74 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='96/7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='09 ANI-1x97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='30/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='67/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='01 DFTB-NNrep98 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='21/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='30 PBE098 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='82/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='48 We then tested the transferability of our DFTB/ChIMES model through comparison to different quantum chemical data that were computed at the wB97X or CCSD(T) level but were not a part of ANI-1x (Table V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' For example, the GDB-10to13 data set97 consists of the molecular energies and forces at the wB97X level of nearly 3000 molecules containing 10-13 C, N, or O atoms for a total of 47,670 configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Our DFTB/ChIMES model exhibits a 60% reduction in the mean average error (MAE) and RMSE error in the energies and a 45 % decrease in the forces over 24 standard DFTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The accuracy of DFTB/ChIMES is similar to values from the ANI-1 and ANI- 1x neural network interatomic potentials97 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=', stand-alone potentials without explicit quantum mechanical elements), and are smaller than the variations between wB97X itself and higher levels of theory such as CCSD(T) and MP2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='9/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='9 kcal/mol for energies and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='6/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='9 kcal/mol- ˚A for forces)93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Our DFTB/ChIMES model is validated against additional CCSD(T) reference data from the ISO34 data set99, which consists of energies of 34 isomers containing the elements C, H, N, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We observe that the accuracy of DFTB/ChIMES is much better than that for standard DFTB, is slightly improved over that from DFTB-NNrep, and approaches the PBE0 data given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' To test the performance of our model on high accuracy force data specifically, we compare DFTB/ChIMES with the CCSD(T)/cc-pVTZ data for 2000 configurations of ethanol in the GDML data set100 (54,000 data points total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Again our DFTB/ChIMES gives an improvement over standard DFTB as MAE and RMSE are both reduced by ∼40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' A direct force comparison to DFTB-NNrep or the ISO34 reference was unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Additional validation of our model included calculation of the n-butane dihedral potential and correct prediction of the energetic ordering of coumarin molecular crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We have also validated DFTB/ChIMES against vibrational frequencies of 342 gas-phase molecules from the Computational Chemistry Comparison and Benchmark Database or CC- CBDB (https://cccbdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='gov/), computed with MP2/cc-pVTZ and wB97XD (with dispersion correction), amongst other methods (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Here, DFTB/ChIMES yields errors in the frequency prediction of MAE/RMSE = 36/61 cm−1, indicating improved accuracy over PBE and with similar accuracy to accuracy to wB97XD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In all of our validation tests, DFTB/ChIMES shows marked improvement over standard DFTB and PBE, and shows similar accuracy to results from wB97X or other higher-levels of theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Further details of all validation calculations are provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Lastly, though the DFTB/ChIMES model developed here is trained on gas phase molecular data, we have also tested its performance in reproducing the structural properties of bulk graphite and diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' We compare predicted density and lattice parameters from different methods in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' For graphite, all computational models considered here give an accurate descrip- tion of the in-plane lattice parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' DFTB and PBE overestimate the interlayer separation (c/2) by over 25% and 30%, respectively, due to their under-prediction of dispersion interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' DFTB/ChIMES predicts the lattice parameters and density in excellent agreement with the exper- imental value, with a deviation of less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' For diamond, the computed values using DFTB, 25 0 1000 2000 3000 4000 Frequency (cm 1) Distribution MP2 ωB97XD DFTB/ChIMES PBE DFTB FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 8: The distribution of the calculated frequency values using DFTB and DFTB/ChIMES for 342 neutral molecules taken from the CCCBDB database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The MP2 and DFT (PBE and wB97XD) calculations using cc-pVTZ basis set in the CCCBDB are selected for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Reprinted with permission from J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 2022 13 (13), 2934-2942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Copyright 2022, American Chemical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' DFTB/ChIMES, and PBE-DFT differ by ∼1% from experimental values for lattice parameters and ∼3% for the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Ultimately, we have shown that ChIMES can be used to extend DFTB to hybrid functional accuracy or greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' DFTB/ChIMES has the capability of reproducing vast quantities of high-level reference data while requiring only a small fraction of it for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' On the basis of the results presented here, DFTB/ChIMES represents a promising direction for developing general purpose quantum models that are applicable to a wide range of materials and conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The small training set required as well as the small number of potential parameters relative to neural network methods could yield significant advantages for future development of computational efficient models with up to coupled cluster accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The ease of parameterization and transferability of DFTB/ChIMES 26 TABLE VI: Comparison of predicted density and lattice parameters of graphite and diamond for DFTB, DFTB/ChIMES, PBE-DFT with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Reprinted with permission from J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' 2022 13 (13), 2934-2942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Copyright 2022, American Chemical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' phase method density (g/cm3) a( ˚A) c/2( ˚A) graphite Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='101 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='462 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='356 PBE-DFT102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='71 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='470 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='420 DFTB/ChIMES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='461 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='379 DFTB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='77 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='474 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='248 diamond Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='101 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='51 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='567 PBE-DFT70 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='48 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='580 DFTB/ChIMES 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='600 DFTB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='600 allows for high-level quantum theory accuracy in systems where traditional wavefunction or hybrid functional methods are far too computationally intensive for intensive use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' DISCUSSION AND FUTURE WORK ChIMES was initially developed as a method for creating many-body force fields for molecular dynamics simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' However, it has also proven robust as a repulsive energy for DFTB models, where the standard two-center approach for both quantum mechanical and repulsive terms can be insufficient for many systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The strength in ChIMES as an element of a semi-empirical quantum model or MD model lies in its use of linear combinations of many-body Chebyshev polynomials, where the nearly optimal nature of the polynomials as well as the linear least-squares fitting allow for rapid optimizations that require far fewer parameters and significantly smaller data sets than the neural network models reviewed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' In addition, ChIMES adds very little extra computational time to DFTB calculations, where the matrix diagonalization and SCC convergence use the vast majority of the CPU effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Future work will involve extending ChIMES to systems with four or more elements, where de- velopment of training sets and proper validation approaches remains an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' It is likely that these ChIMES models will require larger data sets and the potentials themselves will have 27 significantly more parameters than those presented in this work due to the combinatorial effect of forming many-body clusters with different possible combinations of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Determination of ERep for these systems will likely yield significant advantages over pure interatomic potentials due to the short-ranged nature of the repulsive energy as well as the general accuracy of the quan- tum mechanical parts of DFTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Both of these considerations make creation of DFTB/ChIMES model in general more tractable than optimizing ChIMES on its own as an atomistic force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' DFTB/ChIMES can serve as either a stand-alone model for running dynamics and determining physical and chemical properties of a system, or as a surrogate for DFT in a “boot-strapping” op- timization, where it can serve to generate reasonably high fidelity training data for pure ChIMES MD models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Overall, our approach can be used to enhance the speed of quantum accurate pre- dictions for both molecular and condensed matter systems, where there is a historic reliance on computationally intensive quantum simulations for predictions of chemical and physical properties related to experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work performed under the auspices of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' Department of Energy by Lawrence Liv- ermore National Laboratory under Contract DE-AC52-07NA27344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfxP4S/content/2301.01733v1.pdf'} +page_content=' The assigned release number is LLNL-JRNL-XXXXXX.' 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moontae.lee, honglak}@lgresearch.ai +Abstract +Despite surprising performance on zero-shot transfer, pre-training a large-scale +multimodal model is often prohibitive as it requires a huge amount of data and +computing resources. In this paper, we propose a method (BeamCLIP) that can +effectively transfer the representations of a large pre-trained multimodal model +(CLIP-ViT) into a small target model (e.g., ResNet-18). For unsupervised transfer, +we introduce cross-modal similarity matching (CSM) that enables a student model +to learn the representations of a teacher model by matching the relative similarity +distribution across text prompt embeddings. To better encode the text prompts, +we design context-based prompt augmentation (CPA) that can alleviate the lexical +ambiguity of input text prompts. Our experiments show that unsupervised represen- +tation transfer of a pre-trained vision-language model enables a small ResNet-18 to +achieve a better ImageNet-1K top-1 linear probe accuracy (66.2%) than vision-only +self-supervised learning (SSL) methods (e.g., SimCLR: 51.8%, SwAV: 63.7%), +while closing the gap with supervised learning (69.8%). +1 +Introduction +Figure 1: ImageNet-1K top-1 linear probe ac- +curacy on ResNet-18 representations. By trans- +ferring CLIP-ViT [32] vision-language represen- +tations to ResNet-18, the BeamCLIP can learn +better visual representations than vision-only self- +supervised learning (SSL) methods in terms of the +linear probe accuracy. +Learning transferable representations is crucial +for successful downstream tasks. Contrastive +learning such as SimCLR [4] and MoCo-v2 [6] +have shown notable success by forcing features +of individual classes to be clustered and suffi- +ciently scattered [41]. But their linear probe +performances are still far behind the supervised +learning as shown in Figure 1. Recently, large- +scale vision and language pre-trained (VLP) +models provide highly transferable visual repre- +sentations via language supervision. However, +learning VLP models from scratch is prohibitive +as it requires large amounts of training data and +computing resources. +For example, training +CLIP [32] requires 400M paired image-text data +and several hundreds of GPUs. ALIGN [22] fur- +ther scales up to leverage alternative texts speci- +fied for descriptions of web images. While these +models are often based on large Transformers +[40], small ConvNets such as ResNet-50 [17] +and MobileNet [20] are still widely used in practice [1] and even more crucial for low-resource +36th Conference on Neural Information Processing Systems (NeurIPS 2022). +arXiv:2301.02903v1 [cs.LG] 7 Jan 2023 + +ImageNet-1Ktop-1 linearprobe on ResNet-18 +70 +60 +accuracy ( +50 +Supervised +40 +BeamCLIP (ours) +Top-1 +SwAV +BYOL-GA +30 +MoCo-v2 +SimCLR +BYOL +20 +100 +150 +200 +250 +0OE +350 +400 +# epochsenvironments. We reformulate representation learning in terms of knowledge transfer from a large +pre-trained model to a small practical model. +Large-scale vision-language pre-trained models exhibit strong alignments between different modali- +ties. CLIP [32] learns visual concepts from natural language supervision, mapping image and text into +the same vector space. As their training data is not only huge but inaccessible, however, conventional +knowledge distillation [19] based on the source training data is no longer a viable option. Instead, +we propose cross-modal similarity matching (CSM). Imagine your goal is to learn a high quality +representation for an input dog image in CIFAR-10 [23] as in Figure 2. Since CLIP was trained +on numerous image-caption pairs, angular distances from the dog image embedding to the caption +embeddings of other anchor prompt texts such as "A photo of cat" or "A photo of horse" +must comprehensively preserve their visual differences. By training the student to preserve angular +relations witnessed from the teacher, our model achieves near benchmark performance without +accessing the original data for training CLIP. +To better encode the text prompts, we design context-based prompt augmentation (CPA) that can +alleviate the lexical ambiguity of the input text prompts. We find that lexical ambiguity in prompt +texts can lead to semantically incorrect text embeddings. This may result in unexpected discrepancies +of image-text alignment in the teacher’s embedding space. Also, it is known that the zero-shot perfor- +mance of CLIP can be improved by designing task-specific prompt texts. Inspired by this, we design +CPA that extends the basic prompt of CLIP to better encode prototypical anchor representations. +Our experimental results show that the BeamCLIP ("beam" means to transmit) achieves the strongest +and near benchmark performance on ImageNet-1K [10] top-1 linear probe accuracy when using most +popular ResNet-18 and ResNet-50 as the student network. We also compare the effectiveness of the +BeamCLIP against zero-shot transfer learning. Further, we provide ablation study results to show +how much each component contributes to the performance. +Contributions of this paper can be summarized as follows: +• We propose a method (BeamCLIP) that can effectively transfer the representations of a +large pre-trained multimodal model (e.g., CLIP-ViT) into a small target model (e.g., ResNet- +18 or ResNet-50). To achieve this, we introduce cross-modal similarity matching (CSM) +and context-based prompt augmentation (CPA). (Figure 2). +• We empirically show that BeamCLIP enables a small target model (e.g., ResNet-18) to +achieve a better ImageNet-1K linear probe accuracy than vision-only self-supervised learning +(SSL) methods, by effectively transferring CLIP-ViT representations. (Figure 1, Table 2, +and Table 3. +• We also explore the zero-shot capability of the BeamCLIP (Table 5) and analyze the +effectiveness of the BeamCLIP on various target datasets (Table 6 and Table 7). +2 +Related Work +Vision and language pre-trainig. +Vision and language pre-training (VLP) aims to jointly learn +vision and language representations that can be transferred to the downstream tasks such as visual +question answering (VQA), image captioning, and vision and language navigation (VLN). There are +BERT-based vision and language models such as VLBERT [35], ViLBERT [26], and UNITER [7]. +Also, there are contrastive learning-based models such as CLIP [32] and ALIGN [22]. These models +use contrastive loss [30] to learn aligned vision and language representations by performing a task of +matching a large-scale image and text pairs. The BeamCLIP aims to transfer the rich representations +of large-scale vision and language pre-trained models such as CLIP and ALIGN to a small target +model. +Self-supervised learning. +Self-supervised learning (SSL) aims to learn highly transferable repre- +sentations by using unlabeled data. In computer vision, at the early stage, task-specific self-supervised +methods were introduced. These include Context Prediction [11], Rotation Prediction [14], and Col- +orization [43]. More recently, contrastive learning-based methods were introduced as a task-agnostic +approach. These include SimCLR [4] and MoCo-v2 [6]. However, since contrastive self-supervised +methods require a large batch size, non-contrastive methods have been introduced. These include +SwAV [3], BYOL [16], and SimSiam [5]. In this paper, we empirically show that the BeamCLIP can +2 + +provide better visual representations than the state-of-the-art SSL methods by leveraging a large-scale +pre-trained multimodal model. +Knowledge distillation. +Knowledge distillation (KD) [19] aims to transfer rich knowledge from a +strong teacher model to a target student model. In a conventional setting, it encourages the student +model to mimic the task-specific prediction of the teacher model. As the student model is trained +to predict the same probability distribution over pre-defined classes as the teacher model’s, using +Kullback-Leibler (KL) divergence is a natural metric to measure the error between the two models. +For a classification task, the loss function can be formulated as follows: +LKD = +� +i +H(pi, qS +i ) + +� +i +KL(pT +i ||pS +i ). +(1) +The first term indicates the supervised loss, where pi denotes the one-hot labels and H(p, q) denotes +cross-entropy. The second term is the distillation loss, where pT +i and pS +i are the softmax predictions +of the teacher and student models, respectively. +Similarity-based knowledge distillation. +Recently, similarity-based knowledge distillation such +as SEED [13], OSS [8], and ISD [37] was introduced in the context of self-supervised learning +(SSL). SEED [13] showed that the linear probe accuracy of a small student (ResNet-18) can be +improved by transferring the representations of a larger teacher (ResNet-50) pre-trained by SSL +methods such as MoCo-v2 [6]. Unlike this, OSS [8] aims to transfer representations of an evolving +teacher (ResNet-50) into a smaller student (ResNet-18) on the fly. Unlike SEED and OSS, ISD [37] +considered the same size student and teacher network (ResNet-18), and showed a student can learn +visual representations by iterativly distilling the similarity of teacher’s representations. These works +are closely related to our work. Unlike these works, the BeamCLIP aims to transfer rich vision and +language representations of large-scale pre-trained models such as CLIP-ViT/16 [32] into a smaller +network such as ResNet-18. +Prompt engineering. +Recently, researchers showed that prompt engineering [2] is surprisingly +effective at improving the performance of large-scale language models (LLMs) on downstream +tasks without fine-tuning. Prompts are input texts of language models that usually consist of a task +description or several examples. To further simplify prompt engineering, prompt tuning [24] proposed +to add k learnable tokens to the input texts, while having language models frozen. Similar to GPT-3, +it is known that the zero-shot performance of CLIP [32] can be improved by designing the prompt +texts to each task. For example, on satellite image classification datasets, "A satellite photo of +a {label}" provides better performance than the default "A photo of a {label}". Inspired by +this, we propose context-based prompt augmentation that extends the basic prompt of CLIP to better +encode prototypical text anchor representations by alleviating the lexical ambiguity of class label +texts. +3 +Method +Problem formulation. +Formally, our problem is to transfer aligned cross-modal representations of +a strong teacher model f T +φ (·) into a target student model f S +θ (·) with unlabeled data Du = {xi}N +i=1. +Given each unlabeled image xi, we formulate representation transfer as a regression task that matches +teacher representations f T +φ (xi) to a student’s f S +θ (xi). As the student network is parameterized by θ, +the learning objective is +arg min +θ +N +� +i +∥f S +θ (xi) − f T +φ (xi)∥2 +2. +(2) +Normalizing the representations via l2-normalization (i.e., qi = f S +θ (xi)/∥f S +θ (xi)∥2 and ki = +f T +φ (xi)/∥f T +φ (xi)∥2) leads to the following simplification: +arg min +θ +N +� +i +∥qi − ki∥2 +2 = arg min +θ +N +� +i +(2 − 2qi · ki). +(3) +The problem now involves maximizing the cosine similarity between l2-normalized representations +from teacher and student models. +3 + +Figure 2: Overview of the BeamCLIP. Representation transfer can be viewed as a task in which, +given a query input, a student model learns to regress a vector representation of a teacher model. +The BeamCLIP first measures the normalized cross-modal similarity of the query image compared +to anchor text representations in the teacher’s embedding space. Then, it encourages the student +to mimic the same cross-modal similarity in the student’s embedding space. To better align image +representations, our method uses self-supervised pre-training of the student model. Finally, to avoid +text ambiguity, we uses context-based prompt augmentation. +Method overview. +The overview of BeamCLIP is shown in Figure 2. The teacher model of +the BeamCLIP consists of an image encoder f T +φ (·) and a text encoder gT +ψ(·). These encoders are +pre-trained under a simple task of matching images to texts with large-scale corpora. Image represen- +tations f T +φ (xi) and text representations gT +ψ(ti) are thus well-aligned within a cross-modal embedding +space. We provide the details of the BeamCLIP in the following sections. More specifically, we +describe how to extend the basic problem setting by leveraging the unique features of CLIP where +vision and language representations are precisely aligned. Throughout the paper, we use the notation +CLIP-ViT/16 to denote the CLIP [32] model that uses Vision Transformer (ViT) [12] with the patch +size of 16x16 as the image encoder. Similar to this, CLIP-RN50 denotes the CLIP model with +ResNet-50 [17] as the image encoder. +3.1 +Similarity-based cross-modal representation transfer +To effectively distill cross-modal representations, we use similarity-based matching as described +above. Our similarity-based representation transfer utilizes two carefully designed loss functions: (1) +instance similarity matching (ISM) loss and (2) cross-modal similarity matching (CSM) loss. +Instance similarity matching. +This objective is directly derived from Eq. 3. Given a query image +xi, it encourages the student image encoder f S +θ (·) to regress the representation of the teacher image +encoder f T +φ (·). We apply conventional image augmentations (see Appendix B.1) on a query image +xi, and the same augmented image ˆxi is fed to both the teacher and student image encoders. Given +unlabeled query images Du = {xi}N +i=1, it is formulated as follows: +LISM = − +N +� +i=1 +( +f S +θ (ˆxi) +∥f S +θ (ˆxi)∥2 +· +f T +φ (ˆxi) +∥f T +φ (ˆxi)∥2 +) = − +N +� +i=1 +(qi · ki). +(4) +However, the similarity signal from a single instance is not enough to constraint the student represen- +tations. For example, the topological ambiguity may occur in image encoding, since two symmetric +representations have the same cosine similarity compared to a single teacher representation (see +Appendix B.2). We conjecture that this can be mitigated by incorporating multiple anchor points to +the query points. Based on this idea, we introduce cross-modal similarity matching loss. +Cross-modal similarity matching. +To better align a student representation qi with the teacher +representation ki, we introduce cross-modal similarity matching (CSM) loss. We use multiple anchor +points to cope with the ambiguity problem mentioned above. Further, we use text representations as +anchor points, since we can easily generate prototypical anchor points by using text prompts and class +4 + +Image encoder fs +Instance Similarity Matching +Cross-modal +Similarity Matching +Image encoder ft +Text encoder gyTable 1: Examples of context-based prompt augmentation for ambiguous class labels on Flowers102. +Label Index +Label Name +Text Prompt +7 +bird of paradise +"A photo of {bird of paradise}. +{Strelitzia is a +genus of five species of perennial plants, native +to South Africa. +It belongs to the plant family +Strelitziaceae}." +10 +snapdragon +"A photo of {snapdragon}. +{Antirrhinum is a genus of +plants commonly known as dragon flowers or snapdragons +because of the flowers’ fancied resemblance to the +face of a dragon that opens and closes its mouth when +laterally squeezed}." +labels. Since image and text representations are precisely aligned in CLIP, we can effectively apply +this approach. More specifically, the BeamCLIP first measures the normalized image-text similarity +of the query image compared to prototypical text points in the teacher’s embedding space. Then, it +encourages the student to mimic the same image-text similarity in the student’s embedding space. +More formally, we generate multiple anchor representations A = {aj}M +j=1 by encoding class texts +C = {cj}M +j=1 with the teacher text encoder gT +ψ(·) (in other words, aj = gT +ψ(cj)). To measure the +similarity regarding to multiple anchor representations A, we define the normalized cross-modal +similarity as follows: +sj(ki, A) = +exp ((ki · aj)/τ) +�M +m=1 exp ((ki · am)/τ) +(5) +where τ is a temperature hyperparameter that is set to 0.01 in our experiments. +Then, we evaluate the cross-modal similarity distribution by using a set of normalized cross-modal +similarities: +P(ki|A) = [s1(ki, A), ..., sM(ki, A)]. +(6) +Then, the student model is optimized to mimic the normalized cross-modal similarity of the teacher’s +embedding space by minimizing the cross entropy, i.e., H(P(ki|A), P(qi|A)). +We further minimize the entropy of normalized cross-modal similarities in the student embedding +space i.e., H(P(qi|A)). This minimization helps the student provide query representations qi that are +more attracted to anchor representations A = {aj}M +j=1. This entropy minimization is also known to +be effective in other domains such as semi-supervised learning [15, 29]. +Altogether, the CSM loss is formulated as follows: +LCSM = +N +� +i=1 +H(P(ki|A), P(qi|A)) + +N +� +i=1 +H(P(qi|A)). +(7) +Final Loss. The final loss of the BeamCLIP is formulated as follows: +LBeamCLIP = LCSM + λISMLISM +(8) +where λISM is the scale hyperparameter that is set to 10 in our experiments. +3.2 +Context-based prompt augmentation +We found that lexical ambiguity in prompt texts can lead to semantically incorrect text embeddings. +This may result in an unexpected discrepancy of image-text alignment in the teacher’s embedding +space. For example, Flowers102 [28] dataset has some classes with unusual and ambiguous flower +names, such as “snapdragon”, “bird of paradise”, and “colt’s foot”.1 Therefore, incorrect prototypical +anchor points might be compared with a query image. To address this issue of semantic ambiguities +in the text, we introduce context-based prompt augmentation (CPA), a data-driven approach that +augments basic prompts with contextual text such as Wikipedia descriptions or hierarchical labels. +1Examples can be found at: 102 Category Flower Dataset. +5 + +For prompt tuning with Wikipedia descriptions, we use the template "A photo of a {label}. +{Wikipedia description}". We use this template for Flowers102 and Pets37 in our experiments. +We provide some examples from the Flowers102 dataset in Table 1. For prompt tuning with hierar- +chical labels, we use the template "A photo of a {fine label}, categorized as {coarse +label}". We use this template for CIFAR100 and ImageNet in our experiments. Analogous examples +from CIFAR100 can be found in Table 11 in Appendix B.3. +3.3 +Other details +Self-supervised pre-training of student. +To help the student mimic the teacher’s cross-modal +embedding space better, we pre-train the student image encoder with a self-supervised method. Since +self-supervised pre-training such as SimCLR [4], MoCo-v2 [6], and SwAV [3] provides a weakly +clustered embedding space based on similarities, it can be used as a better initial state for the student +to mimic the teacher’s embedding space. The details can be found in B.4. We show the effect of SSL +pre-training of the student in the experiment section (see Table 4 and 7). +Optimization. +For optimization we use SGD with cosine annealing schedule (SGDR) [25]. To +stabilize training, we use a momentum encoder that updates its weights via exponential moving +average (EMA) [18, 16]. The momentum encoder of a student θ ˆS is updated using the following rule: +θ ˆS ←− mθ ˆS + (1 − m)θS +(9) +where θS is the image encoder of a student model and m is a momentum hyperparameter that is set +0.99 in our experiments. The model hyperparameters are summarized in Table 12 in Appendix B.5. +4 +Experiments +Downstream datasets. +We evaluate the BeamCLIP on six standard benchmark datasets: CIFAR10 +[23], CIFAR100 [23], STL10 [9], Flowers102 [28], Pets37 [31], and ImageNet-1K [10]. Following +convention, we split the datasets into train, validation, and test sets. Then, we use train set for transfer, +and test set for evaluation. For ImageNet, we use the validation set as a test set, since its test set does +not provide labels. More details on the datasets are summarized in Table 8 in Appendix A. +4.1 +Representation transfer with unlabeled target data +Setting. We compare the BeamCLIP with various self-supervised methods in terms of linear probe +accuracy on ImageNet-1K. Following the conventional protocol, we use ResNet-18 and ResNet-50 +[17] as the base encoder and evaluate the learned representations by using logistic regression. We +use LBFGS algorithm [44] for logistic regression. Its hyperparameter C is determined through +coarse-grained hyperparameter search on the validation split. And, the accuracy is evaluated in the +test split. We found that it provides the best linear probe accuracy when C is set to 30. We perform +our experiments on 8 NVIDA A100 GPUs and it takes about 30 hours for 200 epoch training. +Table 2: ImageNet-1K top-1 linear probe accuracy on ResNet-50. We compare the BeamCLIP +with vision-only self-supervised methods in terms of linear probe accuracy on ImageNet-1K. The +BeamCLIP representations provide higher linear probe accuracy than self-supervised methods. This +means better transferability. The values are quoted from the original paper, and n/a means "not +available" from the paper. +Epochs +Method +Teacher +Student +Batch +200 +400 +800 +Supervised + +RN50 +256 +76.2 +SimCLR [4] + +RN50 +512 +65.6 +66.7 +67.4 +MoCo-v2 [6] + +RN50 +256 +67.5 +70.1 +71.1 +BYOL-GA [16] + +RN50 +4096 +70.6 +n/a +n/a +SwAV [3] + +RN50 +256 +72.0 +74.3 +n/a +BeamCLIP (ours) +CLIP ViT-B/16 +RN50 +512 +74.8 +75.1 +75.0 +6 + +Table 3: ImageNet-1K top-1 linear probe accuracy on ResNet-18. +Epochs +Method +Teacher +Student +Batch +100 +200 +400 +Supervised + +RN18 +256 +69.8 +SimCLR [4] + +RN18 +256 +47.1 +49.9 +51.8 +MoCo-V2 [6] + +RN18 +256 +48.6 +49.9 +51.9 +BYOL [16] + +RN18 +256 +44.2 +47.5 +46.8 +BYOL-GA [16] + +RN18 +256 +54.2 +56.9 +61.4 +SwAV [3] + +RN18 +256 +57.7 +61.2 +63.7 +OSS [8] +SSL RN50 +RN18 +256 +60.0 +64.1 +65.8 +BeamCLIP (ours) +CLIP ViT-B/16 +RN18 +256 +63.8 +64.8 +66.2 +Table 4: Effect of self-supervised pre-training of student. +Method +Teacher +Student +Pre-training of Student +Batch +Epoch +Linear Probe +Supervised + +RN50 + +256 +- +76.2 +BeamCLIP (ours) +CLIP ViT-B/16 +RN50 +SimCLR [4] +512 +200 +74.8 +BeamCLIP (ours) +CLIP ViT-B/16 +RN50 +SwAV [3] +512 +200 +75.8 +Transfer to ResNet-50. Table 2 shows the comparison of the BeamCLIP with vision-only self- +supervised methods such as SimCLR [4], MoCo-v2 [6], SwAV [3], BYOL [16], and SimSiam +[5]. The BeamCLIP provides better visual representation by achieving 74.8% top-1 linear probe +accuracy on ImageNet-1K [10]. While self-supervised methods take long training epochs to achieve +comparable accuracy, BeamCLIP-RN50 achieves better accuracy with less training epochs. Also, note +that BeamCLIP-RN50’s representations provide better accuracy than CLIP-RN50’s representations. +Transfer to ResNet-18. To check if the BeamCLIP can transfer CLIP representations into smaller +models than ResNet-50 (24M), we also measure ImageNet-1K top-1 linear probe accuracy on +ResNet-18 (11M). ResNet-18 is trained from scratch (not self-supervised pre-trained with SimCLR), +while transferring CLIP ViT-B/16 representations. As shown in Table 3, BeamCLIP learns better +representations than SSL methods such as SimCLR [4], MoCo-v2 [6], BYOL [16], and SwAV [3]. +More importantly, the BeamCLIP provides better performance than OSS [8] that simultaneously +learns and transfers representations from ResNet-50. The learning curve is presented in Figure 8 in +Appendix C.1. +Effect of self-supervised pre-training. Table 4 shows ImageNet-1K top-1 linear probe accuracy on +BeamCLIP-RN50 representations by using different SSL pre-training. With the better SSL method +(SwAV [3] > SimCLR [4]), the BeamCLIP can learn better representations with an increased linear +probe accuracy. +4.2 +Representation transfer with unlabeled non-target data +To check if the BeamCLIP also inherits the powerful zero-shot capability of CLIP, we compare zero- +shot accuracy of CLIP variants on ImageNet-1K. For zero-shot measure, we use CC-3M [34] and +ImageNet-21K (12M samples) [33] that do not have overlap with ImageNet-1K. Table 5 shows the +comparison of zero-shot accuracy. The BeamCLIP-RN50 achieves about 57.5% zero-shot accuracy +that is highly comparable with CLIP RN-50 (59.6%). The learning curve is presented in Figure 9 of +Appendix C.1. +4.3 +Transfer learning accuracy on various target datasets +Setting. We evaluate how effectively the BeamCLIP transfers CLIP-ViT representations into a +student model by evaluating classification accuracy on various datasets (see Table 6). We choose +ResNet-50 [17] as the student model to compare the distilled target model with CLIP-RN50. Also, +ResNet-50 is conventionally used in evaluating linear probe accuracy on ImageNet-1K. However, the +BeamCLIP can adopt any architecture, not just ResNet-50. +7 + +Table 5: Comparison of zero-shot accuracy on ImageNet-1K. On ImageNet-1K, the BeamCLIP +RN50 achieves about 57.5% zero-shot accuracy that is higly comparable with CLIP RN-50 (59.6%). +To achieve such a high zero-shot accuracy, CLIP uses very large image-text pair data (WIT-400M). +Instead, the BeamCLIP can achieve the comparable zero-shot accuracy by effectively transferring the +teacher’s representations, while using only 3% data (ImageNet-21K (12M)). Note that OpenCLIP +provides only about 36.5% zero-shot accuracy with the similar amount of data (CC-12M). +Method +Image +Training +Teacher +Text +Batch +Epochs +ImageNet +Encoder +Data +Model +Prompts +Size +Zero-shot +CLIP [32] +ViT-B/16 +WIT-400M + + +32,768 +32 +68.6 +CLIP [32] +RN50 +WIT-400M + + +32,768 +32 +59.6 +OpenCLIP [21] +RN50 +YFCC-15M + + +256 ∗ 8 +32 +32.7 +OpenCLIP [21] +RN50 +CC-12M + + +256 ∗ 8 +32 +36.5 +BeamCLIP (ours) +RN50 +CC-3M +CLIP ViT-B/16 +IN-1K +64 ∗ 8 +100 +49.5 +BeamCLIP (ours) +RN50 +IN-21K (12M) +CLIP ViT-B/16 +IN-1K +64 ∗ 8 +50 +53.6 +BeamCLIP (ours) +RN50 +IN-21K (12M) +CLIP ViT-B/16 +IN-1K +64 ∗ 8 +200 +57.5 +As baselines, we choose two representative distillation methods among many methods: (1) conven- +tional knowledge distillation (KD) [19] and (2) contrastive representation distillation (CRD) [38]. +Since conventional KD aims to mimic the task-specific predictions of the teacher model unlike the +BeamCLIP , we apply the KL divergence on minimizing the cross-modal similarity distribution (i.e., +P(qi|A) and P(ki|A)), instead of the Cross-entropy (CE). CRD proposes a variant of InfoNCE loss +for representation distillation, which we apply on normalized representations (i.e., qi and ki). The +details of each method can be found in the related work section 2. +Results. Table 6 shows a comparison of teacher and student accuracy on various datasets. We empiri- +cally demonstrate that the BeamCLIP can effectively transfer vision and language representations +of a large teacher model (CLIP ViT-B/16) into a small student model (ResNet-50). We find that +the KL divergence used in conventional knowledge distillation (KD) is not effective in transferring +CLIP-ViT representations. Also, the contrastive learning-based approach is not effective. Unlike this, +the BeamCLIP can effectively transfer CLIP ViT-B/16 representations into ResNet-50, achieving very +high accuracy that is comparable or better than the teacher accuracy. KD simply minimizes the error +between single instances. We conjecture that the cross-modal similarity to multiple anchor points +introduced in the BeamCLIP helps the student preserve the topology of the teacher’s embedding +space. +Also, note that context-based prompt augmentation helps achieve better accuracy after representation +transfer. Since Flowers102 has many ambiguous labels, our experiment shows that text prompt +augmentation significantly increases the student’s accuracy compared to the teacher’s accuracy. +Ablation study. Table 7 shows the ablation study results of the BeamCLIP . Our empirical findings +are as follows: (1) Instance similarity matching (ISM) is not enough by itself to preserve the topology +Table 6: Comparison of teacher and student accuracy on various datasets. Conventional knowl- +edge distillation with the KL divergence is not effective in transferring CLIP-ViT representations. In +contrast, the BeamCLIP effectively transfers CLIP ViT-B/16 representations into ResNet-50 by using +unlabeled query data. We denoted with ∗ in cases our student model surpasses the accuracy of the +teacher model. +Method +Model Type +Img. Enc. +Param. Size +CIFAR10 +CIFAR100 +STL10 +Flowers102 +Pets37 +ImageNet-1K +CLIP [32] (zero-shot) +T +ViT-B/16 +76M +91.6 +68.7 +98.2 +70.4 +88.9 +68.6 +CRD [38] +S +RN50 +24M +76.78 +38.83 +81.41 +26.15 +62.22 +30.90 +KD [19] +S +RN50 +24M +90.89 +58.02 +93.28 +49.01 +77.51 +56.38 +BeamCLIP (ours) +S +RN50 +24M +92.10∗ +67.35 +97.45 +75.86∗ +86.94 +66.17 +8 + +Table 7: Ablation study results. The acronym denotes the sub-methods introduces in the BeamCLIP : +(1-1) ISM means instance similarity matching loss, (1-2) CSM means cross-modal similarity matching +loss, (2) CPA means context-based prompt augmentation, and (3) SSL PT means self-supervised +pre-training of student, . All the technical components contribute to the improvements of transfer +learning accuracy at the student. Note that we only perform CPA on datasets with more than 100 of +class labels. +Method +Type +Img. Enc. +(1-1) ISM +(1-2) CSM +(2) CPA +(3) SSL PT +CIFAR10 +CIFAR100 +Flowers102 +ImageNet-1K +CLIP [32] (zero-shot) +T +ViT-B/16 +- +- +- +- +91.6 +68.7 +70.4 +68.6 +Unsupervised +Representation +Transfer +(BeamCLIP) +S +RN50 +Cosine + + + +39.07 +6.04 +1.43 +21.83 +S +RN50 +Cosine + + +SimCLR +87.00 +48.26 +3.59 +51.83 +S +RN50 + +CE + + +91.28 +58.90 +12.94 +63.30 +S +RN50 + +CE + +SimCLR +91.53 +65.64 +62.18 +65.45 +S +RN50 + +CE+EntMin + +SimCLR +91.71 +66.14 +63.96 +66.23 +S +RN50 +Cosine +CE+EntMin + +SimCLR +92.10∗ +66.18 +64.14 +65.76 +S +RN50 +Cosine +CE+EntMin + +SimCLR +- +67.35 +75.86∗ +66.17 +Figure 3: Comparison of CLIP-RN50 and BeamCLIP-RN50. This figure shows the top-5 text- +image retrieval results. A red rectangle denotes an incorrect result. CLIP-RN50 provides many +incorrect results, since its zero-shot accuracy is relatively low. In contrast, BeamCLIP-RN50 provides +much improved results, since it is transferred from CLIP-ViT/16 with higher zero-shot accuracy. +of the teacher’s embedding space. (2) Cross-modal similarity matching (SCM) compared to multiple +anchor points helps the student mimic the teacher’s embedding space. (3) Self-supervised pre- +training of the student (SSL PT) helps the student mimic the teacher’s embedding space. (4) Entropy +minimization (EntMin) helps to improve the accuracy. (5) Context-based prompt augmentation (CPA) +helps measure the similarity more precisely. As shown in the table, Flowers102 dataset is sensitive to +self-supervised pre-training of student. We conjecture that since Flowers102 dataset has only 1020 +training samples for the 102 classes, it is not enough to probe the teacher’s representation space. +Qualitative result. To see the quality of the transferred representations, we analysed text-image +retrieval results on the Flowers102 dataset. Figure 3 compares the top-5 text-image retrieval results +between CLIP-RN50 and BeamCLIP-RN50. A red rectangle denotes an incorrect result. Compared +to CLIP-RN50, BeamCLIP-RN50 provides much improved results, since its representations are +transferred from CLIP-ViT/16 with higher zero-shot accuracy. More interestingly, BeamCLIP-RN50 +provides surprisingly good text-image retrieval results, even though unseen text prompts such as "a +photo of {pink rose}" or "a photo of {yellow rose}" are given. +9 + +BeamCLIP-RN50 +sim: 0.5786, +sim: 0.5679, +sim: 0.5430, +sim: 0.5386, +sim: 0.5308, +sim: 0.3534, +sim: 0.3479, +sim: 0.3444, +sim: 0.3420, +sim: 0.3343, +label: 10 +label: 10 +label:95 +label: 10 +label: 10 +label: 10 +label: 10 +label: 10 +label: 10 +label: 10 +sim: 0.6338, +sim: 0.5659, +sim: 0.5459, +sim: 0.5449, +sim: 0.5435, +sim: 0.3429, +sim: 0.3397, +sim: 0.3303, +sim: 0.3282, +sim: 0.3277, +label: 73 +label: 73 +label: 73 +label: 86 +label: 73 +label: 73 +label: 73 +label: 73 +label: 73 +label: 73 +sim: 0.6372, +sim: 0.5889, +sim: 0.5737, +sim: 0.5649, +sim: 0.5562, +sim: 0.3468, +sim: 0.3272, +sim: 0.3255, +sim: 0.3241, +sim: 0.3239, +label: 73 +label: 73 +label:87 +label: 73 +label: 73 +label: 73 +label: 73 +label: 73 +label: 73 +sim: 0.6304, +sim: 0.5898, +sim: 0.5835, +sim: 0.5605, +sim: 0.5381, +sim: 0.3436, +sim: 0.3375, +sim: 0.3310, + sim: 0.3255 +sim: 0.3147, +label: 73 +label: 73 +label: 15 +label: 15 +label: 47 +label: 73 +label: 73 +label: 73 +label: 15 +label: 15(a) Subset text prompts from CIFAR100 classes +(b) Random text prompts from ImageNet classes +Figure 4: The effect of random text prompts on CIFAR100. (a) The text prompts are randomly +sampled from the set of 100 class names of CIFAR100. The red dotted line denotes the teacher’s +accuracy as an upper bound. It is more efficient as it is closer to this line. As shown in the blue line, +the BeamCLIP (CE+EntMin) can effectively transfer the CLIP representations, even when the class +names of the target dataset are partially given. (b) The text prompts are randomly sampled from the +1000 class names of ImageNet. The BeamCLIP (CE+EntMin) is still effective, even though the class +names are randomly sampled from a non-target dataset (ImageNet-1K). +4.4 +Effect of random text prompts +We measured how effective the BeamCLIP is in cases where the class names of the target dataset are +not perfectly given. Figure 4 shows the effect of the randomly sampled text prompts on CIFAR100. +We can see that the BeamCLIP is still effective, even when (a) the subset of the 100 class names +of CIFAR100 are given as the text prompts, or (b) the text prompts are randomly sampled from a +non-target dataset (ImageNet-1K). The exact values in Figure 4 are presented in Table 15 and Table +16 in Appendix C.2. Also, the additional results on CIFAR10 are provided in Appendix C.2. +5 +Limitations and Conclusion +Limitations. With the help of rich representations of pre-trained CLIP, the BeamCLIP can learn +better representations than SSL methods. However, since SSL methods can increase the performance +at longer training epochs, the performance margin may be decreased in such a setting. Another +shortcoming is that context-based prompt augmentations may require additional engineering efforts. +Conclusion. In this paper, we provide the BeamCLIP that can effectively transfer large pre-trained +vision-language model (e.g., CLIP-ViT) into a small target model (e.g., ResNet-18) with cross-modal +similarity matching (CSM) and context-based prompt augmentation (CPA). We empirically show that +the BeamCLIP can learn better visual representations than vision-only self-supervised learning (SSL) +methods, by leveraging a pre-trained vision-language model (CLIP). The BeamCLIP is not intended +to be another CLIP, but an effective CLIP student. +Broader impact +This research aims to provide a simple and effective way to leverage CLIP for representation learning. +With the help of CLIP, the BeamCLIP can learn better representations than self-supervised learning +(SSL) methods. Since training CLIP requires very large data and hundreds of GPUs, it is important to +provide a way to effectively reuse the pre-trained CLIP rather than training from scratch on a target +model. We believe that the BeamCLIP can help to save cost and time. +Acknowledgements +We thank anonymous reviewers for their valuable comments. This work was fully supported by LG +AI Research. +10 + +TransferlearningonCFAR1oo +(withsubsetpromptsfromCIFAR1ooclasses) +70 +60 +(%) +50 +Accuracy +40 +30 +20 +CLIP ViT-B/16 (zero-shot) +BeamCLIP-RN50 (CE+EntMin) +10 +BeamCLIP-RN50 (KL) +CLIP-RN50 (zero-shot) +0 +20 +30 +40 + 50 +60 +70 +80 +90 +100 +# of text promptsTransferlearingonCiFAR1oo +(withrandompromptsfromImageNetclasses) +70 +60 +(%) +50 +Accuracy +40 +30 +20 +CLIP ViT-B/16 (zero-shot) +BeamCLIP-RN50 (CE+EntMin) +10 +BeamCLIP-RN50 (KL) +CLIP-RN50 (zero-shot) +0 +200 +400 +600 +800 +1000 +# of text promptsReferences +[1] L. Beyer, X. Zhai, A. Royer, L. Markeeva, R. Anil, and A. Kolesnikov. Knowledge distillation: +A good teacher is patient and consistent. arXiv preprint arXiv:2106.05237, 2021. +[2] T. Brown, B. Mann, N. Ryder, M. Subbiah, J. D. Kaplan, P. Dhariwal, A. Neelakantan, P. 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If you are including theoretical results... +(a) Did you state the full set of assumptions of all theoretical results? [No] Our work does +not include theoretical results. +(b) Did you include complete proofs of all theoretical results? [No] Our work does not +include theoretical results. +3. If you ran experiments... +(a) Did you include the code, data, and instructions needed to reproduce the main experi- +mental results (either in the supplemental material or as a URL)? [Yes] We provide the +code, data, and instructions in the supplemental material. +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they +were chosen)? [Yes] See Section 4, Appendix A, and Appendix B. +(c) Did you report error bars (e.g., with respect to the random seed after running experi- +ments multiple times)? [Yes] See Section 4. +(d) Did you include the total amount of compute and the type of resources used (e.g., type +of GPUs, internal cluster, or cloud provider)? [Yes] See Section 4. +4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... +(a) If your work uses existing assets, did you cite the creators? [Yes] See Section 2, +Section 3, and Section 4. +(b) Did you mention the license of the assets? [No] We only used public benchmark +datasets and open-sourced software. See Section 4. +13 + +(c) Did you include any new assets either in the supplemental material or as a URL? [No] +We did not use new assets. +(d) Did you discuss whether and how consent was obtained from people whose data you’re +using/curating? [Yes] We only used public benchmark datasets. See Section 4. +(e) Did you discuss whether the data you are using/curating contains personally identifiable +information or offensive content? [No] We did not use any data containing personally +identifiable information or offensive content. +5. If you used crowdsourcing or conducted research with human subjects... +(a) Did you include the full text of instructions given to participants and screenshots, +if applicable? [No] We did not use crowdsourcing or conduct research with human +subjects. +(b) Did you describe any potential participant risks, with links to Institutional Review +Board (IRB) approvals, if applicable? [No] We did not use crowdsourcing or conduct +research with human subjects. +(c) Did you include the estimated hourly wage paid to participants and the total amount +spent on participant compensation? [No] We did not use crowdsourcing or conduct +research with human subjects. +14 + +A +Datasets Details +We demonstrate the effectiveness of the BeamCLIP by using six downstream datasets. Table 8 shows +the details of the downstream datasets. +Table 8: Details of datasets used for the BeamCLIP evaluation. +Dataset +Image Size +Classes +Train Size +Val Size +Test Size +CIFAR10 [23] +32x32 +10 +40,000 +10,000 +10,000 +CIFAR100 [23] +32x32 +100 +40,000 +10,000 +10,000 +STL10 [9] +128x128 +10 +4,000 +1,000 +8,000 +Flowers102 [28] +224x224 +102 +1,020 +1,020 +6,149 +Pets37 [31] +224x224 +37 +2,944 +736 +3,669 +ImageNet [10] +224x224 +1,000 +1,231,167 +50,000 +50,000 +B +Method Details +In this section, we provide some details of the BeamCLIP . More specifically, we provide the details +of two main contributions that are (1) cross-modal similarity matching (CSM) and (2) context-based +prompt augmentation (CPA). Also, we provide the other implementation details such as image +augmentation, similarity smoothing, model hyperparameters, etc. +B.1 +Image augmentation details +We use conventional image augmentation when performing representation transfer by using unlabeled +images in downstream datasets. Table 9 provides a list of image augmentation used for unsupervised +representation transfer on downstream datasets. +Table 9: A list of image augmentations used in the BeamCLIP . +Mode +Augmentation +Parameters +Train +RandomResizedCrop +- +RandomHorizontalFlip +p=0.5 +RandomColorJitter +p=0.8 +GaussianBlur +p=0.5, min=0.1, miax=2.0 +Normalize +- +Val +Resize +input_size + 0.1 * input_size +CenterCrop +input_size +Normalize +- +B.2 +Cross-modal similarity matching details +Cross-modal similarity matching (CSM) is the main method of the BeamCLIP . To make the concept +of CSM clearer, we provide an illustration of CSM in Figure 5. +B.3 +Context-based prompt augmentation details +To prepare for better text anchor embeddings for unsupervised representation transfer, we introduce +context-based prompt augmentation (CPA). To make the concept of CPA clearer, we provide an +illustration of CPA in Figure 6. +Also, we provide an example of the hierarchical class labels in Table 10 and an example context-based +prompt augmentation for CIFAR100 in Table 11. +15 + +Figure 5: Illustration of cross-modal similarity matching. Topological ambiguity may occur in +image encoding, since query image embedding qi1 and qi2 can have the same cosine similarity +compared to a single teacher image embedding k, while heading towards different directions. To +mitigate this problem, we introduce cross-modal similarity matching that encourage the student to +mimic the same cross-modal similarity distribution (measured against multiple anchor text points) in +teacher’s embedding space. +Figure 6: Illustration of context-based prompt augmentation. The lexical ambiguity may occur +in text encoding, since the same text may have multiple different meanings. To mitigate this problem, +we introduce context-based prompt augmentation that helps resolve the ambiguity with contextual +texts such as Wikipedia descriptions. +B.4 +Other implementation details +Self-supervised pre-training of student. +For self-supervised pre-training, we adopt SimCLR, +since it is simple and effective. SimCLR learns transferable visual representations by using InfoNCE +16 + +Table 10: Coarse and fine labels for CIFAR100. +Coarse Label +Fine Label +aquatic mammals +beaver, dolphin, otter, seal, whale +fish +aquarium fish, flatfish, ray, shark, trout +flowers +orchid, poppy, rose, sunflower, tulip +food containers +bottle, bowl, can, cup, plate +household electrical devices +clock, keyboard, lamp, telephone, television +household furniture +bed, chair, couch, table, wardrobe +insects +bee, beetle, butterfly, caterpillar, cockroach +large carnivores +bear, leopard, lion, tiger, wolf +large man-made outdoor things +bridge, castle, house, road, skyscraper +large natural outdoor scenes +cloud, forest, mountain, plain, sea +large omnivores and herbivores +camel, cattle, chimpanzee, elephant, kangaroo +medium mammals +fox, porcupine, possum, raccoon, skunk +non-insect invertebrates +crab, lobster, snail, spider, worm +people +baby, boy, girl, man, woman +reptiles +crocodile, dinosaur, lizard, snake, turtle +small mammals +hamster, mouse, rabbit, shrew, squirrel +trees +maple tree, oak tree, palm tree, pine tree, willow tree +vehicles 1 +bicycle, bus, motocycle, pickup truck, train +vehicles 2 +lawn mower, rocket, streetcar, tank, tractor +Table 11: Examples of prompt augmentation with hierarchical labels for CIFAR100. +Label Name +Text Prompt +baby +"A photo of a {baby}, categorized as {people}." +beaver +"A photo of a {beaver}, categorized as {aquatic mammals}." +bee +"A photo of a {bee}, categorized as {insect}." +loss [30, 39] which encourages agreement between multiple views of the same image. More specifi- +cally, InfoNCE maximizes the similarity between multiple views of the same image (i.e., positive +samples) and minimizes the similarity to multiple views of all other images in a training batch (i.e., +negative samples). InfoNCE loss of SimCLR can be formulated as follows: +LInfoNCE = − log +exp ((hS +i · hS +i′)/τ) +�2B +k=1 1[k̸=i] exp ((hS +i · hS +k )/τ) +(10) +where hS +i ∈ R128 is a projection of a student representation qi ∈ R512, τ is a temperature hyperpa- +rameter that is set to 0.1, 1[k̸=i] is an indicator function whose value is 1 if k ̸= i, and B is a batch +size. Here, hi and hi′ are projections of multiple views of the same input images xi. +Similarity Smoothing. +To improve the effectiveness of distillation, we apply Label Smoothing +(LS) [36] to the cross-modal similarity distillation loss. Recent works [27, 42] show that Label +Smoothing helps knowledge distillation. To apply Label Smoothing, we determine the most similar +anchor representation as follows: +j∗ = arg max +j +sj(ki, A). +(11) +Then, we generate a modified cross-modal similarity distribution: +sj(ki, A)LS = 1[j=j∗](1 − α) + α/M +(12) +where 1[j=j∗] is the indicator function whose value is 1 if j = j∗, M is the number of anchors, and +α is the smoothing hyperparameter that is set to 0.2 in our experiments. +17 + +B.5 +Model hyperparameters +Table 12 provides the summary of model hyperparamters. We use the same hyperparameters on all +downstream datasets if not explicitly declared. +Table 12: BeamCLIP hyperparameters. +Hyperparameter +Value +CSM loss temperature +0.01 +ISM loss scale +{0.1, 1.0, 10.0} +similarity smoothing (LS) sacle +0.2 +optimizer +SGDR [25] +initial learning rate +0.5 +weight decay +1e-6 +EMA momentum +0.99 +batch size +{256, 512} +epochs +200 +C +Additional Experiment Results +In this section, we provide additional experiment results. First, we provide the learning curves that are +generated while training the BeamCLIP . Second, we provide some experiment results on the effects +of random text prompts. Third, we provide an example qualitative result that shows the advantage of +the BeamCLIP . +C.1 +Learning curves of the BeamCLIP +We provide the learning curve of the BeamCLIP for the experiment section. Figure 7 shows the +learning curve for ImageNet-1K validation accuracy of BeamCLIP-RN50 representations trained +with unlabeled ImageNet-1K. Figure 8 shows the learning curve for ImageNet-1K validation accuracy +of BeamCLIP-RN18 representations trained with unlabeled ImageNet-1K. Figure 9 shows the +learning curve for ImageNet-1K zero-shot accuracy of BeamCLIP-RN50 representations trained with +unlabeled non-target data (ImageNet-21K). +Figure 7: ImageNet-1K top-1 validation accuracy of BeamCLIP-RN50 representations learned with +unlabeled target data (ImageNet-1K). +C.2 +The effect of random text prompts +In this section, we further analyze the effect of text prompts from the perspective of unsupervised +learning. Before that, we briefly review the proposed method. In this paper, we propose the BeamCLIP +, an unsupervised representation transfer method of a large pre-trained multimodal model such as +CLIP. The BeamCLIP can transfer the visual representations of CLIP by using unlabeled images on a +downstream dataset. To achieve this, we propose cross-modal similarity matching (CSM). In CSM, +18 + +trained with ImageNet-1K +67.5 +val_acc +65.0 +62.5 +(%) +60.0 +Accuracy +57.5 +55.0 +52.5 +50.0 +47.5 +0 +25 + 50 +75 +100 +125 +150 +175 +200 +EpochsFigure 8: ImageNet-1K top-1 validation accuracy of BeamCLIP-RN18 representations learned with +unlabeled target data (ImageNet-1K). +Figure 9: Zero-shot ImageNet-1K top-1 accuracy of BeamCLIP-RN50 representations learned with +unlabeled non-target data (ImageNet-21K). +at first, given an unlabeled image, cross-modal similarity distribution is measured from multiple text +prompt embeddings in the teacher’s embedding space. Then, a student model is encouraged to mimic +the cross-modal similarity distribution of the teacher model by matching these similarity distributions. +To achieve effective transfer, we use anchor text embeddings by encoding text prompts. For example, +on CIFAR10, we use ten text prompts in the form of "a photo of {class name}". Note that the +text prompts are not paired with each image. +CIFAR10. +We measured how effective the BeamCLIP is in cases where the class names of the +target dataset are not perfectly given. Table 10 shows the effect of the randomly sampled text prompts +on CIFAR10 [23]. The values in Figure 10 are also presented in Table 13 and Table 14. +CIFAR100. +Table 4 shows the effect of the randomly sampled text prompts on CIFAR100 [23]. +The values in Figure 4 are also presented in Table 15 and Table 16. +Table 13: Effect of the partial text prompts on CIFAR10. +Prompts +Method +Type +Img. Enc. +3 +5 +7 +9 +10 +- +CLIP [32] (zero-shot) +T +ViT-B/16 +- +- +- +- +- +91.6 +CLIP [32] (zero-shot) +- +RN50 +- +- +- +- +- +75.6 +BeamCLIP (CE+EntMin) +S +RN50 +83.47 +83.25 +88.26 +91.84 +92.10∗ +- +BeamCLIP (KL) +S +RN50 +89.36 +90.43 +90.15 +90.54 +90.85 +- +19 + +ImageNet-1Kzero-shotaccuracyofBeamCLip-RN50 +trained with ImageNet-21K (no overlap) +58 +val_acc +56 +54 + 52 + 50 +48 +46 +25 + 50 +75 +100 +125 +150 +175 +200 +EpochsImageNet-1K validation accuracy of BeamCLiP-RN18 +trained with ImageNet-1K +60 - +val_acc +50 +(%)/ +40 +20 +10 +0 - +0 +20 +40 +60 +80 +100 +Epochs(a) Subset prompts from CIFAR10 +(b) Random prompts from ImageNet +Figure 10: Effect of the random text prompts on CIFAR10. (a) The text prompts are randomly +sampled from the 10 class names of CIFAR10. The red dotted line denotes the teacher’s accuracy as an +upper bound. It is more efficient as it is closer to this line. As shown in the blue line, the BeamCLIP is +still effective, even when the class names of the target dataset are partially given. The BeamCLIP (KL) +means to use the KL-divergence for matching cross-modal similarity distribution. The BeamCLIP +(CE+EntMin) is more effective, as more text prompts are given. (b) The text prompts are randomly +selected from the 1000 class names of ImageNet. The BeamCLIP (CE+EntMin) is still effective, +even though the class names are randomly sampled from a non-target dataset (ImageNet-1K). +Table 14: Effect of the random text prompts on CIFAR10. +Prompts +Method +Type +Img. Enc. +10 +20 +30 +40 +- +CLIP [32] (zero-shot) +T +ViT-B/16 +- +- +- +- +91.6 +CLIP [32] (zero-shot) +- +RN50 +- +- +- +- +75.6 +BeamCLIP (CE+EntMin) +S +RN50 +81.49 +82.05 +84.09 +84.51 +- +BeamCLIP (KL) +S +RN50 +87.76 +87.67 +87.83 +87.88 +- +Table 15: Effect of the partial text prompts on CIFAR100. +Prompts +Method +Type +Img. Enc. +20 +40 +60 +80 +100 +- +CLIP [32] (zero-shot) +T +ViT-B/16 +- +- +- +- +- +68.7 +CLIP [32] (zero-shot) +- +RN50 +- +- +- +- +- +41.6 +BeamCLIP (CE+EntMin) +S +RN50 +21.72 +32.19 +45.38 +56.93 +67.35 +- +BeamCLIP (KL) +S +RN50 +52.10 +52.93 +53.88 +55.15 +56.12 +- +Table 16: Effect of the random text prompts on CIFAR100. +Prompts +Method +Type +Img. Enc. +100 +500 +1000 +- +CLIP [32] (zero-shot) +T +ViT-B/16 +- +- +- +68.7 +CLIP [32] (zero-shot) +- +RN50 +- +- +- +41.6 +BeamCLIP (CE+EntMin) +S +RN50 +45.36 +52.44 +58.93 +- +BeamCLIP (KL) +S +RN50 +48.68 +46.82 +46.15 +- +20 + +TransferlearningonCiFAR10 +(withsubsetpromptsfromCIFAR1o classes) +95 +90 +85 +(%) +Accuracy +80 +75 +70 +CLIP ViT-B/16 (zero-shot) +BeamCLIP-RN50 (CE+EntMin) +65 +BeamCLIP-RN50 (KL) +CLIP-RN50 (zero-shot) +60 +w - +4 +6 +7 +00 +9 +10 +# of text promptsTransferlearing on CIFAR1o +(withrandompromptsfrom ImageNetclasses) +95 +90 +85 +(%) +Accuracy +80 +75 +70 +CLIP ViT-B/16 (zero-shot) +BeamCLIP-RN50 (CE+EntMin) +65 +BeamCLIP-RN50 (KL) +CLIP-RN50 (zero-shot) +60 +10 +15 +20 +25 +30 +35 +40 +# of text prompts \ No newline at end of file diff --git a/HtE1T4oBgHgl3EQfFgOa/content/tmp_files/load_file.txt b/HtE1T4oBgHgl3EQfFgOa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d33e509a08cc1f3fb90393738586a0542123bbbe --- /dev/null +++ b/HtE1T4oBgHgl3EQfFgOa/content/tmp_files/load_file.txt @@ -0,0 +1,1263 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf,len=1262 +page_content='Transferring Pre-trained Multimodal Representations with Cross-modal Similarity Matching Byoungjip Kim1, Sungik Choi1, Dasol Hwang1, Moontae Lee1,2, Honglak Lee1 LG AI Research1, University of Illinois Chicago2 {bjkim, sungik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='choi, dasol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='hwang, moontae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='lee, honglak}@lgresearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='ai Abstract Despite surprising performance on zero-shot transfer, pre-training a large-scale multimodal model is often prohibitive as it requires a huge amount of data and computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' In this paper, we propose a method (BeamCLIP) that can effectively transfer the representations of a large pre-trained multimodal model (CLIP-ViT) into a small target model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', ResNet-18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' For unsupervised transfer, we introduce cross-modal similarity matching (CSM) that enables a student model to learn the representations of a teacher model by matching the relative similarity distribution across text prompt embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To better encode the text prompts, we design context-based prompt augmentation (CPA) that can alleviate the lexical ambiguity of input text prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Our experiments show that unsupervised represen- tation transfer of a pre-trained vision-language model enables a small ResNet-18 to achieve a better ImageNet-1K top-1 linear probe accuracy (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2%) than vision-only self-supervised learning (SSL) methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', SimCLR: 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='8%, SwAV: 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='7%), while closing the gap with supervised learning (69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='8%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 1 Introduction Figure 1: ImageNet-1K top-1 linear probe ac- curacy on ResNet-18 representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' By trans- ferring CLIP-ViT [32] vision-language represen- tations to ResNet-18, the BeamCLIP can learn better visual representations than vision-only self- supervised learning (SSL) methods in terms of the linear probe accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Learning transferable representations is crucial for successful downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Contrastive learning such as SimCLR [4] and MoCo-v2 [6] have shown notable success by forcing features of individual classes to be clustered and suffi- ciently scattered [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' But their linear probe performances are still far behind the supervised learning as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Recently, large- scale vision and language pre-trained (VLP) models provide highly transferable visual repre- sentations via language supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' However, learning VLP models from scratch is prohibitive as it requires large amounts of training data and computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' For example, training CLIP [32] requires 400M paired image-text data and several hundreds of GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' ALIGN [22] fur- ther scales up to leverage alternative texts speci- fied for descriptions of web images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' While these models are often based on large Transformers [40], small ConvNets such as ResNet-50 [17] and MobileNet [20] are still widely used in practice [1] and even more crucial for low-resource 36th Conference on Neural Information Processing Systems (NeurIPS 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='02903v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='LG] 7 Jan 2023 ImageNet-1Ktop-1 linearprobe on ResNet-18 70 60 accuracy ( 50 Supervised 40 BeamCLIP (ours) Top-1 SwAV BYOL-GA 30 MoCo-v2 SimCLR BYOL 20 100 150 200 250 0OE 350 400 # epochsenvironments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We reformulate representation learning in terms of knowledge transfer from a large pre-trained model to a small practical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Large-scale vision-language pre-trained models exhibit strong alignments between different modali- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' CLIP [32] learns visual concepts from natural language supervision, mapping image and text into the same vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' As their training data is not only huge but inaccessible, however, conventional knowledge distillation [19] based on the source training data is no longer a viable option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Instead, we propose cross-modal similarity matching (CSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Imagine your goal is to learn a high quality representation for an input dog image in CIFAR-10 [23] as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Since CLIP was trained on numerous image-caption pairs, angular distances from the dog image embedding to the caption embeddings of other anchor prompt texts such as "A photo of cat" or "A photo of horse" must comprehensively preserve their visual differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' By training the student to preserve angular relations witnessed from the teacher, our model achieves near benchmark performance without accessing the original data for training CLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To better encode the text prompts, we design context-based prompt augmentation (CPA) that can alleviate the lexical ambiguity of the input text prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We find that lexical ambiguity in prompt texts can lead to semantically incorrect text embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' This may result in unexpected discrepancies of image-text alignment in the teacher’s embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Also, it is known that the zero-shot perfor- mance of CLIP can be improved by designing task-specific prompt texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Inspired by this, we design CPA that extends the basic prompt of CLIP to better encode prototypical anchor representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Our experimental results show that the BeamCLIP ("beam" means to transmit) achieves the strongest and near benchmark performance on ImageNet-1K [10] top-1 linear probe accuracy when using most popular ResNet-18 and ResNet-50 as the student network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We also compare the effectiveness of the BeamCLIP against zero-shot transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Further, we provide ablation study results to show how much each component contributes to the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Contributions of this paper can be summarized as follows: We propose a method (BeamCLIP) that can effectively transfer the representations of a large pre-trained multimodal model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', CLIP-ViT) into a small target model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', ResNet- 18 or ResNet-50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To achieve this, we introduce cross-modal similarity matching (CSM) and context-based prompt augmentation (CPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We empirically show that BeamCLIP enables a small target model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', ResNet-18) to achieve a better ImageNet-1K linear probe accuracy than vision-only self-supervised learning (SSL) methods, by effectively transferring CLIP-ViT representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (Figure 1, Table 2, and Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We also explore the zero-shot capability of the BeamCLIP (Table 5) and analyze the effectiveness of the BeamCLIP on various target datasets (Table 6 and Table 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 2 Related Work Vision and language pre-trainig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Vision and language pre-training (VLP) aims to jointly learn vision and language representations that can be transferred to the downstream tasks such as visual question answering (VQA), image captioning, and vision and language navigation (VLN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' There are BERT-based vision and language models such as VLBERT [35], ViLBERT [26], and UNITER [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Also, there are contrastive learning-based models such as CLIP [32] and ALIGN [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' These models use contrastive loss [30] to learn aligned vision and language representations by performing a task of matching a large-scale image and text pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The BeamCLIP aims to transfer the rich representations of large-scale vision and language pre-trained models such as CLIP and ALIGN to a small target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Self-supervised learning (SSL) aims to learn highly transferable repre- sentations by using unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' In computer vision, at the early stage, task-specific self-supervised methods were introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' These include Context Prediction [11], Rotation Prediction [14], and Col- orization [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' More recently, contrastive learning-based methods were introduced as a task-agnostic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' These include SimCLR [4] and MoCo-v2 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' However, since contrastive self-supervised methods require a large batch size, non-contrastive methods have been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' These include SwAV [3], BYOL [16], and SimSiam [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' In this paper, we empirically show that the BeamCLIP can 2 provide better visual representations than the state-of-the-art SSL methods by leveraging a large-scale pre-trained multimodal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Knowledge distillation (KD) [19] aims to transfer rich knowledge from a strong teacher model to a target student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' In a conventional setting, it encourages the student model to mimic the task-specific prediction of the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' As the student model is trained to predict the same probability distribution over pre-defined classes as the teacher model’s, using Kullback-Leibler (KL) divergence is a natural metric to measure the error between the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' For a classification task, the loss function can be formulated as follows: LKD = � i H(pi, qS i ) + � i KL(pT i ||pS i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (1) The first term indicates the supervised loss, where pi denotes the one-hot labels and H(p, q) denotes cross-entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The second term is the distillation loss, where pT i and pS i are the softmax predictions of the teacher and student models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Similarity-based knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Recently, similarity-based knowledge distillation such as SEED [13], OSS [8], and ISD [37] was introduced in the context of self-supervised learning (SSL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' SEED [13] showed that the linear probe accuracy of a small student (ResNet-18) can be improved by transferring the representations of a larger teacher (ResNet-50) pre-trained by SSL methods such as MoCo-v2 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Unlike this, OSS [8] aims to transfer representations of an evolving teacher (ResNet-50) into a smaller student (ResNet-18) on the fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Unlike SEED and OSS, ISD [37] considered the same size student and teacher network (ResNet-18), and showed a student can learn visual representations by iterativly distilling the similarity of teacher’s representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' These works are closely related to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Unlike these works, the BeamCLIP aims to transfer rich vision and language representations of large-scale pre-trained models such as CLIP-ViT/16 [32] into a smaller network such as ResNet-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Prompt engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Recently, researchers showed that prompt engineering [2] is surprisingly effective at improving the performance of large-scale language models (LLMs) on downstream tasks without fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Prompts are input texts of language models that usually consist of a task description or several examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To further simplify prompt engineering, prompt tuning [24] proposed to add k learnable tokens to the input texts, while having language models frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Similar to GPT-3, it is known that the zero-shot performance of CLIP [32] can be improved by designing the prompt texts to each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' For example, on satellite image classification datasets, "A satellite photo of a {label}" provides better performance than the default "A photo of a {label}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Inspired by this, we propose context-based prompt augmentation that extends the basic prompt of CLIP to better encode prototypical text anchor representations by alleviating the lexical ambiguity of class label texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 3 Method Problem formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Formally, our problem is to transfer aligned cross-modal representations of a strong teacher model f T φ (·) into a target student model f S θ (·) with unlabeled data Du = {xi}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Given each unlabeled image xi, we formulate representation transfer as a regression task that matches teacher representations f T φ (xi) to a student’s f S θ (xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' As the student network is parameterized by θ, the learning objective is arg min θ N � i ∥f S θ (xi) − f T φ (xi)∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (2) Normalizing the representations via l2-normalization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', qi = f S θ (xi)/∥f S θ (xi)∥2 and ki = f T φ (xi)/∥f T φ (xi)∥2) leads to the following simplification: arg min θ N � i ∥qi − ki∥2 2 = arg min θ N � i (2 − 2qi · ki).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (3) The problem now involves maximizing the cosine similarity between l2-normalized representations from teacher and student models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 3 Figure 2: Overview of the BeamCLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Representation transfer can be viewed as a task in which, given a query input, a student model learns to regress a vector representation of a teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The BeamCLIP first measures the normalized cross-modal similarity of the query image compared to anchor text representations in the teacher’s embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Then, it encourages the student to mimic the same cross-modal similarity in the student’s embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To better align image representations, our method uses self-supervised pre-training of the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Finally, to avoid text ambiguity, we uses context-based prompt augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Method overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The overview of BeamCLIP is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The teacher model of the BeamCLIP consists of an image encoder f T φ (·) and a text encoder gT ψ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' These encoders are pre-trained under a simple task of matching images to texts with large-scale corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Image represen- tations f T φ (xi) and text representations gT ψ(ti) are thus well-aligned within a cross-modal embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We provide the details of the BeamCLIP in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' More specifically, we describe how to extend the basic problem setting by leveraging the unique features of CLIP where vision and language representations are precisely aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Throughout the paper, we use the notation CLIP-ViT/16 to denote the CLIP [32] model that uses Vision Transformer (ViT) [12] with the patch size of 16x16 as the image encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Similar to this, CLIP-RN50 denotes the CLIP model with ResNet-50 [17] as the image encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1 Similarity-based cross-modal representation transfer To effectively distill cross-modal representations, we use similarity-based matching as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Our similarity-based representation transfer utilizes two carefully designed loss functions: (1) instance similarity matching (ISM) loss and (2) cross-modal similarity matching (CSM) loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Instance similarity matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' This objective is directly derived from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Given a query image xi, it encourages the student image encoder f S θ (·) to regress the representation of the teacher image encoder f T φ (·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We apply conventional image augmentations (see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1) on a query image xi, and the same augmented image ˆxi is fed to both the teacher and student image encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Given unlabeled query images Du = {xi}N i=1, it is formulated as follows: LISM = − N � i=1 ( f S θ (ˆxi) ∥f S θ (ˆxi)∥2 f T φ (ˆxi) ∥f T φ (ˆxi)∥2 ) = − N � i=1 (qi · ki).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (4) However, the similarity signal from a single instance is not enough to constraint the student represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' For example, the topological ambiguity may occur in image encoding, since two symmetric representations have the same cosine similarity compared to a single teacher representation (see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We conjecture that this can be mitigated by incorporating multiple anchor points to the query points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Based on this idea, we introduce cross-modal similarity matching loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Cross-modal similarity matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To better align a student representation qi with the teacher representation ki, we introduce cross-modal similarity matching (CSM) loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We use multiple anchor points to cope with the ambiguity problem mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Further, we use text representations as anchor points, since we can easily generate prototypical anchor points by using text prompts and class 4 Image encoder fs Instance Similarity Matching Cross-modal Similarity Matching Image encoder ft Text encoder gyTable 1: Examples of context-based prompt augmentation for ambiguous class labels on Flowers102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Label Index Label Name Text Prompt 7 bird of paradise "A photo of {bird of paradise}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' {Strelitzia is a genus of five species of perennial plants, native to South Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' It belongs to the plant family Strelitziaceae}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='" 10 snapdragon "A photo of {snapdragon}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' {Antirrhinum is a genus of plants commonly known as dragon flowers or snapdragons because of the flowers’ fancied resemblance to the face of a dragon that opens and closes its mouth when laterally squeezed}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='" labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Since image and text representations are precisely aligned in CLIP, we can effectively apply this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' More specifically, the BeamCLIP first measures the normalized image-text similarity of the query image compared to prototypical text points in the teacher’s embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Then, it encourages the student to mimic the same image-text similarity in the student’s embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' More formally, we generate multiple anchor representations A = {aj}M j=1 by encoding class texts C = {cj}M j=1 with the teacher text encoder gT ψ(·) (in other words, aj = gT ψ(cj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To measure the similarity regarding to multiple anchor representations A, we define the normalized cross-modal similarity as follows: sj(ki, A) = exp ((ki · aj)/τ) �M m=1 exp ((ki · am)/τ) (5) where τ is a temperature hyperparameter that is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='01 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Then, we evaluate the cross-modal similarity distribution by using a set of normalized cross-modal similarities: P(ki|A) = [s1(ki, A), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', sM(ki, A)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (6) Then, the student model is optimized to mimic the normalized cross-modal similarity of the teacher’s embedding space by minimizing the cross entropy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', H(P(ki|A), P(qi|A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We further minimize the entropy of normalized cross-modal similarities in the student embedding space i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', H(P(qi|A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' This minimization helps the student provide query representations qi that are more attracted to anchor representations A = {aj}M j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' This entropy minimization is also known to be effective in other domains such as semi-supervised learning [15, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Altogether, the CSM loss is formulated as follows: LCSM = N � i=1 H(P(ki|A), P(qi|A)) + N � i=1 H(P(qi|A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (7) Final Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The final loss of the BeamCLIP is formulated as follows: LBeamCLIP = LCSM + λISMLISM (8) where λISM is the scale hyperparameter that is set to 10 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2 Context-based prompt augmentation We found that lexical ambiguity in prompt texts can lead to semantically incorrect text embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' This may result in an unexpected discrepancy of image-text alignment in the teacher’s embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' For example, Flowers102 [28] dataset has some classes with unusual and ambiguous flower names, such as “snapdragon”, “bird of paradise”, and “colt’s foot”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1 Therefore, incorrect prototypical anchor points might be compared with a query image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To address this issue of semantic ambiguities in the text, we introduce context-based prompt augmentation (CPA), a data-driven approach that augments basic prompts with contextual text such as Wikipedia descriptions or hierarchical labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 1Examples can be found at: 102 Category Flower Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 5 For prompt tuning with Wikipedia descriptions, we use the template "A photo of a {label}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' {Wikipedia description}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We use this template for Flowers102 and Pets37 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We provide some examples from the Flowers102 dataset in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' For prompt tuning with hierar- chical labels, we use the template "A photo of a {fine label}, categorized as {coarse label}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We use this template for CIFAR100 and ImageNet in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Analogous examples from CIFAR100 can be found in Table 11 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='3 Other details Self-supervised pre-training of student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To help the student mimic the teacher’s cross-modal embedding space better, we pre-train the student image encoder with a self-supervised method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Since self-supervised pre-training such as SimCLR [4], MoCo-v2 [6], and SwAV [3] provides a weakly clustered embedding space based on similarities, it can be used as a better initial state for the student to mimic the teacher’s embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The details can be found in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We show the effect of SSL pre-training of the student in the experiment section (see Table 4 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' For optimization we use SGD with cosine annealing schedule (SGDR) [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To stabilize training, we use a momentum encoder that updates its weights via exponential moving average (EMA) [18, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The momentum encoder of a student θ ˆS is updated using the following rule: θ ˆS ←− mθ ˆS + (1 − m)θS (9) where θS is the image encoder of a student model and m is a momentum hyperparameter that is set 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='99 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The model hyperparameters are summarized in Table 12 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 4 Experiments Downstream datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We evaluate the BeamCLIP on six standard benchmark datasets: CIFAR10 [23], CIFAR100 [23], STL10 [9], Flowers102 [28], Pets37 [31], and ImageNet-1K [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Following convention, we split the datasets into train, validation, and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Then, we use train set for transfer, and test set for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' For ImageNet, we use the validation set as a test set, since its test set does not provide labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' More details on the datasets are summarized in Table 8 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1 Representation transfer with unlabeled target data Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We compare the BeamCLIP with various self-supervised methods in terms of linear probe accuracy on ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Following the conventional protocol, we use ResNet-18 and ResNet-50 [17] as the base encoder and evaluate the learned representations by using logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We use LBFGS algorithm [44] for logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Its hyperparameter C is determined through coarse-grained hyperparameter search on the validation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' And, the accuracy is evaluated in the test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We found that it provides the best linear probe accuracy when C is set to 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We perform our experiments on 8 NVIDA A100 GPUs and it takes about 30 hours for 200 epoch training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Table 2: ImageNet-1K top-1 linear probe accuracy on ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We compare the BeamCLIP with vision-only self-supervised methods in terms of linear probe accuracy on ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The BeamCLIP representations provide higher linear probe accuracy than self-supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' This means better transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The values are quoted from the original paper, and n/a means "not available" from the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Epochs Method Teacher Student Batch 200 400 800 Supervised \x17 RN50 256 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2 SimCLR [4] \x17 RN50 512 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='4 MoCo-v2 [6] \x17 RN50 256 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1 BYOL-GA [16] \x17 RN50 4096 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 n/a n/a SwAV [3] \x17 RN50 256 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='3 n/a BeamCLIP (ours) CLIP ViT-B/16 RN50 512 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='0 6 Table 3: ImageNet-1K top-1 linear probe accuracy on ResNet-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Epochs Method Teacher Student Batch 100 200 400 Supervised \x17 RN18 256 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='8 SimCLR [4] \x17 RN18 256 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='9 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='8 MoCo-V2 [6] \x17 RN18 256 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='9 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='9 BYOL [16] \x17 RN18 256 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='8 BYOL-GA [16] \x17 RN18 256 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='9 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='4 SwAV [3] \x17 RN18 256 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='7 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='7 OSS [8] SSL RN50 RN18 256 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='8 BeamCLIP (ours) CLIP ViT-B/16 RN18 256 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='8 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2 Table 4: Effect of self-supervised pre-training of student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Method Teacher Student Pre-training of Student Batch Epoch Linear Probe Supervised \x17 RN50 \x17 256 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2 BeamCLIP (ours) CLIP ViT-B/16 RN50 SimCLR [4] 512 200 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='8 BeamCLIP (ours) CLIP ViT-B/16 RN50 SwAV [3] 512 200 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='8 Transfer to ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Table 2 shows the comparison of the BeamCLIP with vision-only self- supervised methods such as SimCLR [4], MoCo-v2 [6], SwAV [3], BYOL [16], and SimSiam [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The BeamCLIP provides better visual representation by achieving 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='8% top-1 linear probe accuracy on ImageNet-1K [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' While self-supervised methods take long training epochs to achieve comparable accuracy, BeamCLIP-RN50 achieves better accuracy with less training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Also, note that BeamCLIP-RN50’s representations provide better accuracy than CLIP-RN50’s representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Transfer to ResNet-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To check if the BeamCLIP can transfer CLIP representations into smaller models than ResNet-50 (24M), we also measure ImageNet-1K top-1 linear probe accuracy on ResNet-18 (11M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' ResNet-18 is trained from scratch (not self-supervised pre-trained with SimCLR), while transferring CLIP ViT-B/16 representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' As shown in Table 3, BeamCLIP learns better representations than SSL methods such as SimCLR [4], MoCo-v2 [6], BYOL [16], and SwAV [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' More importantly, the BeamCLIP provides better performance than OSS [8] that simultaneously learns and transfers representations from ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The learning curve is presented in Figure 8 in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Effect of self-supervised pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Table 4 shows ImageNet-1K top-1 linear probe accuracy on BeamCLIP-RN50 representations by using different SSL pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' With the better SSL method (SwAV [3] > SimCLR [4]), the BeamCLIP can learn better representations with an increased linear probe accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2 Representation transfer with unlabeled non-target data To check if the BeamCLIP also inherits the powerful zero-shot capability of CLIP, we compare zero- shot accuracy of CLIP variants on ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' For zero-shot measure, we use CC-3M [34] and ImageNet-21K (12M samples) [33] that do not have overlap with ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Table 5 shows the comparison of zero-shot accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The BeamCLIP-RN50 achieves about 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5% zero-shot accuracy that is highly comparable with CLIP RN-50 (59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The learning curve is presented in Figure 9 of Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='3 Transfer learning accuracy on various target datasets Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We evaluate how effectively the BeamCLIP transfers CLIP-ViT representations into a student model by evaluating classification accuracy on various datasets (see Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We choose ResNet-50 [17] as the student model to compare the distilled target model with CLIP-RN50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Also, ResNet-50 is conventionally used in evaluating linear probe accuracy on ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' However, the BeamCLIP can adopt any architecture, not just ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 7 Table 5: Comparison of zero-shot accuracy on ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' On ImageNet-1K, the BeamCLIP RN50 achieves about 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5% zero-shot accuracy that is higly comparable with CLIP RN-50 (59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To achieve such a high zero-shot accuracy, CLIP uses very large image-text pair data (WIT-400M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Instead, the BeamCLIP can achieve the comparable zero-shot accuracy by effectively transferring the teacher’s representations, while using only 3% data (ImageNet-21K (12M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Note that OpenCLIP provides only about 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5% zero-shot accuracy with the similar amount of data (CC-12M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Method Image Training Teacher Text Batch Epochs ImageNet Encoder Data Model Prompts Size Zero-shot CLIP [32] ViT-B/16 WIT-400M \x17 \x17 32,768 32 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 CLIP [32] RN50 WIT-400M \x17 \x17 32,768 32 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 OpenCLIP [21] RN50 YFCC-15M \x17 \x17 256 ∗ 8 32 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='7 OpenCLIP [21] RN50 CC-12M \x17 \x17 256 ∗ 8 32 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5 BeamCLIP (ours) RN50 CC-3M CLIP ViT-B/16 IN-1K 64 ∗ 8 100 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5 BeamCLIP (ours) RN50 IN-21K (12M) CLIP ViT-B/16 IN-1K 64 ∗ 8 50 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 BeamCLIP (ours) RN50 IN-21K (12M) CLIP ViT-B/16 IN-1K 64 ∗ 8 200 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5 As baselines, we choose two representative distillation methods among many methods: (1) conven- tional knowledge distillation (KD) [19] and (2) contrastive representation distillation (CRD) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Since conventional KD aims to mimic the task-specific predictions of the teacher model unlike the BeamCLIP , we apply the KL divergence on minimizing the cross-modal similarity distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', P(qi|A) and P(ki|A)), instead of the Cross-entropy (CE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' CRD proposes a variant of InfoNCE loss for representation distillation, which we apply on normalized representations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', qi and ki).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The details of each method can be found in the related work section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Table 6 shows a comparison of teacher and student accuracy on various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We empiri- cally demonstrate that the BeamCLIP can effectively transfer vision and language representations of a large teacher model (CLIP ViT-B/16) into a small student model (ResNet-50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We find that the KL divergence used in conventional knowledge distillation (KD) is not effective in transferring CLIP-ViT representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Also, the contrastive learning-based approach is not effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Unlike this, the BeamCLIP can effectively transfer CLIP ViT-B/16 representations into ResNet-50, achieving very high accuracy that is comparable or better than the teacher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' KD simply minimizes the error between single instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We conjecture that the cross-modal similarity to multiple anchor points introduced in the BeamCLIP helps the student preserve the topology of the teacher’s embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Also, note that context-based prompt augmentation helps achieve better accuracy after representation transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Since Flowers102 has many ambiguous labels, our experiment shows that text prompt augmentation significantly increases the student’s accuracy compared to the teacher’s accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Table 7 shows the ablation study results of the BeamCLIP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Our empirical findings are as follows: (1) Instance similarity matching (ISM) is not enough by itself to preserve the topology Table 6: Comparison of teacher and student accuracy on various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Conventional knowl- edge distillation with the KL divergence is not effective in transferring CLIP-ViT representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' In contrast, the BeamCLIP effectively transfers CLIP ViT-B/16 representations into ResNet-50 by using unlabeled query data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We denoted with ∗ in cases our student model surpasses the accuracy of the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Method Model Type Img.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Size CIFAR10 CIFAR100 STL10 Flowers102 Pets37 ImageNet-1K CLIP [32] (zero-shot) T ViT-B/16 76M 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='7 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='9 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 CRD [38] S RN50 24M 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='78 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='83 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='41 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='15 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='22 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='90 KD [19] S RN50 24M 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='89 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='02 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='28 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='01 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='51 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='38 BeamCLIP (ours) S RN50 24M 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='10∗ 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='35 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='45 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='86∗ 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='94 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='17 8 Table 7: Ablation study results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The acronym denotes the sub-methods introduces in the BeamCLIP : (1-1) ISM means instance similarity matching loss, (1-2) CSM means cross-modal similarity matching loss, (2) CPA means context-based prompt augmentation, and (3) SSL PT means self-supervised pre-training of student, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' All the technical components contribute to the improvements of transfer learning accuracy at the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Note that we only perform CPA on datasets with more than 100 of class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Method Type Img.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (1-1) ISM (1-2) CSM (2) CPA (3) SSL PT CIFAR10 CIFAR100 Flowers102 ImageNet-1K CLIP [32] (zero-shot) T ViT-B/16 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='4 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 Unsupervised Representation Transfer (BeamCLIP) S RN50 Cosine \x17 \x17 \x17 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='07 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='43 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='83 S RN50 Cosine \x17 \x17 SimCLR 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='00 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='59 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='83 S RN50 \x17 CE \x17 \x17 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='28 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='90 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='94 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='30 S RN50 \x17 CE \x17 SimCLR 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='53 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='64 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='18 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='45 S RN50 \x17 CE+EntMin \x17 SimCLR 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='71 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='14 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='96 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='23 S RN50 Cosine CE+EntMin \x17 SimCLR 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='10∗ 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='18 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='14 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='76 S RN50 Cosine CE+EntMin \x13 SimCLR 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='35 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='86∗ 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='17 Figure 3: Comparison of CLIP-RN50 and BeamCLIP-RN50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' This figure shows the top-5 text- image retrieval results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' A red rectangle denotes an incorrect result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' CLIP-RN50 provides many incorrect results, since its zero-shot accuracy is relatively low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' In contrast, BeamCLIP-RN50 provides much improved results, since it is transferred from CLIP-ViT/16 with higher zero-shot accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' of the teacher’s embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (2) Cross-modal similarity matching (SCM) compared to multiple anchor points helps the student mimic the teacher’s embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (3) Self-supervised pre- training of the student (SSL PT) helps the student mimic the teacher’s embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (4) Entropy minimization (EntMin) helps to improve the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (5) Context-based prompt augmentation (CPA) helps measure the similarity more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' As shown in the table, Flowers102 dataset is sensitive to self-supervised pre-training of student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We conjecture that since Flowers102 dataset has only 1020 training samples for the 102 classes, it is not enough to probe the teacher’s representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Qualitative result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To see the quality of the transferred representations, we analysed text-image retrieval results on the Flowers102 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Figure 3 compares the top-5 text-image retrieval results between CLIP-RN50 and BeamCLIP-RN50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' A red rectangle denotes an incorrect result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Compared to CLIP-RN50, BeamCLIP-RN50 provides much improved results, since its representations are transferred from CLIP-ViT/16 with higher zero-shot accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' More interestingly, BeamCLIP-RN50 provides surprisingly good text-image retrieval results, even though unseen text prompts such as "a photo of {pink rose}" or "a photo of {yellow rose}" are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 9 BeamCLIP-RN50 sim: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5786, sim: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5679, sim: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5430, sim: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5386, sim: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5308, sim: 0.' metadata={'source': 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sim: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='3310, sim: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='3255 sim: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='3147, label: 73 label: 73 label: 15 label: 15 label: 47 label: 73 label: 73 label: 73 label: 15 label: 15(a) Subset text prompts from CIFAR100 classes (b) Random text prompts from ImageNet classes Figure 4: The effect of random text prompts on CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (a) The text prompts are randomly sampled from the set of 100 class names of CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The red dotted line denotes the teacher’s accuracy as an upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' It is more efficient as it is closer to this line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' As shown in the blue line, the BeamCLIP (CE+EntMin) can effectively transfer the CLIP representations, even when the class names of the target dataset are partially given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (b) The text prompts are randomly sampled from the 1000 class names of ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The BeamCLIP (CE+EntMin) is still effective, even though the class names are randomly sampled from a non-target dataset (ImageNet-1K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='4 Effect of random text prompts We measured how effective the BeamCLIP is in cases where the class names of the target dataset are not perfectly given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Figure 4 shows the effect of the randomly sampled text prompts on CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We can see that the BeamCLIP is still effective, even when (a) the subset of the 100 class names of CIFAR100 are given as the text prompts, or (b) the text prompts are randomly sampled from a non-target dataset (ImageNet-1K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The exact values in Figure 4 are presented in Table 15 and Table 16 in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Also, the additional results on CIFAR10 are provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 5 Limitations and Conclusion Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' With the help of rich representations of pre-trained CLIP, the BeamCLIP can learn better representations than SSL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' However, since SSL methods can increase the performance at longer training epochs, the performance margin may be decreased in such a setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Another shortcoming is that context-based prompt augmentations may require additional engineering efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' In this paper, we provide the BeamCLIP that can effectively transfer large pre-trained vision-language model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', CLIP-ViT) into a small target model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', ResNet-18) with cross-modal similarity matching (CSM) and context-based prompt augmentation (CPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We empirically show that the BeamCLIP can learn better visual representations than vision-only self-supervised learning (SSL) methods, by leveraging a pre-trained vision-language model (CLIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The BeamCLIP is not intended to be another CLIP, but an effective CLIP student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Broader impact This research aims to provide a simple and effective way to leverage CLIP for representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' With the help of CLIP, the BeamCLIP can learn better representations than self-supervised learning (SSL) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Since training CLIP requires very large data and hundreds of GPUs, it is important to provide a way to effectively reuse the pre-trained CLIP rather than training from scratch on a target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We believe that the BeamCLIP can help to save cost and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Acknowledgements We thank anonymous reviewers for their valuable comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' This work was fully supported by LG AI Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='TransferlearningonCFAR1oo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='(withsubsetpromptsfromCIFAR1ooclasses) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='(%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='Accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='CLIP ViT-B/16 (zero-shot) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='BeamCLIP-RN50 (CE+EntMin) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='BeamCLIP-RN50 (KL) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='CLIP-RN50 (zero-shot) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='# of text promptsTransferlearingonCiFAR1oo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='(withrandompromptsfromImageNetclasses) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='(%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='Accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='CLIP ViT-B/16 (zero-shot) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='BeamCLIP-RN50 (CE+EntMin) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='BeamCLIP-RN50 (KL) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='CLIP-RN50 (zero-shot) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='# of text promptsReferences ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='[1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Beyer, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Zhai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Royer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Markeeva, R.' 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Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems (NeurIPS, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [41] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Wang and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Isola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Understanding contrastive representation learning through alignment and uniformity on the hypersphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 9929–9939.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [42] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Yuan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Tay, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Wang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Feng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Revisiting knowledge distillation via la- bel smoothing regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [43] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Isola, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Efros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Colorful image colorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' In European Conference on Computer Vision (ECCV), 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [44] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Zhu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Byrd, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Lu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Nocedal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Algorithm 778: L-bfgs-b: Fortran subroutines for large-scale bound-constrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' ACM Transactions on mathematical software (TOMS), 23(4):550–560, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Checklist 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' For all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [Yes] See Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (b) Did you describe the limitations of your work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [Yes] See Section 4 and Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (c) Did you discuss any potential negative societal impacts of your work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [Yes] See Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [Yes] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' If you are including theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (a) Did you state the full set of assumptions of all theoretical results?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [No] Our work does not include theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (b) Did you include complete proofs of all theoretical results?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [No] Our work does not include theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' If you ran experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [Yes] We provide the code, data, and instructions in the supplemental material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (b) Did you specify all the training details (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', data splits, hyperparameters, how they were chosen)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [Yes] See Section 4, Appendix A, and Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (c) Did you report error bars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', with respect to the random seed after running experi- ments multiple times)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [Yes] See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (d) Did you include the total amount of compute and the type of resources used (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', type of GPUs, internal cluster, or cloud provider)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [Yes] See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' If you are using existing assets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', code, data, models) or curating/releasing new assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (a) If your work uses existing assets, did you cite the creators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [Yes] See Section 2, Section 3, and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (b) Did you mention the license of the assets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [No] We only used public benchmark datasets and open-sourced software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 13 (c) Did you include any new assets either in the supplemental material or as a URL?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [No] We did not use new assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [Yes] We only used public benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [No] We did not use any data containing personally identifiable information or offensive content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' If you used crowdsourcing or conducted research with human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (a) Did you include the full text of instructions given to participants and screenshots, if applicable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [No] We did not use crowdsourcing or conduct research with human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [No] We did not use crowdsourcing or conduct research with human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' [No] We did not use crowdsourcing or conduct research with human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 14 A Datasets Details We demonstrate the effectiveness of the BeamCLIP by using six downstream datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Table 8 shows the details of the downstream datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Table 8: Details of datasets used for the BeamCLIP evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Dataset Image Size Classes Train Size Val Size Test Size CIFAR10 [23] 32x32 10 40,000 10,000 10,000 CIFAR100 [23] 32x32 100 40,000 10,000 10,000 STL10 [9] 128x128 10 4,000 1,000 8,000 Flowers102 [28] 224x224 102 1,020 1,020 6,149 Pets37 [31] 224x224 37 2,944 736 3,669 ImageNet [10] 224x224 1,000 1,231,167 50,000 50,000 B Method Details In this section, we provide some details of the BeamCLIP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' More specifically, we provide the details of two main contributions that are (1) cross-modal similarity matching (CSM) and (2) context-based prompt augmentation (CPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Also, we provide the other implementation details such as image augmentation, similarity smoothing, model hyperparameters, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1 Image augmentation details We use conventional image augmentation when performing representation transfer by using unlabeled images in downstream datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Table 9 provides a list of image augmentation used for unsupervised representation transfer on downstream datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Table 9: A list of image augmentations used in the BeamCLIP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Mode Augmentation Parameters Train RandomResizedCrop RandomHorizontalFlip p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5 RandomColorJitter p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='8 GaussianBlur p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5, min=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1, miax=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='0 Normalize Val Resize input_size + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1 * input_size CenterCrop input_size Normalize B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2 Cross-modal similarity matching details Cross-modal similarity matching (CSM) is the main method of the BeamCLIP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To make the concept of CSM clearer, we provide an illustration of CSM in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='3 Context-based prompt augmentation details To prepare for better text anchor embeddings for unsupervised representation transfer, we introduce context-based prompt augmentation (CPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To make the concept of CPA clearer, we provide an illustration of CPA in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Also, we provide an example of the hierarchical class labels in Table 10 and an example context-based prompt augmentation for CIFAR100 in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 15 Figure 5: Illustration of cross-modal similarity matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Topological ambiguity may occur in image encoding, since query image embedding qi1 and qi2 can have the same cosine similarity compared to a single teacher image embedding k, while heading towards different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To mitigate this problem, we introduce cross-modal similarity matching that encourage the student to mimic the same cross-modal similarity distribution (measured against multiple anchor text points) in teacher’s embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Figure 6: Illustration of context-based prompt augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The lexical ambiguity may occur in text encoding, since the same text may have multiple different meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To mitigate this problem, we introduce context-based prompt augmentation that helps resolve the ambiguity with contextual texts such as Wikipedia descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='4 Other implementation details Self-supervised pre-training of student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' For self-supervised pre-training, we adopt SimCLR, since it is simple and effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' SimCLR learns transferable visual representations by using InfoNCE 16 Table 10: Coarse and fine labels for CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Coarse Label Fine Label aquatic mammals beaver,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' dolphin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' otter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' seal,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' woman reptiles crocodile,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' dinosaur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' lizard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' snake,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' turtle small mammals hamster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' mouse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' rabbit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' shrew,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' squirrel trees maple tree,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' oak tree,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' palm tree,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' pine tree,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' willow tree vehicles 1 bicycle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' bus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' motocycle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' pickup truck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' train vehicles 2 lawn mower,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' rocket,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' streetcar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' tank,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' tractor Table 11: Examples of prompt augmentation with hierarchical labels for CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Label Name Text Prompt baby "A photo of a {baby}, categorized as {people}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='" beaver "A photo of a {beaver}, categorized as {aquatic mammals}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='" bee "A photo of a {bee}, categorized as {insect}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='" loss [30, 39] which encourages agreement between multiple views of the same image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' More specifi- cally, InfoNCE maximizes the similarity between multiple views of the same image (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', positive samples) and minimizes the similarity to multiple views of all other images in a training batch (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=', negative samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' InfoNCE loss of SimCLR can be formulated as follows: LInfoNCE = − log exp ((hS i · hS i′)/τ) �2B k=1 1[k̸=i] exp ((hS i · hS k )/τ) (10) where hS i ∈ R128 is a projection of a student representation qi ∈ R512, τ is a temperature hyperpa- rameter that is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1, 1[k̸=i] is an indicator function whose value is 1 if k ̸= i, and B is a batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Here, hi and hi′ are projections of multiple views of the same input images xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Similarity Smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To improve the effectiveness of distillation, we apply Label Smoothing (LS) [36] to the cross-modal similarity distillation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Recent works [27, 42] show that Label Smoothing helps knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To apply Label Smoothing, we determine the most similar anchor representation as follows: j∗ = arg max j sj(ki, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (11) Then, we generate a modified cross-modal similarity distribution: sj(ki, A)LS = 1[j=j∗](1 − α) + α/M (12) where 1[j=j∗] is the indicator function whose value is 1 if j = j∗, M is the number of anchors, and α is the smoothing hyperparameter that is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 17 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5 Model hyperparameters Table 12 provides the summary of model hyperparamters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We use the same hyperparameters on all downstream datasets if not explicitly declared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Table 12: BeamCLIP hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Hyperparameter Value CSM loss temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='01 ISM loss scale {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='0, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='0} similarity smoothing (LS) sacle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2 optimizer SGDR [25] initial learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5 weight decay 1e-6 EMA momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='99 batch size {256, 512} epochs 200 C Additional Experiment Results In this section, we provide additional experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' First, we provide the learning curves that are generated while training the BeamCLIP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Second, we provide some experiment results on the effects of random text prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Third, we provide an example qualitative result that shows the advantage of the BeamCLIP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='1 Learning curves of the BeamCLIP We provide the learning curve of the BeamCLIP for the experiment section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Figure 7 shows the learning curve for ImageNet-1K validation accuracy of BeamCLIP-RN50 representations trained with unlabeled ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Figure 8 shows the learning curve for ImageNet-1K validation accuracy of BeamCLIP-RN18 representations trained with unlabeled ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Figure 9 shows the learning curve for ImageNet-1K zero-shot accuracy of BeamCLIP-RN50 representations trained with unlabeled non-target data (ImageNet-21K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Figure 7: ImageNet-1K top-1 validation accuracy of BeamCLIP-RN50 representations learned with unlabeled target data (ImageNet-1K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='2 The effect of random text prompts In this section, we further analyze the effect of text prompts from the perspective of unsupervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Before that, we briefly review the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' In this paper, we propose the BeamCLIP , an unsupervised representation transfer method of a large pre-trained multimodal model such as CLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The BeamCLIP can transfer the visual representations of CLIP by using unlabeled images on a downstream dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To achieve this, we propose cross-modal similarity matching (CSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' In CSM, 18 trained with ImageNet-1K 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5 val_acc 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='0 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5 (%) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='0 Accuracy 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='5 0 25 50 75 100 125 150 175 200 EpochsFigure 8: ImageNet-1K top-1 validation accuracy of BeamCLIP-RN18 representations learned with unlabeled target data (ImageNet-1K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Figure 9: Zero-shot ImageNet-1K top-1 accuracy of BeamCLIP-RN50 representations learned with unlabeled non-target data (ImageNet-21K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' at first, given an unlabeled image, cross-modal similarity distribution is measured from multiple text prompt embeddings in the teacher’s embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Then, a student model is encouraged to mimic the cross-modal similarity distribution of the teacher model by matching these similarity distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' To achieve effective transfer, we use anchor text embeddings by encoding text prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' For example, on CIFAR10, we use ten text prompts in the form of "a photo of {class name}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Note that the text prompts are not paired with each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' We measured how effective the BeamCLIP is in cases where the class names of the target dataset are not perfectly given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Table 10 shows the effect of the randomly sampled text prompts on CIFAR10 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The values in Figure 10 are also presented in Table 13 and Table 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Table 4 shows the effect of the randomly sampled text prompts on CIFAR100 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The values in Figure 4 are also presented in Table 15 and Table 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Table 13: Effect of the partial text prompts on CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Prompts Method Type Img.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 3 5 7 9 10 CLIP [32] (zero-shot) T ViT-B/16 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 CLIP [32] (zero-shot) RN50 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 BeamCLIP (CE+EntMin) S RN50 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='47 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='25 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='26 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='84 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='10∗ BeamCLIP (KL) S RN50 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='36 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='43 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='15 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='54 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='85 19 ImageNet-1Kzero-shotaccuracyofBeamCLip-RN50 trained with ImageNet-21K (no overlap) 58 val_acc 56 54 52 50 48 46 25 50 75 100 125 150 175 200 EpochsImageNet-1K validation accuracy of BeamCLiP-RN18 trained with ImageNet-1K 60 - val_acc 50 (%)/ 40 20 10 0 - 0 20 40 60 80 100 Epochs(a) Subset prompts from CIFAR10 (b) Random prompts from ImageNet Figure 10: Effect of the random text prompts on CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (a) The text prompts are randomly sampled from the 10 class names of CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The red dotted line denotes the teacher’s accuracy as an upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' It is more efficient as it is closer to this line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' As shown in the blue line, the BeamCLIP is still effective, even when the class names of the target dataset are partially given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The BeamCLIP (KL) means to use the KL-divergence for matching cross-modal similarity distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The BeamCLIP (CE+EntMin) is more effective, as more text prompts are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' (b) The text prompts are randomly selected from the 1000 class names of ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' The BeamCLIP (CE+EntMin) is still effective, even though the class names are randomly sampled from a non-target dataset (ImageNet-1K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Table 14: Effect of the random text prompts on CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Prompts Method Type Img.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 10 20 30 40 CLIP [32] (zero-shot) T ViT-B/16 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 CLIP [32] (zero-shot) RN50 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 BeamCLIP (CE+EntMin) S RN50 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='49 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='05 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='09 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='51 BeamCLIP (KL) S RN50 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='76 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='67 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='83 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='88 Table 15: Effect of the partial text prompts on CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Prompts Method Type Img.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 20 40 60 80 100 CLIP [32] (zero-shot) T ViT-B/16 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='7 CLIP [32] (zero-shot) RN50 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 BeamCLIP (CE+EntMin) S RN50 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='72 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='19 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='38 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='93 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='35 BeamCLIP (KL) S RN50 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='10 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='93 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='88 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='15 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='12 Table 16: Effect of the random text prompts on CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Prompts Method Type Img.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' Enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content=' 100 500 1000 CLIP [32] (zero-shot) T ViT-B/16 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='7 CLIP [32] (zero-shot) RN50 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 BeamCLIP (CE+EntMin) S RN50 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='36 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='44 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='93 BeamCLIP (KL) S RN50 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='68 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='82 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='15 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='TransferlearningonCiFAR10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='(withsubsetpromptsfromCIFAR1o classes) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='95 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='(%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='Accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='CLIP ViT-B/16 (zero-shot) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='BeamCLIP-RN50 (CE+EntMin) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='65 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='BeamCLIP-RN50 (KL) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='CLIP-RN50 (zero-shot) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='w - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='# of text promptsTransferlearing on CIFAR1o ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='(withrandompromptsfrom ImageNetclasses) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='95 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='(%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='Accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='CLIP ViT-B/16 (zero-shot) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='BeamCLIP-RN50 (CE+EntMin) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='65 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='BeamCLIP-RN50 (KL) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='CLIP-RN50 (zero-shot) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} +page_content='# of text prompts' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfFgOa/content/2301.02903v1.pdf'} diff --git a/J9E3T4oBgHgl3EQfAAmq/content/tmp_files/2301.04254v1.pdf.txt b/J9E3T4oBgHgl3EQfAAmq/content/tmp_files/2301.04254v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..db71089d23dd7d25042e25f11b257556d488ada3 --- /dev/null +++ b/J9E3T4oBgHgl3EQfAAmq/content/tmp_files/2301.04254v1.pdf.txt @@ -0,0 +1,1917 @@ +Impact of a Rapid Diluted Energy Density on the halo mass function +Dante V. Gomez-Navarro,1, ∗ Alejandro Aviles,2, 3, † and Axel de la Macorra1, ‡ +1Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México C.P. 04510, México. +2Departamento de Física, Instituto Nacional de Investigaciones Nucleares, +Apartado Postal 18-1027, Col. Escandón, Ciudad de México,11801, México. +3Consejo Nacional de Ciencia y Tecnología, Av. +Insurgentes Sur 1582, +Colonia Crédito Constructor, Del. Benito Juárez, 03940, Ciudad de México, México. +We study dark energy cosmological models, extensions of the standard model of particles, characterised by +having an extra relativistic energy density at very early times, and that rapidly dilute after a phase transition +occurs. These models generate well localized features (or bumps) in the matter power spectrum for modes +crossing the horizon around and before the phase transition epoch. This is because the presence of the additional +energy component enhances the growth of matter fluctuations during the radiation dominated epoch. Instead of +considering a particular model, we focus on a parametric family of Gaussian bumps in the matter power spectrum, +which otherwise would be a ΛCDM one. We study the evolution of such bump cosmologies and their effects +in the halo mass function and halo power spectrum using N-body simulations, the halo-model based HMcode +method, and the peak background split framework. The bumps are subject to different nonlinear effects that +become physically well understood, and from them we are able to predict that the most distinctive features will +show up for intermediate halo masses 1012.3 ℎ−1𝑀⊙ < 𝑀 < 1013.6 ℎ−1𝑀⊙. Out of this range, we expect halos +are not significantly affected regardless of the location of the primordial bump in the matter power spectrum. +Our analytical results are accurate and in very satisfactory agreement with the simulated data. +I. +INTRODUCTION +Recent cosmological and astrophysical observations have +consolidated our picture of the concordance ΛCDM model [1– +5], which corresponds to a nearly homogeneous and isotropic +expanding universe filled with the particles of the Standard +Model (SM) [6], and supplemented by dark matter and a cos- +mological constant. Despite this success, the two dark com- +ponents have yet to be thoroughly tested and understood, since +their fundamental nature is still a puzzle [7]. At present time, +they accounts for about 96% of the energy budget of the cos- +mos, and so alternative models look for plausible explanations. +In particular, scalar fields have been proposed to describe dark +energy [8–10], whose nature could be that of a fundamental +particle not contained in the SM (Higgs-type particles) or can +be a composite one, as for example a dark meson pion 𝜋-like +particle [11]. +In addition to the fact that we do not know the nature of the +dark sector, some strains in the ΛCDM model began to appear +as the accuracy of cosmological observations improved, and +recently, some interesting tensions have emerged. Perhaps the +most famous is the discrepancy between early times and local +measurements of the Hubble constant [12–14]. The increasing +statistical tension on its value obtained using different observa- +tions has revived interest in alternative cosmological models. +Henceforth, extensions to the standard model of particles have +been proposed to alleviate the 𝐻0 crisis or even simply to de- +scribe the origin of dark energy, for example by introducing +additional particles. +As an example of the interest of this work, the Bound Dark +Energy model (BDE) cosmological model [11, 15] is charac- +∗ dantegomezn@gmail.com +† avilescervantes@gmail.com +‡ macorra@fisica.unam.mx +terised by a supersymmetric Dark Gauge Group (DG), in which +the fundamental particles are massless during early times and +their energy density evolved as radiation. However, at low en- +ergies the postulated gauge interaction becomes strong enough +to bind the elementary dark particles together and form massive +bound states, dark mesons and dark baryons. This process is +similar to the strong QCD interaction in the SM where quarks +are bound together to form baryons and mesons (e.g. protons, +neutrons or pions). +In the BDE scenario, the dark energy +corresponds to the lightest meson scalar particle 𝜙 formed +at a phase transition scale Λ𝑇 , at a scale factor 𝑎𝑇 . Before +the transition, the energy density of the DG particles behaves +as radiation decaying with the expansion of the Universe as +1/𝑎4. Just after the phase transition occurs, for a scale factor +𝑎 > 𝑎𝑇 and lasting a long period of time, its energy density +decays very fast, as 1/𝑎6. During this epoch, there is an abrupt +decrease in the DG cosmic abundance and rapidly becomes +subdominant. We refer to such a behavior very generically as +Rapid Diluted Energy Density (RDED). The existence of these +type of bound particles modifies the evolution of the Hubble +parameter 𝐻 and have also an important impact in the evolu- +tion of density perturbations, leaving distinctive signatures on +cosmological distances, and in the matter power spectrum and +other summary statistics around the corresponding transition +scale 𝑘𝑇 = 𝐻(𝑎𝑇 )𝑎𝑇 [11, 15, 16]. In particular, adding extra +relativistic particles increases the growth rate of matter density +fluctuations. +In this work we are motivated by the impact of a RDED in +the matter power spectrum, which becomes enhanced around +𝑘𝑇 [16]. Since there are several theoretical frameworks that +can lead to similar mechanisms, we will work in a model- +independent way with the introduction of a family of parame- +terized bumps into the linear matter power spectra 𝑃ΛCDMex,1 +1 The suffix “ex” in the notational label “ΛCDMex” makes allusion to the +arXiv:2301.04254v1 [astro-ph.CO] 11 Jan 2023 + +2 +which otherwise would be a ΛCDM one, where we will vary +their positions and widths. By using these linear spectra as +input, we will work beyond the linear regime using different +complementary schemes, and focus on the consequences that +the nonlinear evolution of the injected bumps has on halo clus- +tering and halo abundance using the Peak-Background Split +(PBS) framework [17, 18] and the Sheth-Tormen Halo Mass +Function (HMF) [19, 20]. +Although full N-body simulations successfully describe the +nonlinearities of the matter clustering, they have the disad- +vantage of being computationally expensive. Hence, in this +work we use the approximated particle mesh N-body solver +FastPM2 [21], where the linear growth of displacements, the +Zeldovich approximation solution, is enforced at 𝑘 → 0 by +choosing an appropriate set of kick and drift factors, and hence +very large scale are treated exactly. Further, we use the HM- +code3 [22–24] to describe the nonlinear dark matter power +spectrum. This is a halo-model based method, and as such, +it describes the nonlinear power spectrum as a sum of two +pieces, the 2-halo term that models the correlation between +particles hosted by different halos reducing to the linear the- +ory at large scales, and the 1-halo piece to model the small, +intra-halo clustering scales [25–27]. +Halo clustering is crucial in the study of the large-scale +structure of the Universe, since it is governed by gravitational +instability, responsible for the formation of dark matter halos +and their distribution [28]. Furthermore, well physically moti- +vated models often assume that galaxy formation is the result +of the condensation of baryonic matter in already collapsed +and virialized dark matter halos [19, 27, 29–31]. Therefore, +the HMF is an important tool for studying the formation and +evolution of galaxies. The analytical understanding of these +processes is also desirable, both to obtain a physical intuition, +and for the study of different models and their wide range of +parameters. In this work, we use the Sheth-Tormen HMF to +describe the evolution of the number density of dark matter +halos of a given mass and for our alternative cosmological +models. However, since we are working beyond ΛCDM, we +let free their parameters and fit them to data extracted from +our N-body FastPM simulations. On the other hand, galaxy +surveys show that at large scales, the number density fluctua- +tion is roughly proportional to the matter density overdensity +field, with a multiplicative factor called the bias 𝑏 [32, 33]. +We study this large-scale bias using PBS, which in addition to +its utility, allows a physical interpretation of the halo bias [34]. +Summarizing, we are interested in the distribution of non- +linear virialized objects because it allows us to check the evolu- +tion and final fate of the small primordial density fluctuations +that have undergone gravitational collapse. Consequently, it is +necessary to review the properties of halo statistics in models +that have undergone a phase transition in early times. This the +main topic of this work. +extra energy density component not present in the ΛCDM model at early +times, and referred below as 𝜌ex. +2 https://github.com/fastpm/fastpm +3 https://github.com/alexander-mead/HMcode +The rest of the paper is organized as follows. In Sec. II, we +briefly present how cosmic phase transitions lead to different +cosmological signatures. In Sec. III, we introduce the bump +in the power spectrum 𝑃ΛCDMex and parameterize its width +and position, in this section we also show the specifications of +our N-body simulations suite. In Sec. IV we review the HMF +formalism and the large-scale bias. We present our results +and details for our bump cosmologies in Sec. V, with added +supplementary material in Appendix A. Finally, in Sec. VI we +present our conclusions. +II. +RAPIDLY DILUTION ENERGY DENSITY (RDED) +Models beyond ΛCDM may leave different distinctive fea- +tures in the matter power spectrum. +As mentioned in the +Introduction, we are interested in models that have an extra +energy density 𝜌𝑒𝑥, beyond the standard ΛCDM and before a +transition scale, which dilutes rapidly after the transition takes +place at the scale factor 𝑎𝑇 . The dark matter mode that is +crossing the horizon at that time is 𝑘𝑇 = 𝑎𝑇 𝐻(𝑎𝑇 ). Some +of these models are generally referred as Early Dark Energy +(EDE) [35–37], where the basic idea is to postulate an ex- +tra component that contributes non negligible to the energy +density before recombination, and then it decays faster than +radiation at later times, at a transition scale factor 𝑎𝑐. The +presence of this extra fluid at early times enhances the total +energy density before the last scattering surface leading to a +higher expansion rate 𝐻0, with the potential to resolve the Hub- +ble tension [36–40], and as well leaving other cosmological +fingerprints [41, 42] +In order to describe the effects of a RDED in the structure +formation we appeal to the mechanism of the BDE model +[11, 15]. At very early times, the light particles of the DG +are ultrarelativistic and the extra energy density 𝜌𝑒𝑥 evolves +with an effective equation state parameter 𝑤 = 1/3. Long +after, but still well inside the radiation dominated epoch, at +𝑎𝑇 ∼ 10−6, a phase transition occurs because the coupling +of the DG particles becomes strong and binds the elementary +particles of DG, forming composite particles, dark mesons, +that can be described as a scalar field with an inverse power +law potential [43, 44] +𝑉(𝜙) = Λ4+𝑛 +𝑇 +𝜙−𝑛, +(1) +with 𝑛 = 2/3 and with a condensation energy scale Λ𝑇 ∼ 40 +eV in BDE model [11, 15]. At this phase transition epoch, the +effective equation of state abruptly changes from 𝑤 = 1/3 to +𝑤 = 1. During this epoch, the dark mesons energy density +rapidly dilutes as 𝜌 ∝ 𝑎−6 and lasting for a long period of +time. However, at 𝑎 ∼ 1/1000, the equation of state swiftly +goes from 𝑤 = 1 to 𝑤 = −1 to finally ending up today at +𝑤 ∼ −0.93, because slow-rolling starts to fail around 𝑧 ∼ 0.3. +Notice that despite the evident similarities between BDE and +EDE in its best-known form (see e.g.[37]), these are different +in nature and, moreover, the former may have a greater impact +on the early universe due to that its energy density decays +as that of a radiation component, instead of being constant. +However, in a broader sense BDE can be considered also an + +3 +Name +𝑁𝑠𝑖𝑚𝑠 +𝐴𝑇 +𝜎𝑇 +𝑘𝑇 [ℎMpc−1] 𝐿𝑏𝑜𝑥 [ℎ−1Mpc] +medbump-k1 +5 +0.15 +0.3 +1.0 +1024 +thinbump-k1 +5 +0.15 +0.1 +1.0 +1024 +medbump-k0p5 +5 +0.15 +0.3 +0.5 +1024 +thinbump-k0p5 +5 +0.15 +0.1 +0.5 +1024 +ΛCDM +5 +− +− +− +1024 +TABLE I. Specifications of our N-body simulation suite. The background cosmological parameters are the same for all the simulations: +Ω𝑚 = 0.3, Ω𝑏 = 0.05, ΩΛ = 0.7, Ω𝜈 = 0, ℎ = 0.7, 𝑛𝑠 = 0.96, 𝜎8 = 0.8. Each simulation uses 10243 particles distributed over 𝑁𝑔𝑟𝑖𝑑 = 10243 +cells. We consider the redshifts 𝑧 = 0, 0.5, 1, 2. +EDE scenario, because it constitutes a non-negligible dark +energy component in early epochs. +Now, during radiation domination, matter perturbations +grow only logarithmically with the growth rate +𝑓 += +𝑑 ln 𝛿𝑚/𝑑 ln 𝑎 ∝ 1/𝛿𝑚. +On the other hand, matter pertur- +bations in the model containing 𝜌𝑒𝑥 are initially suppressed +compared to a ΛCDM model since the initial amplitude de- +pends on the fraction of relativistic particles [16], and because +𝑓ΛCDMex > 𝑓ΛCDM, the ratio 𝛿𝑚,ΛCDMex/𝛿𝑚,ΛCDM increases +and is further boosted by the extra relativistic component 𝜌𝑒𝑥 +before the phase transition occurs at 𝑎𝑇 . Notice that this boost +affects only the modes crossing the horizon before 𝑎𝑇 , i.e. +𝑘 ≥ 𝑘𝑇 . This is the characteristic signature on the matter fluc- +tuations because of the presence of an early times RDED and +was presented in [11], and further developed in [15, 16, 45, 46]. +III. +MODELLING THE POWER SPECTRUM +In the following, we will characterize the effect of having a +RDED by choosing a parametrization that we refer throughout +as the bump cosmology, where the linear power spectrum is +a modification to that of a standard ΛCDM cosmology one +given by +𝑃bump(𝑘, 𝑧) = +� +1 + 𝐹(𝑘) +� +𝑃ΛCDM(𝑘, 𝑧), +(2) +with the 𝐹(𝑘) parametric function describing the bump, +𝐹(𝑘) = 𝐴𝑇 exp +� +− [ln(𝑘/𝑘T)]2 +𝜎2 +𝑇 +� +. +(3) +The parameters 𝐴𝑇 , 𝑘𝑇 and 𝜎𝑇 are the amplitude, scale, and +width of the bump, respectively. The width of the bump corre- +sponds to how fast the rapid diluted energy density phase takes +place, whereas 𝑘𝑇 represents the mode entering the horizon +about the phase transition time. +We consider four different bump cosmologies, each with +fixed amplitude 𝐴𝑇 = 0.15, as motivated by the BDE mod- +els.4 We choose two different widths of the bump 𝜎𝑇 = 0.3, +4 In the original BDE model, the extra energy density has an abundance +ΩBDE(𝑎𝑇 ) ∼ 0.11 before the phase transition that occurs at 𝑎𝑇 += +2.48 × 10−6, and hence the mode entering the horizon at that time is +𝑘𝑇 ≈ 0.92 ℎ Mpc−1 [15]. +and 0.1, and locate the bump at two different scales: 𝑘𝑇 = 0.5, +and 1.0 ℎ Mpc−1 (see Table I). We consider these bump cos- +mologies at different redshifts: 𝑧 = 0.0, 0.5, 1.0, and 2.0. +To generate the ΛCDM power spectrum we use the cosmo- +logical parameters reported in Sec. III A, or in the caption of +Table I, which are the same for all the bump and standard cos- +mologies, and as such, the only difference between the models +is the presence of the bump parametrized by Eq. (2). +In the following two subsections we briefly describe the +methods we use to study the nonlinear evolution of the bump +in the power spectrum. +A. +N-body simulations +We generate 25 rapid N-body simulations using the code +FastPM, 5 for each of the cosmologies detailed in Table I. +Zeldovich initial conditions were generated at 𝑧𝑖 = 99, and we +use 100 linearly space steps up to redshift 𝑧 = 0. Each simula- +tion uses 10243 particles to approximate the density field. The +box sizes of the simulations are 𝐿𝑏𝑜𝑥 = 1024 ℎ−1Mpc. We +analyze four snapshots at 𝑧 = 0.0, 0.5, 1.0, 2.0. +Our baseline cosmology is ΛCDM with dark matter density +Ω𝑐𝑑𝑚 = 0.25, baryon density Ω𝑏 = 0.05, fluctuation variance +𝜎8 = 0.8, dimensionless Hubble constant ℎ = 0.7, and spectral +index 𝑛𝑠 = 0.96. Neutrinos are considered massless. +We identify and construct halo catalogs with the Friends- +of-Friends algorithm [47], already implemented in the N- +BodyKit package5 [48], for which we use a linking length +𝑙 = 0.2 and where each halo is formed by at least 20 particles. +B. +HMcode +As a complementary tool to N-body simulations, we make +use of the HMcode [22, 49], which is an augmented version +of the standard halo-model scheme for the nonlinear matter +power spectrum. The starting point is a standard halo-model +calculation, where the power spectrum is split into two terms: +one that accounts for the clustering arising within individual +5 http://nbodykit.readthedocs.io + +4 +halos, and the second that accounts for the clustering of dark +matter between two different halos and that follows closely the +linear theory at large scales. We use the most recent HMcode +version [24], which further accounts for the BAO damping +into the two-halo term. In this new scheme, HMcode adds +a smoothing parameter for the transition region between the +1- and 2-halo terms when constructing full halo-model power +spectra. In the nonlinear regime at low redshifts, we expect +HMcode to match adequately the N-body simulations. +IV. +MODELLING THE HALO ABUNDANCE AND +CLUSTERING +Quantitative comparisons between theoretical predictions +and observations allow us to compute constraints on cosmo- +logical parameters, e.g. the abundance of halo identified as +clusters of galaxies. The bias of halo dark matter contains com- +plementary information on their abundance since data survey +is understood through the bias of the halos which they form +[50, 51]. In this section, we describe the analytical techniques +to study the halo abundance and halo bias using the Sheth- +Tormen mass function and PBS prescriptions. +A. +The halo mass function +The HMF gives the number density of dark matter halos as +a function of their masses (per unit comoving volume). How- +ever, occasionally a good start point to the HMF discussion is +by introducing the scaled differential mass function 𝑓 (𝜎) as +the fraction of the total mass ¯𝜌𝑉 (when the volume 𝑉 is very +large) hosted by halos in a logarithmic interval of 𝜎−1, +𝑓 (𝜎) = 𝑑(𝜌/ ¯𝜌) +𝑑 ln 𝜎−1 , +(4) +with 𝜎(𝑀) the variance of linear fluctuations when smoothed +over a scale 𝑅(𝑀) = (3𝑀/4𝜋 ¯𝜌)1/3, +𝜎2(𝑀, 𝑧) = 𝐷2(𝑧) +2𝜋2 +∫ ∞ +0 +𝑑𝑘 𝑘2𝑃𝐿(𝑘)𝑊2(𝑘𝑅), +(5) +where 𝑃𝐿 is the linear power spectrum, 𝐷 the linear growth +function and 𝑊 the top-hat filter in Fourier space +𝑊(𝑘𝑅) = +3 +(𝑘𝑅)3 +� sin(𝑘𝑅) − 𝑘𝑅 cos(𝑘𝑅)�. +(6) +The utility of 𝑓 (𝜎) is that apparently this quantity, when prop- +erly scaled, is nearly universal throughout a wide range of halo +masses, and for cosmologies both within the ΛCDM as well as +dark energy models [52] or even in Modified Gravity theories +[53].6 In contrast, the HMF is very sensitive to the cosmo- +6 This is more evident by comparing between different models the multiplicity +𝜈 𝑓 (𝜈) against the rescaled variance, or peak significance 𝜈 = 𝛿𝑐/𝜎 +introduced below; see, e.g., Fig. 7 of Ref. [53]. Notice also that 𝛿𝑐 is mass +dependent in theories that introduce new additional scales as in warm dark +matter or modified gravity and hence comparing against 𝜈 and 𝜎−1 are not +equivalent. +logical parameters and the specific theory through the power +spectrum dependence of the variance. +The precise connection between 𝑓 (𝜎) and the HMF is given +by +𝑑𝑛 +𝑑 log 𝑀 = 𝑓 (𝜎) ¯𝜌 +𝑀 +𝑑 ln 𝜎−1 +𝑑 log 𝑀 , +(7) +where one assumes that all of the matter in the universe is +hosted by halos. The simplest HMF is the Press-Schechter +mass function [54], which analytical form can be obtained ex- +actly based on the spherical collapse model and the hypothesis +that the mass in collapsed objects is related to the volume with +density above a certain threshold. However, data from simu- +lations show that the Press-Schechter HMF is not accurate at +the low and the high mass ends. Hence, several other mass +functions have been proposed based on different assumptions +or as purely empirical fits [55–58]. The Sheth-Torman HMF +[19, 20] is perhaps the best known and most used alternative to +Press-Shechter, it is based on the more realistic ellipsoidal col- +lapse and reproduce N-body simulations better. It is defined +through +𝑓𝑆𝑇 (𝜈) = 𝐴(𝑝) +√︂ +2𝑞 +𝜋 +� +1 + +� 1 +𝑞𝜈2 +� 𝑝� +exp +� +−𝑞𝜈2 +2 +� +, +(8) +with 𝐴(𝑝) = [1 + 𝜋−1/22−𝑝Γ(0.5 − 𝑝)]−1 a normalization +factor coming from the assumption that all the dark matter in +the Universe is contained within halos. The variable 𝜈 = 𝛿𝑐/𝜎 +is the peak significance, and 𝛿𝑐 is the critical overdensity for +collapse. Notice that we are abusing notation because 𝑓 (𝜎) +and 𝑓 (𝜈) represent the same function. For a matter-dominated +universe, 𝛿𝑐 = 1.686, while for Ω𝑚 < 1 the numerical value +varies slightly with the redshift but with no significant impact +on the cosmological outputs. From Eq. 8, one recovers the +Press-Schecter mass function by choosing 𝑞 = 1.0 and 𝑝 = 0.0. +The standard values 𝑝 = 0.3 and 𝑞 = 0.707 for the Sheth- +Tormen HMF are obtained by fitting to ΛCDM simulations. +For the bump cosmologies, in Sect. V B we will rely on Eq. (8) +for computing the HMF, but we will fit the 𝑝 and 𝑞 parameters +to our N-body simulations. +B. +Biasing on large scales +To compute the halo bias we use the PBS formalism [17– +20] with the aid of the Sheth-Tormen mass function. In this +picture, long- and small-wavelength density fluctuations are +split, with the crests of long wavelength overdensities serv- +ing as locations where the average density is higher than the +background cosmological density, and on top of that small- +wavelengths fluctuations, the peaks, collapse to form halos on +which ultimately galaxy formation takes place. The main idea +to obtain the halo bias is that more massive halos are formed in +locations where the average (over large regions) local density +is high, or in other words the PBS biases are the responses +of the mean abundance of tracers against small changes in the +background density [20, 34]. That is, under the PBS formal- +ism the Lagrangian (𝐿) biases can be written in terms of the + +5 +multiplicity function 𝜈 𝑓 (𝜈) through +𝑏𝐿 +𝑛 = +(−1)𝑛 +𝜎𝑛(𝑀, 𝑧) +1 +𝜈 𝑓 (𝜈) +𝑑𝑛𝜈 𝑓 (𝜈) +𝑑𝜈𝑛 +. +(9) +It turned out that the bias computed in this way are the physical, +renormalized local biases, up to subdominant factors intro- +duced by the artificial smoothing scale of density fluctuations +[34, 59, 60]. For the Sheth-Tormen HMF given by Eq. (8), the +linear Lagrangian bias is +𝑏𝐿 +1 (𝑀) = 1 +𝛿𝑐 +� +𝑞𝜈2 − 1 + +2𝑝 +1 + (𝑞𝜈2) 𝑝 +� +. +(10) +To obtain the biases over a mass range [𝑀𝑚𝑖𝑛, 𝑀𝑚𝑎𝑥] one has +to average over. That is, the large-scale bias within a mass +range becomes [59] +𝑏𝐿 +1 = +1 +𝐼𝑑𝑀 +∫ +𝑀max +𝑀min +𝑑𝑀 1 +𝛿𝑐 +𝜈 +𝑀 +𝜕 𝑓 +𝜕𝜈 +𝑑 ln 𝜈 +𝑑𝑀 , +(11) +with +𝐼𝑑𝑀 = +∫ +𝑀max +𝑀min +𝑑𝑀 𝑓 +𝑀 +𝑑 ln 𝜈 +𝑑𝑀 . +(12) +The corresponding linear Eulerian bias is given by +𝑏1 = 1 + 𝑏𝐿 +1 . +(13) +We will use these expressions when compute the halo biases +for bump cosmologies in Sect. V C. +V. +ANALYSIS AND RESULTS +The evolution, dilution and shift of the bump is studied via +our suite of FastPM N-body simulations and the HMcode +results. We first focus on response functions, computed as the +ratio of statistics between a model containing a bump and one +without it. For the power spectrum this is given by +𝑅(𝑘) = 𝑃bump(𝑘) +𝑃ΛCDM(𝑘) . +(14) +The importance of this analysis is that once a good under- +standing and modeling of the response function is acquired for +a given alternative cosmology, its power spectrum can be com- +puted by multiplying it by an as well modeled power spectrum +for the ΛCDM one, which has been studied comprehensively. +This response function analysis has shown to give fruitful re- +sults in different contexts beyond ΛCDM [49, 61–64]. +Our numerical results are contrasted with the linear the- +ory, for which the response in the power spectrum is simply +𝑅𝐿(𝑘) = 1 + 𝐹(𝑘) at all 𝑧 since the linear growth is scale +independent, well after the phase transition had occurred. +The details of our simulations are enlisted in Table I. The +bumps are located at scales 𝑘𝑇 = 0.5, 1.0 ℎ Mpc−1 and have +widths 𝜎𝑇 = 0.3 and 0.1. Hence, they are inside the full non- +linear regime, where a perturbative treatment is not adequate, +as it is for the cases 𝑘𝑇 = 0.05 and 0.1 ℎ Mpc−1 covered in +our previous work [16]. Hence, here we rely on the HMcode +and FastPM to model the nonlinearities and not in perturba- +tion theory. In [16] we also studied the case with transition +scale 𝑘𝑇 = 1 ℎ Mpc−1, although only for the dark matter power +spectrum and correlation function, while here we augmented +the analysis for the case of biased tracers and put emphasis in +the HMF and halo bias. When there is overlap with the above +mentioned reference our results agree, despite that in that work +we use full N-body simulations (but with a smaller number of +particles: 𝑁𝑝 = 2563). +A. +Matter power spectrum +To extract the power spectrum data we use the cloud-in- +cell (CIC) mass-assignment scheme implemented in the N- +BodyKit package. +The grid in our simulations is divided +into 𝑁𝑔𝑟𝑖𝑑 = 2048 cells and the size of the box in all cases +is 𝐿 = 1024 ℎ−1Mpc. The power spectrum ranges are binned +in 80 log-spaced 𝑘-points over the interval [𝑘𝑚𝑖𝑛, 𝑘Ny], where +𝑘𝑚𝑖𝑛 = 2𝜋/𝐿 and 𝑘Ny = 𝑁𝑔𝑟𝑖𝑑𝜋/𝐿 is the Nyquist frequency. +Usually, power spectra in CIC approach are considered to be +accurate up to half of the Nyquist frequency, and as such, this +is the upper limit we show in our plots. +In Figs. 1 and 2 we show the matter power spectra using the +bump cosmology located at the scales 𝑘𝑇 = 0.5 and 1 ℎ Mpc−1, +respectively. The matter power spectra are computed using our +different nonlinear methods and then divided by their counter- +parts in the ΛCDM model to show the response function; see +Eq. (14). These analyses compare how the bump cosmology +power spectra are modified by nonlinearities within the differ- +ent schemes. The squares correspond to the FastPM synthetic +data and the error bars are the scattering over the 5 simulated +boxes for each model. +As expected, at higher redshifts nonlinear effects are smaller +and the responses for all approaches are very similar. The +observed features in the plots can be described through two +important nonlinear effects +1.- There exists a generation of a second bump. This non- +linear effect was observed first in [16] and is due to +that the primordial bump enhances the amplitude of the +long wavelength perturbation where peaks in the density +fluctuation locate, which magnify them, and ultimately +forming a second bump. In other, simpler words, the +generation of the second bump is a consequence of the +development of structures on top of the first, primordial +bump. This effect is even more pronounced for wider +bumps because these provide a greater enhancement of +linear power and then, in the language of halo-based +models, a broader range of interaction with the 1-halo +term. For the case of 𝑘𝑇 = 0.5 ℎ Mpc−1, as can be ob- +served in Fig. 1, this nonlinear second bump is well mod- +elled by the HMcode, and reaches a maximum relative +to the first bump value, at 𝑧 = 0. For 𝑘𝑇 = 1 ℎ Mpc−1 the +generation of this second bump is still present, although +less evident because the Nyquist frequency coincides +with the onset of the bump; see Fig. 2. + +6 +10 +1 +100 +1.0 +1.05 +1.1 +1.15 +1.2 +Pmm/P CDM +mm +z = 2 +Linear +HMcode +N-body +10 +1 +100 +z = 1 +10 +1 +100 +z = 0.5 +10 +1 +100 +z = 0 +10 +1 +100 +k [h Mpc +1] +1.0 +1.05 +1.1 +1.15 +1.2 +Pmm/P CDM +mm +10 +1 +100 +k [h Mpc +1] +10 +1 +100 +k [h Mpc +1] +10 +1 +100 +k [h Mpc +1] +FIG. 1. Matter power spectrum response functions for bump cosmologies at 𝑘𝑇 = 0.5 ℎ Mpc−1 with widths 𝜎𝑇 = 0.3 (top panel) and 𝜎𝑇 = 0.1 +(bottom panel). From left to right, cyan curves are for 𝑧 = 2; green for 𝑧 = 1; red for 𝑧 = 0.5; and blue for 𝑧 = 0. Dot-dashed curves correspond +to linear theory; solid to the HMcode model; and squares are the data extracted from the FastPM N-body simulations. Error bars that are not +visible are within the mark size. +2.- The primordial bumps tend to vanish with the gravi- +tational collapse. This effect can be better understood +under a configuration space description, where localized +features become oscillations. Since bulk displacements +of matter tend to partially move out from overdense +regions and populate underdense regions, these oscilla- +tions are erased with the collapse. In fact, the character- +istic scale for this to happen is given by two times the +Lagrangian displacements variance +2𝜎Ψ = 2𝐷(𝑧) +�∫ ∞ +0 +𝑑𝑘 +6𝜋2 𝑃𝐿(𝑘, 𝑧 = 0) +�1/2 +∼ 10𝐷(𝑧) ℎ−1Mpc, +(15) +with 𝐷(𝑧) the linear growth function. As can be seen by +comparing Figs. 1 and 2, as higher is the transition mode +𝑘𝑇 , the more the bump is damped. This is expected be- +cause higher-𝑘 locations of the bump in Fourier space +correspond to higher frequency of oscillations in config- +uration space, and since particles travel in random paths +an average distance 𝜎Ψ, then with high probability, parti- +cles will tend to displace out from the overdense regions +if the oscillations wavelength are smaller. Of course +this is a consequence of having a comoving distance +between peaks and troughs in the 2-point configuration +space correlation function comparable or smaller than +2 × 𝜎Ψ. So, it is not expected to happen for bumps lo- +cated at low 𝑘 values, as those studied in [16], which +oscillations in configuration space have very large wave- +lengths. Notice this effect is completely nonlinear in the +matter overdensity, i.e. it is non-perturbative in the Eu- +lerian theory, however, it can be completely described by +Lagrangian perturbation theory, even at the linear order, +Zeldovich approximation because bulk, large scale dis- +placement fields are responsible for erasing it. Indeed, +we notice this effect has the same origin as the smearing +of the BAO peak. + +7 +10 +1 +100 +1.0 +1.05 +1.1 +1.15 +1.2 +Pmm/P CDM +mm +z = 2 +Linear +HMcode +N-body +10 +1 +100 +z = 1 +10 +1 +100 +z = 0.5 +10 +1 +100 +z = 0 +10 +1 +100 +k [h Mpc +1] +1.0 +1.05 +1.1 +1.15 +1.2 +Pmm/P CDM +mm +10 +1 +100 +k [h Mpc +1] +10 +1 +100 +k [h Mpc +1] +10 +1 +100 +k [h Mpc +1] +FIG. 2. Same as Fig. 1, but for the bump cosmologies with 𝑘𝑇 = 1 ℎ Mpc−1. +Name +𝑞 +𝑝 +medbump-k1 +0.727 +0.314 +thinbump-k1 +0.734 +0.326 +medbump-k0p5 +0.723 +0.343 +thinbump-k0p5 +0.691 +0.334 +ΛCDM +0.711 +0.317 +TABLE II. Best-fit for 𝑝 and 𝑞 Sheth-Tormen parameters. These results were obtained by using the criterion that minimizes Eq. 17 at redshift +𝑧 = 0. +B. +Halo mass function +From our N-body simulations, we have constructed halo +catalogs with the Friends-of-Friends algorithm as described +in Sec. III A. From them, we obtain the HMF for the different +bump cosmologies. The number of halos per logarithmic mass +intervals 𝑑𝑛/𝑑 log 𝑀 in a simulation of volume 𝐿3 is given by +𝑑𝑛 +𝑑 log 𝑀 = 𝑀 +𝐿3 +Δ𝑁 +Δ log 𝑀 , +(16) +for which we construct logarithmic bins in mass with size +Δ log 𝑀 = 0.2. Complementary, in Appendix A we show a +histogram of our halos within the interval 𝑀 = [1012.3, 𝑀 = +1015] ℎ−1𝑀⊙ as well as a Table with the mean number of halos +of our catalog suite. +Since we are working in cosmologies beyond the ΛCDM, +and anticipating possible deviations in the scaled differential +mass function 𝑓 (𝜎) function, we recalibrate the model param- +eters of the Sheth-Tormen functional form of Eq. (8). That is, +we compute the best fit of Sheth-Tormen 𝑝 and 𝑞 parameters + +8 +10 +9 +10 +8 +10 +7 +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +dn/dlogM [h3 Mpc +3] +z = 2 +z = 1 +z = 0.5 +z = 0 +13 +14 +15 +logM [h +1M ] +0.95 +1.0 +1.05 +1.1 +1.15 +1.2 +1.25 +1.3 +1.35 +1.4 +(dn/dlogM)/(dn/dlogM) CDM +13 +14 +15 +logM [h +1M ] +13 +14 +15 +logM [h +1M ] +13 +14 +15 +logM [h +1M ] +10 +9 +10 +8 +10 +7 +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +dn/dlogM [h3 Mpc +3] +z = 2 +z = 1 +z = 0.5 +z = 0 +13 +14 +15 +logM [h +1M ] +0.95 +1.0 +1.05 +1.1 +1.15 +(dn/dlogM)/(dn/dlogM) CDM +13 +14 +15 +logM [h +1M ] +13 +14 +15 +logM [h +1M ] +13 +14 +15 +logM [h +1M ] +FIG. 3. Halo mass function 𝑑𝑛/𝑑 log 𝑀 for the bump cosmologies at 𝑘𝑇 = 0.5 ℎ Mpc−1 (top panel) and 𝑘𝑇 = 1 ℎ Mpc−1 (bottom panel). The +dashed curves are for the Sheth-Tormen mass function of the bump cosmologies with 𝜎𝑇 = 0.3; solid for the widths 𝜎𝑇 = 0.1; stars are for the +measurement from N-body simulations with 𝜎𝑇 = 0.3; and squares for 𝜎𝑇 = 0.1. Error bars that are not visible are within the mark size. + +9 +using the criterion that minimizes the quantity +∑︁ +𝑖 +���� +𝑛𝑠𝑖𝑚𝑠(𝑀𝑖) +𝑛𝑚𝑜𝑑𝑒𝑙(𝑀𝑖, 𝑝, 𝑞) − 1 +���� . +(17) +where the sum runs over all the Δ log 𝑀 intervals. To perform +the minimization we used the set of halo counts at redshift +𝑧 = 0.0. +The best fit values are detailed in Table II. The +results seem to be compatible with Sheth-Tormen since the +ΛCDM best fit parameters departs as much (to be fair, maybe +a little less) from 𝑞 = 0.707 and 𝑝 = 0.3 as the parameters for +the bump cosmologies, which nonetheless are small deviation. +This analysis shows the universality of the Sheth-Tormen HMF, +or more precisely of the scaled differential mass function, or +multiplicity 𝑓 (𝜎), in these scenarios. Also for this reason, +when we use the HMcode we do not change the Sheth-Tormen +parameters. +Even though, we do expect the HMF to be sensitive to the +nature of dark energy for bump cosmologies since the de- +pendence of the variance 𝜎 on the mass is different to that +of ΛCDM. In particular, with distinctive signatures in halos +with intermediate masses for the reasons we discuss below. In +Fig. 3 we show the HMFs measured from the simulated data, +as well as the ratio between them in the bump and ΛCDM cos- +mologies, both at 𝑘𝑇 = 0.5 (top panel) and 𝑘𝑇 = 1.0 ℎ Mpc−1 +(bottom panel). We do this for different redshifts, from left to +right, these are 𝑧 = 2.0, 1.0, 0.5, 0.0. It is evident and expected +that the largest deviations from the standard model occurs in +the medbump cosmologies (𝜎𝑇 = 0.3) since these provide the +largest enhancement of the linear power spectrum at the bump +location. +The upper panel of Fig. 3 shows the HMF for the tran- +sition mode 𝑘𝑇 += 0.5 ℎ Mpc−1. +The largest deviation +from ΛCDM occurs for halos with intermediate mass 𝑀 ∼ +[1013.4, 1013.9] ℎ−1𝑀⊙, reaching a difference of 38% (10%) +at 𝑧 = 2.0 (𝑧 = 0.0) for medbump-k0p5 cosmologies. At the +lowest redshifts snapshots (𝑧 = 0.5, 0.0), there are more small +halos in the ΛCDM model than in the bump cosmologies. +On the other hand, for the thinbump-k0p5 cosmology we see +that for very massive halos, the HMF is the same as that in +ΛCDM at late times. We notice from the panels in Fig. 3 +showing the absolute HMF 𝑑𝑛/𝑑 log 𝑀, instead of its ratio to +the ΛCDM one, that the number of halos per mass interval in- +creases with redshift for both ΛCDM and bump cosmological +models, which means that massive structures are been formed +from these initial density fields. Nevertheless, the differences +between the HMFs diminish with time, as expected because +the nonlinear evolution tends to erase the bumps, as explained +in detail in the previous section. +For the bumps located at 𝑘𝑇 = 1.0 ℎ Mpc−1, qualitatively +similar results are shown in the bottom panel of Fig. 3, +but now the differences with ΛCDM are located at smaller +halo masses 𝑀 ∼ [1012, 1013] ℎ−1𝑀⊙. +The HMF of the +medbump-k1 cosmologies (thinbump-k1) reaches a differ- +ence with respect of ΛCDM of ∼ 9% (∼ 3%) for halos of +mass 12.5 ℎ−1𝑀⊙, which decreases at late times for the same +reasons explained above in the case of the transition scale +𝑘𝑇 = 0.5 ℎ Mpc−1. These differences between medbump-k1 +cosmologies and ΛCDM fall below the 1 per cent for massive +halos with mass 𝑀 > 1014 ℎ−1𝑀⊙. +The above mentioned departures from ΛCDM are purely a +consequence of the RDED in the bump cosmologies. These are +the consequence in the increase of the halo formation relative +to a cosmology with no bump, because the HMF is related +to the standard deviation in the density field when smoothed +over the Lagrangian radius 𝑅(𝑀). When the enhancement of +the matter fluctuations appears a higher 𝑘, it affects smaller +scales and then smaller massive halos are expected to be more +abundant than in ΛCDM. On the other hand we have seen +that as higher is the transition mode 𝑘𝑇 , the bump is more +susceptible to be erased by the nonlinear evolution, and hence +the above picture cannot continue indefinitely and on the lower +mass tail of the HMF range we always expect to obtain back +the ΛCDM results. That is, we expect to be able to test these +kinds of cosmologies with intermediate massive halos, being +these the most useful to constraint dark energy models. This +can be put in contrast to alternative dark matter models, where +the largest differences from CDM are expected to show up for +the lowest massive halos, as e.g. in warm or fuzzy dark matter +models in which the abundance of very low massive halos is +suppressed, if they can be formed at all [65–67]. +C. +Halo power spectrum and large scale bias +Now, we show results for the clustering of dark matter halos +in Fourier space. For this analysis we first focus in the mass in- +terval 1012.3 ℎ−1𝑀⊙ < 𝑀 < 1013.0 ℎ−1𝑀⊙. In Figs. 4 and 5 we +show the halo power spectrum for bump cosmologies located +at 𝑘𝑇 = 0.5 ℎ Mpc−1 and at 𝑘𝑇 = 1 ℎ Mpc−1, respectively, with +𝜎𝑇 = 0.3 (top panels) and 𝜎𝑇 = 0.1 (bottom panels). On each +panel, we show the response function given by the ratio of +the halo power spectra in bump and ΛCDM cosmologies; see +Eq. (14). We notice a weaker clustering at large scales in the +bump cosmologies compared to ΛCDM. However, for small +scales, for modes that entered the horizon before the transition, +the situation changes drastically and the clustering is enhanced +in the bump cosmology. The latter effect is expected because +small scales tend to grow larger since they lie on top of modes +around 𝑘𝑇 that were affected by the phase transition of the +RDED. However, the former, the fact that at large scales one +obtains a weaker clustering is less intuitive, and as we see be- +low, is due to a different mass function and hence a different +large scale bias. +Indeed, as a consequence of the biasing being different be- +tween the different models, the response is not equal to unity +at large scales. To understand this, in Fig. 6 (solid lines) we +show the theoretical dark matter halo bias as function of halo +mass 𝑏1(𝑀), as obtained from the PBS formalism and the +Sheth-Tormen HMF using Eq. 10. We do this for the bumps +located at 𝑘𝑇 = 0.5 ℎ Mpc−1 (top panel) and 𝑘𝑇 = 1.0 ℎ Mpc−1 +(bottom panel), and we plot their ratios to the ΛCDM biases. +This analysis shows that the large scale offsets (departing from +unity) observed in Figs. 4 and 5 are due to different HMFs, and +that the computed values with the PBS recipe match very ac- +curate the simulations. We further notice this is a consequence + +10 +10 +1 +100 +0.8 +0.85 +0.9 +0.95 +1.0 +1.05 +1.1 +Phh/P CDM +hh +b1 = 2.74 +z = 2 +Linear +N-body +10 +1 +100 +b1 = 1.65 +z = 1 +10 +1 +100 +b1 = 1.235 +z = 0.5 +10 +1 +100 +b1 = 0.935 +z = 0 +10 +1 +100 +k [h Mpc +1] +0.85 +0.9 +0.95 +1.0 +1.05 +1.1 +Phh/P CDM +hh +b1 = 2.85 +10 +1 +100 +k [h Mpc +1] +b1 = 1.705 +10 +1 +100 +k [h Mpc +1] +b1 = 1.27 +10 +1 +100 +k [h Mpc +1] +b1 = 0.958 +FIG. 4. Halo power spectrum for the bump cosmologies at 𝑘𝑇 = 0.5 ℎ Mpc−1 for 𝜎𝑇 = 0.3 (top panel) and 𝜎𝑇 = 0.1 (bottom panel). We +consider halos in the mass interval [1012.3, 1013] ℎ−1𝑀⊙. From left to right, cyan curves are for redshift 𝑧 = 2; green for 𝑧 = 1; red for 𝑧 = 0.5; +and blue for 𝑧 = 0. The dot-dashed curve corresponds to the linear theory; and squares are for the measurement from N-body simulations. +Error bars that are not visible are within the mark size. +mainly to the dependence of the variance of fluctuations with +the mass, 𝜎(𝑀), and not to the difference in the obtained 𝑝 +and 𝑞 parameters for each models, which is actually small, +certainly not enough to explain this large discrepancy on the +biases. +Together with the theoretical results for 𝑏(𝑀), in Fig. 6 +we show the bias estimated by taking the ratio between the +simulated nonlinear and linear power spectrum at the smallest +wave-numbers 𝑘, +𝑏(𝑀) = +√︄ +𝑃FastPM (𝑘) +𝑃Linear(𝑘) +����� +𝑘 → 0 +. +(18) +We do this for two halo mass intervals: the one already used +above 1012.3 ℎ−1𝑀⊙ < 𝑀 < 1013.0 ℎ−1𝑀⊙ to show the power +spectrum plots, and in addition we choose 1013 ℎ−1𝑀⊙ < 𝑀 < +1013.6 ℎ−1𝑀⊙ that we use to compare to the analytical out- +comes. These simulated results are displayed with square and +star markers in Fig. 6, where the error bars denote the stan- +dard deviations of the 5 different realizations for each model. +On the other hand, the filled circle markers in the plots show +the theoretical results when averaged over the corresponding +mass intervals, obtained using Eq. 11. As it is clear from the +plots, the match between simulations and theory is very good. +As expected, the largest differences between the biases are lo- +cated at smaller masses for the 𝑘𝑇 = 1.0 ℎ Mpc−1 case (bottom +panel) than for the 𝑘𝑇 = 0.5 ℎ Mpc−1 case (top panel). Further, +the wider bumps, those with 𝜎𝑇 = 0.3 (red dot-dashed lines), +show larger deviations than the bumps with 𝜎𝑇 = 0.1 (blue +lines), which is expected because in the former case there is +stronger gravitational interactions and consequently a greater +clustering leading to a lower bias. +VI. +CONCLUSIONS +Bump cosmologies are inspired by models beyond ΛCDM +that have dark sector energy densities that suffer phase transi- +tions, leaving distinctive features in abundance and the clus- +tering data due to a Rapid Diluted Energy Density (RDED). + +11 +10 +1 +100 +0.85 +0.9 +0.95 +1.0 +1.05 +1.1 +Phh/P CDM +hh +b1 = 2.835 +z = 2 +Linear +N-body +10 +1 +100 +b1 = 1.695 +z = 1 +10 +1 +100 +b1 = 1.26 +z = 0.5 +10 +1 +100 +b1 = 0.95 +z = 0 +10 +1 +100 +k [h Mpc +1] +0.90 +0.95 +1.00 +1.05 +1.10 +1.15 +Phh/P CDM +hh +b1 = 2.885 +10 +1 +100 +k [h Mpc +1] +b1 = 1.723 +10 +1 +100 +k [h Mpc +1] +b1 = 1.28 +10 +1 +100 +k [h Mpc +1] +b1s = 0.964 +FIG. 5. Same as Fig. 4 but for the bump cosmologies with 𝑘𝑇 = 1 ℎ Mpc−1. +In such scenarios, the power spectrum is enhanced at scales +where otherwise the power would be smooth, originating a +primordial bump at nonlinear scales relative to a model with +no phase transition. This can be understood as adding an extra +relativistic energy density 𝜌𝑒𝑥 during the radiation dominated +epoch increases the growth rate of the linear matter fluctua- +tions impacting modes 𝑘 ≥ 𝑘𝑇 entering the horizon before the +phase transition occurs, i.e. for 𝑎 < 𝑎𝑇 . +In this work, we have studied halo abundance and cluster- +ing in bump cosmologies. Instead of considering any specific +BDE model, we have used a parametric family of bumps, al- +lowing us to explore a wider range of theoretical models. We +have run FastPM N-body simulations [21], which are com- +plemented by nonlinear halo model approximations from the +HMcode model [22]. We noticed that the nonlinear effects +shift the peaks and originates a second bump at smaller scales +because the primordial, original linear bumps serve as re- +gions where average densities are higher than the background, +the gravitational collapse becomes more rapid and efficient, +and peaks can cross the critical density threshold for collapse +more often. We have focused on the abundance and clustering +statistics and mainly on the responses as given by the ratio of +summary statistics in a bump cosmology to a ΛCDM cosmol- +ogy with no bump. This analysis is useful since ΛCDM is well +known, and then good models for the response translate into +good models for the statistics themselves. We have studied the +nonlinearities in the matter and halo power spectrum and how +these fingerprints are translated to large-scale halo bias. We +also have studied how the number of halos are affected by the +phenomenology of the bump cosmology. +We have analysed and compared to ΛCDM, four bump +parametrized cosmologies. These are the combinations of two +locations 𝑘𝑇 = 0.5 and 1 ℎ Mpc−1 and two widths 𝜎𝑇 = 0.3 +and 0.1. The location of the bump corresponds to the mode +reentering the horizon at the phase transition redshift, while +the width to the duration of the phase transition. +We have confirmed that the power spectrum in the RDED +cosmologies is affected by two important nonlinear effects. +First, the bumps are erased because the large scale random bulk +matter motions tend to populate underdense regions, while +moving out of regions with less matter than the average. This +effect has a similar origin than the BAO damping and is more +evident for bumps located at higher 𝑘 modes because these +translate in oscillations in configuration space with smaller +distances between crests and troughs. The second effect is the +appearance of a second bump due to the fact that the primordial + +12 +12 +13 +14 +15 +0.93 +0.94 +0.95 +0.96 +0.97 +0.98 +0.99 +1.0 +b(M)/b(M) CDM +z = 2 +12 +13 +14 +15 +z = 1 +12 +13 +14 +15 +z = 0.5 +12 +13 +14 +15 +z = 0 +12 +13 +14 +15 +logM [h +1M ] +0.96 +0.97 +0.98 +0.99 +1.0 +b(M)/b(M) CDM +12 +13 +14 +15 +logM [h +1M ] +12 +13 +14 +15 +logM [h +1M ] +12 +13 +14 +15 +logM [h +1M ] +FIG. 6. Comparison between the halo bias from N-body simulations (star and square markers) and from theoretical prediction using PBS +approach (solid and dot-dashed curves) for bump cosmologies at 𝑘𝑇 = 0.5 ℎ Mpc−1 (top panel) and at 𝑘𝑇 = 1.0 ℎ Mpc−1 (bottom panel). Stars +are for the bump cosmologies with 𝜎𝑇 = 0.3; and squares for 𝜎𝑇 = 0.1. Dashed curves are for 𝜎𝑇 = 0.3; and solid are for 𝜎𝑇 = 0.1. Circles +are for the effective bias defined in Eq. (11) in the mean ranges of [1012.3 ℎ−1𝑀⊙, 1013 ℎ−1𝑀⊙] and [1013 ℎ−1𝑀⊙, 1013.6 ℎ−1𝑀⊙]. +one serves as a location with high density and where even +smaller structures are more prone to form. +We have seen that the presence of localized bumps in the +matter power spectrum have consequences on the halo statis- +tics at all scales. The halo power spectrum suffers an offset +with respect to the ΛCDM one because the large scale bias is +sensitive to the variance of density fluctuations 𝜎(𝑀), which +is affected by the bump. The differences with the ΛCDM can +be considerable even for small bumps located well inside the +nonlinear region. +We have computed the Halo Mass Function (HMF) for our +bump cosmologies using the Sheth-Tormen recipe. +How- +ever, while the Sheth-Tormen HMF has fixed parameters, +we fit them to the simulations anticipating the possibility +of more optimal parameters for bump cosmologies. +How- +ever, we found their values to be very close to the standard +𝑝 = 0.3 and 𝑞 = 0.707. +Despite this, the HMFs do dif- +fer considerably for the bump and ΛCDM cosmologies, but +this is because of different matter power spectra and then dif- +ferent variances 𝜎(𝑀). We compared our analytical results +to the outcomes of the N-body simulations finding excellent +agreement. +Further, we were capable of correctly predict +the large scale halo bias by applying the Peak-Background- +Split formalism to our Sheth-Tormen HMF description, match- +ing the simulated power spectrum data at low 𝑘 for two +ranges of masses: 1012.3 ℎ−1𝑀⊙ < 𝑀 < 1013.0 ℎ−1𝑀⊙ and +1013 ℎ−1𝑀⊙ < 𝑀 < 1013.6 ℎ−1𝑀⊙. +In general, for the bump cosmologies and within the ade- +quate mass ranges, that is, within the ranges where the forma- +tion of halos of certain masses are more enhanced, we found +smaller biases than in ΛCDM. This is because in those regions +the presence of the primordial bump yields more clustering +and then stronger gravity effects, which translate in a more +efficient relaxation of the bias, which tends toward unity more +rapidly than in a cosmology without the bump. +In summary, this distinctive fingerprint, named as a bump, +has been studied modelling the statistics of biased tracer of +the density, such as halos, which have the potential to be +detectable by current and future galaxy surveys [68–71] al- +lowing to put tight constraints on cosmological constraints. +Future interesting routes to continue the analysis of this work +include the investigation of the halo-galaxy connection and the +clustering in mock galaxy catalogs that can shed light on the +expected observables for galaxies surveys such as the Dark +Energy Spectroscopic Instrument (DESI) [72]. Also, since +weak lensing depends critically on the real-space matter power +spectrum, whose nonlinear effects are easy to understood and +well modeled by the halo-based HMcode, we forecast that the + +13 +consequences of bump cosmologies on statistics measured by +photometric surveys (e.g. the 3 × 2 point correlation func- +tions) are relatively straightforward to obtain, with a view on +the upcoming Rubin Observatory Legacy of Space and Time +(LSST) Survey [73]. +ACKNOWDLEDGMENTS +DGN thanks support from a CONACyT PhD fellowship. +AM and DGN acknowledges support from PAPIIT- DGPA +(UNAM) IN101415. A. A. is supported by Ciencia de Frontera +grant No. 319359, and also acknowledges partial support to +grants Ciencia de Frontera 102958 and CONACyT 283151. +Appendix A: Halos per mass interval +In Fig. 7, we show histograms for the counts of halos as a +function of their mass within the interval 𝑀 = 1012.3 ℎ−1𝑀⊙ to +𝑀 = 1015 ℎ−1𝑀⊙. 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Zu, The DESI Ex- +periment Part I: Science,Targeting, and Survey Design, arXiv +e-prints , arXiv:1611.00036 (2016), arXiv:1611.00036 [astro- +ph.IM]. +[73] LSST Dark Energy Science Collaboration, Large Synoptic Sur- +vey Telescope: +Dark Energy Science Collaboration, arXiv +e-prints , arXiv:1211.0310 (2012), arXiv:1211.0310 [astro- +ph.CO]. + diff --git a/J9E3T4oBgHgl3EQfAAmq/content/tmp_files/load_file.txt b/J9E3T4oBgHgl3EQfAAmq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..649e874edc0a16f0f9e170cfc169ba59c6e34adc --- /dev/null +++ b/J9E3T4oBgHgl3EQfAAmq/content/tmp_files/load_file.txt @@ -0,0 +1,2158 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf,len=2157 +page_content='Impact of a Rapid Diluted Energy Density on the halo mass function Dante V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Gomez-Navarro,1, ∗ Alejandro Aviles,2, 3, † and Axel de la Macorra1, ‡ 1Instituto de Física, Universidad Nacional Autónoma de México, Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' de México C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 04510, México.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 2Departamento de Física, Instituto Nacional de Investigaciones Nucleares, Apartado Postal 18-1027, Col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Escandón, Ciudad de México,11801, México.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 3Consejo Nacional de Ciencia y Tecnología, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Insurgentes Sur 1582, Colonia Crédito Constructor, Del.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Benito Juárez, 03940, Ciudad de México, México.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We study dark energy cosmological models, extensions of the standard model of particles, characterised by having an extra relativistic energy density at very early times, and that rapidly dilute after a phase transition occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' These models generate well localized features (or bumps) in the matter power spectrum for modes crossing the horizon around and before the phase transition epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This is because the presence of the additional energy component enhances the growth of matter fluctuations during the radiation dominated epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Instead of considering a particular model, we focus on a parametric family of Gaussian bumps in the matter power spectrum, which otherwise would be a ΛCDM one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We study the evolution of such bump cosmologies and their effects in the halo mass function and halo power spectrum using N-body simulations, the halo-model based HMcode method, and the peak background split framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The bumps are subject to different nonlinear effects that become physically well understood, and from them we are able to predict that the most distinctive features will show up for intermediate halo masses 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 ℎ−1𝑀⊙ < 𝑀 < 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='6 ℎ−1𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Out of this range, we expect halos are not significantly affected regardless of the location of the primordial bump in the matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Our analytical results are accurate and in very satisfactory agreement with the simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' INTRODUCTION Recent cosmological and astrophysical observations have consolidated our picture of the concordance ΛCDM model [1– 5], which corresponds to a nearly homogeneous and isotropic expanding universe filled with the particles of the Standard Model (SM) [6], and supplemented by dark matter and a cos- mological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Despite this success, the two dark com- ponents have yet to be thoroughly tested and understood, since their fundamental nature is still a puzzle [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' At present time, they accounts for about 96% of the energy budget of the cos- mos, and so alternative models look for plausible explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In particular, scalar fields have been proposed to describe dark energy [8–10], whose nature could be that of a fundamental particle not contained in the SM (Higgs-type particles) or can be a composite one, as for example a dark meson pion 𝜋-like particle [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In addition to the fact that we do not know the nature of the dark sector, some strains in the ΛCDM model began to appear as the accuracy of cosmological observations improved, and recently, some interesting tensions have emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Perhaps the most famous is the discrepancy between early times and local measurements of the Hubble constant [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The increasing statistical tension on its value obtained using different observa- tions has revived interest in alternative cosmological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Henceforth, extensions to the standard model of particles have been proposed to alleviate the 𝐻0 crisis or even simply to de- scribe the origin of dark energy, for example by introducing additional particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' As an example of the interest of this work, the Bound Dark Energy model (BDE) cosmological model [11, 15] is charac- ∗ dantegomezn@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='com † avilescervantes@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='com ‡ macorra@fisica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='unam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='mx terised by a supersymmetric Dark Gauge Group (DG), in which the fundamental particles are massless during early times and their energy density evolved as radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' However, at low en- ergies the postulated gauge interaction becomes strong enough to bind the elementary dark particles together and form massive bound states, dark mesons and dark baryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This process is similar to the strong QCD interaction in the SM where quarks are bound together to form baryons and mesons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' protons, neutrons or pions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In the BDE scenario, the dark energy corresponds to the lightest meson scalar particle 𝜙 formed at a phase transition scale Λ𝑇 , at a scale factor 𝑎𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Before the transition, the energy density of the DG particles behaves as radiation decaying with the expansion of the Universe as 1/𝑎4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Just after the phase transition occurs, for a scale factor 𝑎 > 𝑎𝑇 and lasting a long period of time, its energy density decays very fast, as 1/𝑎6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' During this epoch, there is an abrupt decrease in the DG cosmic abundance and rapidly becomes subdominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We refer to such a behavior very generically as Rapid Diluted Energy Density (RDED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The existence of these type of bound particles modifies the evolution of the Hubble parameter 𝐻 and have also an important impact in the evolu- tion of density perturbations, leaving distinctive signatures on cosmological distances, and in the matter power spectrum and other summary statistics around the corresponding transition scale 𝑘𝑇 = 𝐻(𝑎𝑇 )𝑎𝑇 [11, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In particular, adding extra relativistic particles increases the growth rate of matter density fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In this work we are motivated by the impact of a RDED in the matter power spectrum, which becomes enhanced around 𝑘𝑇 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Since there are several theoretical frameworks that can lead to similar mechanisms, we will work in a model- independent way with the introduction of a family of parame- terized bumps into the linear matter power spectra 𝑃ΛCDMex,1 1 The suffix “ex” in the notational label “ΛCDMex” makes allusion to the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='04254v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='CO] 11 Jan 2023 2 which otherwise would be a ΛCDM one, where we will vary their positions and widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' By using these linear spectra as input, we will work beyond the linear regime using different complementary schemes, and focus on the consequences that the nonlinear evolution of the injected bumps has on halo clus- tering and halo abundance using the Peak-Background Split (PBS) framework [17, 18] and the Sheth-Tormen Halo Mass Function (HMF) [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Although full N-body simulations successfully describe the nonlinearities of the matter clustering, they have the disad- vantage of being computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Hence, in this work we use the approximated particle mesh N-body solver FastPM2 [21], where the linear growth of displacements, the Zeldovich approximation solution, is enforced at 𝑘 → 0 by choosing an appropriate set of kick and drift factors, and hence very large scale are treated exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Further, we use the HM- code3 [22–24] to describe the nonlinear dark matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This is a halo-model based method, and as such, it describes the nonlinear power spectrum as a sum of two pieces, the 2-halo term that models the correlation between particles hosted by different halos reducing to the linear the- ory at large scales, and the 1-halo piece to model the small, intra-halo clustering scales [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Halo clustering is crucial in the study of the large-scale structure of the Universe, since it is governed by gravitational instability, responsible for the formation of dark matter halos and their distribution [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Furthermore, well physically moti- vated models often assume that galaxy formation is the result of the condensation of baryonic matter in already collapsed and virialized dark matter halos [19, 27, 29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Therefore, the HMF is an important tool for studying the formation and evolution of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The analytical understanding of these processes is also desirable, both to obtain a physical intuition, and for the study of different models and their wide range of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In this work, we use the Sheth-Tormen HMF to describe the evolution of the number density of dark matter halos of a given mass and for our alternative cosmological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' However, since we are working beyond ΛCDM, we let free their parameters and fit them to data extracted from our N-body FastPM simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' On the other hand, galaxy surveys show that at large scales, the number density fluctua- tion is roughly proportional to the matter density overdensity field, with a multiplicative factor called the bias 𝑏 [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We study this large-scale bias using PBS, which in addition to its utility, allows a physical interpretation of the halo bias [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Summarizing, we are interested in the distribution of non- linear virialized objects because it allows us to check the evolu- tion and final fate of the small primordial density fluctuations that have undergone gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Consequently, it is necessary to review the properties of halo statistics in models that have undergone a phase transition in early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This the main topic of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' extra energy density component not present in the ΛCDM model at early times, and referred below as 𝜌ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 2 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='com/fastpm/fastpm 3 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='com/alexander-mead/HMcode The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' II, we briefly present how cosmic phase transitions lead to different cosmological signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' III, we introduce the bump in the power spectrum 𝑃ΛCDMex and parameterize its width and position, in this section we also show the specifications of our N-body simulations suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' IV we review the HMF formalism and the large-scale bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We present our results and details for our bump cosmologies in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' V, with added supplementary material in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' VI we present our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' RAPIDLY DILUTION ENERGY DENSITY (RDED) Models beyond ΛCDM may leave different distinctive fea- tures in the matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' As mentioned in the Introduction, we are interested in models that have an extra energy density 𝜌𝑒𝑥, beyond the standard ΛCDM and before a transition scale, which dilutes rapidly after the transition takes place at the scale factor 𝑎𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The dark matter mode that is crossing the horizon at that time is 𝑘𝑇 = 𝑎𝑇 𝐻(𝑎𝑇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Some of these models are generally referred as Early Dark Energy (EDE) [35–37], where the basic idea is to postulate an ex- tra component that contributes non negligible to the energy density before recombination, and then it decays faster than radiation at later times, at a transition scale factor 𝑎𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The presence of this extra fluid at early times enhances the total energy density before the last scattering surface leading to a higher expansion rate 𝐻0, with the potential to resolve the Hub- ble tension [36–40], and as well leaving other cosmological fingerprints [41, 42] In order to describe the effects of a RDED in the structure formation we appeal to the mechanism of the BDE model [11, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' At very early times, the light particles of the DG are ultrarelativistic and the extra energy density 𝜌𝑒𝑥 evolves with an effective equation state parameter 𝑤 = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Long after, but still well inside the radiation dominated epoch, at 𝑎𝑇 ∼ 10−6, a phase transition occurs because the coupling of the DG particles becomes strong and binds the elementary particles of DG, forming composite particles, dark mesons, that can be described as a scalar field with an inverse power law potential [43, 44] 𝑉(𝜙) = Λ4+𝑛 𝑇 𝜙−𝑛, (1) with 𝑛 = 2/3 and with a condensation energy scale Λ𝑇 ∼ 40 eV in BDE model [11, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' At this phase transition epoch, the effective equation of state abruptly changes from 𝑤 = 1/3 to 𝑤 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' During this epoch, the dark mesons energy density rapidly dilutes as 𝜌 ∝ 𝑎−6 and lasting for a long period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' However, at 𝑎 ∼ 1/1000, the equation of state swiftly goes from 𝑤 = 1 to 𝑤 = −1 to finally ending up today at 𝑤 ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='93, because slow-rolling starts to fail around 𝑧 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Notice that despite the evident similarities between BDE and EDE in its best-known form (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' [37]), these are different in nature and, moreover, the former may have a greater impact on the early universe due to that its energy density decays as that of a radiation component, instead of being constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' However, in a broader sense BDE can be considered also an 3 Name 𝑁𝑠𝑖𝑚𝑠 𝐴𝑇 𝜎𝑇 𝑘𝑇 [ℎMpc−1] 𝐿𝑏𝑜𝑥 [ℎ−1Mpc] medbump-k1 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 1024 thinbump-k1 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 1024 medbump-k0p5 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 1024 thinbump-k0p5 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 1024 ΛCDM 5 − − − 1024 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Specifications of our N-body simulation suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The background cosmological parameters are the same for all the simulations: Ω𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3, Ω𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='05, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='7, Ω𝜈 = 0, ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='7, 𝑛𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='96, 𝜎8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Each simulation uses 10243 particles distributed over 𝑁𝑔𝑟𝑖𝑑 = 10243 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We consider the redshifts 𝑧 = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' EDE scenario, because it constitutes a non-negligible dark energy component in early epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Now, during radiation domination, matter perturbations grow only logarithmically with the growth rate 𝑓 = 𝑑 ln 𝛿𝑚/𝑑 ln 𝑎 ∝ 1/𝛿𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' On the other hand, matter pertur- bations in the model containing 𝜌𝑒𝑥 are initially suppressed compared to a ΛCDM model since the initial amplitude de- pends on the fraction of relativistic particles [16], and because 𝑓ΛCDMex > 𝑓ΛCDM, the ratio 𝛿𝑚,ΛCDMex/𝛿𝑚,ΛCDM increases and is further boosted by the extra relativistic component 𝜌𝑒𝑥 before the phase transition occurs at 𝑎𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Notice that this boost affects only the modes crossing the horizon before 𝑎𝑇 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 𝑘 ≥ 𝑘𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This is the characteristic signature on the matter fluc- tuations because of the presence of an early times RDED and was presented in [11], and further developed in [15, 16, 45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' MODELLING THE POWER SPECTRUM In the following, we will characterize the effect of having a RDED by choosing a parametrization that we refer throughout as the bump cosmology, where the linear power spectrum is a modification to that of a standard ΛCDM cosmology one given by 𝑃bump(𝑘, 𝑧) = � 1 + 𝐹(𝑘) � 𝑃ΛCDM(𝑘, 𝑧), (2) with the 𝐹(𝑘) parametric function describing the bump, 𝐹(𝑘) = 𝐴𝑇 exp � − [ln(𝑘/𝑘T)]2 𝜎2 𝑇 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (3) The parameters 𝐴𝑇 , 𝑘𝑇 and 𝜎𝑇 are the amplitude, scale, and width of the bump, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The width of the bump corre- sponds to how fast the rapid diluted energy density phase takes place, whereas 𝑘𝑇 represents the mode entering the horizon about the phase transition time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We consider four different bump cosmologies, each with fixed amplitude 𝐴𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='15, as motivated by the BDE mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='4 We choose two different widths of the bump 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3, 4 In the original BDE model, the extra energy density has an abundance ΩBDE(𝑎𝑇 ) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='11 before the phase transition that occurs at 𝑎𝑇 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='48 × 10−6, and hence the mode entering the horizon at that time is 𝑘𝑇 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='92 ℎ Mpc−1 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1, and locate the bump at two different scales: 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 ℎ Mpc−1 (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We consider these bump cos- mologies at different redshifts: 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' To generate the ΛCDM power spectrum we use the cosmo- logical parameters reported in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' III A, or in the caption of Table I, which are the same for all the bump and standard cos- mologies, and as such, the only difference between the models is the presence of the bump parametrized by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In the following two subsections we briefly describe the methods we use to study the nonlinear evolution of the bump in the power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' N-body simulations We generate 25 rapid N-body simulations using the code FastPM, 5 for each of the cosmologies detailed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Zeldovich initial conditions were generated at 𝑧𝑖 = 99, and we use 100 linearly space steps up to redshift 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Each simula- tion uses 10243 particles to approximate the density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The box sizes of the simulations are 𝐿𝑏𝑜𝑥 = 1024 ℎ−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We analyze four snapshots at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Our baseline cosmology is ΛCDM with dark matter density Ω𝑐𝑑𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='25, baryon density Ω𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='05, fluctuation variance 𝜎8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='8, dimensionless Hubble constant ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='7, and spectral index 𝑛𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Neutrinos are considered massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We identify and construct halo catalogs with the Friends- of-Friends algorithm [47], already implemented in the N- BodyKit package5 [48], for which we use a linking length 𝑙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='2 and where each halo is formed by at least 20 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' HMcode As a complementary tool to N-body simulations, we make use of the HMcode [22, 49], which is an augmented version of the standard halo-model scheme for the nonlinear matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The starting point is a standard halo-model calculation, where the power spectrum is split into two terms: one that accounts for the clustering arising within individual 5 http://nbodykit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='io 4 halos, and the second that accounts for the clustering of dark matter between two different halos and that follows closely the linear theory at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We use the most recent HMcode version [24], which further accounts for the BAO damping into the two-halo term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In this new scheme, HMcode adds a smoothing parameter for the transition region between the 1- and 2-halo terms when constructing full halo-model power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In the nonlinear regime at low redshifts, we expect HMcode to match adequately the N-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' MODELLING THE HALO ABUNDANCE AND CLUSTERING Quantitative comparisons between theoretical predictions and observations allow us to compute constraints on cosmo- logical parameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' the abundance of halo identified as clusters of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The bias of halo dark matter contains com- plementary information on their abundance since data survey is understood through the bias of the halos which they form [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In this section, we describe the analytical techniques to study the halo abundance and halo bias using the Sheth- Tormen mass function and PBS prescriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The halo mass function The HMF gives the number density of dark matter halos as a function of their masses (per unit comoving volume).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' How- ever,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' occasionally a good start point to the HMF discussion is by introducing the scaled differential mass function 𝑓 (𝜎) as the fraction of the total mass ¯𝜌𝑉 (when the volume 𝑉 is very large) hosted by halos in a logarithmic interval of 𝜎−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 𝑓 (𝜎) = 𝑑(𝜌/ ¯𝜌) 𝑑 ln 𝜎−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (4) with 𝜎(𝑀) the variance of linear fluctuations when smoothed over a scale 𝑅(𝑀) = (3𝑀/4𝜋 ¯𝜌)1/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 𝜎2(𝑀,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 𝑧) = 𝐷2(𝑧) 2𝜋2 ∫ ∞ 0 𝑑𝑘 𝑘2𝑃𝐿(𝑘)𝑊2(𝑘𝑅),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (5) where 𝑃𝐿 is the linear power spectrum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 𝐷 the linear growth function and 𝑊 the top-hat filter in Fourier space 𝑊(𝑘𝑅) = 3 (𝑘𝑅)3 � sin(𝑘𝑅) − 𝑘𝑅 cos(𝑘𝑅)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (6) The utility of 𝑓 (𝜎) is that apparently this quantity, when prop- erly scaled, is nearly universal throughout a wide range of halo masses, and for cosmologies both within the ΛCDM as well as dark energy models [52] or even in Modified Gravity theories [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='6 In contrast, the HMF is very sensitive to the cosmo- 6 This is more evident by comparing between different models the multiplicity 𝜈 𝑓 (𝜈) against the rescaled variance, or peak significance 𝜈 = 𝛿𝑐/𝜎 introduced below;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 7 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Notice also that 𝛿𝑐 is mass dependent in theories that introduce new additional scales as in warm dark matter or modified gravity and hence comparing against 𝜈 and 𝜎−1 are not equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' logical parameters and the specific theory through the power spectrum dependence of the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The precise connection between 𝑓 (𝜎) and the HMF is given by 𝑑𝑛 𝑑 log 𝑀 = 𝑓 (𝜎) ¯𝜌 𝑀 𝑑 ln 𝜎−1 𝑑 log 𝑀 , (7) where one assumes that all of the matter in the universe is hosted by halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The simplest HMF is the Press-Schechter mass function [54], which analytical form can be obtained ex- actly based on the spherical collapse model and the hypothesis that the mass in collapsed objects is related to the volume with density above a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' However, data from simu- lations show that the Press-Schechter HMF is not accurate at the low and the high mass ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Hence, several other mass functions have been proposed based on different assumptions or as purely empirical fits [55–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The Sheth-Torman HMF [19, 20] is perhaps the best known and most used alternative to Press-Shechter, it is based on the more realistic ellipsoidal col- lapse and reproduce N-body simulations better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' It is defined through 𝑓𝑆𝑇 (𝜈) = 𝐴(𝑝) √︂ 2𝑞 𝜋 � 1 + � 1 𝑞𝜈2 � 𝑝� exp � −𝑞𝜈2 2 � , (8) with 𝐴(𝑝) = [1 + 𝜋−1/22−𝑝Γ(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 − 𝑝)]−1 a normalization factor coming from the assumption that all the dark matter in the Universe is contained within halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The variable 𝜈 = 𝛿𝑐/𝜎 is the peak significance, and 𝛿𝑐 is the critical overdensity for collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Notice that we are abusing notation because 𝑓 (𝜎) and 𝑓 (𝜈) represent the same function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' For a matter-dominated universe, 𝛿𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='686, while for Ω𝑚 < 1 the numerical value varies slightly with the redshift but with no significant impact on the cosmological outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 8, one recovers the Press-Schecter mass function by choosing 𝑞 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 and 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The standard values 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 and 𝑞 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='707 for the Sheth- Tormen HMF are obtained by fitting to ΛCDM simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' For the bump cosmologies, in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' V B we will rely on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (8) for computing the HMF, but we will fit the 𝑝 and 𝑞 parameters to our N-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Biasing on large scales To compute the halo bias we use the PBS formalism [17– 20] with the aid of the Sheth-Tormen mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In this picture, long- and small-wavelength density fluctuations are split, with the crests of long wavelength overdensities serv- ing as locations where the average density is higher than the background cosmological density, and on top of that small- wavelengths fluctuations, the peaks, collapse to form halos on which ultimately galaxy formation takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The main idea to obtain the halo bias is that more massive halos are formed in locations where the average (over large regions) local density is high, or in other words the PBS biases are the responses of the mean abundance of tracers against small changes in the background density [20, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' That is, under the PBS formal- ism the Lagrangian (𝐿) biases can be written in terms of the 5 multiplicity function 𝜈 𝑓 (𝜈) through 𝑏𝐿 𝑛 = (−1)𝑛 𝜎𝑛(𝑀, 𝑧) 1 𝜈 𝑓 (𝜈) 𝑑𝑛𝜈 𝑓 (𝜈) 𝑑𝜈𝑛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (9) It turned out that the bias computed in this way are the physical, renormalized local biases, up to subdominant factors intro- duced by the artificial smoothing scale of density fluctuations [34, 59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' For the Sheth-Tormen HMF given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (8), the linear Lagrangian bias is 𝑏𝐿 1 (𝑀) = 1 𝛿𝑐 � 𝑞𝜈2 − 1 + 2𝑝 1 + (𝑞𝜈2) 𝑝 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (10) To obtain the biases over a mass range [𝑀𝑚𝑖𝑛, 𝑀𝑚𝑎𝑥] one has to average over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' That is, the large-scale bias within a mass range becomes [59] 𝑏𝐿 1 = 1 𝐼𝑑𝑀 ∫ 𝑀max 𝑀min 𝑑𝑀 1 𝛿𝑐 𝜈 𝑀 𝜕 𝑓 𝜕𝜈 𝑑 ln 𝜈 𝑑𝑀 , (11) with 𝐼𝑑𝑀 = ∫ 𝑀max 𝑀min 𝑑𝑀 𝑓 𝑀 𝑑 ln 𝜈 𝑑𝑀 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (12) The corresponding linear Eulerian bias is given by 𝑏1 = 1 + 𝑏𝐿 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (13) We will use these expressions when compute the halo biases for bump cosmologies in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' V C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' ANALYSIS AND RESULTS The evolution, dilution and shift of the bump is studied via our suite of FastPM N-body simulations and the HMcode results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We first focus on response functions, computed as the ratio of statistics between a model containing a bump and one without it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' For the power spectrum this is given by 𝑅(𝑘) = 𝑃bump(𝑘) 𝑃ΛCDM(𝑘) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (14) The importance of this analysis is that once a good under- standing and modeling of the response function is acquired for a given alternative cosmology, its power spectrum can be com- puted by multiplying it by an as well modeled power spectrum for the ΛCDM one, which has been studied comprehensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This response function analysis has shown to give fruitful re- sults in different contexts beyond ΛCDM [49, 61–64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Our numerical results are contrasted with the linear the- ory, for which the response in the power spectrum is simply 𝑅𝐿(𝑘) = 1 + 𝐹(𝑘) at all 𝑧 since the linear growth is scale independent, well after the phase transition had occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The details of our simulations are enlisted in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The bumps are located at scales 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 ℎ Mpc−1 and have widths 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Hence, they are inside the full non- linear regime, where a perturbative treatment is not adequate, as it is for the cases 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 ℎ Mpc−1 covered in our previous work [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Hence, here we rely on the HMcode and FastPM to model the nonlinearities and not in perturba- tion theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In [16] we also studied the case with transition scale 𝑘𝑇 = 1 ℎ Mpc−1, although only for the dark matter power spectrum and correlation function, while here we augmented the analysis for the case of biased tracers and put emphasis in the HMF and halo bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' When there is overlap with the above mentioned reference our results agree, despite that in that work we use full N-body simulations (but with a smaller number of particles: 𝑁𝑝 = 2563).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Matter power spectrum To extract the power spectrum data we use the cloud-in- cell (CIC) mass-assignment scheme implemented in the N- BodyKit package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The grid in our simulations is divided into 𝑁𝑔𝑟𝑖𝑑 = 2048 cells and the size of the box in all cases is 𝐿 = 1024 ℎ−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The power spectrum ranges are binned in 80 log-spaced 𝑘-points over the interval [𝑘𝑚𝑖𝑛, 𝑘Ny], where 𝑘𝑚𝑖𝑛 = 2𝜋/𝐿 and 𝑘Ny = 𝑁𝑔𝑟𝑖𝑑𝜋/𝐿 is the Nyquist frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Usually, power spectra in CIC approach are considered to be accurate up to half of the Nyquist frequency, and as such, this is the upper limit we show in our plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 1 and 2 we show the matter power spectra using the bump cosmology located at the scales 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 and 1 ℎ Mpc−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The matter power spectra are computed using our different nonlinear methods and then divided by their counter- parts in the ΛCDM model to show the response function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' These analyses compare how the bump cosmology power spectra are modified by nonlinearities within the differ- ent schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The squares correspond to the FastPM synthetic data and the error bars are the scattering over the 5 simulated boxes for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' As expected, at higher redshifts nonlinear effects are smaller and the responses for all approaches are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The observed features in the plots can be described through two important nonlinear effects 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='- There exists a generation of a second bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This non- linear effect was observed first in [16] and is due to that the primordial bump enhances the amplitude of the long wavelength perturbation where peaks in the density fluctuation locate, which magnify them, and ultimately forming a second bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In other, simpler words, the generation of the second bump is a consequence of the development of structures on top of the first, primordial bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This effect is even more pronounced for wider bumps because these provide a greater enhancement of linear power and then, in the language of halo-based models, a broader range of interaction with the 1-halo term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' For the case of 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 ℎ Mpc−1, as can be ob- served in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 1, this nonlinear second bump is well mod- elled by the HMcode, and reaches a maximum relative to the first bump value, at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' For 𝑘𝑇 = 1 ℎ Mpc−1 the generation of this second bump is still present, although less evident because the Nyquist frequency coincides with the onset of the bump;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 6 10 1 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='2 Pmm/P CDM mm z = 2 Linear HMcode N-body 10 1 100 z = 1 10 1 100 z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 10 1 100 z = 0 10 1 100 k [h Mpc 1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='2 Pmm/P CDM mm 10 1 100 k [h Mpc 1] 10 1 100 k [h Mpc 1] 10 1 100 k [h Mpc 1] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Matter power spectrum response functions for bump cosmologies at 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 ℎ Mpc−1 with widths 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 (top panel) and 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' From left to right, cyan curves are for 𝑧 = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' green for 𝑧 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' red for 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' and blue for 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Dot-dashed curves correspond to linear theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' solid to the HMcode model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' and squares are the data extracted from the FastPM N-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Error bars that are not visible are within the mark size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='- The primordial bumps tend to vanish with the gravi- tational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This effect can be better understood under a configuration space description, where localized features become oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Since bulk displacements of matter tend to partially move out from overdense regions and populate underdense regions, these oscilla- tions are erased with the collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In fact, the character- istic scale for this to happen is given by two times the Lagrangian displacements variance 2𝜎Ψ = 2𝐷(𝑧) �∫ ∞ 0 𝑑𝑘 6𝜋2 𝑃𝐿(𝑘, 𝑧 = 0) �1/2 ∼ 10𝐷(𝑧) ℎ−1Mpc, (15) with 𝐷(𝑧) the linear growth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' As can be seen by comparing Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 1 and 2, as higher is the transition mode 𝑘𝑇 , the more the bump is damped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This is expected be- cause higher-𝑘 locations of the bump in Fourier space correspond to higher frequency of oscillations in config- uration space, and since particles travel in random paths an average distance 𝜎Ψ, then with high probability, parti- cles will tend to displace out from the overdense regions if the oscillations wavelength are smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Of course this is a consequence of having a comoving distance between peaks and troughs in the 2-point configuration space correlation function comparable or smaller than 2 × 𝜎Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' So, it is not expected to happen for bumps lo- cated at low 𝑘 values, as those studied in [16], which oscillations in configuration space have very large wave- lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Notice this effect is completely nonlinear in the matter overdensity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' it is non-perturbative in the Eu- lerian theory, however, it can be completely described by Lagrangian perturbation theory, even at the linear order, Zeldovich approximation because bulk, large scale dis- placement fields are responsible for erasing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Indeed, we notice this effect has the same origin as the smearing of the BAO peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 7 10 1 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='2 Pmm/P CDM mm z = 2 Linear HMcode N-body 10 1 100 z = 1 10 1 100 z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 10 1 100 z = 0 10 1 100 k [h Mpc 1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='2 Pmm/P CDM mm 10 1 100 k [h Mpc 1] 10 1 100 k [h Mpc 1] 10 1 100 k [h Mpc 1] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 1, but for the bump cosmologies with 𝑘𝑇 = 1 ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Name 𝑞 𝑝 medbump-k1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='727 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='314 thinbump-k1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='326 medbump-k0p5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='723 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='343 thinbump-k0p5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='691 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='334 ΛCDM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='711 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='317 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Best-fit for 𝑝 and 𝑞 Sheth-Tormen parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' These results were obtained by using the criterion that minimizes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 17 at redshift 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Halo mass function From our N-body simulations, we have constructed halo catalogs with the Friends-of-Friends algorithm as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' From them, we obtain the HMF for the different bump cosmologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The number of halos per logarithmic mass intervals 𝑑𝑛/𝑑 log 𝑀 in a simulation of volume 𝐿3 is given by 𝑑𝑛 𝑑 log 𝑀 = 𝑀 𝐿3 Δ𝑁 Δ log 𝑀 , (16) for which we construct logarithmic bins in mass with size Δ log 𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Complementary, in Appendix A we show a histogram of our halos within the interval 𝑀 = [1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3, 𝑀 = 1015] ℎ−1𝑀⊙ as well as a Table with the mean number of halos of our catalog suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Since we are working in cosmologies beyond the ΛCDM, and anticipating possible deviations in the scaled differential mass function 𝑓 (𝜎) function, we recalibrate the model param- eters of the Sheth-Tormen functional form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' That is, we compute the best fit of Sheth-Tormen 𝑝 and 𝑞 parameters 8 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 dn/dlogM [h3 Mpc 3] z = 2 z = 1 z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 z = 0 13 14 15 logM [h 1M ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='4 (dn/dlogM)/(dn/dlogM) CDM 13 14 15 logM [h 1M ] 13 14 15 logM [h 1M ] 13 14 15 logM [h 1M ] 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 dn/dlogM [h3 Mpc 3] z = 2 z = 1 z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 z = 0 13 14 15 logM [h 1M ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='15 (dn/dlogM)/(dn/dlogM) CDM 13 14 15 logM [h 1M ] 13 14 15 logM [h 1M ] 13 14 15 logM [h 1M ] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Halo mass function 𝑑𝑛/𝑑 log 𝑀 for the bump cosmologies at 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 ℎ Mpc−1 (top panel) and 𝑘𝑇 = 1 ℎ Mpc−1 (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The dashed curves are for the Sheth-Tormen mass function of the bump cosmologies with 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' solid for the widths 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' stars are for the measurement from N-body simulations with 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' and squares for 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Error bars that are not visible are within the mark size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 9 using the criterion that minimizes the quantity ∑︁ 𝑖 ���� 𝑛𝑠𝑖𝑚𝑠(𝑀𝑖) 𝑛𝑚𝑜𝑑𝑒𝑙(𝑀𝑖, 𝑝, 𝑞) − 1 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (17) where the sum runs over all the Δ log 𝑀 intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' To perform the minimization we used the set of halo counts at redshift 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The best fit values are detailed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The results seem to be compatible with Sheth-Tormen since the ΛCDM best fit parameters departs as much (to be fair, maybe a little less) from 𝑞 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='707 and 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 as the parameters for the bump cosmologies, which nonetheless are small deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This analysis shows the universality of the Sheth-Tormen HMF, or more precisely of the scaled differential mass function, or multiplicity 𝑓 (𝜎), in these scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Also for this reason, when we use the HMcode we do not change the Sheth-Tormen parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Even though, we do expect the HMF to be sensitive to the nature of dark energy for bump cosmologies since the de- pendence of the variance 𝜎 on the mass is different to that of ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In particular, with distinctive signatures in halos with intermediate masses for the reasons we discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 3 we show the HMFs measured from the simulated data, as well as the ratio between them in the bump and ΛCDM cos- mologies, both at 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 (top panel) and 𝑘𝑇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 ℎ Mpc−1 (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We do this for different redshifts, from left to right, these are 𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' It is evident and expected that the largest deviations from the standard model occurs in the medbump cosmologies (𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3) since these provide the largest enhancement of the linear power spectrum at the bump location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 3 shows the HMF for the tran- sition mode 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The largest deviation from ΛCDM occurs for halos with intermediate mass 𝑀 ∼ [1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='4, 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='9] ℎ−1𝑀⊙, reaching a difference of 38% (10%) at 𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 (𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0) for medbump-k0p5 cosmologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' At the lowest redshifts snapshots (𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0), there are more small halos in the ΛCDM model than in the bump cosmologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' On the other hand, for the thinbump-k0p5 cosmology we see that for very massive halos, the HMF is the same as that in ΛCDM at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We notice from the panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 3 showing the absolute HMF 𝑑𝑛/𝑑 log 𝑀, instead of its ratio to the ΛCDM one, that the number of halos per mass interval in- creases with redshift for both ΛCDM and bump cosmological models, which means that massive structures are been formed from these initial density fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Nevertheless, the differences between the HMFs diminish with time, as expected because the nonlinear evolution tends to erase the bumps, as explained in detail in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' For the bumps located at 𝑘𝑇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 ℎ Mpc−1, qualitatively similar results are shown in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 3, but now the differences with ΛCDM are located at smaller halo masses 𝑀 ∼ [1012, 1013] ℎ−1𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The HMF of the medbump-k1 cosmologies (thinbump-k1) reaches a differ- ence with respect of ΛCDM of ∼ 9% (∼ 3%) for halos of mass 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 ℎ−1𝑀⊙, which decreases at late times for the same reasons explained above in the case of the transition scale 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' These differences between medbump-k1 cosmologies and ΛCDM fall below the 1 per cent for massive halos with mass 𝑀 > 1014 ℎ−1𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The above mentioned departures from ΛCDM are purely a consequence of the RDED in the bump cosmologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' These are the consequence in the increase of the halo formation relative to a cosmology with no bump, because the HMF is related to the standard deviation in the density field when smoothed over the Lagrangian radius 𝑅(𝑀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' When the enhancement of the matter fluctuations appears a higher 𝑘, it affects smaller scales and then smaller massive halos are expected to be more abundant than in ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' On the other hand we have seen that as higher is the transition mode 𝑘𝑇 , the bump is more susceptible to be erased by the nonlinear evolution, and hence the above picture cannot continue indefinitely and on the lower mass tail of the HMF range we always expect to obtain back the ΛCDM results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' That is, we expect to be able to test these kinds of cosmologies with intermediate massive halos, being these the most useful to constraint dark energy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This can be put in contrast to alternative dark matter models, where the largest differences from CDM are expected to show up for the lowest massive halos, as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' in warm or fuzzy dark matter models in which the abundance of very low massive halos is suppressed, if they can be formed at all [65–67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Halo power spectrum and large scale bias Now, we show results for the clustering of dark matter halos in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' For this analysis we first focus in the mass in- terval 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 ℎ−1𝑀⊙ < 𝑀 < 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 ℎ−1𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 4 and 5 we show the halo power spectrum for bump cosmologies located at 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 ℎ Mpc−1 and at 𝑘𝑇 = 1 ℎ Mpc−1, respectively, with 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 (top panels) and 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 (bottom panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' On each panel, we show the response function given by the ratio of the halo power spectra in bump and ΛCDM cosmologies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We notice a weaker clustering at large scales in the bump cosmologies compared to ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' However, for small scales, for modes that entered the horizon before the transition, the situation changes drastically and the clustering is enhanced in the bump cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The latter effect is expected because small scales tend to grow larger since they lie on top of modes around 𝑘𝑇 that were affected by the phase transition of the RDED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' However, the former, the fact that at large scales one obtains a weaker clustering is less intuitive, and as we see be- low, is due to a different mass function and hence a different large scale bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Indeed, as a consequence of the biasing being different be- tween the different models, the response is not equal to unity at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' To understand this, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 6 (solid lines) we show the theoretical dark matter halo bias as function of halo mass 𝑏1(𝑀), as obtained from the PBS formalism and the Sheth-Tormen HMF using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We do this for the bumps located at 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 ℎ Mpc−1 (top panel) and 𝑘𝑇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 ℎ Mpc−1 (bottom panel), and we plot their ratios to the ΛCDM biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This analysis shows that the large scale offsets (departing from unity) observed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 4 and 5 are due to different HMFs, and that the computed values with the PBS recipe match very ac- curate the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We further notice this is a consequence 10 10 1 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 Phh/P CDM hh b1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='74 z = 2 Linear N-body 10 1 100 b1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='65 z = 1 10 1 100 b1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='235 z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 10 1 100 b1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='935 z = 0 10 1 100 k [h Mpc 1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 Phh/P CDM hh b1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='85 10 1 100 k [h Mpc 1] b1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='705 10 1 100 k [h Mpc 1] b1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='27 10 1 100 k [h Mpc 1] b1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='958 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Halo power spectrum for the bump cosmologies at 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 ℎ Mpc−1 for 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 (top panel) and 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We consider halos in the mass interval [1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3, 1013] ℎ−1𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' From left to right, cyan curves are for redshift 𝑧 = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' green for 𝑧 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' red for 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' and blue for 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The dot-dashed curve corresponds to the linear theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' and squares are for the measurement from N-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Error bars that are not visible are within the mark size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' mainly to the dependence of the variance of fluctuations with the mass, 𝜎(𝑀), and not to the difference in the obtained 𝑝 and 𝑞 parameters for each models, which is actually small, certainly not enough to explain this large discrepancy on the biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Together with the theoretical results for 𝑏(𝑀), in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 6 we show the bias estimated by taking the ratio between the simulated nonlinear and linear power spectrum at the smallest wave-numbers 𝑘, 𝑏(𝑀) = √︄ 𝑃FastPM (𝑘) 𝑃Linear(𝑘) ����� 𝑘 → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (18) We do this for two halo mass intervals: the one already used above 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 ℎ−1𝑀⊙ < 𝑀 < 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 ℎ−1𝑀⊙ to show the power spectrum plots, and in addition we choose 1013 ℎ−1𝑀⊙ < 𝑀 < 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='6 ℎ−1𝑀⊙ that we use to compare to the analytical out- comes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' These simulated results are displayed with square and star markers in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 6, where the error bars denote the stan- dard deviations of the 5 different realizations for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' On the other hand, the filled circle markers in the plots show the theoretical results when averaged over the corresponding mass intervals, obtained using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' As it is clear from the plots, the match between simulations and theory is very good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' As expected, the largest differences between the biases are lo- cated at smaller masses for the 𝑘𝑇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 ℎ Mpc−1 case (bottom panel) than for the 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 ℎ Mpc−1 case (top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Further, the wider bumps, those with 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 (red dot-dashed lines), show larger deviations than the bumps with 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 (blue lines), which is expected because in the former case there is stronger gravitational interactions and consequently a greater clustering leading to a lower bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' CONCLUSIONS Bump cosmologies are inspired by models beyond ΛCDM that have dark sector energy densities that suffer phase transi- tions, leaving distinctive features in abundance and the clus- tering data due to a Rapid Diluted Energy Density (RDED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 11 10 1 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 Phh/P CDM hh b1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='835 z = 2 Linear N-body 10 1 100 b1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='695 z = 1 10 1 100 b1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='26 z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 10 1 100 b1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='95 z = 0 10 1 100 k [h Mpc 1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='15 Phh/P CDM hh b1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='885 10 1 100 k [h Mpc 1] b1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='723 10 1 100 k [h Mpc 1] b1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='28 10 1 100 k [h Mpc 1] b1s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='964 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 4 but for the bump cosmologies with 𝑘𝑇 = 1 ℎ Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In such scenarios, the power spectrum is enhanced at scales where otherwise the power would be smooth, originating a primordial bump at nonlinear scales relative to a model with no phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This can be understood as adding an extra relativistic energy density 𝜌𝑒𝑥 during the radiation dominated epoch increases the growth rate of the linear matter fluctua- tions impacting modes 𝑘 ≥ 𝑘𝑇 entering the horizon before the phase transition occurs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' for 𝑎 < 𝑎𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In this work, we have studied halo abundance and cluster- ing in bump cosmologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Instead of considering any specific BDE model, we have used a parametric family of bumps, al- lowing us to explore a wider range of theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We have run FastPM N-body simulations [21], which are com- plemented by nonlinear halo model approximations from the HMcode model [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We noticed that the nonlinear effects shift the peaks and originates a second bump at smaller scales because the primordial, original linear bumps serve as re- gions where average densities are higher than the background, the gravitational collapse becomes more rapid and efficient, and peaks can cross the critical density threshold for collapse more often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We have focused on the abundance and clustering statistics and mainly on the responses as given by the ratio of summary statistics in a bump cosmology to a ΛCDM cosmol- ogy with no bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This analysis is useful since ΛCDM is well known, and then good models for the response translate into good models for the statistics themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We have studied the nonlinearities in the matter and halo power spectrum and how these fingerprints are translated to large-scale halo bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We also have studied how the number of halos are affected by the phenomenology of the bump cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We have analysed and compared to ΛCDM, four bump parametrized cosmologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' These are the combinations of two locations 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 and 1 ℎ Mpc−1 and two widths 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The location of the bump corresponds to the mode reentering the horizon at the phase transition redshift, while the width to the duration of the phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We have confirmed that the power spectrum in the RDED cosmologies is affected by two important nonlinear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' First, the bumps are erased because the large scale random bulk matter motions tend to populate underdense regions, while moving out of regions with less matter than the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This effect has a similar origin than the BAO damping and is more evident for bumps located at higher 𝑘 modes because these translate in oscillations in configuration space with smaller distances between crests and troughs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The second effect is the appearance of a second bump due to the fact that the primordial 12 12 13 14 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 b(M)/b(M) CDM z = 2 12 13 14 15 z = 1 12 13 14 15 z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 12 13 14 15 z = 0 12 13 14 15 logM [h 1M ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 b(M)/b(M) CDM 12 13 14 15 logM [h 1M ] 12 13 14 15 logM [h 1M ] 12 13 14 15 logM [h 1M ] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Comparison between the halo bias from N-body simulations (star and square markers) and from theoretical prediction using PBS approach (solid and dot-dashed curves) for bump cosmologies at 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 ℎ Mpc−1 (top panel) and at 𝑘𝑇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 ℎ Mpc−1 (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Stars are for the bump cosmologies with 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' and squares for 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Dashed curves are for 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' and solid are for 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Circles are for the effective bias defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' (11) in the mean ranges of [1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 ℎ−1𝑀⊙, 1013 ℎ−1𝑀⊙] and [1013 ℎ−1𝑀⊙, 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='6 ℎ−1𝑀⊙].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' one serves as a location with high density and where even smaller structures are more prone to form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We have seen that the presence of localized bumps in the matter power spectrum have consequences on the halo statis- tics at all scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The halo power spectrum suffers an offset with respect to the ΛCDM one because the large scale bias is sensitive to the variance of density fluctuations 𝜎(𝑀), which is affected by the bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' The differences with the ΛCDM can be considerable even for small bumps located well inside the nonlinear region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We have computed the Halo Mass Function (HMF) for our bump cosmologies using the Sheth-Tormen recipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' How- ever, while the Sheth-Tormen HMF has fixed parameters, we fit them to the simulations anticipating the possibility of more optimal parameters for bump cosmologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' How- ever, we found their values to be very close to the standard 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 and 𝑞 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Despite this, the HMFs do dif- fer considerably for the bump and ΛCDM cosmologies, but this is because of different matter power spectra and then dif- ferent variances 𝜎(𝑀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' We compared our analytical results to the outcomes of the N-body simulations finding excellent agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Further, we were capable of correctly predict the large scale halo bias by applying the Peak-Background- Split formalism to our Sheth-Tormen HMF description, match- ing the simulated power spectrum data at low 𝑘 for two ranges of masses: 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 ℎ−1𝑀⊙ < 𝑀 < 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 ℎ−1𝑀⊙ and 1013 ℎ−1𝑀⊙ < 𝑀 < 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='6 ℎ−1𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In general, for the bump cosmologies and within the ade- quate mass ranges, that is, within the ranges where the forma- tion of halos of certain masses are more enhanced, we found smaller biases than in ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' This is because in those regions the presence of the primordial bump yields more clustering and then stronger gravity effects, which translate in a more efficient relaxation of the bias, which tends toward unity more rapidly than in a cosmology without the bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' In summary, this distinctive fingerprint, named as a bump, has been studied modelling the statistics of biased tracer of the density, such as halos, which have the potential to be detectable by current and future galaxy surveys [68–71] al- lowing to put tight constraints on cosmological constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Future interesting routes to continue the analysis of this work include the investigation of the halo-galaxy connection and the clustering in mock galaxy catalogs that can shed light on the expected observables for galaxies surveys such as the Dark Energy Spectroscopic Instrument (DESI) [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Also, since weak lensing depends critically on the real-space matter power spectrum, whose nonlinear effects are easy to understood and well modeled by the halo-based HMcode, we forecast that the 13 consequences of bump cosmologies on statistics measured by photometric surveys (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' the 3 × 2 point correlation func- tions) are relatively straightforward to obtain, with a view on the upcoming Rubin Observatory Legacy of Space and Time (LSST) Survey [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' ACKNOWDLEDGMENTS DGN thanks support from a CONACyT PhD fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' AM and DGN acknowledges support from PAPIIT- DGPA (UNAM) IN101415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' is supported by Ciencia de Frontera grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 319359, and also acknowledges partial support to grants Ciencia de Frontera 102958 and CONACyT 283151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Appendix A: Halos per mass interval In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 7, we show histograms for the counts of halos as a function of their mass within the interval 𝑀 = 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 ℎ−1𝑀⊙ to 𝑀 = 1015 ℎ−1𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Complementary to this figure, in Table III we show the mean number of halos of our catalog suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Both in the Figure and in the Table, the errors show the standard deviations over the 5 simulations for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' At higher redshift, the mean number of halos in bump cosmologies is larger than in ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Meanwhile, at present time the mean number of halos of ΛCDM is larger than bump cosmologies for the cases 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 ℎ Mpc−1, but not for 𝑘𝑇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 ℎ Mpc−1 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Riess, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' V.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 80, 1582 (1998), arXiv:astro-ph/9708069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 14 13 14 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='4 N/N CDM z = 2 13 14 15 z = 1 13 14 15 z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 13 14 15 z = 0 13 14 15 logM [h 1M ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1 N/N CDM 13 14 15 logM [h 1M ] 13 14 15 logM [h 1M ] 13 14 15 logM [h 1M ] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Histogram of halos for the bump cosmologies at 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5 ℎ Mpc−1 (top panel) and 𝑘𝑇 = 1 ℎ Mpc−1 (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Stars are for the measurement from N-body simulations with 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' and squares for 𝜎𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Error bars that are not visible are within the mark size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Name 𝑁(𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0) 𝑁(𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0) 𝑁(𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5) 𝑁(𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='0) medbump-k1 870050 ± 1239 1703776 ± 2872 2116618 ± 3923 2382548 ± 4320 thinbump-k1 835498 ± 1538 1658230 ± 3460 2072237 ± 4096 2344930 ± 4590 medbump-k0p5 882950 ± 1468 1673741 ± 2985 2048335 ± 4240 2280528 ± 3735 thinbump-k0p5 837524 ± 1574 1640229 ± 3557 2037594 ± 4161 2294898 ± 4134 ΛCDM 816183 ± 1639 1632756 ± 3499 2047768 ± 4369 2323036 ± 4598 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Mean number of halos of our halo catalog suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Column I corresponds to the models’ names;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' column II to mean number 𝑁 at 𝑧 = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' column III to 𝑧 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' column IV to 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' and column V to 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' de la Macorra and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Almaraz, Theoretical and Observational Constraints of Bound Dark Energy with Precision Cosmological Data, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E3T4oBgHgl3EQfAAmq/content/2301.04254v1.pdf'} +page_content=' Rev.' metadata={'source': 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Properties of practices in industrial use are rarely studied. Security workers satisfice. There +is a widespread perception that security work must be cumbersome, and thus there’s no value to +assessing levels of effort. This is complemented by a belief that the nth day of work will produce +value equal to the first. These perceptions impact both practice and research. This paper expands the +acceptable paradigms for security analysis to include the fast, cheap and good enough. “Nothing” +is often enough for industry. This paper makes a case for valuing lightweight (“fast and cheap”) +methods, presents a set of case studies and evaluation criteria for such tools, including card decks and +role playing games. +Keywords security, threat modeling, security engineering, usable security +1 +Introduction +“Copying and Pasting code from Stack Exchange” is a derisive meme, built on the idea that there’s a Proper Solution, +and ... there’s what you find on Stack Exchange. This behavior – looking for quick solutions – demonstrates two +realities. The first is even skilled technical professionals find themselves needing help in unfamiliar situations. Second, +efficiency is often scorned. +This paper explores a new frame for security methodologies: fast and cheap are good.1 This is in intentionally +provocative contrast to the engineering truism of “fast, cheap, good: choose any two.” We are not arguing these tradeoffs +don’t exist, but the aphorism exists because demands for such tradeoffs are common. Research into fast and cheap +tools is undervalued relative to mathematically or otherwise “sophisticated” tools. Approaches that prioritize speed or +simplicity are frequently derided as trivial or obvious, rather than praised as elegant, or recognized as what security +workers need. +The new paradigm leads us to ask a new question about the distribution of security issues (vulnerabilities or flaws). If +such issues are very similar to one another, then knowledge of the system is important, because the primary effort to +find the issues is search – where might this crop up? If such issues are unique, then security knowledge or even ‘an +adversarial mindset’ may be paramount. +I and others create low-friction threat modeling methodologies. This work grounds and investigates the intuition that +lower friction is a worthwhile goal for security workers. This paper explains threat modeling and tools in response to +constraints. We analyze three tools we label fast, cheap and good (Section 3). From there, we construct a framework, +a flaw space and an Flaw Oracle which reliably finds them (Section 4). The juxtaposition enables us to ask how we +should judge them. We propose initial criteria of cost, expertise requirements and quality (Section 5). +1Forthcoming as Nothing is Good Enough IEEE S+P Magazine, 2023 +arXiv:2301.03593v1 [cs.CR] 10 Dec 2022 + +Fast, Cheap, Good: Lightweight Methods are Undervalued +(PREPRINT) +1.1 +Contribution +This work: +1. Collects and organizes a set of new techniques that are rarely discussed coherently in the literature. +2. Produces a framework, motivated by those new techniques, that explicitly considers the difficulty of threat +modeling practice. +3. Provide evaluation criteria for new techniques this framework allows. +4. Ask if vulnerabilities are more like products of a factory, very similar to each other, or artisanal, each unique +and special. +2 +Background +People satisfice. They accept “good enough” except where there are unavoidable pressures to go further. By way of +example, “we do not use any of our limited citation count for the idea of satisficing.” +Threat modeling is a family of techniques to anticipate future problems with a system, so they can be addressed. There +are many competing methodologies and evaluation criteria vary widely. In practice, threat modeling students often +express concerns that it will fail to find all relevant threats or complete in appropriate time. 2 +Is one possible explanation—an explanation that could implicitly undergird all of these observations—that threat +modeling is or appears harder than it needs to be? Perhaps threat modeling practices, designed to be comprehensive, +become so comprehensive that they are too tedious to learn or execute? (This is a “perfect is the enemy of the good” +theory.) +What happens when a threat modeling practice trades precision for ease, becoming less burdensome, but easier to +perform and more accessible to a wide range of practitioners? Prior work, which we discuss in section 3, suggests that +the answer can be that practices become more effective. Can these practices, or ones similar to them, actually improve +threat modeling overall, leveraging their relative light ‘weight’ to help practitioners perform threat modeling more +regularly than they would otherwise? Through our review of this literature, we argue that the answer to this question is +also yes. Building on this argument, the next section builds a conceptual framework for “fast, cheap, and good enough” +threat modeling practices. +2.1 +Related work +There is a rich literature on the use of serious games or games with a purpose outside the scope of this paper. For +example, (Prensky [2006]) and (Abt [1987]). There now is a well-understood set of challenges for usable security, +including that security is often a secondary task. For this preprint, we assume that software engineering and systems +operations are security workers, at least some of the time. We assume that those engineers have a compliance budget +(Beautement et al. [2008]), and that faster and cheaper techniques are more likely to fit within it. We assume that for +those workers, the larger the request, the more resistance it will meet, and conversely, it is easier to get them to do easier +work. That is: fast and cheap are good because security workers satisfice. (Garfinkel [2014]) +2.2 +Software is used in diverse ways +There are many dimensions of diversity of software: +1. Development lifecycles range from continuous development and deployment through agile sprints to waterfall. +2. Programming languages range from assembler to Javascript or even codeless systems. +3. Deployment environments range from cloud systems through downloaded software through IoT and even +interplanetary spacecraft. +4. Impact of failure ranges from the very low impact of a local game failing through medical device software +failure potentially costing many lives. +Perhaps threat modeling practices, designed to address the highest risk systems, become so comprehensive that they are +too tedious to learn or execute? What happens when a threat modeling practice trades precision for ease, becoming less +2Personal observation as an instructor +2 + +Fast, Cheap, Good: Lightweight Methods are Undervalued +(PREPRINT) +burdensome, but easier to perform and more accessible to a wide range of practitioners? Prior work, which we discuss +in section 3, suggests that the answer can be that practices become more effective. These sorts of practice may improve +threat modeling overall, leveraging their relative light by helping more people threat model or threat model more often. +The next section builds a conceptual framework for “fast, cheap, and good enough” threat modeling practices. +2.3 +Threat modeling: Responding to diverse needs +Given the diverse goals of software and the diverse contexts in which software tools are used, comprehensive “security +checklists” become untenably tedious. As a result, threat modeling seeks to imagine possible threats through speculative +processes. For example, STRIDE was originally proposed as a model of threats to products (STRIDE is an acrostic of +Spoofing, Tampering, Repudiation, Information Disclosure, Denial of Service and Elevation of Privilege) (Kohnfelder +and Garg [1999]). Engineers would consider those threats to a product as a whole. A derivative, STRIDE per element, +examines each component of a system for relevant STRIDE threats (Shostack [2008]). Clearly, the later use is both +more time consuming than the former, and less likely to miss issues. Systems such as STRIDE are intended to help +security workers search for attacks by providing prompts. (Shostack [2014]) Board and card games sometimes assist in +the speculative process of threat modeling (Denning et al. [2013] Frey et al. [2019] ). For example, the game Elevation +of Privilege stemmed from Microsoft’s development of the STRIDE methodology (Shostack [2014]). Even the US +Central Intelligence Agency (CIA) developed a (now-declassified) game for helping developers generate threat models +(Masnick [2018]). +There are some common threads in critiques of existing methods: +1. They are difficult to learn, especially for those who are not security specialists (Weir et al. [2018], Denning +et al. [2013]). Developers who are not specifically trained in computer security might write more secure +software if they could better participate in threat identification. However, existing practices are generally +aimed at security specialists working at technology producers. +2. They require significant time, even for specialists, to carry out. Threat modeling methods require signifi- +cant, devoted time. Even games like Elevation of Privilege require at least 30 minutes; more detailed threat +modeling practices can take hours, days or weeks to complete. +3. They may systematically fail to identify particular types of threats, particularly those that arise from social +factors or relational characteristics such as gender, race, age, or disability; or those whose impacts fall on end +users or those who are invisibly impacted by it. (For example, the use of facial recognition on city streets.) +(Pierce et al. [2018], Freed et al. [2018], Nissenbaum [2005], Hong et al. [2004], Friedman and Hendry [2012], +Suchman et al. [2017], Coles-Kemp [2009]). +Given the variety of software systems and delivery methods, it seems unreasonable to assume that there will be a single +approach to modeling threats, any more than there will be a Unified Development Process. Many threat modeling +approaches have been proposed and are coalescing around an organizing set of questions: ‘what are we working on,’ +’what can go wrong’, ‘what are we going to do’ and ’did we do a good job.’ (A set of industrial practitioners converged +on these in a “Threat Modeling Manifesto” in 20203.) +3 +Emerging techniques serve a broader audience +Many organizations are not doing security work at all. They are not threat modeling, using static analysis, or fuzzing. +“Nothing” is good enough for them. In that context, fast and cheap methods may literally be better than that ‘nothing.’ +A more nuanced view is the adoption and use of security improvements is a complex tradeoff space, and we’re not +good at evaluating these lightweight tools (Hollnagel et al. [2006]). This work provides three short case studies of +lightweight approaches to threat modeling presented elsewhere in the literature. These are intended to show approaches +to security in product development that are lighter than commonly cited approaches. We can compare these to ‘STRIDE +per element,’ a more structured approach. Similarly, for static analysis we compare it to sound approaches. (We include +an example of a static analysis tool to show that present work is not restricted to threat modeling.) +3.1 +Security Fictions +3.1.1 +What is Security Fictions? +Security Fictions is an “improvisational role-playing game” that takes place between an interlocutor (in the case of the +study, the researcher) and an engineer. The interlocutor poses as a system’s enemy: someone who wants to accomplish +3Available at threatmodelingmanifesto.org +3 + +Fast, Cheap, Good: Lightweight Methods are Undervalued +(PREPRINT) +a particular goal using the system. Examples from the study include (1) impersonating another user on the platform, (2) +finding a user’s geographic location, and (3) identifying all the users who have particular political goals. The engineer +then role-plays as someone trying to help the enemy—an “insider threat”—and brainstorms with the interlocutor about +how best to achieve the nefarious goal. (Merrill [2020]) +3.1.2 +Why was Security Fictions designed this way? +Security Fictions was designed to respond to critiques (1) and (2) in section 2.3. To make the game easier for non- +specialists to learn, it looked toward another methodological tradition for surfacing potential future issues: speculative +design. (In building this analogy, the authors observe “that threat identification is an intrinsically speculative practice: +it requires imagining possible futures.”) To synthesize these two research traditions, Security Fictions draws the +threat modeling practice of redteaming together with Elsden et al.’s Speculative Enactments, which engage research +participants in improvisational play with real-world consequences (Elsden et al. [2017]). From this role-playing tradition, +the practice was designed to be quick to learn, and better at surfacing sociotechnical threats; from this redteaming +tradition, it was designed to identify actionable and specific security issues. +3.1.3 +How well does Security Fictions work? +In an exploratory deployment of the game among software developers (some of whom did not specialize in security), +developers enjoyed playing the game, and used it to come up with realistic threats. Some of the threats that non-security +specialists devised were not strictly technical (for example, a UX designer discovered a social attack that relied on a +misleading feature in a user interface). +Security Fictions provokes useful insights through two related mechanisms: gamification and permission. The +gamification is the explicit framing as a role playing game. This is accompanied by an invitation to “set ethical concerns +aside,” giving social permission to discuss unwanted behaviors. This allows those with less explicit security knowledge +to explore possible security flaws, even if they lack terminology, structure or other explicit security knowledge. +3.1.4 +Analysis +Security Fictions are inexpensive to run (from 15-60 minutes per session) once a set of scenarios has been created or +adopted. The open-ended nature of the framework allows the use of fewer or more questions in a given session. There is +no particular expertise required to create scenarios, but a degree of either storytelling or dungeon-mastering experience +may be helpful in effective execution, and more research may reveal criteria for good prompts. +3.2 +Elevation of Privilege +3.2.1 +What is EoP? +Elevation of Privilege is a threat modeling approach embedded into a card game. The deck consists of 6 suits, based on +the STRIDE model of threats. Within each suit there are cards 2 through Ace. Each card has a hint, for example the 8 +of Information Disclosure says: ‘An attacker can access information through a search indexer, logger, or other such +mechanism.’ The deck is used with a system diagram to ‘draw developers into threat modeling,’ using familiar card +game mechanics. +3.2.2 +Why was EoP designed this way? +The game was designed to be an enticing, supportive and non-threatening way to draw people into threat modeling, +to bring in people with different skill sets and knowledge. I wrote ‘If it were possible to have developers do basic +threat modeling, then experts could be used more effectively to find the really unusual problems with a design,’ and +‘Those who are new to threat modeling or those who threat model occasionally require a more procedural approach, and +procedural approaches are generally at odds with creativity. Elevation of Privilege was created in this constraint space +in part to expose non-security experts to the enjoyment that security experts bring to threat modeling.’ +3.2.3 +How well does EoP work? +The game has been available for a dozen years, and I reported on a variety of anecdotes to its success ranging from +six-year-olds to companies that report having weekly sessions (Shostack [2014]. There are at least three substantiative +variants, including OWASP Cornucopia, F-Secure’s Elevation of Privacy (with 4 new suits) and LogMeIn has added a +privacy suit, making STRIPED. Elevation of Privilege enables contributions from people who lack security knowledge +via the hints on the cards, and requires it through the turn-taking game structure. The hints are augmented by game-table +4 + +Fast, Cheap, Good: Lightweight Methods are Undervalued +(PREPRINT) +conversation and banter, offering a chance for participants to ask questions about their hints in a lower-pressure +environment. +3.2.4 +Analysis +When Elevation of Privilege was created, it was intended to be the simplest, lowest cost structured introduction to threat +modeling possible. Since then, faster and simpler approaches have appeared, putting the game at the more expensive +end of fast, cheap approaches, taking about an hour. +3.3 +Semgrep +We include discussion of semgrep because it may be to static analysis what Elevation of Privilege is to threat modeling: +a lighter, more accessible approach, and it may be helpful to consider fast, cheap, good approaches beyond threat +modeling. +3.3.1 +What it is? +Semgrep is an open source, lightweight static analysis tool that’s designed to be fast, easy to deploy and easy to program. +3.3.2 +Why was it designed this way? +“Semgrep is designed for the security engineer or developer who wants to automate code review.” and “We think +modern static analysis tools should run on every keystroke in the editor, without needing network access.” (Both +https://semgrep.dev/docs/faq/) +These properties of easy to deploy, fast, and easy to write rules for seem intuitively useful, but they are less prioritized +than soundness and completeness. Semgrep explicitly rejects the need to be able to handle an abstract syntax tree, +stating in their FAQ “if you can write code, you can write a Semgrep rule — no program analysis PhD required!”. +3.3.3 +How well does it work? +Semgrep has a ‘trophy case’ on their website with 8 CVEs, and a set of useful contributions, but its use seems to be +largely internal to enterprises, which makes assessment more challenging. Further, the team behind it seems to have +a frame that value is evidenced by continued use, rather than publications. As such, the rate of rule publication may +be helpful evidence of engagement. As of April, 2021, Semgrep’s core was being released approximately weekly, in +addition to roughly 25 new rules contributed each week. This indicates that at least a few people are motivated to +continue participating. +3.3.4 +Relevant reflections +Semgrep explicitly focuses on being inexpensive to learn and use, without specifying what those mean in practice. They +also focus on being accessible to those without a PhD. +The relationship between applied and academic static analysis work was the subject of reflection by the Coverity team. +Their paper on the subject discusses many of the challenges that semgrep has taken on, including troubles in deploying +their tool, speed, and managing multiple languages (Bessey et al. [2010]). +4 +A Flaw Space and A Flaw Oracle +The distribution of flaws by the system knowledge and security expertise required to find them (Darker areas have more +flaws). Issues in the lower left are what we describe as light flaws. Lightweight threat modeling practices can help find +them more effectively than the widely-recommended practices common today. +Flaws require different levels of system knowledge, and different levels of security expertise, to discover. Some flaws, +which we will herein refer to as light flaws, require a less system knowledge and/or less security expertise to uncover +(Figure 1). We argue that the methods we describe above work because they help to catch light flaws, and do so with a +minimum of time and training on the part of developers. +Existing threat modeling practices, we argue, aim to capture the entire flaw space. In doing so, they undervalue the time, +energy, and organizational costs required to do so. By providing lighter practices by which non-specialists can capture +and solve light flaws, security experts would be relatively less burdened, able to focus only on a subset of the flaw space +5 + +Fast, Cheap, Good: Lightweight Methods are Undervalued +(PREPRINT) +Figure 1: Figure 1 +(represented by the upper-right in Figure 1). The remainder of this section uses a thought experiment to expound on our +reasoning. +Imagine a perfect examiner who instantly and reliably finds flaw in a system, given the correct inputs. This Flaw Oracle +takes as input first, knowledge of the system and second, knowledge of security (This is a thought experiment, we don’t +propose such an examiner exists, nor do we claim such knowledge can be codified). We are focused on what flaws will +be found, without specifying an underlying distribution, or requiring knowledge of the actual distribution. We can also +imagine that an examiner requires intensive security knowledge. Informally, some theorem provers might be analogous +to such an examiner. (The term ‘flaw’ is often used for the design subset of security issues, complementing bugs or +vulnerabilities. We use it to encompass both, as security oracle is too broad. We can also consider a Laggard Flaw +Oracle (LFO), who finds flaws and emits information at a later date. +The knowledge needed by the examiner can be seen as analogous to the participants in system engineering. The software +engineers may have a great deal of knowledge of the system, and less knowledge of security. Security engineers, +especially consultants, will bring knowledge of security, and less knowledge of the system. Knowledge is expensive. +Each step of gathering it, checking it, encoding it into human- or machine-readable forms, checking the encoding didn’t +introduce problems takes effort. Similarly, keeping such information up-to-date as systems evolve requires effort. +We can visualize the distribution of flaws in systems, and simple visualizations such as Figure 1 can capture intuitions. +Another possible distribution (not shown) is that flaws are distributed randomly and evenly. +Thus systems that require fewer expensive inputs can have value, if there are many issues that an examiner can find +with system knowledge and a modest amount of security knowledge. For example, once you know that file APIs will +often accept network paths, someone with knowledge of where a system opens files can easily check for remote file +inclusion (RFI) bugs. The birthday problem is easily taught, but seeing how it applies to a system can be harder, and +subtle cryptographic bugs may require both security and system knowledge. +This account of the flaw space explains why the lightweight threat modeling practices we describe above may have +worked. Further, it provides a theoretical backdrop that motivates further work on developing and evaluating lightweight +threat modeling practices. Readers may be concerned that this work will do more harm than good. An analog of +Gresham’s Law of “Bad money drives out good” may be relevant: bad security may drive out the good. We disagree. +The availability of inexpensive clothes does not stop people from paying for designer clothes; each delivers a different +value. +6 + +RFI +Subtle crypto +System +knowledge +required +Birthday +problem +Security expertise requiredFast, Cheap, Good: Lightweight Methods are Undervalued +(PREPRINT) +5 +Evaluating Practices +Having put forth the idea that light flaws exist, we ask: how best can we judge practices that aim to catch these +flaws? We propose three, primary dimensions: cost, expertise requirements, and quality of output. These are important +dimensions which make up a multidimensional design space; we expect there are other useful criteria beyond these. +These criteria exclude the impact of the output, which has surprised some early readers. The usefulness of a technique +is in a context, and the tradeoffs which that context demands are independent of the tool (We do not judge a hammer +useless because we have a lot of hex nuts). +5.1 +Cost +Cost include money, time, and organizational energy. Time includes training, preparation, and execution. Some systems +require only minutes of training or setup. For example, Security Fictions takes a few minutes to explain and set the +stage. Elevation of Privilege usually involves a few minutes of stage-setting, 5-15 minutes to draw a system diagram, +and a round of test play for people to ‘get the hang of it.’ This is in contrast to a half day or even days of training in +deeper threat modeling approaches. Some systems, like Security Fictions, requires some prep work in crafting the +scenarios. (It is reasonable to think that a compendium of scenarios could be created, but none has.) Organizational +costs include coordination overhead associated with communicating flaws across functional teams, prioritizing issues to +fix, and so on. The more that lightweight processes reduce coordination overhead (e.g., by empowering UX designers to +solve problem on their own), the lower their cost. The LFO adds cost by delay. It is well understood that fixing issues is +more expensive after code is written, and more expensive yet when the code has dependencies which need to be at least +checked and possibly fixed, or when the code is already deployed possibly at many sites, each of which must deploy the +fixes. Thus, the slower the LFO, the more cost it allows to creep into the system. +5.2 +Expertise requirement +Figure 1 shows a model where discovering flaws require different levels of system-specific and security-specific +expertise. Effective lightweight methods can minimize reliance on security-specific expertise, compensating by relying +instead on system-specific expertise. In some empirical work (Merrill and Weber [2019]), a UX designer used misleading +unicode symbols to perform an impersonation attack, effectively using their systems knowledge to compensate for a +lack of security-specific knowledge. The ability of various practices to facilitate this tradeoff is critical for catching +light flaws. Characterizing and defining the exact security knowledge required to make good use of a tool is beyond the +scope of this paper, but we’ll note briefly that we have never seen a characterization of skill except “this tool requires a +PhD in static analysis.” +5.3 +Quality +The most challenging but most crucial dimension to measure is the quality of practices’ output. Students often seek to +find “good threats.” What makes a “good threat”? There is an argument that a “good threat” discovered by a threat +modeling approach is one that the organization cares about enough to fix. That is, the time spent to fix it is evidence of +a useful output. But there can be reasons to not fix a worrisome problem. The organization may be fixing other more +pressing issues, there may be issues of application compatibility or usability of a fix. In other words, the perceived +relevance of flaws is contextual. The exact same issue, discovered at (for example) Amazon, Microsoft, Bank of +America or a local community bank might be treated differently at each. Bug bounties offered by each are scoped +differently, rewarding different flaw types. +There may be work in assessing quality from the perspective of the users. For example, in (Slupska et al. [2021]) an +argument is put forth for threat modeling techniques that are easier for those other than cybersecurity professionals to +use. There are important dimensions of quality, such as avoiding creepy advertisements or stalking, that traditional +information-security centered approaches may not prioritize. Future work should delve into these questions as they +apply to the flaw space described here. +6 +Future work +We hope to create a framework in which future approaches to threat modeling can be understood, evaluated and +compared. We hope to enable work in the area of fast and cheap tools to help people analyze security, and to enable +rigorous and thoughtful discussion of those tools. Ease of use, speed, scope, suitability for non-technical staff at +organizations building systems, suitability for use by the public, advocates, activists, and other factors are all important. +7 + +Fast, Cheap, Good: Lightweight Methods are Undervalued +(PREPRINT) +By defining the nature of tradeoffs that threat modeling tools make, we hope to enable exploration of new tools that +more strategically and purposefully approach different portions of the flaw space we describe in Figure 1. +While this work is highly preliminary, it raises one critical point of reflection for future research: our observation that +a flaw space, distributed across system and security expertise, might exist begs a cultural shift in the way that we, as +security professionals, evaluate security processes and outcomes. +How do we judge the quality of processes and outcomes? Analyses, typically received from academia, focus on the +soundness of analysis, formality of proofs, precision and recall of algorithms, and so on. But perhaps quality in security +is an emergent characteristic of an organization. Banks and falafel stops have not only different security needs, but +different ways of evaluating what kinds of security make sense for them to approach, and different competing constraints +for their time and attention. +Sensitivity to quality as an emergent and situated characteristic of organizations allows us to judge threat modeling +practices better. It centers broad outcomes, and diverse notions of what it means to be secure. +The cultural shift here, for us, is to make less formal and less analytical processes acceptable by the cultural standards +of security research (The author has seen a paper rejected with the comment “This is great, and it needs equations”). +Moving toward these more situated notions will help us understand the work that, for example, a local falafel shop does +to protect themselves as relevant and important, assessing the quality of their output that isn’t overly tied to narrow +analytical frameworks. I return, this will help us build practices that make security more relevant and actionable to the +people who need it. +Our call to action: security needs better ways of meeting people where they are, understanding what they need. An +analogy to economics helps illustrate the conceptual shift required here. In economics, a broad and useful assumption is +that people are rational; that they have a reason to do the things that they do. When economists observe people doing +something seemingly irrational (for example, not saving as much as they “should”), economists look for the rationality +that underlies that decision (for example, perhaps people don’t save because they believe they’ll die having never spent +their money; this makes saving contextually irrational). +As security researchers, we can do something similar. What are people doing, and what is the rationality that underlies +those actions? When people ignore our security advice, perhaps doing so is rational for them because, for example, it +costs them too much in time or energy relative to their other goals (see our design dimensions above and Herley [2009]). +How can we best use those observations to design more relevant and actionable security practices? +7 +Acknowledgements +This article comes from deep conversation with Nick Merrill about why these lightweight approaches are meaningful. +The author appreciates thoughtful comments from Mary Ellen Zurko, which improved this draft. +References +Marc Prensky. Don’t bother me, mom, i’m learning!: How computer and video games are preparing kids for 21st +century success and how you can help!. Paragon house, 2006. +Clark C. Abt. Serious games. University press of America, 1987. +Adam Beautement, M Angela Sasse, and Mike Wonham. The compliance budget: managing security behaviour in +organisations. 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In Proceedings of +the 2019 ACM Conference on Computer Human Interaction (CHI ’19), Glasgow, UK, 2019. +Julia Slupska, Scarlet Dawson Dawson Duckworth, Linda Ma, and Gina Neff. Participatory threat modelling: Exploring +paths to reconfigure cybersecurity. In Extended Abstracts of the 2021 CHI Conference on Human Factors in +Computing Systems, pages 1–6, 2021. +Cormac Herley. So long, and no thanks for the externalities: the rational rejection of security advice by users. In +Proceedings of the 2009 workshop on New security paradigms workshop, pages 133–144, 2009. +9 + diff --git a/KtE2T4oBgHgl3EQfAgbT/content/tmp_files/load_file.txt b/KtE2T4oBgHgl3EQfAgbT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..408e131fd05eea63203ad51002bde9b886502be7 --- /dev/null +++ b/KtE2T4oBgHgl3EQfAgbT/content/tmp_files/load_file.txt @@ -0,0 +1,468 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf,len=467 +page_content='FAST, CHEAP, GOOD: LIGHTWEIGHT METHODS ARE UNDERVALUED (PREPRINT) Adam Shostack Shostack + Associates, Inc and University of Washington Seattle, Washington 98195 adam@shostack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='org January 11, 2023 ABSTRACT Engineering techniques to address the endless parade of security issues are an important area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Properties of practices in industrial use are rarely studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Security workers satisfice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' There is a widespread perception that security work must be cumbersome, and thus there’s no value to assessing levels of effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' This is complemented by a belief that the nth day of work will produce value equal to the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' These perceptions impact both practice and research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' This paper expands the acceptable paradigms for security analysis to include the fast, cheap and good enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' “Nothing” is often enough for industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' This paper makes a case for valuing lightweight (“fast and cheap”) methods, presents a set of case studies and evaluation criteria for such tools, including card decks and role playing games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Keywords security, threat modeling, security engineering, usable security 1 Introduction “Copying and Pasting code from Stack Exchange” is a derisive meme, built on the idea that there’s a Proper Solution, and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' there’s what you find on Stack Exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' This behavior – looking for quick solutions – demonstrates two realities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The first is even skilled technical professionals find themselves needing help in unfamiliar situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Second, efficiency is often scorned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' This paper explores a new frame for security methodologies: fast and cheap are good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='1 This is in intentionally provocative contrast to the engineering truism of “fast, cheap, good: choose any two.” We are not arguing these tradeoffs don’t exist, but the aphorism exists because demands for such tradeoffs are common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Research into fast and cheap tools is undervalued relative to mathematically or otherwise “sophisticated” tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Approaches that prioritize speed or simplicity are frequently derided as trivial or obvious, rather than praised as elegant, or recognized as what security workers need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The new paradigm leads us to ask a new question about the distribution of security issues (vulnerabilities or flaws).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' If such issues are very similar to one another, then knowledge of the system is important, because the primary effort to find the issues is search – where might this crop up?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' If such issues are unique, then security knowledge or even ‘an adversarial mindset’ may be paramount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' I and others create low-friction threat modeling methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' This work grounds and investigates the intuition that lower friction is a worthwhile goal for security workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' This paper explains threat modeling and tools in response to constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' We analyze three tools we label fast, cheap and good (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' From there, we construct a framework, a flaw space and an Flaw Oracle which reliably finds them (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The juxtaposition enables us to ask how we should judge them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' We propose initial criteria of cost, expertise requirements and quality (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 1Forthcoming as Nothing is Good Enough IEEE S+P Magazine, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='03593v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='CR] 10 Dec 2022 Fast, Cheap, Good: Lightweight Methods are Undervalued (PREPRINT) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='1 Contribution This work: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Collects and organizes a set of new techniques that are rarely discussed coherently in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Produces a framework, motivated by those new techniques, that explicitly considers the difficulty of threat modeling practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Provide evaluation criteria for new techniques this framework allows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Ask if vulnerabilities are more like products of a factory, very similar to each other, or artisanal, each unique and special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 2 Background People satisfice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' They accept “good enough” except where there are unavoidable pressures to go further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' By way of example, “we do not use any of our limited citation count for the idea of satisficing.” Threat modeling is a family of techniques to anticipate future problems with a system, so they can be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' There are many competing methodologies and evaluation criteria vary widely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' In practice, threat modeling students often express concerns that it will fail to find all relevant threats or complete in appropriate time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 2 Is one possible explanation—an explanation that could implicitly undergird all of these observations—that threat modeling is or appears harder than it needs to be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Perhaps threat modeling practices, designed to be comprehensive, become so comprehensive that they are too tedious to learn or execute?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' (This is a “perfect is the enemy of the good” theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=') What happens when a threat modeling practice trades precision for ease, becoming less burdensome, but easier to perform and more accessible to a wide range of practitioners?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Prior work, which we discuss in section 3, suggests that the answer can be that practices become more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Can these practices, or ones similar to them, actually improve threat modeling overall, leveraging their relative light ‘weight’ to help practitioners perform threat modeling more regularly than they would otherwise?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Through our review of this literature, we argue that the answer to this question is also yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Building on this argument, the next section builds a conceptual framework for “fast, cheap, and good enough” threat modeling practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='1 Related work There is a rich literature on the use of serious games or games with a purpose outside the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' For example, (Prensky [2006]) and (Abt [1987]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' There now is a well-understood set of challenges for usable security, including that security is often a secondary task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' For this preprint, we assume that software engineering and systems operations are security workers, at least some of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' We assume that those engineers have a compliance budget (Beautement et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' [2008]), and that faster and cheaper techniques are more likely to fit within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' We assume that for those workers, the larger the request, the more resistance it will meet, and conversely, it is easier to get them to do easier work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' That is: fast and cheap are good because security workers satisfice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' (Garfinkel [2014]) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='2 Software is used in diverse ways There are many dimensions of diversity of software: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Development lifecycles range from continuous development and deployment through agile sprints to waterfall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Programming languages range from assembler to Javascript or even codeless systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Deployment environments range from cloud systems through downloaded software through IoT and even interplanetary spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Impact of failure ranges from the very low impact of a local game failing through medical device software failure potentially costing many lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Perhaps threat modeling practices, designed to address the highest risk systems, become so comprehensive that they are too tedious to learn or execute?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' What happens when a threat modeling practice trades precision for ease, becoming less 2Personal observation as an instructor 2 Fast, Cheap, Good: Lightweight Methods are Undervalued (PREPRINT) burdensome, but easier to perform and more accessible to a wide range of practitioners?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Prior work, which we discuss in section 3, suggests that the answer can be that practices become more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' These sorts of practice may improve threat modeling overall, leveraging their relative light by helping more people threat model or threat model more often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The next section builds a conceptual framework for “fast, cheap, and good enough” threat modeling practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='3 Threat modeling: Responding to diverse needs Given the diverse goals of software and the diverse contexts in which software tools are used, comprehensive “security checklists” become untenably tedious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' As a result, threat modeling seeks to imagine possible threats through speculative processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' For example, STRIDE was originally proposed as a model of threats to products (STRIDE is an acrostic of Spoofing, Tampering, Repudiation, Information Disclosure, Denial of Service and Elevation of Privilege) (Kohnfelder and Garg [1999]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Engineers would consider those threats to a product as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' A derivative, STRIDE per element, examines each component of a system for relevant STRIDE threats (Shostack [2008]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Clearly, the later use is both more time consuming than the former, and less likely to miss issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Systems such as STRIDE are intended to help security workers search for attacks by providing prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' (Shostack [2014]) Board and card games sometimes assist in the speculative process of threat modeling (Denning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' [2013] Frey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' [2019] ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' For example, the game Elevation of Privilege stemmed from Microsoft’s development of the STRIDE methodology (Shostack [2014]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Even the US Central Intelligence Agency (CIA) developed a (now-declassified) game for helping developers generate threat models (Masnick [2018]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' There are some common threads in critiques of existing methods: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' They are difficult to learn, especially for those who are not security specialists (Weir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' [2018], Denning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' [2013]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Developers who are not specifically trained in computer security might write more secure software if they could better participate in threat identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' However, existing practices are generally aimed at security specialists working at technology producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' They require significant time, even for specialists, to carry out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Threat modeling methods require signifi- cant, devoted time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Even games like Elevation of Privilege require at least 30 minutes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' more detailed threat modeling practices can take hours, days or weeks to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' They may systematically fail to identify particular types of threats, particularly those that arise from social factors or relational characteristics such as gender, race, age, or disability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' or those whose impacts fall on end users or those who are invisibly impacted by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' (For example, the use of facial recognition on city streets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=') (Pierce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' [2018], Freed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' [2018], Nissenbaum [2005], Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' [2004], Friedman and Hendry [2012], Suchman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' [2017], Coles-Kemp [2009]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Given the variety of software systems and delivery methods, it seems unreasonable to assume that there will be a single approach to modeling threats, any more than there will be a Unified Development Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Many threat modeling approaches have been proposed and are coalescing around an organizing set of questions: ‘what are we working on,’ ’what can go wrong’, ‘what are we going to do’ and ’did we do a good job.’ (A set of industrial practitioners converged on these in a “Threat Modeling Manifesto” in 20203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=') 3 Emerging techniques serve a broader audience Many organizations are not doing security work at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' They are not threat modeling, using static analysis, or fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' “Nothing” is good enough for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' In that context, fast and cheap methods may literally be better than that ‘nothing.’ A more nuanced view is the adoption and use of security improvements is a complex tradeoff space, and we’re not good at evaluating these lightweight tools (Hollnagel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' [2006]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' This work provides three short case studies of lightweight approaches to threat modeling presented elsewhere in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' These are intended to show approaches to security in product development that are lighter than commonly cited approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' We can compare these to ‘STRIDE per element,’ a more structured approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Similarly, for static analysis we compare it to sound approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' (We include an example of a static analysis tool to show that present work is not restricted to threat modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=') 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='1 Security Fictions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='1 What is Security Fictions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Security Fictions is an “improvisational role-playing game” that takes place between an interlocutor (in the case of the study, the researcher) and an engineer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The interlocutor poses as a system’s enemy: someone who wants to accomplish 3Available at threatmodelingmanifesto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='org 3 Fast, Cheap, Good: Lightweight Methods are Undervalued (PREPRINT) a particular goal using the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Examples from the study include (1) impersonating another user on the platform, (2) finding a user’s geographic location, and (3) identifying all the users who have particular political goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The engineer then role-plays as someone trying to help the enemy—an “insider threat”—and brainstorms with the interlocutor about how best to achieve the nefarious goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' (Merrill [2020]) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='2 Why was Security Fictions designed this way?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Security Fictions was designed to respond to critiques (1) and (2) in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' To make the game easier for non- specialists to learn, it looked toward another methodological tradition for surfacing potential future issues: speculative design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' (In building this analogy, the authors observe “that threat identification is an intrinsically speculative practice: it requires imagining possible futures.”) To synthesize these two research traditions, Security Fictions draws the threat modeling practice of redteaming together with Elsden et al.’s Speculative Enactments, which engage research participants in improvisational play with real-world consequences (Elsden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' [2017]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' From this role-playing tradition, the practice was designed to be quick to learn, and better at surfacing sociotechnical threats;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' from this redteaming tradition, it was designed to identify actionable and specific security issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='3 How well does Security Fictions work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' In an exploratory deployment of the game among software developers (some of whom did not specialize in security), developers enjoyed playing the game, and used it to come up with realistic threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Some of the threats that non-security specialists devised were not strictly technical (for example, a UX designer discovered a social attack that relied on a misleading feature in a user interface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Security Fictions provokes useful insights through two related mechanisms: gamification and permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The gamification is the explicit framing as a role playing game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' This is accompanied by an invitation to “set ethical concerns aside,” giving social permission to discuss unwanted behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' This allows those with less explicit security knowledge to explore possible security flaws, even if they lack terminology, structure or other explicit security knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='4 Analysis Security Fictions are inexpensive to run (from 15-60 minutes per session) once a set of scenarios has been created or adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The open-ended nature of the framework allows the use of fewer or more questions in a given session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' There is no particular expertise required to create scenarios, but a degree of either storytelling or dungeon-mastering experience may be helpful in effective execution, and more research may reveal criteria for good prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='2 Elevation of Privilege 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='1 What is EoP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Elevation of Privilege is a threat modeling approach embedded into a card game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The deck consists of 6 suits, based on the STRIDE model of threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Within each suit there are cards 2 through Ace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Each card has a hint, for example the 8 of Information Disclosure says: ‘An attacker can access information through a search indexer, logger, or other such mechanism.’ The deck is used with a system diagram to ‘draw developers into threat modeling,’ using familiar card game mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='2 Why was EoP designed this way?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The game was designed to be an enticing, supportive and non-threatening way to draw people into threat modeling, to bring in people with different skill sets and knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' I wrote ‘If it were possible to have developers do basic threat modeling, then experts could be used more effectively to find the really unusual problems with a design,’ and ‘Those who are new to threat modeling or those who threat model occasionally require a more procedural approach, and procedural approaches are generally at odds with creativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Elevation of Privilege was created in this constraint space in part to expose non-security experts to the enjoyment that security experts bring to threat modeling.’ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='3 How well does EoP work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The game has been available for a dozen years, and I reported on a variety of anecdotes to its success ranging from six-year-olds to companies that report having weekly sessions (Shostack [2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' There are at least three substantiative variants, including OWASP Cornucopia, F-Secure’s Elevation of Privacy (with 4 new suits) and LogMeIn has added a privacy suit, making STRIPED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Elevation of Privilege enables contributions from people who lack security knowledge via the hints on the cards, and requires it through the turn-taking game structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The hints are augmented by game-table 4 Fast, Cheap, Good: Lightweight Methods are Undervalued (PREPRINT) conversation and banter, offering a chance for participants to ask questions about their hints in a lower-pressure environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='4 Analysis When Elevation of Privilege was created, it was intended to be the simplest, lowest cost structured introduction to threat modeling possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Since then, faster and simpler approaches have appeared, putting the game at the more expensive end of fast, cheap approaches, taking about an hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='3 Semgrep We include discussion of semgrep because it may be to static analysis what Elevation of Privilege is to threat modeling: a lighter, more accessible approach, and it may be helpful to consider fast, cheap, good approaches beyond threat modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='1 What it is?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Semgrep is an open source, lightweight static analysis tool that’s designed to be fast, easy to deploy and easy to program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='2 Why was it designed this way?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' “Semgrep is designed for the security engineer or developer who wants to automate code review.” and “We think modern static analysis tools should run on every keystroke in the editor, without needing network access.” (Both https://semgrep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='dev/docs/faq/) These properties of easy to deploy, fast, and easy to write rules for seem intuitively useful, but they are less prioritized than soundness and completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Semgrep explicitly rejects the need to be able to handle an abstract syntax tree, stating in their FAQ “if you can write code, you can write a Semgrep rule — no program analysis PhD required!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='3 How well does it work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Semgrep has a ‘trophy case’ on their website with 8 CVEs, and a set of useful contributions, but its use seems to be largely internal to enterprises, which makes assessment more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Further, the team behind it seems to have a frame that value is evidenced by continued use, rather than publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' As such, the rate of rule publication may be helpful evidence of engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' As of April, 2021, Semgrep’s core was being released approximately weekly, in addition to roughly 25 new rules contributed each week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' This indicates that at least a few people are motivated to continue participating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='4 Relevant reflections Semgrep explicitly focuses on being inexpensive to learn and use, without specifying what those mean in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' They also focus on being accessible to those without a PhD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The relationship between applied and academic static analysis work was the subject of reflection by the Coverity team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Their paper on the subject discusses many of the challenges that semgrep has taken on, including troubles in deploying their tool, speed, and managing multiple languages (Bessey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' [2010]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 4 A Flaw Space and A Flaw Oracle The distribution of flaws by the system knowledge and security expertise required to find them (Darker areas have more flaws).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Issues in the lower left are what we describe as light flaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Lightweight threat modeling practices can help find them more effectively than the widely-recommended practices common today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Flaws require different levels of system knowledge, and different levels of security expertise, to discover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Some flaws, which we will herein refer to as light flaws, require a less system knowledge and/or less security expertise to uncover (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' We argue that the methods we describe above work because they help to catch light flaws, and do so with a minimum of time and training on the part of developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Existing threat modeling practices, we argue, aim to capture the entire flaw space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' In doing so, they undervalue the time, energy, and organizational costs required to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' By providing lighter practices by which non-specialists can capture and solve light flaws, security experts would be relatively less burdened, able to focus only on a subset of the flaw space 5 Fast, Cheap, Good: Lightweight Methods are Undervalued (PREPRINT) Figure 1: Figure 1 (represented by the upper-right in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The remainder of this section uses a thought experiment to expound on our reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Imagine a perfect examiner who instantly and reliably finds flaw in a system, given the correct inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' This Flaw Oracle takes as input first, knowledge of the system and second, knowledge of security (This is a thought experiment, we don’t propose such an examiner exists, nor do we claim such knowledge can be codified).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' We are focused on what flaws will be found, without specifying an underlying distribution, or requiring knowledge of the actual distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' We can also imagine that an examiner requires intensive security knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Informally, some theorem provers might be analogous to such an examiner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' (The term ‘flaw’ is often used for the design subset of security issues, complementing bugs or vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' We use it to encompass both, as security oracle is too broad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' We can also consider a Laggard Flaw Oracle (LFO), who finds flaws and emits information at a later date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The knowledge needed by the examiner can be seen as analogous to the participants in system engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The software engineers may have a great deal of knowledge of the system, and less knowledge of security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Security engineers, especially consultants, will bring knowledge of security, and less knowledge of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Knowledge is expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Each step of gathering it, checking it, encoding it into human- or machine-readable forms, checking the encoding didn’t introduce problems takes effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Similarly, keeping such information up-to-date as systems evolve requires effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' We can visualize the distribution of flaws in systems, and simple visualizations such as Figure 1 can capture intuitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Another possible distribution (not shown) is that flaws are distributed randomly and evenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Thus systems that require fewer expensive inputs can have value, if there are many issues that an examiner can find with system knowledge and a modest amount of security knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' For example, once you know that file APIs will often accept network paths, someone with knowledge of where a system opens files can easily check for remote file inclusion (RFI) bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The birthday problem is easily taught, but seeing how it applies to a system can be harder, and subtle cryptographic bugs may require both security and system knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' This account of the flaw space explains why the lightweight threat modeling practices we describe above may have worked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Further, it provides a theoretical backdrop that motivates further work on developing and evaluating lightweight threat modeling practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Readers may be concerned that this work will do more harm than good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' An analog of Gresham’s Law of “Bad money drives out good” may be relevant: bad security may drive out the good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' We disagree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The availability of inexpensive clothes does not stop people from paying for designer clothes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' each delivers a different value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 6 RFI Subtle crypto System knowledge required Birthday problem Security expertise requiredFast, Cheap, Good: Lightweight Methods are Undervalued (PREPRINT) 5 Evaluating Practices Having put forth the idea that light flaws exist, we ask: how best can we judge practices that aim to catch these flaws?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' We propose three, primary dimensions: cost, expertise requirements, and quality of output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' These are important dimensions which make up a multidimensional design space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' we expect there are other useful criteria beyond these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' These criteria exclude the impact of the output, which has surprised some early readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The usefulness of a technique is in a context, and the tradeoffs which that context demands are independent of the tool (We do not judge a hammer useless because we have a lot of hex nuts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='1 Cost Cost include money, time, and organizational energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Time includes training, preparation, and execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Some systems require only minutes of training or setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' For example, Security Fictions takes a few minutes to explain and set the stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Elevation of Privilege usually involves a few minutes of stage-setting, 5-15 minutes to draw a system diagram, and a round of test play for people to ‘get the hang of it.’ This is in contrast to a half day or even days of training in deeper threat modeling approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Some systems, like Security Fictions, requires some prep work in crafting the scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' (It is reasonable to think that a compendium of scenarios could be created, but none has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=') Organizational costs include coordination overhead associated with communicating flaws across functional teams, prioritizing issues to fix, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The more that lightweight processes reduce coordination overhead (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=', by empowering UX designers to solve problem on their own), the lower their cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The LFO adds cost by delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' It is well understood that fixing issues is more expensive after code is written, and more expensive yet when the code has dependencies which need to be at least checked and possibly fixed, or when the code is already deployed possibly at many sites, each of which must deploy the fixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Thus, the slower the LFO, the more cost it allows to creep into the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='2 Expertise requirement Figure 1 shows a model where discovering flaws require different levels of system-specific and security-specific expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Effective lightweight methods can minimize reliance on security-specific expertise, compensating by relying instead on system-specific expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' In some empirical work (Merrill and Weber [2019]), a UX designer used misleading unicode symbols to perform an impersonation attack, effectively using their systems knowledge to compensate for a lack of security-specific knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The ability of various practices to facilitate this tradeoff is critical for catching light flaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Characterizing and defining the exact security knowledge required to make good use of a tool is beyond the scope of this paper, but we’ll note briefly that we have never seen a characterization of skill except “this tool requires a PhD in static analysis.” 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='3 Quality The most challenging but most crucial dimension to measure is the quality of practices’ output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Students often seek to find “good threats.” What makes a “good threat”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' There is an argument that a “good threat” discovered by a threat modeling approach is one that the organization cares about enough to fix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' That is, the time spent to fix it is evidence of a useful output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' But there can be reasons to not fix a worrisome problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The organization may be fixing other more pressing issues, there may be issues of application compatibility or usability of a fix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' In other words, the perceived relevance of flaws is contextual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The exact same issue, discovered at (for example) Amazon, Microsoft, Bank of America or a local community bank might be treated differently at each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Bug bounties offered by each are scoped differently, rewarding different flaw types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' There may be work in assessing quality from the perspective of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' For example, in (Slupska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' [2021]) an argument is put forth for threat modeling techniques that are easier for those other than cybersecurity professionals to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' There are important dimensions of quality, such as avoiding creepy advertisements or stalking, that traditional information-security centered approaches may not prioritize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Future work should delve into these questions as they apply to the flaw space described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 6 Future work We hope to create a framework in which future approaches to threat modeling can be understood, evaluated and compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' We hope to enable work in the area of fast and cheap tools to help people analyze security, and to enable rigorous and thoughtful discussion of those tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Ease of use, speed, scope, suitability for non-technical staff at organizations building systems, suitability for use by the public, advocates, activists, and other factors are all important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 7 Fast, Cheap, Good: Lightweight Methods are Undervalued (PREPRINT) By defining the nature of tradeoffs that threat modeling tools make, we hope to enable exploration of new tools that more strategically and purposefully approach different portions of the flaw space we describe in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' While this work is highly preliminary, it raises one critical point of reflection for future research: our observation that a flaw space, distributed across system and security expertise, might exist begs a cultural shift in the way that we, as security professionals, evaluate security processes and outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' How do we judge the quality of processes and outcomes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Analyses, typically received from academia, focus on the soundness of analysis, formality of proofs, precision and recall of algorithms, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' But perhaps quality in security is an emergent characteristic of an organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Banks and falafel stops have not only different security needs, but different ways of evaluating what kinds of security make sense for them to approach, and different competing constraints for their time and attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Sensitivity to quality as an emergent and situated characteristic of organizations allows us to judge threat modeling practices better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' It centers broad outcomes, and diverse notions of what it means to be secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The cultural shift here, for us, is to make less formal and less analytical processes acceptable by the cultural standards of security research (The author has seen a paper rejected with the comment “This is great, and it needs equations”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Moving toward these more situated notions will help us understand the work that, for example, a local falafel shop does to protect themselves as relevant and important, assessing the quality of their output that isn’t overly tied to narrow analytical frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' I return, this will help us build practices that make security more relevant and actionable to the people who need it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Our call to action: security needs better ways of meeting people where they are, understanding what they need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' An analogy to economics helps illustrate the conceptual shift required here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' In economics, a broad and useful assumption is that people are rational;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' that they have a reason to do the things that they do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' When economists observe people doing something seemingly irrational (for example, not saving as much as they “should”), economists look for the rationality that underlies that decision (for example, perhaps people don’t save because they believe they’ll die having never spent their money;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' this makes saving contextually irrational).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' As security researchers, we can do something similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' What are people doing, and what is the rationality that underlies those actions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' When people ignore our security advice, perhaps doing so is rational for them because, for example, it costs them too much in time or energy relative to their other goals (see our design dimensions above and Herley [2009]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' How can we best use those observations to design more relevant and actionable security practices?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 7 Acknowledgements This article comes from deep conversation with Nick Merrill about why these lightweight approaches are meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The author appreciates thoughtful comments from Mary Ellen Zurko, which improved this draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' References Marc Prensky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Don’t bother me, mom, i’m learning!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' : How computer and video games are preparing kids for 21st century success and how you can help!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='. Paragon house, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Clark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Abt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Serious games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' University press of America, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Adam Beautement, M Angela Sasse, and Mike Wonham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The compliance budget: managing security behaviour in organisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' In Proceedings of the 2008 New Security Paradigms Workshop, pages 47–58, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Heather Richter Lipford Garfinkel, Simson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' "usable security: History, themes, and challenges, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Loren Kohnfelder and Praerit Garg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' The threats to our products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Microsoft Interface, Microsoft Corporation, 33, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Adam Shostack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Security briefs-reinvigorate your threat modeling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' MSDN magazine, page 117, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Adam Shostack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Elevation of Privilege: Drawing Developers 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' ACM, 53(2):66–75, February 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' ISSN 0001-0782.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='1145/1646353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='1646374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='1145/1646353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content='1646374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Nick Merrill and Steven Weber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Threat Fictions: Speculative Enactments on the Edges of Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' In Proceedings of the 2019 ACM Conference on Computer Human Interaction (CHI ’19), Glasgow, UK, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Julia Slupska, Scarlet Dawson Dawson Duckworth, Linda Ma, and Gina Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Participatory threat modelling: Exploring paths to reconfigure cybersecurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' In Extended Abstracts of the 2021 CHI Conference on Human Factors in Computing Systems, pages 1–6, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' Cormac Herley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' So long, and no thanks for the externalities: the rational rejection of security advice by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' In Proceedings of the 2009 workshop on New security paradigms workshop, pages 133–144, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} +page_content=' 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE2T4oBgHgl3EQfAgbT/content/2301.03593v1.pdf'} diff --git a/L9AzT4oBgHgl3EQfyv7I/content/tmp_files/2301.01759v1.pdf.txt b/L9AzT4oBgHgl3EQfyv7I/content/tmp_files/2301.01759v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7f34b5c4d85027db5952e8f0740d4a458c370f45 --- /dev/null +++ b/L9AzT4oBgHgl3EQfyv7I/content/tmp_files/2301.01759v1.pdf.txt @@ -0,0 +1,761 @@ +Microgrid Optimal Energy Scheduling +with Risk Analysis +Ali Siddique +Department of Electrical and Computer +Engineering +University of Houston +Houston, TX, USA +asiddique2@uh.edu +Cunzhi Zhao +Department of Electrical and Computer +Engineering +University of Houston +Houston, TX, USA +czhao20@uh.edu +Xingpeng Li +Department of Electrical and Computer +Engineering +University of Houston +Houston, TX, USA +xli82@uh.edu + +Abstract—Risk analysis is currently not quantified in +microgrid resource scheduling optimization. This paper conducts +a conditional value at risk (cVaR) analysis on a grid-disconnected +residential microgrid with distributed energy resources (DER). +We assume the infrastructure to set up an ad-hoc microgrid is +already in place for a residential neighborhood with power +sources such as photovoltaic (PV), diesel, and battery energy +storage system (BESS). With this scenario in mind, we solve day- +ahead scheduling to optimally allocate various resources to match +demand in scenarios where neighborhoods, especially residential, +are disconnected from the overall grid such as in flooding, +hurricanes, winter storms, or operational failures. The goal is to +provide an alternative framework to optimize power availability +for priority customers and strengthen the overall grid against +dips in power outside of normal operating considerations. The +focus of this paper will be taking in renewable energy sources +from PV combined with diesel and BESS while minimizing cost. +Case studies demonstrate that with the proposed energy +management system, microgrids can be implemented to be more +resilient against new challenges. +Keywords— Battery degradation, Conditional value at risk, +Day-ahead scheduling, Energy management system, Microgrid, +Risk management, Optimization. +Nomenclature +������������������������������������ +Priority customer demand defined as customers where +electricity cannot be curtailed. +������������������������������������ +Essential customer demand. Defined as residential +customers whose electricity can be curtailed. +������������������������������������������������ +Essential customer curtailed. Defined as residential +customers whose electricity is curtailed. +������������������������������������������������ ������������������������������������������������������������ Demand load of the system subtracted from any +residential PV that is generated. +������������������������������������������������������������������������ +Demand total for all customers in microgrid. +������������������������������������������������������������������������ +Power of the battery energy storage system. +������������������������������������������������ +Power value of photovoltaic residential solar panels. +������������������������������������������������ +Power output of diesel generator. +������������������������������������������������������������������������ +Power output minimum for diesel generator. +������������������������������������������������������������������������ +Power output maximum for diesel generator. +������������������������������������������������������������������������������������ +Total available power. +������������������������������������������������������������ +������������ + +The maximum discharge power of the battery. +������������������������������������ +������������ +The discharging power of the battery. +������������������������������������ +������������ +The charging power of the battery. +������������������������ ������������������������������������ +������������ + +The maximum charge power of the battery. +������������������������������������������������������������,������������������������������������ +The additional price ($) of the battery cost when the +battery state of charge is outside the green zone. +������������������������������������������������������������ +Fuel cost ($/kW) of diesel generation. +������������������������������������������������ +Degradation cost ($) of the battery. +������������������������������������������������������������������������������������ +The capital cost ($) of the battery. +������������������������������������ +Cost ($/kW) of curtailing essential customers. +������������������������ +Binary value indicating whether the battery state of +charge is outside the green zone. +������������������������������������ +Binary value for charging status. +������������������������������������ +Binary value for discharging status. +������������������������������������������������ +Binary state determining if diesel generator is on or off. +������������������������������������������������ +The energy state of the battery. +������������������������������������������������������������������������ +������������������������������������������������������������ Minimum state of charge at which normal battery +degradation can occurs. +������������������������������������������������������������������������ +������������������������������������������������������������ Maximum state of charge at which normal battery +degradation can occurs. +������������������������������������������������������������������������ +Minimum possible state of charge for battery. +������������������������������������������������������������������������ +Maximum possible state of charge for battery. +������������������������������������������������ +Depth of discharge for battery. +������������������������������������������������ +Depth of charge for battery. +������������ +Number of scenarios. +������������������������������������������������,������������������������������������������������ +Rated maximum number of cycles of battery system. +������������������������������������������������������������ +Battery cycle count. +������������������������������������������������ +Conditional value at risk formulation used to calculate +risk at high-risk low probability scenarios. +������������������������������������ +Possible value at risk. +������������ +Smallest possible cost for admissible loss. +������������(������������, ������������) +Unmet demand after generation is accounted. +������������ +Confidence level. +������������������������ +BESS power and diesel power of the specific interval. +������������ +Time segment per analysis. +∆λ +Change in degradation between two-time intervals. +������������������������������������������������������������������������ +The capacity factor loss at the Nth cycle. +∆������������ +The length of time segment. +������������ +Price normalizer value ($/cycle). +������������ +Relationship between essential and priority +customers. + +I. INTRODUCTION +here have been 500 weather events in North America +impacting 50,000 customers for each event from +2005-2015 [1]. Similar increased electricity outages due to +weather have been reported on other continents. These +increases in the severity of natural disasters are due to the +forces of climate change [2]. Also, blackouts have occurred +due to operational errors resulting in millions of customers +losing power [3]. Lastly, attacks against the grid have become +more common from foreign actors [4]. Both trends have +emphasized the need for a more distributed and decentralized +electric grid which should function to some extent even if +disconnected from the overall electric utility. +A microgrid is defined by the Department of Energy as “a +group of interconnected loads and distributed energy +resources within clearly defined electrical boundaries that acts +as a single controllable entity with respect to the grid” [5]. +T + +Microgrid technology has become increasingly more common +in the past few decades due to its ability to supply areas with +geographical constraints, disaster prone issues, and rural areas. +It is also an effective tool for electricity distribution and +reliability. Additionally, a microgrid has the capacity to +disconnect from the main grid and be self-sufficient for a +period of time but it can also remain connected and function +alongside a larger grid system in normal operations. This is +essential in a blackout or disaster scenario since a microgrid +can disconnect from the other supply issues or even equipment +damage that could be occurring elsewhere. This allows the +microgrid to avoid cascading failures and provide reliable +power in its specific service area [6]. +The focus of this paper will be on the microgrid’s ability to +disconnect from the larger electric grid in a time of outages and +be able to reliably provide power to a specific section +otherwise referred as an island state. However, this requires +that a microgrid have its own energy management system +(EMS) and far more refined control methods than a traditional +EMS since both the energy demand and consumption is at a far +more granular level [7]-[10]. These enhanced requirements are +implemented in this paper with two systems. Firstly, the day- +ahead scheduling is used to optimize resource allocation since +an emergency usually unfolds on a day-to-day basis. This +system also makes sure that demand is being met. Lastly, it +also allows cost approximation to allocate the correct energy +supply ensuring effectiveness and ideal dispatching [11]. +In addition to physical infrastructure, new forms of EMS +including intermittent energy such as solar panels must be +considered +for +resource +allocation +[11]. +Microgrid +functionality must be built into the system as more microgrids +are being integrated or being developed alongside the main +grid. This will have far reaching consequences in energy +management systems as large changes in both the generation +and consumption of energy are rapidly shifting. +The energy management system in a regular electrical +system has incredible reliability and is a marvel of the modern +world. Unfortunately, this reliability and interconnectedness is +only guaranteed for normal conditions. The electric grid’s +ability to respond to issues under abnormal conditions such as +storms, flooding, or other disasters may be reduced [6]. +This paper primarily focus on such circumstances where +the normal standards for reliability are not available. The high +standard is only possible due to a vast and durable +interconnected system which relies on large-scale generation +transmitted +to +distributed +residential +systems. +These +infrastructure advantages are guaranteed in a natural disaster +where due to damage, the system can be disconnected into +multiple sections. When this happens, individual residential +homes or industrial systems must have previously installed +redundant energy resources such as diesel generation or BESS. +Otherwise, their ability to receive electricity is entirely +dependent on the speed at which the whole system can be +reintegrated into a default state [12]. +Therefore, advanced EMS software is necessary along with +more resilient physical assets to harden the overall grid [1]. +There are also new forms of distributed generation which +change the dynamics of power transmission. All these factors +require a rethinking of acceptable risk which currently is not +acknowledged for existing systems. This paper utilizes day- +ahead scheduling with specific time segments by assigning +certain cost objectives to various resources including solar +power, load curtailment, BESS, and diesel generation. This +allows the model to create the most effective mix of resources +to supply a load while minimizing resource usage throughout +the day. +This paper presents one such approach to reduce +unreliability by looking at day-ahead scheduling resource +allocation which is then analyzed through a risk management +method specifically a conditional value at risk (cVaR) analysis +method to determine the risk factor of load curtailment +throughout the day. This framework points out how +intermittent resources and non-critical load curtailment can +increase reliability [13]–[14]. The goal is to understand that not +only load curtailment can be necessary in certain situations but +how to quantify this necessity to ensure that system reliability +is maximized in an emergency. It also creates a starting point +to discuss instances where property that is currently controlled +by individual use can be used in a more communal manner. +This will allow a more sophisticated conversation about non- +critical load curtailment instead of the current reality of +demand reduction occurring haphazardly [15]. +The remainder of the paper is organized as follows. Section +II presents and describes the mathematic model for microgrid +optimal scheduling. Section III presents the proposed cVaR +analysis framework. Case study is presented in Section IV. +Finally, Section V concludes the paper. +II. MATHEMATICAL MODEL +The objective function in this paper is to maximize power +availability for priority customers by minimizing risk and cost +of volatile power generation sources. +min �{������������������������������������������������������������������������������������������������������������������������ + ������������������������������������������������������������������������������������������������������������ + ������������������������������������������������������������������������������������ ++ ������������������������������������������������������������������������������������,������������������������������������} +(1) +The objective function represented by (1) is a variation of +the cost function of traditional unit commitment models +showing resource allocation for BESS, diesel, and load +curtailment while balancing demand and PV generation. +������������������������������������������������������������������������ − ������������������������������������������������ = ������������������������������������������������ + ������������������������������������������������������������������������ + ������������������������������������������������ +(2) +Constraint (2) represents a basic requirement for all electric +grid operations ensuring that demand meets supply. The usage +of ������������������������������������������������ to minimize demand will be explained in the Load +Curtailment section. Diesel systems are a useful fuel source +around the world in grid operations as a DER alongside BESS +and residential PV [3]. As a base constraint, there is a +maximum discharge and charge capacity for diesel generators +to meet technical limitations as +������������������������������������������������������������������������ ≤ ������������������������������������������������ ≤ ������������������������������������������������������������������������ +(3) +Equation (4) defines the fundamental connection between how +demand is configured in the system. Constraint (5) sets the +grouping of priority customers and essential customers. +Priority customers are a fraction defined by ������������ of the essential +customers. In the essential customer group, only ������������������������������������������������ is +defined as essential customer curtailed are removed from the +system as shown in (6). Equation (7) limits the ������������������������������������ and ������������������������������������������������. +(8) – (10) enforces the BESS status to be charging, discharging +or idle. Constraints (11)-(13) limit the charging and + +discharging power. Equation (14) defines the cost factor for +any usage of the battery outside the green zone. +������������������������������������������������ ������������������������������������������������������������ = ������������������������������������ + ������������������������������������ − ������������������������������������������������ +(4) +������������������������������������ = ������������������������������������������������ +(5) +������������������������������������ + ������������������������������������ − ������������������������������������������������ = ������������������������������������������������������������������������������������ +(6) +������������������������������������ ≥ ������������������������������������������������ ≥ 0 +(7) +������������������������������������{1, ������������ℎ������������������������������������������������������������������������ ������������������������������������������������������������. 0, ������������������������������������ ������������ℎ������������������������������������������������������������������������ } +(8) +������������������������������������ {1, ������������������������������������������������ℎ������������������������������������������������������������������������ ������������������������������������������������������������ 0, ������������������������������������ ������������������������������������������������ℎ������������������������������������������������������������������������} +(9) +������������������������������������ + ������������������������������������ ≤ 1 +(10) +������������������������������������������������������������������������ = ������������������������������������ +������������ − ������������������������������������ +������������ +(11) +0 ≤ ������������������������������������ +������������ ≤ ������������������������������������������������������������ ������������������������������������ +������������ + +(12) +0 ≤ ������������������������������������ +������������ ≤ ������������������������������������������������������������ ������������������������������������ +������������ + +(13) +� +������������������������������������������������������������������������ +������������������������������������������������������������ ≤ ������������������������������������������������ ≤ ������������������������������������������������������������������������ +������������������������������������������������������������ , ������������������������ = 0 +������������������������������������������������������������������������ +������������������������������������������������������������ > ������������������������������������������������ ������������������������ ������������������������������������������������ > ������������������������������������������������������������������������ +������������������������������������������������������������ , + ������������������������ = 1 + +(14) +The cost of the battery system is connected to the maximum +life cycle to calculate the overall cost of the battery as +connected to cycle count. This allows us to take a specific +portion of battery usage such as one day and connect it to the +overall cost of the battery by (15). ������������������������������������������������������������ in (16) is the number +of cycles the battery is at while ������������������������������������������������������������������������ is the capacity factor +loss at the Nth cycle. Equation (17) defines the total cost of the +BESS and (18) represents the difference of the degradation +cost between different time intervals. +������������������������������������������������������������ = ∑ +1 +2 (������������������������������������������������ + ������������������������������������������������) +������������ +������������=0 + +(15) +������������������������������������������������,������������������������������������������������ − ������������������������������������������������������������ = +������������������������������������������������,������������������������������������������������(1 − ������������������������������������������������������������������������) +������������������������������������������������������������������������ ∗ ������������ + +(16) +������������������������������������������������ = ������������������������������������������������������������������������ ∗ ������������������������������������������������������������������������������������ +(17) +∆������������ = ������������������������������������������������������������������������ − ������������������������������������������������������������������������−1 +(18) +III. PROPOSED CVAR FRAMEWORK +This section explains how the costs defined in the model for +day-ahead scheduling is used in the cVaR framework. Fig. 1 +presents the process from day-ahead scheduling to cVaR +analysis. First, all the scenarios in the day-ahead scheduling +must be completed. This means that for one time interval, t, +there will be hundreds of scenarios operating with different +demand constraints and PV generation. Then, when all N +scenarios have been completed, they will create a large set of +data points of cost optimized resource allocation including any +possible +load +curtailment. +These +load +curtailment +measurements can be tested for stability and resiliency and +used to create a risk profile using cVaR analysis. + + +Figure 1. Procedure of the proposed microgrid scheduling and risk analysis. + +A. cVaR Formulation +The use of risk-constrained scenarios in financial models +and utilities is to maximize profit with an internal pricing +mechanism [16]. cVaR is a popular risk calculation algorithm. +It is built on the work of value at risk (VaR) which calculates +how to reduce risk within a certain confidence level (β) by +minimizing loss due to the uncertainty in specific variables +[16] otherwise defined as equation (19). The ������������(������������, ������������) factor in +(19) is defined as the losses calculated. The x denotes the +variables available to fine tune and reduce risk where y +represents the volatile uncertainty inherent in our system. By +minimizing the worst-case scenario of y, the system could +create an expected risk profile. This is calculated by taking the +smallest possible cost (α) that is greater or equal to ������������(������������, ������������) and +then calculated the risk factor over β. This can be used to +calculate the level of risk inherent in investing in certain +markets and diversification tools (such as cash or bond +hedging). +������������������������������������ = ������������������������������������{������������ ∈ ������������: ������������{������������(������������, ������������) ≤ ������������} ≥ ������������} ������������������������������������ 0 +≤ ������������ ≤ 1 +(19) +Unfortunately, VaR suffers from two key issues. +Mathematically, it has a lack of convexity and subadditivity +making it non-ideal for intensive calculation operations. +Secondly, VaR only minimizes losses within a given +confidence level and does not consider losses occurring at a +confidence level outside of its boundaries at 1-β. cVaR allows +a better grasp for situations where a small likelihood of risk +could have a huge effect [17]. cVaR as a financial constraint is +seen in equation (20). +������������������������������������������������ = ������������������������(������������(������������, ������������)|������������(������������, ������������) ≥ ������������������������������������) +(20) +In this evolution of the original VaR equation, the cVaR is +now taking the expected value of random variables above its +VaR consideration. In other words, it is taking the loss factors +inherent in the system and calculating them in situations of 1- +β or above the standard confidence interval. This is a much +more robust and flexible system since it allows forecasting of +situations where non-likely events outside of the confidence +interval occur. Additionally, a higher cVaR means the system +is inherently less stable because in non-normal situations, the +losses can be considerably higher. To transition from the above +equations to models with samples, (20) can be converted into +equation (21). +������������������������������������������������ = min � ������������ + +1 +������������(1 − ������������) �[������������(������������, ������������) +������������ +������������=1 +− ������������]+ � +(21) +The first shift here is the addition of N moving the model +from continuous to samples with N scenarios. The second +change is that the positive component of our losses taken by +known x and volatile y subtracted by α as our hedging cost. For +our formulation, we can then replace [������������(������������, ������������) − ������������]+ with ������������������������ as +shown in (22). The cVaR equation can now be redefined with +������������������������as seen in (23). +������������������������ = [������������(������������, ������������) − ������������]+ +(22) +������������������������������������ = ������������������������������������ � ������������ + +1 +������������(1−������������) ∑ +������������������������ +������������ +������������=1 +� +(23) +B. cVaR Application in Microgrid +This section explains how cVaR will be used to maximize +power reliability for priority customers. cVaR gives a weighted +average of risk above the normal confidence level. This allows +a calculation of the risk in high-demand scenarios that can +occur in emergency situations. Equation (24) takes ������������(������������, ������������) + +No +Demand +EdgeCaseSummation +ResidentialPV +DayAhead +NScenario +Yes +NScenarios +cVaRAnalysis +SystemRisk +Scheduling +Completed? +Evaluation +t Time Segmentsfrom (19) and defines the combined losses as demand +subtracted from diesel, PV, and BESS [14]. ������������ in (25) is set as +the smallest load curtailment while maintaining stability at the +confidence level β [16]. The α value will be measured in units +of kilowatts. While keeping with cVaR convention, demand +load that is curtailed will be referred to as α moving forward. +All this can be represented as: +������������(������������, ������������) = ������������������������������������������������ ������������������������������������������������������������ − ������������������������������������������������������������������������ − ������������������������������������������������ +(24) +������������������������ = [������������������������������������������������ ������������������������������������������������������������ − ������������������������������������������������������������������������ − ������������������������������������������������ − ������������]+ +(25) +IV. CASE STUDIES +The test residential microgrid is designed with currently +available commercial products. It is designed with a battery +system made up of twenty Tesla Powerwall batteries with a +capacity of 15 kWh each that starts at an initial value of 10 +kWh for each battery. ������������������������ is the capital cost of the BESS system +at $10,000 [18]. The standard rooftop residential solar output +is at 4 kW during peak solar generation [19]. There are ten +residential homes in need of power all with installed solar +panels. ������������ in (5) is set to 0.5 therefore priority customers were a +total of 33% of total demand. This means that a maximum of +66% of customers can be essential customers [11]. A +simulation of all 187 possible scenarios, N, is run and the +battery, diesel, and PV combination are recorded for each +specific segment. ������������ is defined as 5% for the confidence level +in this analysis. +The input data for the scenarios including load demand and +PV generation is graciously provided by Pecan Street. This is +part of Pecan Street’s Dataport Project [19] which includes the +world’s largest resource for residential energy use data, electric +transportation and has been expanded to include residential +water usage, electric transportation, and regenerative +agriculture [20]. Electricity demand as well as PV generation +will have expected statistical deviation from historical data. +������������������������������������������������������������������������ +������������������������������������������������������������ is defined as 20% and ������������������������������������������������������������������������ +������������������������������������������������������������ as 80% in the +model using values from previous research [21]. ������������������������������������������������������������������������ is set +as 0 and ������������������������������������������������������������������������ is set as 3.75 kW for the system generator. The +diesel generator is assumed to have sufficient fuel to operate +during the whole course of the day. ������������������������ ������������������������������������ +������������ + and ������������������������ ������������������������������������ +������������ + is +defined as 5 kW for one Tesla Powerwall [18]. ∆T represents +the length of time segment which is 15 minutes in this paper. +There are 96 segments for a 24-hour period. The load +curtailment if any for each fifteen-minute interval is recorded. +The load curtailment is divided by the total demand supplied +and recorded in a matrix. Python is used to take these values +and calculate the conditional value at risk for the most +demanding and highest load curtailment of the five percent of +scenarios (nine scenarios) of the total set of 187 scenarios for +all time segments. +The results highlight the cVaR analysis on the microgrid +system for one full day or 96 segments on a total of 187 +scenarios. Fig. 2 shows in how many instances curtailment was +necessary in the model. This showcases a high level of self- +sufficient reliability that above would be a boon to the existing +electrical grid infrastructure. The system had zero instances of +load curtailment for 90% of scenarios. It had a maximum of 13 +instances of load curtailment in the most challenging 5% of +cases for all segments. +From a system wide load curtailment view, now we can +take a more in depth look at the 5% of challenging scenarios in +terms of balancing generation and demand. The standard +deviation shown in Fig. 3 presents the difference in values of +the dataset for each segment. + +Figure 2. Time segments with Active Curtailment. + + +Figure 3. Standard deviation of load curtailment in cVaR analysis. + +Within each segment, there is a 20-30% standard deviation +indicating the model is robust. These results prove that the +model can take in very different demand constraints and +respond appropriately to the need of the specific scenario. +Interestingly, the standard deviation is largely consistent +throughout the day, indicating that the load curtailment +deviation is not too different between sample segments. An +exception to this is late mornings to end of the afternoon when +the generation of residential PV are sufficient, there are far less +load curtailments and therefore the standard deviation is lower. + + +Figure 4. Active curtailment during an entire day. + +Fig. 4 presents the twenty-six scenarios or 13.9% of the +entire scenario dataset that was responsible for all load +curtailment. This is expected since the model was tested on a +robust dataset which has microgrid scenarios with larger than +expected demands. This is very likely in emergency situations +due to weather conditions, and it is important to note how the +microgrid would react in these scenarios. + +0 +2 +4 +6 +8 +10 +12 +14 +0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 +Number of scenarios +Hour of Day +0 +5 +10 +15 +20 +25 +30 +35 +0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 +Standard Deviation +(%) +Hour of Day +19 +23 +48 +43 +20 +16 +33 +2 +15 +24 +14 +5 +7 +15 +1 +6 +37 +12 +25 +1 +10 +11 +21 +27 +16 +15 +A B C D E F G H I +J K L M N O P Q R S T U V W X Y Z +INSTANCE +SCENARIOS + + +Figure 5. The cVAR analysis at a 5% confidence level. + +As shown in Fig. 5, the behavior of the case study matched +expectations in the following ways. The risk when calculating +real time energy management for the hours of 10 AM to 4 PM +were reduced and in some time-segments brought to zero. This +means residential solar times matched demand at these times +and reduced risk of load curtailment. This is one of the main +benefits of residential solar. It especially helps microgrids in +providing a power source for a part of the day. Load +curtailment was expected to be used in a small percentage of +the case. The correct and targeted load curtailments can +improve the system’s reliability for priority customers. This is +complementary to grid hardening efforts but has the advantage +of lower costs because it can be built with existing +infrastructure. +V. CONCLUSIONS +cVaR analysis is conducted in a stand-alone microgrid +alongside day ahead scheduling in this paper. The proposed +energy management system demonstrates the adaptability of a +multitude of generation sources being utilized along with load +curtailment in different demand-constraint scenarios. The +objective was to conduct a risk assessment on a microgrid +system to assess likelihood of load curtailment. 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Sirouspour, “Optimal Control of Energy Storage +in a Microgrid by Minimizing Conditional Value-at-Risk,” in IEEE +Transactions on Sustainable Energy, vol. 7, no. 3, pp. 1264-1273, July +2016. +[18] “Powerwall,” +Tesla +Inc., +11-Jun-2019. +[Online]. +Available: +https://www.tesla.com/sites/default/files/pdfs/powerwall/Powerwall%2 +02_AC_Datasheet_en_northamerica.pdf. +[19] “ Pecan Street Dataport,” Pecan Street Inc., 02-Dec-2021. [Online]. +Available: https://www.pecanstreet.org/dataport/. [Accessed: 21-Mar- +2022]. +[20] B. McCracken, M. Crosby, C. Holcomb, S. Russo, and C. Smithson, +Data-Driven Insights From the Nations Deepest Ever Research on +Customer Energy Use, Pecan Res. Inst., Austin, TX, USA, 2013. +[21] M. Koller, T. Borsche, A. Ulbig and G. Andersson, “Defining a +degradation cost function for optimal control of a battery energy storage +system,” 2013 IEEE Grenoble Conference, 2013, pp. 1-6. +0% +10% +20% +30% +40% +50% +60% +0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 +Percentage of Load +Curtailment +Hour of Day + diff --git a/L9AzT4oBgHgl3EQfyv7I/content/tmp_files/load_file.txt b/L9AzT4oBgHgl3EQfyv7I/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..82b6a5bfb7a2bfd938539615c5b400a52f975936 --- /dev/null +++ b/L9AzT4oBgHgl3EQfyv7I/content/tmp_files/load_file.txt @@ -0,0 +1,449 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf,len=448 +page_content='Microgrid Optimal Energy Scheduling with Risk Analysis Ali Siddique Department of Electrical and Computer Engineering University of Houston Houston, TX, USA asiddique2@uh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='edu Cunzhi Zhao Department of Electrical and Computer Engineering University of Houston Houston, TX, USA czhao20@uh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='edu Xingpeng Li Department of Electrical and Computer Engineering University of Houston Houston, TX, USA xli82@uh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='edu Abstract—Risk analysis is currently not quantified in microgrid resource scheduling optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This paper conducts a conditional value at risk (cVaR) analysis on a grid-disconnected residential microgrid with distributed energy resources (DER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' We assume the infrastructure to set up an ad-hoc microgrid is already in place for a residential neighborhood with power sources such as photovoltaic (PV), diesel, and battery energy storage system (BESS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' With this scenario in mind, we solve day- ahead scheduling to optimally allocate various resources to match demand in scenarios where neighborhoods, especially residential, are disconnected from the overall grid such as in flooding, hurricanes, winter storms, or operational failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The goal is to provide an alternative framework to optimize power availability for priority customers and strengthen the overall grid against dips in power outside of normal operating considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The focus of this paper will be taking in renewable energy sources from PV combined with diesel and BESS while minimizing cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Case studies demonstrate that with the proposed energy management system, microgrids can be implemented to be more resilient against new challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Keywords— Battery degradation, Conditional value at risk, Day-ahead scheduling, Energy management system, Microgrid, Risk management, Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Nomenclature ������������������������������������ Priority customer demand defined as customers where electricity cannot be curtailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������ Essential customer demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Defined as residential customers whose electricity can be curtailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������ Essential customer curtailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Defined as residential customers whose electricity is curtailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������ ������������������������������������������������������������ Demand load of the system subtracted from any residential PV that is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������������������ Demand total for all customers in microgrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������������������ Power of the battery energy storage system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������ Power value of photovoltaic residential solar panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������ Power output of diesel generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������������������ Power output minimum for diesel generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������������������ Power output maximum for diesel generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������������������������������ Total available power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������ ������������ The maximum discharge power of the battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������ ������������ The discharging power of the battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������ ������������ The charging power of the battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������ ������������������������������������ ������������ The maximum charge power of the battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������,������������������������������������ The additional price ($) of the battery cost when the battery state of charge is outside the green zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������ Fuel cost ($/kW) of diesel generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������ Degradation cost ($) of the battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������������������������������ The capital cost ($) of the battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������ Cost ($/kW) of curtailing essential customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������ Binary value indicating whether the battery state of charge is outside the green zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������ Binary value for charging status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������ Binary value for discharging status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������ Binary state determining if diesel generator is on or off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������ The energy state of the battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������������������ ������������������������������������������������������������ Minimum state of charge at which normal battery degradation can occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������������������ ������������������������������������������������������������ Maximum state of charge at which normal battery degradation can occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������������������ Minimum possible state of charge for battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������������������ Maximum possible state of charge for battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������ Depth of discharge for battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������ Depth of charge for battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������ Number of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������,������������������������������������������������ Rated maximum number of cycles of battery system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������ Battery cycle count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������ Conditional value at risk formulation used to calculate risk at high-risk low probability scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������ Possible value at risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������ Smallest possible cost for admissible loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������(������������, ������������) Unmet demand after generation is accounted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������ Confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������ BESS power and diesel power of the specific interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������ Time segment per analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ∆λ Change in degradation between two-time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������������������ The capacity factor loss at the Nth cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ∆������������ The length of time segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������ Price normalizer value ($/cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������ Relationship between essential and priority customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' INTRODUCTION here have been 500 weather events in North America impacting 50,000 customers for each event from 2005-2015 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Similar increased electricity outages due to weather have been reported on other continents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' These increases in the severity of natural disasters are due to the forces of climate change [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Also, blackouts have occurred due to operational errors resulting in millions of customers losing power [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Lastly, attacks against the grid have become more common from foreign actors [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Both trends have emphasized the need for a more distributed and decentralized electric grid which should function to some extent even if disconnected from the overall electric utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' A microgrid is defined by the Department of Energy as “a group of interconnected loads and distributed energy resources within clearly defined electrical boundaries that acts as a single controllable entity with respect to the grid” [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' T Microgrid technology has become increasingly more common in the past few decades due to its ability to supply areas with geographical constraints, disaster prone issues, and rural areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' It is also an effective tool for electricity distribution and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Additionally, a microgrid has the capacity to disconnect from the main grid and be self-sufficient for a period of time but it can also remain connected and function alongside a larger grid system in normal operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This is essential in a blackout or disaster scenario since a microgrid can disconnect from the other supply issues or even equipment damage that could be occurring elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This allows the microgrid to avoid cascading failures and provide reliable power in its specific service area [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The focus of this paper will be on the microgrid’s ability to disconnect from the larger electric grid in a time of outages and be able to reliably provide power to a specific section otherwise referred as an island state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' However, this requires that a microgrid have its own energy management system (EMS) and far more refined control methods than a traditional EMS since both the energy demand and consumption is at a far more granular level [7]-[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' These enhanced requirements are implemented in this paper with two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Firstly, the day- ahead scheduling is used to optimize resource allocation since an emergency usually unfolds on a day-to-day basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This system also makes sure that demand is being met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Lastly, it also allows cost approximation to allocate the correct energy supply ensuring effectiveness and ideal dispatching [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' In addition to physical infrastructure, new forms of EMS including intermittent energy such as solar panels must be considered for resource allocation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Microgrid functionality must be built into the system as more microgrids are being integrated or being developed alongside the main grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This will have far reaching consequences in energy management systems as large changes in both the generation and consumption of energy are rapidly shifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The energy management system in a regular electrical system has incredible reliability and is a marvel of the modern world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Unfortunately, this reliability and interconnectedness is only guaranteed for normal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The electric grid’s ability to respond to issues under abnormal conditions such as storms, flooding, or other disasters may be reduced [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This paper primarily focus on such circumstances where the normal standards for reliability are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The high standard is only possible due to a vast and durable interconnected system which relies on large-scale generation transmitted to distributed residential systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' These infrastructure advantages are guaranteed in a natural disaster where due to damage, the system can be disconnected into multiple sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' When this happens, individual residential homes or industrial systems must have previously installed redundant energy resources such as diesel generation or BESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Otherwise, their ability to receive electricity is entirely dependent on the speed at which the whole system can be reintegrated into a default state [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Therefore, advanced EMS software is necessary along with more resilient physical assets to harden the overall grid [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' There are also new forms of distributed generation which change the dynamics of power transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' All these factors require a rethinking of acceptable risk which currently is not acknowledged for existing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This paper utilizes day- ahead scheduling with specific time segments by assigning certain cost objectives to various resources including solar power, load curtailment, BESS, and diesel generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This allows the model to create the most effective mix of resources to supply a load while minimizing resource usage throughout the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This paper presents one such approach to reduce unreliability by looking at day-ahead scheduling resource allocation which is then analyzed through a risk management method specifically a conditional value at risk (cVaR) analysis method to determine the risk factor of load curtailment throughout the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This framework points out how intermittent resources and non-critical load curtailment can increase reliability [13]–[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The goal is to understand that not only load curtailment can be necessary in certain situations but how to quantify this necessity to ensure that system reliability is maximized in an emergency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' It also creates a starting point to discuss instances where property that is currently controlled by individual use can be used in a more communal manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This will allow a more sophisticated conversation about non- critical load curtailment instead of the current reality of demand reduction occurring haphazardly [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Section II presents and describes the mathematic model for microgrid optimal scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Section III presents the proposed cVaR analysis framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Case study is presented in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Finally, Section V concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' MATHEMATICAL MODEL The objective function in this paper is to maximize power availability for priority customers by minimizing risk and cost of volatile power generation sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' min �{������������������������������������������������������������������������������������������������������������������������ + ������������������������������������������������������������������������������������������������������������ + ������������������������������������������������������������������������������������ + ������������������������������������������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������������������������������} (1) The objective function represented by (1) is a variation of the cost function of traditional unit commitment models showing resource allocation for BESS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' diesel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' and load curtailment while balancing demand and PV generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������������������ − ������������������������������������������������ = ������������������������������������������������ + ������������������������������������������������������������������������ + ������������������������������������������������ (2) Constraint (2) represents a basic requirement for all electric grid operations ensuring that demand meets supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The usage of ������������������������������������������������ to minimize demand will be explained in the Load Curtailment section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Diesel systems are a useful fuel source around the world in grid operations as a DER alongside BESS and residential PV [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' As a base constraint, there is a maximum discharge and charge capacity for diesel generators to meet technical limitations as ������������������������������������������������������������������������ ≤ ������������������������������������������������ ≤ ������������������������������������������������������������������������ (3) Equation (4) defines the fundamental connection between how demand is configured in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Constraint (5) sets the grouping of priority customers and essential customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Priority customers are a fraction defined by ������������ of the essential customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' In the essential customer group, only ������������������������������������������������ is defined as essential customer curtailed are removed from the system as shown in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Equation (7) limits the ������������������������������������ and ������������������������������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' (8) – (10) enforces the BESS status to be charging, discharging or idle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Constraints (11)-(13) limit the charging and discharging power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Equation (14) defines the cost factor for any usage of the battery outside the green zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������������������������������������������ ������������������������������������������������������������ = ������������������������������������ + ������������������������������������ − ������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������������������������������ = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='(5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������������������������������ + ������������������������������������ − ������������������������������������������������ = ������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='(6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������������������������������ ≥ ������������������������������������������������ ≥ 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='(7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������������������������������{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������ℎ������������������������������������������������������������������������ ������������������������������������������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������ ������������ℎ������������������������������������������������������������������������ } (8) ������������������������������������ {1,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='0 ≤ ������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������ ≤ ������������������������������������������������������������ ������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='(12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='0 ≤ ������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������ ≤ ������������������������������������������������������������ ������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='(13) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������������������������������������������������������ ≤ ������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='≤ ������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������ = 0 ������������������������������������������������������������������������ ������������������������������������������������������������ > ������������������������������������������������ ������������������������ ������������������������������������������������ > ������������������������������������������������������������������������ ������������������������������������������������������������ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������ = 1 (14) The cost of the battery system is connected to the maximum life cycle to calculate the overall cost of the battery as connected to cycle count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This allows us to take a specific portion of battery usage such as one day and connect it to the overall cost of the battery by (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������ in (16) is the number of cycles the battery is at while ������������������������������������������������������������������������ is the capacity factor loss at the Nth cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Equation (17) defines the total cost of the BESS and (18) represents the difference of the degradation cost between different time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������ = ∑ 1 2 (������������������������������������������������ + ������������������������������������������������) ������������ ������������=0 (15) ������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������������������������������������������ − ������������������������������������������������������������ = ������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������������������������������������������(1 − ������������������������������������������������������������������������) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������������������������������������������������������������������ ∗ ������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='(16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������������������������������������������ = ������������������������������������������������������������������������ ∗ ������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='(17) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='∆������������ = ������������������������������������������������������������������������ − ������������������������������������������������������������������������−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='(18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' PROPOSED CVAR FRAMEWORK This section explains how the costs defined in the model for day-ahead scheduling is used in the cVaR framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' 1 presents the process from day-ahead scheduling to cVaR analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' First, all the scenarios in the day-ahead scheduling must be completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This means that for one time interval, t, there will be hundreds of scenarios operating with different demand constraints and PV generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Then, when all N scenarios have been completed, they will create a large set of data points of cost optimized resource allocation including any possible load curtailment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' These load curtailment measurements can be tested for stability and resiliency and used to create a risk profile using cVaR analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Procedure of the proposed microgrid scheduling and risk analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' cVaR Formulation The use of risk-constrained scenarios in financial models and utilities is to maximize profit with an internal pricing mechanism [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' cVaR is a popular risk calculation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' It is built on the work of value at risk (VaR) which calculates how to reduce risk within a certain confidence level (β) by minimizing loss due to the uncertainty in specific variables [16] otherwise defined as equation (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The ������������(������������, ������������) factor in (19) is defined as the losses calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The x denotes the variables available to fine tune and reduce risk where y represents the volatile uncertainty inherent in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' By minimizing the worst-case scenario of y, the system could create an expected risk profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This is calculated by taking the smallest possible cost (α) that is greater or equal to ������������(������������, ������������) and then calculated the risk factor over β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This can be used to calculate the level of risk inherent in investing in certain markets and diversification tools (such as cash or bond hedging).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������ = ������������������������������������{������������ ∈ ������������: ������������{������������(������������, ������������) ≤ ������������} ≥ ������������} ������������������������������������ 0 ≤ ������������ ≤ 1 (19) Unfortunately, VaR suffers from two key issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Mathematically, it has a lack of convexity and subadditivity making it non-ideal for intensive calculation operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Secondly, VaR only minimizes losses within a given confidence level and does not consider losses occurring at a confidence level outside of its boundaries at 1-β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' cVaR allows a better grasp for situations where a small likelihood of risk could have a huge effect [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' cVaR as a financial constraint is seen in equation (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������ = ������������������������(������������(������������, ������������)|������������(������������, ������������) ≥ ������������������������������������) (20) In this evolution of the original VaR equation, the cVaR is now taking the expected value of random variables above its VaR consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' In other words, it is taking the loss factors inherent in the system and calculating them in situations of 1- β or above the standard confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This is a much more robust and flexible system since it allows forecasting of situations where non-likely events outside of the confidence interval occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Additionally, a higher cVaR means the system is inherently less stable because in non-normal situations, the losses can be considerably higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' To transition from the above equations to models with samples, (20) can be converted into equation (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������ = min � ������������ + 1 ������������(1 − ������������) �[������������(������������, ������������) ������������ ������������=1 − ������������]+ � (21) The first shift here is the addition of N moving the model from continuous to samples with N scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The second change is that the positive component of our losses taken by known x and volatile y subtracted by α as our hedging cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' For our formulation, we can then replace [������������(������������, ������������) − ������������]+ with ������������������������ as shown in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The cVaR equation can now be redefined with ������������������������as seen in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������ = [������������(������������, ������������) − ������������]+ (22) ������������������������������������ = ������������������������������������ � ������������ + 1 ������������(1−������������) ∑ ������������������������ ������������ ������������=1 � (23) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' cVaR Application in Microgrid This section explains how cVaR will be used to maximize power reliability for priority customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' cVaR gives a weighted average of risk above the normal confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This allows a calculation of the risk in high-demand scenarios that can occur in emergency situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Equation (24) takes ������������(������������, ������������) No Demand EdgeCaseSummation ResidentialPV DayAhead NScenario Yes NScenarios cVaRAnalysis SystemRisk Scheduling Completed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Evaluation t Time Segmentsfrom (19) and defines the combined losses as demand subtracted from diesel, PV, and BESS [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������ in (25) is set as the smallest load curtailment while maintaining stability at the confidence level β [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The α value will be measured in units of kilowatts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' While keeping with cVaR convention, demand load that is curtailed will be referred to as α moving forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' All this can be represented as: ������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������) = ������������������������������������������������ ������������������������������������������������������������ − ������������������������������������������������������������������������ − ������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='(24) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='������������������������ = [������������������������������������������������ ������������������������������������������������������������ − ������������������������������������������������������������������������ − ������������������������������������������������ − ������������]+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='(25) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' CASE STUDIES The test residential microgrid is designed with currently available commercial products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' It is designed with a battery system made up of twenty Tesla Powerwall batteries with a capacity of 15 kWh each that starts at an initial value of 10 kWh for each battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������ is the capital cost of the BESS system at $10,000 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The standard rooftop residential solar output is at 4 kW during peak solar generation [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' There are ten residential homes in need of power all with installed solar panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������ in (5) is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='5 therefore priority customers were a total of 33% of total demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This means that a maximum of 66% of customers can be essential customers [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' A simulation of all 187 possible scenarios, N, is run and the battery, diesel, and PV combination are recorded for each specific segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������ is defined as 5% for the confidence level in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The input data for the scenarios including load demand and PV generation is graciously provided by Pecan Street.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This is part of Pecan Street’s Dataport Project [19] which includes the world’s largest resource for residential energy use data, electric transportation and has been expanded to include residential water usage, electric transportation, and regenerative agriculture [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Electricity demand as well as PV generation will have expected statistical deviation from historical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������������������ ������������������������������������������������������������ is defined as 20% and ������������������������������������������������������������������������ ������������������������������������������������������������ as 80% in the model using values from previous research [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������������������������������������������������������ is set as 0 and ������������������������������������������������������������������������ is set as 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='75 kW for the system generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The diesel generator is assumed to have sufficient fuel to operate during the whole course of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ������������������������ ������������������������������������ ������������ and ������������������������ ������������������������������������ ������������ is defined as 5 kW for one Tesla Powerwall [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' ∆T represents the length of time segment which is 15 minutes in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' There are 96 segments for a 24-hour period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The load curtailment if any for each fifteen-minute interval is recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The load curtailment is divided by the total demand supplied and recorded in a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Python is used to take these values and calculate the conditional value at risk for the most demanding and highest load curtailment of the five percent of scenarios (nine scenarios) of the total set of 187 scenarios for all time segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The results highlight the cVaR analysis on the microgrid system for one full day or 96 segments on a total of 187 scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' 2 shows in how many instances curtailment was necessary in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This showcases a high level of self- sufficient reliability that above would be a boon to the existing electrical grid infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The system had zero instances of load curtailment for 90% of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' It had a maximum of 13 instances of load curtailment in the most challenging 5% of cases for all segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' From a system wide load curtailment view, now we can take a more in depth look at the 5% of challenging scenarios in terms of balancing generation and demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The standard deviation shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' 3 presents the difference in values of the dataset for each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Time segments with Active Curtailment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Standard deviation of load curtailment in cVaR analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Within each segment, there is a 20-30% standard deviation indicating the model is robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' These results prove that the model can take in very different demand constraints and respond appropriately to the need of the specific scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Interestingly, the standard deviation is largely consistent throughout the day, indicating that the load curtailment deviation is not too different between sample segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' An exception to this is late mornings to end of the afternoon when the generation of residential PV are sufficient, there are far less load curtailments and therefore the standard deviation is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Active curtailment during an entire day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' 4 presents the twenty-six scenarios or 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content='9% of the entire scenario dataset that was responsible for all load curtailment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This is expected since the model was tested on a robust dataset which has microgrid scenarios with larger than expected demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This is very likely in emergency situations due to weather conditions, and it is important to note how the microgrid would react in these scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Number of scenarios Hour of Day 0 5 10 15 20 25 30 35 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Standard Deviation (%) Hour of Day 19 23 48 43 20 16 33 2 15 24 14 5 7 15 1 6 37 12 25 1 10 11 21 27 16 15 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z INSTANCE SCENARIOS Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The cVAR analysis at a 5% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' 5, the behavior of the case study matched expectations in the following ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The risk when calculating real time energy management for the hours of 10 AM to 4 PM were reduced and in some time-segments brought to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This means residential solar times matched demand at these times and reduced risk of load curtailment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This is one of the main benefits of residential solar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' It especially helps microgrids in providing a power source for a part of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Load curtailment was expected to be used in a small percentage of the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The correct and targeted load curtailments can improve the system’s reliability for priority customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This is complementary to grid hardening efforts but has the advantage of lower costs because it can be built with existing infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' CONCLUSIONS cVaR analysis is conducted in a stand-alone microgrid alongside day ahead scheduling in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The proposed energy management system demonstrates the adaptability of a multitude of generation sources being utilized along with load curtailment in different demand-constraint scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' The objective was to conduct a risk assessment on a microgrid system to assess likelihood of load curtailment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' This allows for evaluating the risk of existing system infrastructure facing controlled load curtailment in a disaster scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Instead of proposing a brand new microgrid installation, existing electrical infrastructure in neighborhoods particularly those with high residential penetration can be retrofitted with additional diesel generation and battery storage services alongside its own energy management system with the proposed 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+page_content=' Crosby, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Holcomb, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Russo, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Smithson, Data-Driven Insights From the Nations Deepest Ever Research on Customer Energy Use, Pecan Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=', Austin, TX, USA, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Koller, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Borsche, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Ulbig and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' Andersson, “Defining a degradation cost function for optimal control of a battery energy storage system,” 2013 IEEE Grenoble Conference, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' 1-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} +page_content=' 0% 10% 20% 30% 40% 50% 60% 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Percentage of Load Curtailment Hour of Day' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfyv7I/content/2301.01759v1.pdf'} diff --git a/N9AyT4oBgHgl3EQf6_rk/content/tmp_files/2301.00833v1.pdf.txt b/N9AyT4oBgHgl3EQf6_rk/content/tmp_files/2301.00833v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2829603f772c44234e3cb7d280b6164d2bd9b284 --- /dev/null +++ b/N9AyT4oBgHgl3EQf6_rk/content/tmp_files/2301.00833v1.pdf.txt @@ -0,0 +1,1124 @@ +Hyperuniform disordered parametric loudspeaker array +Kun Tang1, Yuqi Wang1, Shaobo Wang1, Da Gao1, Haojie Li2, +Xindong Liang2, Patrick Sebbah3, Jin Zhang1∗ and Junhui Shi1† +1 Research Center for Humanoid Sensing, Zhejiang Lab, Hangzhou 311100, China. +2 Taiji Laboratory for Gravitational Wave Universe, +School of Physics and Optoelectronic Engineering, +Hangzhou Institute for Advanced Study, Hangzhou, 311100, China and +3 Department of Physics, The Jack and Pearl Resnick Institute for Advanced Technology, +Bar-Ilan University, Ramat-Gan 5290002, Israel. +(Dated: January 4, 2023) +A steerable parametric loudspeaker array is known for its directivity and narrow beam width. +However, it often suffers from the grating lobes due to periodic array distributions. Here we propose +the array configuration of hyperuniform disorder, which is short-range random while correlated at +large scales, as a promising alternative distribution of acoustic antennas in phased arrays. Angle- +resolved measurements reveal that the proposed array suppresses grating lobes and maintains a +minimal radiation region in the vicinity of the main lobe for the primary frequency waves. These +distinctive emission features benefit the secondary frequency wave in canceling the grating lobes +regardless of the frequencies of the primary waves. Besides that, the hyperuniform disordered array +is duplicatable, which facilitates extra-large array design without any additional computational +efforts. +I. +INTRODUCTION +The prototype of a parametric loudspeaker was con- +ceived by Westervelt nearly 50 years ago [1]. +It is an +application of the parametric acoustic array in air, which +generates a low-frequency sound beam from the inter- +action of two collimated ultrasonic beams as a result of +nonlinear acoustic effects [2]. The high-frequency ultra- +sonic components are commonly referred as the primary +frequencies and the low-frequency component generated +from the interaction of primary frequencies is thus re- +ferred as secondary (difference) frequency. Owing to the +collimation behavior of ultrasonic beams when propa- +gating in air, the parametric loudspeaker overperforms +the traditional loudspeaker in terms of rendering audible +sound with high directivity, especially at low frequency +range [3, 4]. +To achieve broader applications, such as sound field +reinforcement or personal audio space realization [5–7], +audible sound beam steering using acoustic beamforming +techniques [8] is adopted. In a recent work [9], Shi. et. al +has successfully developed a steerable parametric loud- +speaker array (PLA) using periodic distributions. The +results show that the secondary audible wave partially +inherits the directive features from the primary waves. +Although it is mathematically simple, the periodic con- +figuration suffers from fundamental restraints, so-called +diffraction limits which result in unwanted gratings lobes +in both spatial and angular domains [10]. +The occur- +rence of grating lobes for secondary wave found in the +periodic distributed PLA [9] generates sound transmit- +ting towards undesired directions and cause disturbances, +∗ jinzhang@zhejianglab.com +† junhuishi@zhejianglab.com +which therefore should be eliminated by reconfiguring the +PLA with new configurations. +The concept of hyperuniform disorder (HUD) was first +introduced as an order metric for ranking point patterns +according to their local density fluctuations [11]. +Hy- +peruniform structures cover the intermediate regime be- +tween random and periodic structures and thus exhibit +properties usually associated with both. Hyperuniform +stealthy disordered photonic (phononic) structures ex- +hibit large isotropic photonic (phononic) band gaps for +the light (elastic wave) for all polarizations [12–14] and +possess rather unusual scattering properties for light [15] +and sound waves [16]. Recently, the design of phased an- +tenna arrays with hyperuniform disorder has been pro- +posed in microwave which exhibits directive radiation for +large steering angles and wide operating bandwidths [10]. +The radiation patterns of 16 Vivaldi elements have been +examined due to the limited number of ports in the mea- +surement system. Due to the different nature of the HUD +and the traditional array configurations (linear, circular, +triangular, spiral, and spherical form of arrays) [17–22], +we would like to apply this new concept to the design of +the PLA to benefit from its distinctive features. +Here we implement the HUD distribution in the design +of a PLA [23, 24] to achieve high-directive rendering of +audible sound. Owing to the fact the working frequen- +cies for airborne sound are much lower than microwaves, +around 200 elements can be implemented in our exper- +imental setup with low-cost electronic components and +transducers [25, 26]. We demonstrate that the proposed +array suppresses grating lobes and maintains a minimal +radiation region in the vicinity of the main lobe for the +primary frequency waves. These distinctive sound emis- +sion features benefit the secondary waves in complete +cancellation of grating lobes, regardless of the working +frequencies of the primary waves [9]. Interestingly, we +arXiv:2301.00833v1 [eess.AS] 3 Jan 2023 + +2 +found that the grating lobes of the secondary waves even +at very low frequencies, e.g. a 1kHz audio sinusoidal sig- +nal carried by an ultrasonic beam at a frequency 40kHz, +can be completely canceled. +II. +RADIATION PROPERTIES OF HUD +ACOUSTIC ARRAYS +We start with a brief explanation of radiation proper- +ties for the HUD acoustic array. A hyperuniform point +pattern is a point pattern in real space for which the +number variance σ2(R) within a spherical sampling win- +dow of radius R (in d dimensions) grows more slowly +than the window volume (∝ Rd) for large R. This con- +cept can be well understood within the reciprocal space, +where the structure factor S(⃗k) for a hyperuniform point +pattern can be obtained through a 2D Fourier transform +of the point pattern in real space. The resulting struc- +ture factor S(⃗k) vanishes as the wave vector approaches +zero |⃗k| → 0, +S(⃗k) = 1 +N | +N +� +n=1 +ei⃗k· ⃗ +rn| +(1) +where ⃗k are vectors in reciprocal space and ⃗rn are the +positions of the N particles. +We furthermore consider that the structure factor S(⃗k) +is isotropic and vanishes for a finite range of wave num- +bers 0 < |⃗k| ≤ kc for the cutoff radius kc [14]. The size +of the vanished region for the structure factor can be ex- +pressed by the stealthy parameter +χ = M(kc)/Nd, +(2) +where M(kc) is the number of linearly independent ⃗k +vectors where S(⃗k) = 0. +As an example, we illustrate the characteristics of the +HUD array with a point pattern consisting of 200 points +and χ = 0.5, as shown in the upper panel pf Fig. 1(a), +a being the average nearest neighbor distance among the +points. Fig. 1(b) depicts the structure factor of the hy- +peruniform point pattern shown in Fig. 1(a). The cut-off +radius kc of the region with vanishing structure factor +is estimated with Eq. (2) and shown in Fig. 1(b) with a +red dashed circle. We also compare its structure factor +with two typical arrays which possess the same number of +points within the occupied area, e.g. the 14×14 periodic +array of 196 points and the random arrays of 200 points. +It is observed that periodically distributed peaks show +up in the structure factor of the periodic array (middle +panel) as in the diffraction pattern for a typical crystal, +which reveals high symmetry in the point pattern. +In +contrast, the structure factor of the random array is ho- +mogeneously distributed (lower panel). Thus the HUD +point pattern with χ = 0.5 displays a moderate level of +ordered symmetry other than these two extremes. +The normalized far-field radiation pattern in the recip- +rocal space of an array of identical elements distributed +according to a point pattern can be described by its array +factor +|A(⃗k)|2 = 1 +N | +N +� +j=1 +ei⃗k· ⃗rj|2, +(3) +where N identical elements are located in the z=0 plane +of a Cartesian coordinate system at positions ⃗rj. Here, +the wavevector, k, is associated with the working wave- +length λ and the position of the observer in real space, +which can be expressed by the corresponding elevation +and azimuth angles (θ, φ): +⃗k = 2π +λ sin θ(cos φ, sin φ). +(4) +It’s worth noting that despite the similarities between +the arithmetic expressions of the structure factor in +Eq. (1) and the array factor Eq. (3), the physical mean- +ings of the wavevectors are different. The former ⃗k is re- +lated to the wavevectors in reciprocal space, whereas the +latter ⃗k (Eq. (4)) is associated with the spatial position +of the observer in real space and working frequencies. +In Fig. 1(c)-1(d), we present the normalized magnitude +of the array factors, A(⃗k), in decibel scale at working +frequencies f0 and 3f0, where λ0/2 = a, λ0 is the cor- +responding wavelength of f0. It’s worth noting that no +grating lobes show up in the array factor for the HUD ar- +ray (upper panel of Fig. 1(e)), which is in stark contrast +to the array factor for a normal periodic array (middle +panel). Besides that, one can also observe a circular re- +gion with very weak radiation that surrounds the main +lobe, in contrast to the array factor for a random array +(lower panel). This circular exclusion region can act as +a protective layer for the main lobe and protects it from +exterior interference [10]. The radius of the circular ex- +clusion region is given by +θexc = arcsin kcλ +2π , +(5) +where kc is the structure factor cutoff radius and λ is +the operating wavelength. The estimated cutoff radius +obtained from Eq. (5) for the far-field radiation when +f = 3f0 is plotted with a red dashed line in the upper +panel of Fig. 1(d), which fits the array factor calculated +from Eq. (3). We have presented 5 different HUD point +patterns with increasing stealthy parameter χ =0.1-0.5 +in Appendix A, accompanied by corresponding simulated +structural factors and array factors. As χ increases from +0.1 to 0.5 gradually, M(kc) grows as well, which ends +up with the expansion of the circular exclusion region as +predicted by Eq. (5). Generally, the point pattern is con- +sidered to be in the disordered regime when χ ≤ 0.5, but +this threshold value may vary with the number of points +N [27]. The HUD point pattern with χ = 0.5 still resides +in the disordered regime and maintains a relatively broad +circular exclusion region at the same time. + +3 +FIG. 1. Structure factor & Array factor for different point patterns. (a) Schematics of a HUD pattern in real space +with 200 points and χ=0.5 (upper panel), the periodic array (middle panel), and the random array (lower panel). (b) The +normalized magnitude of the structure factor in reciprocal space with a logarithmic scale. The array factor patterns when +f = f0 (c) and when f = 3f0 (d) and the main lobe is steered towards φ = 0◦, θ = 30◦ direction (e), measured from the array’s +boresight (axis of maximum radiated power), with f0 being the frequency for which the average distance between the nearest +elements in the HUD array equals half of the operating wavelength. The angular and radial coordinates in the polar radiation +plots correspond to the φ and θ angles, respectively. +Moreover, +this unique radiation property of the +HUD acoustic array is preserved during beam steering. +Fig. 1(e) exhibits the array factor at frequency f = 3f0 +when the main lobe is steered towards φ = 0◦, θ = 30◦ di- +rection. The radiation exclusion region surrounding the +main lobe of the HUD array is preserved and it also acts +as an indication of the steering direction (upper panel). +This contrasts with the steered radiation pattern of the +random array (lower panel), where the direction of the +main lobe is barely distinguishable from the surround- +ings. As for the case of the periodic array (middle panel), +the presence of strong grating lobes makes it difficult to +determine the steering direction, which is the so-called +spatial aliasing effect [9, 28]. Therefore, we expect the +HUD array benefits from the advantages of both peri- +odic (exclusion regions) and random arrays (cancellation +of grating lobes) and incorporate them into a single de- +sign, thus performing better than both. +III. +MEASURED DIRECTIVIES OF PLA WITH +HYPERUNIFORM DISORDER +To further validate the theoretical analysis, we conduct +a series of measurements using ultrasound transducer +arrays. +The typical working frequency is in the range +of 20-100kHz and does not require wide bandwidths on +transmission electronics, such that we can employ a much +larger number of array elements than that operating at +microwave frequencies [10]. +Specifically, we fabricate +two configurations: a periodic array with 196 elements +and a HUD array with 200 elements and χ = 0.5, us- +ing 10-mm-diameter piezoelectrically actuated transduc- +ers (MA40S4S, MURATA, Japan) as the array element +for both configurations. +The 200-element HUD trans- +ducer array along with the test environment is shown in +Fig. 2. The overall size of the transducer array is around +200×200 mm2. The far-field radiation directivity of the +transducer array is obtained by rotating the planar array +and measuring the signal using a microphone placed 2m +away from the array center. +A PLA is known for its sharp directivity due to the +ultrasonic carrier wave. An audio signal modulated onto +this carrier wave is reproduced along the beam by intrin- +sic non-linearity of the air [24]. In order to steer the main +lobe towards a specific direction, namely the elevation +angle θs and the azimuth angle φs, we need to multiply +the signal from each element in the array with a complex +phase wj = e−i ⃗ks· ⃗rj, where ⃗ks is the steered wavevector +defined as ⃗ks = +2π +λ sin θ(cos φ, sin φ), ⃗rj is the position +vector of the jth array element, λ being the working +wavelength. +In the experimental setup, the phase dif- +ferences between each element are controlled by signal +delays. As shown in Fig. 3, the desired audio signal is +first compared with a 40kHz triangle wave to generate +a pulse-width modulated (PWM) signal by a compara- + +(a) +(b) +(d) +() +(e) +Hyperuniform +p/ +Cua/2m +dB +6 +6 +-10 +Periodic +-20 +p/L +ya/2m +.3 +-30 +2 +40 +Random +50 +kaa/2 +/a +dB4 +FIG. 2. Experimental setup. The fabricated 200-element +HUD transducer array is mounted on a motorized rotary stage +(marked by the red bending arrow) with a standing pole. The +sound intensity emitted by the transducer array at each spin- +ning angle is measured by a microphone (marked by a red +circle), which is pointed towards the center of the array and +fixed at the same height and 2 meters away from the array. +Inset: enlarged view of the transducer array. All the mea- +surements are conducted within a semi-anechoic chamber to +reduce the unwanted reflections from the surrounding envi- +ronments. +FIG. 3. Block diagram of the PLA. +tor chip (TLV3501AIDR, TI). Then the PWM signal is +input to the Field Programmable Gate Array (FPGA, +EP4CE6E22C8N, Intel), which performs digital delay for +the 200 channels of external PWM signal. The resolu- +tion for the signal delay of each channel is 0.8 µs, which +guarantees the accuracy of beam steering [29]. Each de- +layed PWM signal is amplified from 5V up to 12V using a +dual MOSFET (TC4427AEOA713, Microchip) driver. A +serial-to-parallel chip (74HC595D,118, Nexperia) is em- +ployed to reduce the number of output pins. Through +the above analog modulation, the amplitude-modulated +ultrasound wave has a carrier, upper and lower side-band +components, which generate the audible sound in the air +due to the nonlinear interaction of the carrier component +and each side-band component in the ultrasound beams. +We conduct our experiments near the resonance fre- +FIG. 4. Forward far-field directivity patterns of PLA. +Measured far-field directivity in the azimuth plane (φ = 0◦) +for primary frequencies 39kHz (black dashed line), 40kHz +(black solid line), and 41kHz (black dotted line) for the pe- +riodic array (a) and the HUD array (b). The corresponding +measured directivity of the secondary wave at 1kHz is plotted +in blue solid lines in (c) and (d), respectively. The simulated +directivity of the secondary wave at 1kHz is predicted by dif- +ferent models: product directivity model (black) and convolu- +tion model (red), from the primary frequency waves at 40kHz +& 39kHz (dotted lines), and 40kHz & 41kHz (solid lines). +quency of the transducers f = 40kHz, which is equal +to 3f0. A sinusoidal wave signal at frequency 1kHz is +modulated to the carrier frequency of 40kHz. The mea- +sured far-field directivity patterns in the azimuth plane +(φ = 0◦) for primary frequency waves at three differ- +ent frequencies, 39kHz (black dashed line), 40kHz (black +solid line), and 41kHz (black dotted line) for the periodic +array are illustrated in Fig. 4(a). We can clearly observe +that the main lobe of three different primary frequency +waves coincide in the same direction (θ = 0◦) with each +other, whereas the grating lobes appear in slightly shifted +different directions (near θ = arcsin 2/3 = ±41.8◦) be- +cause of the difference between their wavenumbers. The +corresponding results for the HUD array are given in +Fig. 4(b). As can be seen, the grating lobes of the three +primary frequency waves are completely suppressed, and +an exclusion region with minimal radiation as low as - +30dB (|θ| ≤ 31.8◦) can be observed surrounding the +main lobe. In Fig. 4(c), we present the measured far- +field directivity patterns of the secondary frequency wave +at f=1kHz (blue solid line) for the periodic array. It’s +observed that, besides the main lobe, the grating lobes +occur at the direction where the grating lobes of the pri- +mary frequency waves overlap (Fig. 4(a)). It comes from +the nonlinear interactions of the grating lobes of the car- +rier component (f =40kHz) and each side-band compo- +nent (f =39kHz, 41kHz). +It has been a long-existing problem to accurately com- +pute the directivity of the secondary frequency wave of +the PLA with a low computational complexity [4]. +It + +MicrophoneAudio +TLV3501 +Ultrasonic +PWM +DS1 +[Q1:Q8] +STCP1 +[S1:S8] +74HC595 +TC4427 +SHCP1 +DS13 +[Q97:Q104] +[S97:S104] +74HC595 +TC4427 +FPGA +Transducers +(EP4CE6) +(MA40S4S) +DS14 +[Q105:Q112] +[S105:S112] +STCP2 +74HC595 +TC4427 +SHCP2 +DS25 +[Q193:Q200] +[S193:S200] +74HC595 +TC44270 +39k +(b) +39k +40k +1 Radiation (dB) +40k +10 +-10 +41k +41k +-20 +-20 +-30 +-30 +Far-field J +-40 +-40 +... +-50 +-50 +-90 +-60 +-30 +0 +30 +60 +-90 +-60 +-30 +0 +30 +90 +60 +90 +0 +Product directivity 40k&41k +(d) +c +Convolution 40k&41k +Far-field Radiation (dB) +Measurement +-20 +-20 +.. Product directivity 40k&39k +Convolution 40k&39k +-40 +-40 +60 +-60 +-80 +-80 +-100 +-100 +-90 +-60 +-30 +0 +30 +60 +90 +-90 +-60 +-30 +0 +30 +60 +90 +0(deg) +0(deg)5 +would be interesting to predict the directivity of the sec- +ondary frequency wave of a PLA with periodic and HUD +configuration using two well-known models: the product +directivity model, and the convolution model. The for- +mer is directly calculated using the product of measured +beampatterns of primary frequency waves [2]. Whereas +the latter predicts the directivity of the secondary fre- +quency wave [30] by convoluting the product directivity +of the primary frequency waves D1(θ) and D2(θ) with +the Westervelt’s directivity DW (θ), +Dd(θ) = [D1(θ)D2(θ)] ⊗ DW (θ), +(6) +where ⊗ denotes the linear convolution operation. West- +ervelt’s directivity can be represented by DW (θ) = +αs +√ +(α2s+k2 +d tan4 θ), where αs = α1 + α2, kd = |k1 − k2|, αn +and kn are the attenuation rate and wavenumber of the +primary wave at frequency fn, θ is the off-axis angle be- +tween the observation point and normal direction of the +loudspeaker array, with the center of the array set as the +origin. Notice that due to the double-side modulation +of our electronic devices, the secondary (1kHz difference +frequency) wave can be generated by the nonlinear pro- +cess either between primary frequencies 40kHz and 39kHz +(dotted lines), or 40kHz and 41kHz (solid lines). We ob- +serve that the simulation result using the product direc- +tivity model fits well with the measured width of the main +lobe in the range of |θ| ≤ 5◦ and deviates for other spa- +tial directions. The convolution model performs better +than the product directivity in predicting the outline of +measurement for overall angular dimensions. The corre- +sponding results of the secondary frequency wave for the +HUD array are given in Fig. 4(d). It’s worth noting that +the grating lobes are completely canceled, even for such a +low-frequency (1kHz) secondary frequency wave carried +by an ultrasonic beam (40kHz), which is in stark con- +trast with the periodic configurations (Fig. 4(c)). Again, +the product directivity model fits well within the range +of |θ| ≤ 5◦. The simulation result using the convolution +model roughly agrees with the measurement in the re- +gion surrounding the main lobe (|θ| ≤ 12◦) and deviates +for the other directions. This can be attributed to the +assumption of collimated narrow beams for primary fre- +quency waves in Westervelt’s directivity [1] so that the +theoretical result has limited angular validity near the +main lobe (near the grating lobes in the directivity of +the periodic array as well). +We also steered the main lobe of the secondary wave +at frequency 1kHz towards two different directions in +the azimuth plane, namely the φ = 0◦, θ = 15◦ and +φ = 0◦, θ = 30◦ directions. The measurement results +(blue solid lines) in the azimuth plane (φ = 0◦) are il- +lustrated in Fig. 5(a), 5(c) for the periodic array and in +Fig. 5(b), 5(d) for the HUD array. The far-field directiv- +ity of the secondary frequency wave for the HUD array +is delineated by a main steered lobe, whereas the grat- +ing lobes are completely canceled. This contrasts with +the radiations for the periodic array where the spatial +aliasing effect takes place, e.g. steering direction of the +FIG. 5. Steered Far-field directivity patterns of PLA. +Measured (blue solid line) directivity of the secondary fre- +quency waves at 1kHz in the azimuth plane (φ = 0◦) when the +main lobe is steered towards the φ = 0◦, θ = 15◦ (a, b), and +the φ = 0◦, θ = 30◦ (c, d) directions for the periodic array (a, +c) and the HUD array (b, d). The corresponding simulated +directivity of the secondary frequency wave at 1kHz is pre- +dicted by different methods: product directivity (black) and +convolution model (red), from the primary frequency waves +at 40kHz & 39kHz (dotted lines), and 40kHz & 41kHz (solid +lines). +main lobe is hardly distinguished from the grating lobes. +We also present the simulation results predicted by the +product directivity model (black lines) and convolution +model (red lines) using the measured directivities of the +primary waves at frequencies 41kHz&40kHz (solid lines) +and 40kHz&39kHz (dotted lines). Both models capture +the main features, e.g. main lobe and side lobes, of the +directivity for secondary frequency waves radiated from +different arrays. +Again, the product directivity model +well predicts the measured width of the main lobe, while +the convolution model shows a rough coincidence with +the measurement considering the outline of overall an- +gular dimensions. To summarize the above analysis, the +grating lobes of the secondary frequency wave of PLA can +be completely eliminated by a HUD array configuration, +regardless of frequencies of the primary waves [9]. More- +over, the exclusion regions around the main lobe are well +preserved for the primary frequency waves of the HUD +array distributed PLA, although, which is not evident for +the secondary frequency wave. +Furthermore, extra-large acoustic arrays with a great +number of array elements are required in demanding ap- +plication scenarios, like high-intensity sound radiations +[31] or precise source localization [32]. +Optimization +methods proposed to improve the array performances +[33, 34] are, however, computationally expensive dur- +ing the adaption of an extra-large array. Therefore, it +is important yet necessary to demonstrate another fea- +ture of the HUD array, its duplicability. This behavior +inherits from the generation process of a HUD point pat- + +0 +Far-field Radiation (dB) +b +20 +-20 +-40 +40 +60 +60 +00000000 +A +-80 +80 +-60 +90 +-30 +0 +30 +60 +90 +-90 +09- +-30 +0 +30 +60 +90 +Product directivity 40k&4 +c +Far-field Radiation (dB) +Convolution 40k&41k +Measurement +: Product directivity 40k&39k +-20 +20 +Convolution 40k&39k +-40 +40 +-60 +60 +80 +80 +-60 +-30 +0 +30 +60 +90 +-90 +-60 +-30 +0 +30 +60 +-90 +90 +0(deg) +0(deg)6 +FIG. 6. Far-field directivity patterns for large HUD +array. Simulated directivity for the large HUD array (red +dashed line) and measured directivity for the original HUD +sub-array (black line) in the azimuth plane (φ = 0◦) (a), when +the main lobe is steered towards the φ = 0◦, θ = 30◦ direction +(b), at frequency f =40kHz. +tern where periodic boundary conditions are applied to +the two-dimensional computational domain [35]. Thus a +large array consisting of periodic replication of a HUD +subarray remains hyperuniform disordered and possesses +the same distinctive emission behaviors [10]. We compare +the far-field directivity for primary frequency wave at +f =40kHz of a large array (800 elements) made of a 2×2 +subarray with the measured far-field directivity of the +subarray (200 elements). Throughout the measurements, +we found that the radiation properties of the transduc- +ers can be well captured by the piston source model [26] +(see the agreement between the simulated and measured +directivity for primary frequencies at f =30&40kHz in +Fig. 8&9 of Appendix B). We multiply the radiation of +an individual single element with the array factor of the +large array, and the resulting directivity patterns are il- +lustrated in Fig. 6. We present the case where no steering +is applied in Fig. 6(a) and the case of beam steering to- +wards the φ = 0◦, θ = 30◦ direction in Fig. 6(b). As can +be seen, for both cases, the large array of 800 elements +behaves similarly to the HUD subarray of 200 elements, +as the main lobe is surrounded by the exclusion region +with the absence of the grating lobes. Besides that, the +nulling in the exclusion region of the sound radiation for +the large array is much deeper than that for the original +subarray. With infinite replications of the original sub- +array, the radiation values in the exclusion region would +approach zero as predicted by the structure factor shown +in Fig. 1. It is expected that the large array will suppress +the grating lobes of the secondary frequency wave as the +original HUD subarray considering the above radiation +properties for the primary frequency wave. +IV. +CONCLUSIONS +In conclusion, we developed a PLA which follows a +HUD array configuration. +Unlike existing acoustic ar- +rays, this distribution originates from an order metric +(hyperuniformity) to characterize local density fluctua- +tions of a point pattern. Both the simulated and mea- +sured results reveal the effectiveness of the proposed ar- +ray for the primary frequency waves in suppressing the +grating lobes (like a random array) while maintaining a +minimal radiation region around the main lobe (like a +periodic array), which incorporates them into a single +design and performs better than both its periodic and +random counterparts. These properties benefit the sec- +ondary frequency waves in canceling the grating lobes +regardless of the frequencies of primary waves unlike in +[9]. Moreover, these HUD arrays are duplicatable to gen- +erate extra-large arrays, which avoids exponentially in- +creasing computational costs commonly found by adopt- +ing optimizing algorithms [36, 37]. +The proposed ap- +proach can also be employed for ultrasonic transducer +arrays [38] working in other media, e.g. water and hu- +man tissues, which could bring potential applications in +undersea communication and medical therapies. This de- +sign also opens an interesting route to bionic acoustic ar- +rays inspired by the hidden symmetry from nature [39]. +APPENDIX A: RADIATION PROPERTY OF +HUD ARRAY AS χ VARY +In the experimental investigation and numerical analy- +sis presented in the main text, the stealthy parameter of +the HUD acoustic array was chosen to be χ=0.5. Here, +we check the dependence on χ of the radiation property. +As illustrated in the upper panels of Fig. 7, we present +a point pattern with the same number of points N=200, +but with different stealthy parameters χ, ranging from +0.1 to 0.5. One can observe that as χ grows, the par- +ticle clustering effect disappears gradually and occupy +the entire space uniformly. The structure factor S(⃗k) is +calculated through a 2D Fourier transform of the point +pattern and shown in the middle panels. As χ grows, the +nulling region surrounding the origin k=0 where S(⃗k)=0 +enlarges and shows great agreement with the calculated +kc (red dashed circle). +The corresponding normalized +magnitude of the array factors in decibels is plotted in +the lower panels, with 0 dB meaning the maximum value. +The working frequency is chosen to be 3f0. Similar to +the nulling region in the structure factor, the size of the +circular exclusion region enlarges as χ grows and shows +great correspondence with the estimated θexc (red dashed +circle) given by Eq. (5). It indicates that the point pat- +tern with χ=0.5 possesses the largest exclusion region +and suppresses the grating lobes at the same time. Gen- +erally speaking, the point pattern is considered to be in +the disordered regime when χ ≤0.5. As χ grows further, +the point pattern turns into ordered and the grating lobe +takes place like in the periodic array. + + Original array + Orignal array +ar-field Radiation (dB) +(b) + Large array +Large array +-20 +20 +-40 +-60 +60 +-90 +-60 +-30 +30 +60 +90 +-90 +-60 +-30 +0 +0 +30 +60 +90 +0(deg) +0(deg7 +FIG. 7. Radiation property of HUD array as χ vary. Schematics of HUD point patterns with different stealthy parameters +χ=0.1-0.5 (upper panels); Corresponding structure factor (middle panels), where the red dashed circle indicates the cutoff radius +kc (normalized by 2π/a), the structure factor S(⃗k) vanishes for 0 < |⃗k| ≤ kc; Corresponding array factor (lower panels), where +the red dashed circle indicates the radius of the circular exclusion region θexc. +FIG. 8. Forward far-field directivity patterns. Simu- +lated (black solid line) and measured (red dotted line) far- +field directivity patterns in the azimuth plane (φ = 0◦) when +f=30kHz (upper panel) and f=40kHz (lower panel) for the +periodic transducer array (a, c) and the HUD transducer ar- +ray (b, d). +APPENDIX B: MEASURED AND SIMULATED +DIRECTIVITY FOR PRIMARY FREQUENCY +WAVES +The simulated far-field directivity for the primary fre- +quency wave of different acoustic arrays are obtained us- +ing an analytical model, in which the total radiation pat- +tern of an array with identical source elements can be +expressed by the multiplication of the individual single +source element with the array factor. +Each ultrasonic +transducer is modeled as a piston source [26]. The com- +plex acoustic pressure at point ⃗r due to a piston source +emitting at a single frequency can be modeled as +P(⃗r) = P0ADf(θ) +d +ei(φ+kd), +(7) +where P0 is a constant that defines the transducer ampli- +tude power and A is the peak-to-peak amplitude of the +excitation signal. Df(θ) is a far-field directivity function +that depends on the angle θ between the transducer nor- +mal and ⃗r. Here, Df(θ) = 2J1(ka sin θ)/ka sin θ, which is +the directivity function of a circular piston source, where +J1 is a first-order Bessel function of the first kind and a +is the piston radius. The term 1/d accounts for diver- +gence, where d is the propagation distance in free space. +k = 2π/λ is the wavenumber and λ is the wavelength. φ +is the initial phase of the piston. +We conduct our experiments at two different working +frequencies f = 30kHz and 40kHz, which are larger than +2f0 (26.7kHz) and exactly 3f0. The measured and sim- +ulated far-field directivity patterns in the azimuth plane +(φ = 0◦) for the periodic array are illustrated in Fig. 8(a) +(f=30kHz) and Fig. 8(c) (f=40kHz). The corresponding +results for the HUD array at the same working frequen- +cies are given in Fig. 8(b) and Fig. 8(d). In both cases +for different frequencies, the measurement and simula- +tion results agree well with each other within the main +lobe region. The discrepancies between the measurement +and simulation data outside the main lobe region might +be attributed to the misalignment of the motorized stage +and the transducer array in the experimental setup. We +can observe that, for both working frequencies, the HUD +transducer array suppresses the grating lobes and has + + g +8 +8 +8 +Structure factor (dB) +kya/2元 +kra/2T +kra/21 +Array factor (dB) +13.6° +19.5° +24.1 +28.1 +1.80 +0 +Simulation +(b) +Simulation +a +(dB) +Measurement +: Measurement +-10 +-10 +I Radiation +-20 +-20 +Far-field ) +30 +-30 +-40 +-40 +-50 +-50 +-90 +-60 +-30 +0 +30 +60 +90 +-90 +-60 +-30 +0 +30 +60 +90 +0 +(c) +Simulation +(d) +Simulation +(dB) +Measurement + Measurement +-10 +-10 +Far-field Radiation +-20 +20 +30 +-30 +40 +-40 +-50 +-50 +-90 +-60 +-30 +0 +30 +60 +90 +-90 +-60 +-30 +0 +30 +60 +90 +0 (deg) +θ (deg)8 +FIG. 9. Steered far-field directivity patterns. Simulated +(black solid line) and measured (red dotted line) far-field ra- +diation patterns in the azimuth plane (φ = 0◦) when the main +lobe is steered towards the φ = 0◦, θ = 15◦ (a, b) and the +φ = 0◦, θ = 30◦ (c, d) directions when f =40kHz for the pe- +riodic transducer array (a, c) and the HUD transducer array +(b, d). +significantly reduced peak side lobe level values. Specif- +ically, the measured peak side lobe level (PSLL) value +at frequency 30kHz (40kHz) is reduced from -9.3dB (- +3.8dB) for the periodic array to -21.8dB (-18.7dB) for +the HUD array. +At f =40kHz, we also steered the main lobe towards +two different directions in the azimuth plane, namely the +φ = 0◦, θ = 15◦ and φ = 0◦, θ = 30◦ directions. The +measurement (red dashed lines) and simulation results +(black solid lines) in the azimuth plane (φ = 0◦) are il- +lustrated in Fig. 9(a)&9(c) for the periodic array and in +Fig. 9(b)&9(d) for the HUD array, which shows great +coincidence with each other. As predicted by the simu- +lations for the array factor with steered directions, the +far-field directivity pattern for the HUD array is delin- +eated by the main beam surrounded by a weak emission +exclusion region, whereas outside this exclusion region +the sidelobes are kept at a low level. +This is in stark +contrast to the far-field directivity pattern for the peri- +odic array where several grating lobes can be seen and +steering direction is hardly determined. The maximum +side lobe level is even higher than the main lobe when +steered towards the φ = 0◦, θ = 30◦ direction. It’s worth +noting that the measured PSLL value, when steered to- +wards the φ = 0◦, θ = 15◦ (φ = 0◦, θ = 30◦) direction +at frequency 40kHz is reduced from -1.4dB (2dB) for the +periodic array to -15.4dB (-11.1dB) for the HUD array. +ACKNOWLEDGMENTS +The authors thank Prof. Marian Florescu for gener- +ating the initial hyperuniform disordered point patterns. +This research was supported by the Youth Foundation +Project of Zhejiang Lab (Grant No. 2020MC0AA07). P. +S. is thankful to the Israel Science Foundation (Grants +No. +1871/15, 2074/15, and 2630/20), and the United +States-Israel Binational Science Foundation NSF/BSF +(Grant No. 2015694 and No. 2021811). +[1] P. J. Westervelt, The Journal of the Acoustical Society +of America 35, 535 (1963). +[2] M. Hamilton and D. Blackstock, Nonlinear Acoustics +(Acoustical Society of America, 2008). +[3] W.-S. Gan, J. Yang, and T. Kamakura, Applied Acous- +tics 73, 1211 (2012). +[4] C. Shi, Y. Kajikawa, and W.-S. 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E 89, 022721 (2014). + diff --git a/N9AyT4oBgHgl3EQf6_rk/content/tmp_files/load_file.txt b/N9AyT4oBgHgl3EQf6_rk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6f0e70b339927fc561ab539f12cecd350b840f26 --- /dev/null +++ b/N9AyT4oBgHgl3EQf6_rk/content/tmp_files/load_file.txt @@ -0,0 +1,523 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf,len=522 +page_content='Hyperuniform disordered parametric loudspeaker array Kun Tang1, Yuqi Wang1, Shaobo Wang1, Da Gao1, Haojie Li2, Xindong Liang2, Patrick Sebbah3, Jin Zhang1∗ and Junhui Shi1† 1 Research Center for Humanoid Sensing, Zhejiang Lab, Hangzhou 311100, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 2 Taiji Laboratory for Gravitational Wave Universe, School of Physics and Optoelectronic Engineering, Hangzhou Institute for Advanced Study, Hangzhou, 311100, China and 3 Department of Physics, The Jack and Pearl Resnick Institute for Advanced Technology, Bar-Ilan University, Ramat-Gan 5290002, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' (Dated: January 4, 2023) A steerable parametric loudspeaker array is known for its directivity and narrow beam width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' However, it often suffers from the grating lobes due to periodic array distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Here we propose the array configuration of hyperuniform disorder, which is short-range random while correlated at large scales, as a promising alternative distribution of acoustic antennas in phased arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Angle- resolved measurements reveal that the proposed array suppresses grating lobes and maintains a minimal radiation region in the vicinity of the main lobe for the primary frequency waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' These distinctive emission features benefit the secondary frequency wave in canceling the grating lobes regardless of the frequencies of the primary waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Besides that, the hyperuniform disordered array is duplicatable, which facilitates extra-large array design without any additional computational efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' INTRODUCTION The prototype of a parametric loudspeaker was con- ceived by Westervelt nearly 50 years ago [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' It is an application of the parametric acoustic array in air, which generates a low-frequency sound beam from the inter- action of two collimated ultrasonic beams as a result of nonlinear acoustic effects [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The high-frequency ultra- sonic components are commonly referred as the primary frequencies and the low-frequency component generated from the interaction of primary frequencies is thus re- ferred as secondary (difference) frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Owing to the collimation behavior of ultrasonic beams when propa- gating in air, the parametric loudspeaker overperforms the traditional loudspeaker in terms of rendering audible sound with high directivity, especially at low frequency range [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' To achieve broader applications, such as sound field reinforcement or personal audio space realization [5–7], audible sound beam steering using acoustic beamforming techniques [8] is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' In a recent work [9], Shi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' al has successfully developed a steerable parametric loud- speaker array (PLA) using periodic distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The results show that the secondary audible wave partially inherits the directive features from the primary waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Although it is mathematically simple, the periodic con- figuration suffers from fundamental restraints, so-called diffraction limits which result in unwanted gratings lobes in both spatial and angular domains [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The occur- rence of grating lobes for secondary wave found in the periodic distributed PLA [9] generates sound transmit- ting towards undesired directions and cause disturbances, ∗ jinzhang@zhejianglab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='com † junhuishi@zhejianglab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='com which therefore should be eliminated by reconfiguring the PLA with new configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The concept of hyperuniform disorder (HUD) was first introduced as an order metric for ranking point patterns according to their local density fluctuations [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Hy- peruniform structures cover the intermediate regime be- tween random and periodic structures and thus exhibit properties usually associated with both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Hyperuniform stealthy disordered photonic (phononic) structures ex- hibit large isotropic photonic (phononic) band gaps for the light (elastic wave) for all polarizations [12–14] and possess rather unusual scattering properties for light [15] and sound waves [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Recently, the design of phased an- tenna arrays with hyperuniform disorder has been pro- posed in microwave which exhibits directive radiation for large steering angles and wide operating bandwidths [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The radiation patterns of 16 Vivaldi elements have been examined due to the limited number of ports in the mea- surement system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Due to the different nature of the HUD and the traditional array configurations (linear, circular, triangular, spiral, and spherical form of arrays) [17–22], we would like to apply this new concept to the design of the PLA to benefit from its distinctive features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Here we implement the HUD distribution in the design of a PLA [23, 24] to achieve high-directive rendering of audible sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Owing to the fact the working frequen- cies for airborne sound are much lower than microwaves, around 200 elements can be implemented in our exper- imental setup with low-cost electronic components and transducers [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' We demonstrate that the proposed array suppresses grating lobes and maintains a minimal radiation region in the vicinity of the main lobe for the primary frequency waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' These distinctive sound emis- sion features benefit the secondary waves in complete cancellation of grating lobes, regardless of the working frequencies of the primary waves [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Interestingly, we arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='00833v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='AS] 3 Jan 2023 2 found that the grating lobes of the secondary waves even at very low frequencies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' a 1kHz audio sinusoidal sig- nal carried by an ultrasonic beam at a frequency 40kHz, can be completely canceled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' RADIATION PROPERTIES OF HUD ACOUSTIC ARRAYS We start with a brief explanation of radiation proper- ties for the HUD acoustic array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' A hyperuniform point pattern is a point pattern in real space for which the number variance σ2(R) within a spherical sampling win- dow of radius R (in d dimensions) grows more slowly than the window volume (∝ Rd) for large R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' This con- cept can be well understood within the reciprocal space, where the structure factor S(⃗k) for a hyperuniform point pattern can be obtained through a 2D Fourier transform of the point pattern in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The resulting struc- ture factor S(⃗k) vanishes as the wave vector approaches zero |⃗k| → 0, S(⃗k) = 1 N | N � n=1 ei⃗k· ⃗ rn| (1) where ⃗k are vectors in reciprocal space and ⃗rn are the positions of the N particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' We furthermore consider that the structure factor S(⃗k) is isotropic and vanishes for a finite range of wave num- bers 0 < |⃗k| ≤ kc for the cutoff radius kc [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The size of the vanished region for the structure factor can be ex- pressed by the stealthy parameter χ = M(kc)/Nd, (2) where M(kc) is the number of linearly independent ⃗k vectors where S(⃗k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' As an example, we illustrate the characteristics of the HUD array with a point pattern consisting of 200 points and χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='5, as shown in the upper panel pf Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 1(a), a being the average nearest neighbor distance among the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 1(b) depicts the structure factor of the hy- peruniform point pattern shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The cut-off radius kc of the region with vanishing structure factor is estimated with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' (2) and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 1(b) with a red dashed circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' We also compare its structure factor with two typical arrays which possess the same number of points within the occupied area, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' the 14×14 periodic array of 196 points and the random arrays of 200 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' It is observed that periodically distributed peaks show up in the structure factor of the periodic array (middle panel) as in the diffraction pattern for a typical crystal, which reveals high symmetry in the point pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' In contrast, the structure factor of the random array is ho- mogeneously distributed (lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Thus the HUD point pattern with χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='5 displays a moderate level of ordered symmetry other than these two extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The normalized far-field radiation pattern in the recip- rocal space of an array of identical elements distributed according to a point pattern can be described by its array factor |A(⃗k)|2 = 1 N | N � j=1 ei⃗k· ⃗rj|2, (3) where N identical elements are located in the z=0 plane of a Cartesian coordinate system at positions ⃗rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Here, the wavevector, k, is associated with the working wave- length λ and the position of the observer in real space, which can be expressed by the corresponding elevation and azimuth angles (θ, φ): ⃗k = 2π λ sin θ(cos φ, sin φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' (4) It’s worth noting that despite the similarities between the arithmetic expressions of the structure factor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' (1) and the array factor Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' (3), the physical mean- ings of the wavevectors are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The former ⃗k is re- lated to the wavevectors in reciprocal space, whereas the latter ⃗k (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' (4)) is associated with the spatial position of the observer in real space and working frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 1(c)-1(d), we present the normalized magnitude of the array factors, A(⃗k), in decibel scale at working frequencies f0 and 3f0, where λ0/2 = a, λ0 is the cor- responding wavelength of f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' It’s worth noting that no grating lobes show up in the array factor for the HUD ar- ray (upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 1(e)), which is in stark contrast to the array factor for a normal periodic array (middle panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Besides that, one can also observe a circular re- gion with very weak radiation that surrounds the main lobe, in contrast to the array factor for a random array (lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' This circular exclusion region can act as a protective layer for the main lobe and protects it from exterior interference [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The radius of the circular ex- clusion region is given by θexc = arcsin kcλ 2π , (5) where kc is the structure factor cutoff radius and λ is the operating wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The estimated cutoff radius obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' (5) for the far-field radiation when f = 3f0 is plotted with a red dashed line in the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 1(d), which fits the array factor calculated from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' We have presented 5 different HUD point patterns with increasing stealthy parameter χ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='5 in Appendix A, accompanied by corresponding simulated structural factors and array factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' As χ increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='5 gradually, M(kc) grows as well, which ends up with the expansion of the circular exclusion region as predicted by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Generally, the point pattern is con- sidered to be in the disordered regime when χ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='5, but this threshold value may vary with the number of points N [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The HUD point pattern with χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='5 still resides in the disordered regime and maintains a relatively broad circular exclusion region at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Structure factor & Array factor for different point patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' (a) Schematics of a HUD pattern in real space with 200 points and χ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='5 (upper panel), the periodic array (middle panel), and the random array (lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' (b) The normalized magnitude of the structure factor in reciprocal space with a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The array factor patterns when f = f0 (c) and when f = 3f0 (d) and the main lobe is steered towards φ = 0◦, θ = 30◦ direction (e), measured from the array’s boresight (axis of maximum radiated power), with f0 being the frequency for which the average distance between the nearest elements in the HUD array equals half of the operating wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The angular and radial coordinates in the polar radiation plots correspond to the φ and θ angles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Moreover, this unique radiation property of the HUD acoustic array is preserved during beam steering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 1(e) exhibits the array factor at frequency f = 3f0 when the main lobe is steered towards φ = 0◦, θ = 30◦ di- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The radiation exclusion region surrounding the main lobe of the HUD array is preserved and it also acts as an indication of the steering direction (upper panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' This contrasts with the steered radiation pattern of the random array (lower panel), where the direction of the main lobe is barely distinguishable from the surround- ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' As for the case of the periodic array (middle panel), the presence of strong grating lobes makes it difficult to determine the steering direction, which is the so-called spatial aliasing effect [9, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Therefore, we expect the HUD array benefits from the advantages of both peri- odic (exclusion regions) and random arrays (cancellation of grating lobes) and incorporate them into a single de- sign, thus performing better than both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' MEASURED DIRECTIVIES OF PLA WITH HYPERUNIFORM DISORDER To further validate the theoretical analysis, we conduct a series of measurements using ultrasound transducer arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The typical working frequency is in the range of 20-100kHz and does not require wide bandwidths on transmission electronics, such that we can employ a much larger number of array elements than that operating at microwave frequencies [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Specifically, we fabricate two configurations: a periodic array with 196 elements and a HUD array with 200 elements and χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='5, us- ing 10-mm-diameter piezoelectrically actuated transduc- ers (MA40S4S, MURATA, Japan) as the array element for both configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The 200-element HUD trans- ducer array along with the test environment is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The overall size of the transducer array is around 200×200 mm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The far-field radiation directivity of the transducer array is obtained by rotating the planar array and measuring the signal using a microphone placed 2m away from the array center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' A PLA is known for its sharp directivity due to the ultrasonic carrier wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' An audio signal modulated onto this carrier wave is reproduced along the beam by intrin- sic non-linearity of the air [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' In order to steer the main lobe towards a specific direction, namely the elevation angle θs and the azimuth angle φs, we need to multiply the signal from each element in the array with a complex phase wj = e−i ⃗ks· ⃗rj, where ⃗ks is the steered wavevector defined as ⃗ks = 2π λ sin θ(cos φ, sin φ), ⃗rj is the position vector of the jth array element, λ being the working wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' In the experimental setup, the phase dif- ferences between each element are controlled by signal delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 3, the desired audio signal is first compared with a 40kHz triangle wave to generate a pulse-width modulated (PWM) signal by a compara- (a) (b) (d) () (e) Hyperuniform p/ Cua/2m dB 6 6 10 Periodic 20 p/L ya/2m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='3 30 2 40 Random 50 kaa/2 /a dB4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The fabricated 200-element HUD transducer array is mounted on a motorized rotary stage (marked by the red bending arrow) with a standing pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The sound intensity emitted by the transducer array at each spin- ning angle is measured by a microphone (marked by a red circle), which is pointed towards the center of the array and fixed at the same height and 2 meters away from the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Inset: enlarged view of the transducer array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' All the mea- surements are conducted within a semi-anechoic chamber to reduce the unwanted reflections from the surrounding envi- ronments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Block diagram of the PLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' tor chip (TLV3501AIDR, TI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Then the PWM signal is input to the Field Programmable Gate Array (FPGA, EP4CE6E22C8N, Intel), which performs digital delay for the 200 channels of external PWM signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The resolu- tion for the signal delay of each channel is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='8 µs, which guarantees the accuracy of beam steering [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Each de- layed PWM signal is amplified from 5V up to 12V using a dual MOSFET (TC4427AEOA713, Microchip) driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' A serial-to-parallel chip (74HC595D,118, Nexperia) is em- ployed to reduce the number of output pins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Through the above analog modulation, the amplitude-modulated ultrasound wave has a carrier, upper and lower side-band components, which generate the audible sound in the air due to the nonlinear interaction of the carrier component and each side-band component in the ultrasound beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' We conduct our experiments near the resonance fre- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Forward far-field directivity patterns of PLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Measured far-field directivity in the azimuth plane (φ = 0◦) for primary frequencies 39kHz (black dashed line), 40kHz (black solid line), and 41kHz (black dotted line) for the pe- riodic array (a) and the HUD array (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The corresponding measured directivity of the secondary wave at 1kHz is plotted in blue solid lines in (c) and (d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The simulated directivity of the secondary wave at 1kHz is predicted by dif- ferent models: product directivity model (black) and convolu- tion model (red), from the primary frequency waves at 40kHz & 39kHz (dotted lines), and 40kHz & 41kHz (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' quency of the transducers f = 40kHz, which is equal to 3f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' A sinusoidal wave signal at frequency 1kHz is modulated to the carrier frequency of 40kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The mea- sured far-field directivity patterns in the azimuth plane (φ = 0◦) for primary frequency waves at three differ- ent frequencies, 39kHz (black dashed line), 40kHz (black solid line), and 41kHz (black dotted line) for the periodic array are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' We can clearly observe that the main lobe of three different primary frequency waves coincide in the same direction (θ = 0◦) with each other, whereas the grating lobes appear in slightly shifted different directions (near θ = arcsin 2/3 = ±41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='8◦) be- cause of the difference between their wavenumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The corresponding results for the HUD array are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' As can be seen, the grating lobes of the three primary frequency waves are completely suppressed, and an exclusion region with minimal radiation as low as - 30dB (|θ| ≤ 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='8◦) can be observed surrounding the main lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 4(c), we present the measured far- field directivity patterns of the secondary frequency wave at f=1kHz (blue solid line) for the periodic array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' It’s observed that, besides the main lobe, the grating lobes occur at the direction where the grating lobes of the pri- mary frequency waves overlap (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 4(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' It comes from the nonlinear interactions of the grating lobes of the car- rier component (f =40kHz) and each side-band compo- nent (f =39kHz, 41kHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' It has been a long-existing problem to accurately com- pute the directivity of the secondary frequency wave of the PLA with a low computational complexity [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' It MicrophoneAudio TLV3501 Ultrasonic PWM DS1 [Q1:Q8] STCP1 [S1:S8] 74HC595 TC4427 SHCP1 DS13 [Q97:Q104] [S97:S104] 74HC595 TC4427 FPGA Transducers (EP4CE6) (MA40S4S) DS14 [Q105:Q112] [S105:S112] STCP2 74HC595 TC4427 SHCP2 DS25 [Q193:Q200] [S193:S200] 74HC595 TC44270 39k (b) 39k 40k 1 Radiation (dB) 40k 10 10 41k 41k 20 20 30 30 Far-field J 40 40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 50 50 90 60 30 0 30 60 90 60 30 0 30 90 60 90 0 Product directivity 40k&41k (d) c Convolution 40k&41k Far-field Radiation (dB) Measurement 20 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='. Product directivity 40k&39k Convolution 40k&39k 40 40 60 60 80 80 100 100 90 60 30 0 30 60 90 90 60 30 0 30 60 90 0(deg) 0(deg)5 would be interesting to predict the directivity of the sec- ondary frequency wave of a PLA with periodic and HUD configuration using two well-known models: the product directivity model, and the convolution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The for- mer is directly calculated using the product of measured beampatterns of primary frequency waves [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Whereas the latter predicts the directivity of the secondary fre- quency wave [30] by convoluting the product directivity of the primary frequency waves D1(θ) and D2(θ) with the Westervelt’s directivity DW (θ), Dd(θ) = [D1(θ)D2(θ)] ⊗ DW (θ), (6) where ⊗ denotes the linear convolution operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' West- ervelt’s directivity can be represented by DW (θ) = αs √ (α2s+k2 d tan4 θ), where αs = α1 + α2, kd = |k1 − k2|, αn and kn are the attenuation rate and wavenumber of the primary wave at frequency fn, θ is the off-axis angle be- tween the observation point and normal direction of the loudspeaker array, with the center of the array set as the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Notice that due to the double-side modulation of our electronic devices, the secondary (1kHz difference frequency) wave can be generated by the nonlinear pro- cess either between primary frequencies 40kHz and 39kHz (dotted lines), or 40kHz and 41kHz (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' We ob- serve that the simulation result using the product direc- tivity model fits well with the measured width of the main lobe in the range of |θ| ≤ 5◦ and deviates for other spa- tial directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The convolution model performs better than the product directivity in predicting the outline of measurement for overall angular dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The corre- sponding results of the secondary frequency wave for the HUD array are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' It’s worth noting that the grating lobes are completely canceled, even for such a low-frequency (1kHz) secondary frequency wave carried by an ultrasonic beam (40kHz), which is in stark con- trast with the periodic configurations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 4(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Again, the product directivity model fits well within the range of |θ| ≤ 5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The simulation result using the convolution model roughly agrees with the measurement in the re- gion surrounding the main lobe (|θ| ≤ 12◦) and deviates for the other directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' This can be attributed to the assumption of collimated narrow beams for primary fre- quency waves in Westervelt’s directivity [1] so that the theoretical result has limited angular validity near the main lobe (near the grating lobes in the directivity of the periodic array as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' We also steered the main lobe of the secondary wave at frequency 1kHz towards two different directions in the azimuth plane, namely the φ = 0◦, θ = 15◦ and φ = 0◦, θ = 30◦ directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The measurement results (blue solid lines) in the azimuth plane (φ = 0◦) are il- lustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 5(a), 5(c) for the periodic array and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 5(b), 5(d) for the HUD array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The far-field directiv- ity of the secondary frequency wave for the HUD array is delineated by a main steered lobe, whereas the grat- ing lobes are completely canceled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' This contrasts with the radiations for the periodic array where the spatial aliasing effect takes place, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' steering direction of the FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Steered Far-field directivity patterns of PLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Measured (blue solid line) directivity of the secondary fre- quency waves at 1kHz in the azimuth plane (φ = 0◦) when the main lobe is steered towards the φ = 0◦, θ = 15◦ (a, b), and the φ = 0◦, θ = 30◦ (c, d) directions for the periodic array (a, c) and the HUD array (b, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The corresponding simulated directivity of the secondary frequency wave at 1kHz is pre- dicted by different methods: product directivity (black) and convolution model (red), from the primary frequency waves at 40kHz & 39kHz (dotted lines), and 40kHz & 41kHz (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' main lobe is hardly distinguished from the grating lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' We also present the simulation results predicted by the product directivity model (black lines) and convolution model (red lines) using the measured directivities of the primary waves at frequencies 41kHz&40kHz (solid lines) and 40kHz&39kHz (dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Both models capture the main features, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' main lobe and side lobes, of the directivity for secondary frequency waves radiated from different arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Again, the product directivity model well predicts the measured width of the main lobe, while the convolution model shows a rough coincidence with the measurement considering the outline of overall an- gular dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' To summarize the above analysis, the grating lobes of the secondary frequency wave of PLA can be completely eliminated by a HUD array configuration, regardless of frequencies of the primary waves [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' More- over, the exclusion regions around the main lobe are well preserved for the primary frequency waves of the HUD array distributed PLA, although, which is not evident for the secondary frequency wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Furthermore, extra-large acoustic arrays with a great number of array elements are required in demanding ap- plication scenarios, like high-intensity sound radiations [31] or precise source localization [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Optimization methods proposed to improve the array performances [33, 34] are, however, computationally expensive dur- ing the adaption of an extra-large array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Therefore, it is important yet necessary to demonstrate another fea- ture of the HUD array, its duplicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' This behavior inherits from the generation process of a HUD point pat- 0 Far-field Radiation (dB) b 20 20 40 40 60 60 00000000 A 80 80 60 90 30 0 30 60 90 90 09- 30 0 30 60 90 Product directivity 40k&4 c Far-field Radiation (dB) Convolution 40k&41k Measurement : Product directivity 40k&39k 20 20 Convolution 40k&39k 40 40 60 60 80 80 60 30 0 30 60 90 90 60 30 0 30 60 90 90 0(deg) 0(deg)6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Far-field directivity patterns for large HUD array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Simulated directivity for the large HUD array (red dashed line) and measured directivity for the original HUD sub-array (black line) in the azimuth plane (φ = 0◦) (a), when the main lobe is steered towards the φ = 0◦, θ = 30◦ direction (b), at frequency f =40kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' tern where periodic boundary conditions are applied to the two-dimensional computational domain [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Thus a large array consisting of periodic replication of a HUD subarray remains hyperuniform disordered and possesses the same distinctive emission behaviors [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' We compare the far-field directivity for primary frequency wave at f =40kHz of a large array (800 elements) made of a 2×2 subarray with the measured far-field directivity of the subarray (200 elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Throughout the measurements, we found that the radiation properties of the transduc- ers can be well captured by the piston source model [26] (see the agreement between the simulated and measured directivity for primary frequencies at f =30&40kHz in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 8&9 of Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' We multiply the radiation of an individual single element with the array factor of the large array, and the resulting directivity patterns are il- lustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' We present the case where no steering is applied in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 6(a) and the case of beam steering to- wards the φ = 0◦, θ = 30◦ direction in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' As can be seen, for both cases, the large array of 800 elements behaves similarly to the HUD subarray of 200 elements, as the main lobe is surrounded by the exclusion region with the absence of the grating lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Besides that, the nulling in the exclusion region of the sound radiation for the large array is much deeper than that for the original subarray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' With infinite replications of the original sub- array, the radiation values in the exclusion region would approach zero as predicted by the structure factor shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' It is expected that the large array will suppress the grating lobes of the secondary frequency wave as the original HUD subarray considering the above radiation properties for the primary frequency wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' CONCLUSIONS In conclusion, we developed a PLA which follows a HUD array configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Unlike existing acoustic ar- rays, this distribution originates from an order metric (hyperuniformity) to characterize local density fluctua- tions of a point pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Both the simulated and mea- sured results reveal the effectiveness of the proposed ar- ray for the primary frequency waves in suppressing the grating lobes (like a random array) while maintaining a minimal radiation region around the main lobe (like a periodic array), which incorporates them into a single design and performs better than both its periodic and random counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' These properties benefit the sec- ondary frequency waves in canceling the grating lobes regardless of the frequencies of primary waves unlike in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Moreover, these HUD arrays are duplicatable to gen- erate extra-large arrays, which avoids exponentially in- creasing computational costs commonly found by adopt- ing optimizing algorithms [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The proposed ap- proach can also be employed for ultrasonic transducer arrays [38] working in other media, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' water and hu- man tissues, which could bring potential applications in undersea communication and medical therapies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' This de- sign also opens an interesting route to bionic acoustic ar- rays inspired by the hidden symmetry from nature [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' APPENDIX A: RADIATION PROPERTY OF HUD ARRAY AS χ VARY In the experimental investigation and numerical analy- sis presented in the main text, the stealthy parameter of the HUD acoustic array was chosen to be χ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Here, we check the dependence on χ of the radiation property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' As illustrated in the upper panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 7, we present a point pattern with the same number of points N=200, but with different stealthy parameters χ, ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' One can observe that as χ grows, the par- ticle clustering effect disappears gradually and occupy the entire space uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The structure factor S(⃗k) is calculated through a 2D Fourier transform of the point pattern and shown in the middle panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' As χ grows, the nulling region surrounding the origin k=0 where S(⃗k)=0 enlarges and shows great agreement with the calculated kc (red dashed circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The corresponding normalized magnitude of the array factors in decibels is plotted in the lower panels, with 0 dB meaning the maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The working frequency is chosen to be 3f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Similar to the nulling region in the structure factor, the size of the circular exclusion region enlarges as χ grows and shows great correspondence with the estimated θexc (red dashed circle) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' It indicates that the point pat- tern with χ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='5 possesses the largest exclusion region and suppresses the grating lobes at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Gen- erally speaking, the point pattern is considered to be in the disordered regime when χ ≤0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' As χ grows further, the point pattern turns into ordered and the grating lobe takes place like in the periodic array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Original array Orignal array ar-field Radiation (dB) (b) Large array Large array 20 20 40 60 60 90 60 30 30 60 90 90 60 30 0 0 30 60 90 0(deg) 0(deg7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Radiation property of HUD array as χ vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Schematics of HUD point patterns with different stealthy parameters χ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='5 (upper panels);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Corresponding structure factor (middle panels), where the red dashed circle indicates the cutoff radius kc (normalized by 2π/a), the structure factor S(⃗k) vanishes for 0 < |⃗k| ≤ kc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Corresponding array factor (lower panels), where the red dashed circle indicates the radius of the circular exclusion region θexc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Forward far-field directivity patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Simu- lated (black solid line) and measured (red dotted line) far- field directivity patterns in the azimuth plane (φ = 0◦) when f=30kHz (upper panel) and f=40kHz (lower panel) for the periodic transducer array (a, c) and the HUD transducer ar- ray (b, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' APPENDIX B: MEASURED AND SIMULATED DIRECTIVITY FOR PRIMARY FREQUENCY WAVES The simulated far-field directivity for the primary fre- quency wave of different acoustic arrays are obtained us- ing an analytical model, in which the total radiation pat- tern of an array with identical source elements can be expressed by the multiplication of the individual single source element with the array factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Each ultrasonic transducer is modeled as a piston source [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The com- plex acoustic pressure at point ⃗r due to a piston source emitting at a single frequency can be modeled as P(⃗r) = P0ADf(θ) d ei(φ+kd), (7) where P0 is a constant that defines the transducer ampli- tude power and A is the peak-to-peak amplitude of the excitation signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Df(θ) is a far-field directivity function that depends on the angle θ between the transducer nor- mal and ⃗r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Here, Df(θ) = 2J1(ka sin θ)/ka sin θ, which is the directivity function of a circular piston source, where J1 is a first-order Bessel function of the first kind and a is the piston radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The term 1/d accounts for diver- gence, where d is the propagation distance in free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' k = 2π/λ is the wavenumber and λ is the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' φ is the initial phase of the piston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' We conduct our experiments at two different working frequencies f = 30kHz and 40kHz, which are larger than 2f0 (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='7kHz) and exactly 3f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The measured and sim- ulated far-field directivity patterns in the azimuth plane (φ = 0◦) for the periodic array are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 8(a) (f=30kHz) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 8(c) (f=40kHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The corresponding results for the HUD array at the same working frequen- cies are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 8(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 8(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' In both cases for different frequencies, the measurement and simula- tion results agree well with each other within the main lobe region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The discrepancies between the measurement and simulation data outside the main lobe region might be attributed to the misalignment of the motorized stage and the transducer array in the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' We can observe that, for both working frequencies, the HUD transducer array suppresses the grating lobes and has g 8 8 8 Structure factor (dB) kya/2元 kra/2T kra/21 Array factor (dB) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='6° 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='5° 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='80 0 Simulation (b) Simulation a (dB) Measurement : Measurement 10 10 I Radiation 20 20 Far-field ) 30 30 40 40 50 50 90 60 30 0 30 60 90 90 60 30 0 30 60 90 0 (c) Simulation (d) Simulation (dB) Measurement Measurement 10 10 Far-field Radiation 20 20 30 30 40 40 50 50 90 60 30 0 30 60 90 90 60 30 0 30 60 90 0 (deg) θ (deg)8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Steered far-field directivity patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Simulated (black solid line) and measured (red dotted line) far-field ra- diation patterns in the azimuth plane (φ = 0◦) when the main lobe is steered towards the φ = 0◦, θ = 15◦ (a, b) and the φ = 0◦, θ = 30◦ (c, d) directions when f =40kHz for the pe- riodic transducer array (a, c) and the HUD transducer array (b, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' significantly reduced peak side lobe level values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Specif- ically, the measured peak side lobe level (PSLL) value at frequency 30kHz (40kHz) is reduced from -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='3dB (- 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='8dB) for the periodic array to -21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='8dB (-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='7dB) for the HUD array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' At f =40kHz, we also steered the main lobe towards two different directions in the azimuth plane, namely the φ = 0◦, θ = 15◦ and φ = 0◦, θ = 30◦ directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The measurement (red dashed lines) and simulation results (black solid lines) in the azimuth plane (φ = 0◦) are il- lustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 9(a)&9(c) for the periodic array and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 9(b)&9(d) for the HUD array, which shows great coincidence with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' As predicted by the simu- lations for the array factor with steered directions, the far-field directivity pattern for the HUD array is delin- eated by the main beam surrounded by a weak emission exclusion region, whereas outside this exclusion region the sidelobes are kept at a low level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' This is in stark contrast to the far-field directivity pattern for the peri- odic array where several grating lobes can be seen and steering direction is hardly determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' The maximum side lobe level is even higher than the main lobe when steered towards the φ = 0◦, θ = 30◦ direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' It’s worth noting that the measured PSLL value, when steered to- wards the φ = 0◦, θ = 15◦ (φ = 0◦, θ = 30◦) direction at frequency 40kHz is reduced from -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='4dB (2dB) for the periodic array to -15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='4dB (-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content='1dB) for the HUD array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' Marian Florescu for gener- ating the initial hyperuniform disordered point patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' This research was supported by the Youth Foundation Project of Zhejiang Lab (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 2020MC0AA07).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' is thankful to the Israel Science Foundation (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} +page_content=' 1871/15, 2074/15, and 2630/20), and the United 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+page_content=' E 89, 022721 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQf6_rk/content/2301.00833v1.pdf'} diff --git a/N9E0T4oBgHgl3EQfTQCe/content/tmp_files/2301.02234v1.pdf.txt b/N9E0T4oBgHgl3EQfTQCe/content/tmp_files/2301.02234v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5033a561d1d8ebc3e702ceeb029dfe5c0f140f77 --- /dev/null +++ b/N9E0T4oBgHgl3EQfTQCe/content/tmp_files/2301.02234v1.pdf.txt @@ -0,0 +1,3543 @@ +arXiv:2301.02234v1 [math.DG] 5 Jan 2023 +Geodesics in 3-dimensional Euclidean Space +with One or Two Analytic Obstacles +Chengcheng Yang +(communited by Prof. Robert Hardt) +January 6, 2023 +1 + +Abstract +F. Albrecht and I.D. Berg proved that in an n-dimensional Eu- +clidean space with an analytic obstacle a geodesic is locally an al- +ternating finite union of a boundary segment on the surface of the +obstacle and a line segment. Their proof depends on the initial ve- +locity of the geodesic. At the end of their paper, they conjected that +there is a uniform bound on the number of line segments near a fixed +initial position, so it should be independent of the initial velocity. +In this paper we are going to show that this conjecture is true in +the case when n = 3. Furthermore, we are able to generalize the above +local finiteness property to the union of two analytic obstacles which +intersect transversally in a 3-dimensional Euclidean space. That is to +say a geodesic does not bounce infinitely often between two obstacles +near any point. +2 + +1 +introduction +In the paper [1] F. Albrecht and I.D. Berg proved that a geodesic cannot +have an accumulation of interior line segments in Rn or Cn with an analytic +obstacle. +More precisely, let M be the closure of the complement of an +obstacle in an Euclidean space and let S be its boundary. A geodesic in M +(thought of as a string stretching over some obstacle) is a locally shortest +path consisting of two types of segments: touching the boundary (which is +known as a boundary segment and whose acceleration is outward normal +to S) and not touching the boundary (which is a line segment known as an +interior line segment or an interval). We call the point connecting a boundary +segment and an interior line segment a switch point following the tradition +in the same paper above. A geodesic is necessarily C1 at the switch points +and can be parametrized by arc length; furthermore the acceleration exists +everywhere except at the switch points [2]. When S is analytic, F. Albrecht +and I.D. Berg showed that the switch points do not accumulate. That is to +say, there exists an ǫ > 0 such that γ(s) has no switch point for 0 < s < ǫ. +The result fails if S is just C∞. +In our paper we are going to generalize the result by looking at the union +of two obstacles in a 3-dimensional Euclidean space. Let M1, M2 be the clo- +sure of the complement of two obstacles in an Euclidean space whose surfaces +are S1, S2, respectively. Assume that S1 and S2 intersect transversally. Let +M be the intersection of M1 and M2 which is the closure of the complement +of the union of the two obstacles. Suppose γ is a geodesic in M, then the +same conclusion holds so γ does not bounce back and forth between two +surfaces infinitely often locally. In other words, γ is eventually a boundary +segment in one of the surfaces or a line segment. +The proof consists of two parts. The first part is concerned with the case +of R3 and the angle between S1 and S2 is less than 90◦. The argument uses +symmetry. The second part is still dealing with the case of R3 but the angle +can be more than 90◦. Here the symmetry arguemnt in the previous part +fails, so we need to come up with an asymmetric argument. Lastly R can be +replaced by C but for visualization we stay with the real Euclidean space. +The second half of our paper concerns with the conjecture at end of the +paper [1] by F. Albrecht and I.D. Berg. Let M be the same as above, and +fix one point p on M. They conjectured that there is a uniform bound on +the number of intervals (or switch points) within a neighborhood of p. We +are able to prove this conjecture if the dimension is 3. Namely, there exists +3 + +an ǫ such that for any geodesic γ initiating from p in M, there are at most +two intervals within the ǫ-ball of p. +The proof shows that there are finitely many wedges covering the (x, y)- +plane such that within each wedge such an ǫ exists. +4 + +2 +Part One +Theorem 1. Let M1 and M2 be 3-dimensional analytic manifolds with bound- +ary embedded in R3. Denote the boundary surfaces of M1 and M2 by S1 and +S2, respectively. Assume that S1 and S2 intersect transversally whose angle +is less than 90◦. Let M be the intersection of M1 and M2. If γ is a geodesics +stretching over M parametrized by arc length s, with γ(0) = p ∈ S1 ∩ S2. +Then there exists an ǫ > 0 such that γ has no switch point for 0 < s < ǫ. +Proof. 1. Set the coordinate system. +Without loss of generality we may assume that p is the origin, the x-axis +is tangent to the given geodesic γ at p, and the outward normal vectors to +S1 and S2 at p are (0, −k, 1) and (0, k, 1), respectively. So the normal vec- +tors at p are symmetric with respect to the z-axis. Let 0 < k < +1 +1+β for +some β > 0 so k < 1. The two surfaces S1 and S2 are defined near the +origin by analytic equations of the form z = g(x, y) and z = h(x, y), respec- +tively. It follows that the outward normal vector to S1 at the origin is equal +to (−gx(0, 0), −gy(0, 0), 1), which is also equal to (0, −k, 1) by hypothesis, +therefore +gx(0, 0) = 0, +gy(0, 0) = k. +Similarly, we have +hx(0, 0) = 0, +hy(0, 0) = −k. +Since +∂ +∂y +�� +(0,0)[g(x, y) − h(x, y)] = 2k > 0, +the inverse function theorem implies that the intersection of S1 and S2 near +p is a real analytic curve defined by the following equations: +y = φ(x), +z = g(x, φ(x)) = h(x, φ(x)), +where φ is real analytic and φ(0) = 0, φ′(0) = 0. In fact the tangent vector +to S1 ∩ S2 at the origin is given by +(1, φ′(x), gx(x, φ(x)) + gy(x, φ(x))φ′(x)) +�� +x=0 += +(1, φ′(0), gx(0, 0) + gy(0, 0)φ′(0)) += +(1, φ′(0), kφ′(0)), +which is normal to (0, k, 1), thus +(1, φ′(0), kφ′(0)) · (0, k, 1) = 2kφ′(0) = 0 =⇒ φ′(0) = 0. +5 + +So +φ(x) = aMxM + aM+1xM+1 + · · · , +(1) +where M ≥ 2 and aM ̸= 0. Notice that we assume φ(x) is not identically +zero and g(x, 0) is not identically zero. Otherwise we will have trivial cases +which will be included at the end. Thus the equation defining S1 near p is +of the form +g(x, y) = ky + xNa(x, y) + xyb(x, y) + y2c(y), +(2) +where N ≥ 2, the functions a, b, c are analytic, and a(0, 0) ̸= 0. Likewise +assume that h(x, 0) is not identically zero, the equation defining S2 near p is +of the form +h(x, y) = −ky + x +˜ +N˜a(x, y) + xy˜b(x, y) + y2˜c(y), +(3) +where ˜N ≥ 2, the functions ˜a,˜b, ˜c are analytic, and ˜a(0, 0) ̸= 0. Choose the +orientation of the coordinate system so that γ′(0) = (1, 0, 0), M1 = {z ≤ +g(x, y)}, and M2 = {z ≤ h(x, y)}. +2. Project the S1 ∩ S2 onto the (x, y)-plane. +The projection of the intersection of S1 and S2 onto the (x, y)-plane is a +curve given by (x, φ(x)), where x ∈ (−δ, δ) for some δ > 0. Then it divides +the vertical strip (−δ, δ) × (−∞, ∞) into two disconnected regions: +{x < φ(x)}, +{x > φ(x)}. +Suppose y < 0 = φ(0), then using the linear approximation +g(0, y) − h(0, y) ≈ gy(0, 0)y − hy(0, 0)y = 2ky < 0. +Thus connectedness implies that +{g(x, y) < h(x, y)} = {y < φ(x)}. +(4) +Similarly, +{g(x, y) > h(x, y)} = {y > φ(x)}. +(5) +Given a point z = g(x, y) in S1. If the point lies in M2, then z ≤ h(x, y), +thus g(x, y) ≤ h(x, y). With (4) it follows that the projection of S1 ∩ M2 +near p is +{g(x, y) ≤ h(x, y), −δ < x < δ} = {x ≤ φ(x), −δ < x < δ}. +6 + +Likewise with (5) the projection of S2 ∩ M1 near p is +{g(x, y) ≥ h(x, y), −δ < x < δ} = {x ≥ φ(x), −δ < x < δ}. +In conclusion the graph of φ(x) divides the (x, y)-plane into two parts near +0: the part below the graph corresponding to the projection of the surface +S1 in M and the part above the graph corresponding to the projection of the +surface S2 in M. +3. Concavity of φ, which will become crucial in proving the theorem later. +Since γ′(0) = (1, 0, 0), x′(s) > 0 for 0 ≤ s ≤ ǫ if we choose ǫ small enough. +Therefore x(s) > 0 when 0 ≤ s ≤ ǫ. According to the equation (1), +φ′′(x) = M(M − 1)aMxM−2 + (M + 1)MaM+1xM−1 + · · · . +When aM > 0, φ(x) is concave upward over the interval (0, δ); and when +aM < 0, φ(x) is concave downward over the same interval. Furthermore, we +may also assume that x(s) < δ for 0 ≤ s ≤ ǫ. So there are two cases to +consider: aM > 0 and aM < 0. +4. Approximate y(s) and y′(s) using the normal vectors N1(s), N2(s) to +S1, S2. +We denote +γ(s) = (x(s), y(s), z(s)), for 0 ≤ s ≤ ǫ. +If γ(s) ∈ S1, the normal vector to S1 at γ(s) is +N1(s) = (−gx(x(s), y(s)), −gy(x(s), y(s)), 1). +From (2) it follows that +gx(x(s), y(s)) = x(s)m(x(s), y(s)) + y(s)b(x(s), y(s)); +gy(x(s), y(s)) = k + x(s)k(x(s), y(s)) + y(s)l(y(s)), +where the functions m, b, k, l are bounded near (0, 0). Moreover, since x′(s) > +0 for 0 ≤ s ≤ ǫ, x(s) has a C1-inverse function s(x) for s ∈ [0, ǫ]. Therefore +we can express y(s) as +y(s) = y(s(x)) = α(x), +where α is a C1-function and α(0) = dα +dx(0) = 0. Then one has y(s) = o(x(s)). +Hence +gx(x(s), y(s)) = x(s)[m(x(s), y(s)) + y(s) +x(s)b(x(s), y(s))] = x(s)V1(s); +gy(x(s), y(s)) = k + x(s)[k(x(s), y(s)) + y(s) +x(s)l(y(s))] = k + x(s)V2(s). +7 + +Therefore +N1(s) = (−x(s)V1(s), −k − x(s)V2(s), 1), +where V1(s) and V2(s) are bounded for s near 0. Let s be such that γ′′(s) +exists, then γ′′(s) = z′′(s)N1(s) since the acceleration is outward normal to +S1. This implies that +x′′(s) = −z′′(s)x(s)V1(s), +y′′(s) = −z′′(s)(k + x(s)V2(s)). +(6) +For ǫ sufficiently small, |x(s)V2(s)| ≤ βk. Therefore |y′′(s)| ≤ (1 + β)k|z′′(s)| +from the second equality in (6). +Similarly, if γ(s) ∈ S2 and γ′′(s) exists, the equality in (3) deduces that +the normal vector to S2 at γ(s) is +N2(s) = (−x(s)W1(s), k − x(s)W2(s), 1), +where W1(s) and W2(s) are bounded for s near 0. +Then it follows from +γ′′(s) = z′′(s)N2(s) that +x′′(s) = −z′′(s)x(s)W1(s), +y′′(s) = −z′′(s)(−k + x(s)W2(s)). +(7) +Again for ǫ sufficiently small, one may assume that |x(s)W2(s)| ≤ βk. There- +fore |y′′(s)| ≤ (1 + β)k|z′′(s)| from the second equality in (7). +When γ(s) does not lie on S1 and S2, γ is a line segment so γ′′(s) is equal to +0. Combining with what we’ve found above, one gets |y′′(s)| ≤ (1+β)kz′′(s). +Notice that z(0) = z′(0) = y(0) = y′(0) = 0. Furthermore z′′(s) ≥ 0 (and +hence z′(s) ≥ 0) on the interval [0, ǫ], because the outward normal vectors to +S1 and S2 have a positive z-coordinate of 1 at the origin and γ′′(s) is directed +outward on a boundary segment on S1 or S2. Indeed, γ(s) is a locally shortest +path and if γ(s) lies on the surface of M1 or M2, its acceleration exists +everywhere except at the switch points and is outward normal to the surface +[2]. So for s ∈ [0, ǫ], one can approximate +|y′(s)| = | +� s +0 +y′′(σ)dσ| ≤ (1 + β)k +� s +0 +z′′(σ)dσ = (1 + β)kz′(s). +Integrating again one obtains +|y(s)| ≤ (1 + β)kz(s). +8 + +If γ(s) ∈ S1, the equality in (2) gives +z(s) += +g(x(s), y(s)) += +ky(s) + x(s)Na(x(s), y(s)) + x(s)y(s)b(x(s), y(s)) + y2(s)c(y(s)) +≤ +k|y(s)| + x(s)N|a(x(s), y(s))| + |y(s)||x(s)b(x(s), y(s)) + y(s)c(y(s))| +≤ +k|y(s)| + C1x(s)N + C2|y(s)| +≤ +(k + C2)(1 + β)kz(s) + C1x(s)N, +for some constants C1, C2. Since k < +1 +1+β, one can choose ǫ small enough so +that (k + C2)(1 + β)k < 1. Therefore there exists a positive constant A such +that +z(s) ≤ Ax(s)N ⇒ |y(s)| ≤ (1 + β)kAx(s)N = Bx(s)N. +(8) +Similarly, if γ(s) ∈ S2, the equality in (3) gives us +z(s) ≤ Ax(s) +˜ +N ⇒ |y(s)| ≤ Bx(s) +˜ +N, +(9) +by enlarging A and B if necessary. +Choosing ǫ small enough so that x(s) < 1 and assuming without loss of +generality that N ≤ ˜N, one has x(s) ˜ +N ≤ x(s)N. Thus with (8) and (9) +z(s) ≤ Ax(s)N, |y(s)| ≤ Bx(s)N, +(10) +if γ(s) ∈ S1 or S2. +Next let’s approximate y′(s). If γ(s) ∈ S1, differentiating z(s) = g(x(s), y(s)) +gives +z′(s) += +ky′(s) + x(s)N−1[Nx′(s)a(x(s), y(s)) + x(s)(ax(x(s), y(s))x′(s) + ay(x(s), y(s))y′(s))] ++y′(s)[x(s)b(x(s), y(s)) + x(s)y(s)by(x(s), y(s)) + 2y(s)c(y(s)) + y2(s)c′(y(s))] ++y(s)[x′(s)b(x(s), y(s)) + x(s)x′(s)bx(x(s), y(s))] +≤ +k|y′(s)| + C1x(s)N−1 + C2|y′(s)| + C3|y(s)| +≤ +(k + C2)(1 + β)kz′(s) + C1x(s)N−1 + C3Bx(s)N, +for some constants C1, C2, C3. Again since (1 + β)k < 1, for ǫ sufficiently +small, one can make (k + C2)(1 + β)k < 1, so +z′(s) ≤ Cx(s)N−1 ⇒ |y′(s)| ≤ (1 + β)kGx(s)N−1 = Dx(s)N−1. +(11) +9 + +Similarly, if γ(s) ∈ S2, differentiating z(s) = h(x(s), y(s)) gives +z′(s) ≤ Cx(s) +˜ +N−1 ⇒ |y′(s)| ≤ (1 + β)kGx(s) +˜ +N−1 = Dx(s) +˜ +N−1. +(12) +by enlarging C and D if necessary. Combining (11) and (12), together with +˜N ≥ N, one has +z′(s) ≤ Cx(s)N−1, |y′(s)| ≤ (1 + β)kGx(s)N−1 = Dx(s)N−1. +(13) +Now let’s look at the situation when γ(s) is in an interior line segment. +Considering a line segment in the image of γ with two endpoints γ(s1) and +γ(s2), we can parametrize y(s) for s ∈ [s1, s2] by +y(s) = y(s1) + T(x(s) − x(s1)), where T = dα +dx(x(s1)), +where with (13) +��dα +dx(x(s)) +�� = +��dα +dx(x(s1)) +�� = +��y′(s1) +x′(s2) +�� ≤ 2|y′(s1)| ≤ 2Dx(s1)N−1 ≤ 2Dx(s)N−1, +(14) +if x′ ≥ 1/2 by choosing ǫ small enough and the last inequality holds because +x(s) is increasing. Hence with (10) and (14) one obtains +|y(s)| +≤ +|y(s1)| + |T|(|x(s)| + |x(s1)|) +≤ +Bx(s1)N + 2Dx(s)N−1(x(s) + x(s)) +≤ +(B + 4D)x(s)N. +Replacing B by B + 4D, together with (10), yields that in general, +|y(s)| ≤ Bx(s)N for every s ∈ [0, ǫ]. +(15) +5. Prove M ≥ N. +If the geodesic γ moves from S1 to S2 or from S2 to S1, (x(s), y(s)) must +cross the curve y = φ(x). On the one hand, |φ(x(s))| ≤ Bx(s)N according to +(15); on the other hand, |φ(x(s))| ≥ |aM | +2 x(s)M by (1). Therefore one obtains +the following relation: +|aM| +2 x(s)M ≤ Bx(s)N =⇒ x(s)M−N ≤ 2B +|aM|. +10 + +Suppose M < N the left-hand side converges to infinity as s approaches 0, +a contradiction. This means that if M < N the geodesic γ eventually stops +bouncing between S1 and S2. Therefore it reduces to the case of one obstacle. +Hence we proceed with M ≥ N. +6. Next let’s prove that N = ˜N. +Case 1: M > N. On the intersection of S1 and S2 we’ve shown that +y = φ(x) for x ∈ (−δ, δ) and hence g(x, φ(x)) = h(x, φ(x)) over the interval +(−δ, δ). Using the equalities (1) and (2) one has +g(x, φ(x)) += +kφ(x) + xNa(x, φ(x)) + xφ(x)b(x, φ(x)) + φ2(x)c(φ(x)) += +kxM(aM + aM+1x + . . . ) + xN(a(0, 0) + . . . ) + +xM+1(aMb(0, 0) + . . . ) + a2 +Mx2M(c(0) + . . . ). +Since M > N the first nonzero term in the power serious expansion of +g(x, φ(x)) is a(0, 0)xN. Similarly the equalities (1) and (3) gives +h(x, φ(x)) += +−kφ(x) + x +˜ +N˜a(x, φ(x)) + xφ(x)˜b(x, φ(x)) + φ2(x)˜c(φ(x)) += +−kxM(aM + aM+1x + . . . ) + x +˜ +N(˜a(0, 0) + . . . ) + +xM+1(aM˜b(0, 0) + . . . ) + a2 +Mx2M(˜c(0) + . . . ). +By the uniqueness of the power serious expansion one must have ˜N = N, +otherwise the first nonzero term in the power serious expansion of h(x, φ(x)) +has an order of at least N + 1 (we assumed ˜N ≥ N earlier). +Case 2: M = N. If there is an interior line segment in the image of γ +with two endpoints γ(s1) and γ(s2) such that γ(s1) ∈ S2 and γ(s2) ∈ S1. We +can parametrize y(s) for s ∈ [s1, s2] by +y(s) = y(s1) + T(x(s) − x(s1)), where T = dα +dx(x(s1)). +Since γ(s1) ∈ S2, one can use the second inequality in (12) to estimate +��dα +dx(x(s1)) +�� = +��y′(s1) +x′(s1) +�� ≤ 2|y′(s1)| ≤ 2Dx(s1) +˜ +N−1, +if |x′(s)| ≥ 1 +2 by choosing ǫ small enough. With (9) one gets +|y(s)| +≤ +|y(s1)| + |T|(|x(s)| + |x(s1)|) +≤ +Bx(s1) +˜ +N + 2Dx(s1) +˜ +N−1(x(s) + x(s)) +≤ +(B + 4D)x(s) +˜ +N, +11 + +where the last inequality holds because x(s) is increasing. +For some s ∈ +(s1, s2), we have y(s) = φ(x(s)) and so +|φ(x(s))| ≤ (B + 4D)x(s) +˜ +N. +On the other hand, |φ(x(s))| ≥ |aN| +2 x(s)N if s is sufficiently close to 0 by (1). +Thus the following relation holds: +|aN| +2 x(s)N ≤ (B + 4D)x(s) +˜ +N =⇒ x(s)N− ˜ +N ≤ 2(B + 4D) +|aN| +. +Suppose ˜N > N then the left-hand side converges to infinity as s approaches +0, a contradiction. So γ eventually stops going from S2 to S1 which reduces +to the case of one obstacle. Hence we proceed with ˜N = N. +7. Show a(0, 0) > 0, ˜a(0, 0) > 0. +Let’s prove by contradiction. Suppose that a(0, 0) < 0. Assume γ has a +switch point inside S1 at s = s0. That is to say γ(s0) ∈ S1 and for either +s > s0 nearby or s < s0 nearly, γ(s) is an interior line segment. Denote +(x(s0), y(s0)) by (x0, y0) for simplicity. Consider the intersection of the two- +dimensional plane y = y0 + T(x − x0) with the surface z = g(x, y). Set +f(x) = g(x, y0 + T(x − x0)), where T = dα +dx(x0). +It follows that +d2f +dx2 (x0) = gxx(x0, y0) + 2gxy(x0, y0)T + gyy(x0, y0)T 2. +(16) +Using (2), (15) and choosing ǫ sufficiently small one can estimate +gxx(x0, y0) += +xN−2 +0 +[N(N − 1)a(x0, y0) + x0p(x0, y0)] + y0q(x0, y0) (17) +≤ +xN−2 +0 +N(N − 1)1 +2a(0, 0) + BxN +0 C1 +≤ +xN−2 +0 +N(N − 1)1 +2a(0, 0) − xN−2 +0 +N(N − 1)1 +4a(0, 0) += +xN−2 +0 +N(N − 1)1 +4a(0, 0) +Furthermore, using the inequality in (14) and letting ǫ be small enough one +has +|2gxy(x0, y0)T + gyy(x0, y0)T 2| +(18) +≤ +C2|T| ≤ C2 · 2DxN +0 +≤ +−xN−2 +0 +N(N − 1)1 +8a(0, 0). +12 + +So +d2f +dx2 (x0) ≤ xN−2 +0 +N(N − 1)1 +8a(0, 0) < 0. +Therefore f is concave downward at x0 and the tangent line to the curve at +x0 is above the graph, a contradiction. In other words, γ has no switch point +on S1 near the origin and so initially stays inside S2 or is a line segment. +Similarly γ initially stays inside S1 or is a line segment for ˜a(0, 0) < 0. Hence +this reduces to the case of one obstacle. Thus we proceed with assuming that +a(0, 0) > 0 and ˜a(0, 0) > 0. +8. Show that given ǫ small enough, if γ leaves S1 at a switch point, it will +never enter S1 again. Similarly, when γ leaves S2, it has to enter S1 at the +next switch point. +Indeed we can prove by contradiction. Suppose γ(s) leaves S1 at s = s0 +and dives into the interior of M for increasing s until it enters S1 again at +s = s1. Again set +f(x) = g(x, y0 + T(x − x0)), +where (x0, y0) = (x(s0), y(s0)) and T = dα +dx(x0). It follows that for s ∈ [s0, s1] +d2f +dx2 (x(s)) = gxx(x(s), y(s)) + 2gxy(x(s), y(s))T + gyy(x(s), y(s))T 2. +Note that when s = s0, this is just the expression in (16). Using an analogous +argument as shown in (17) for the case a(0, 0) > 0 one yields +gxx(x(s), y(s)) ≥ x(s)N−2N(N − 1)1 +4a(0, 0). +Moreover in analogy to (18) one has +2gxy(x(s), y(s))T + gyy(x(s), y(s))T 2 +≥ +−xN−2 +0 +N(N − 1)1 +8a(0, 0) +≥ +−x(s)N−2N(N − 1)1 +8a(0, 0). +Hence +d2f +dx2 (x(s)) ≥ x(s)N−2N(N − 1)1 +8a(0, 0) > 0 for all s ∈ [s0, s1]. +13 + +Therefore f ′(x(s)) is increasing as x(s) increases from x(s0) = x0 to x(s1) = +x1. On the other hand, since the interior line segment is tangent to S1 at the +two endpoints, we must have +f ′(x0) = f ′(x1), +a contradiction. Therefore if γ leaves S1 at the switch point γ(s0) for some +s0 ∈ [0, ǫ], the geodesic arc beyond this point is a line segment never meeting +S1 again. Hence the next switch point (if there is one) lies on the surface S2. +The same argument holds for S2 as well. +9. The global behavior of γ. +Lemma 1. Near the origin the geodesic is an alternating sequence of a bound- +ary segment on S1, an interval from S1 to S2, a boundary segment on S2, an +interval from S2 to S1, and so on. +Proof. Each time the projection of γ crosses the graph of φ at time s, there is +l(s) > 0 such that γ is an interior line segment over the interval [s − l(s), s + +l(s)]. The set A of such s with 0 ≤ s ≤ ǫ is therefore countable. Furthermore +if A1 = sup A then A1 is actually the maximum of the set, because there is +no s ∈ A within the l(A1)-distance of A1. Let A2 = sup(A − A1), A3 = +sup(A − {A1, A2}), and so on. +It follows that the set A can be linearly +ordered as +A = {A1 > A2 > A3 > . . . }, +such that between An and An+1 the curve γ lies entirely in S1 or S2 for each +n ≥ 1. +10. If aM > 0, the curve y = φ(x) is concave upward for x > 0 nearby. +We can obtain a contradiction as follows. +• If γ leaves a point in S2 and enters a point in S1, then (x(s), y(s)) +crosses φ from above to below at some s = s1. By concavity one must +have dα +dx(x(s1)) < φ(x(s1)); +• Later (x(s), y(s)) crosses φ from below to above at some s = s2, then +dα +dx(x(s2)) > φ(x(s2)); +• In between γ stays in S1 all the time. +14 + +• Since dα +dx(x(s1)) = y′(s1) +x′(s1) and dα +dx(x(s2)) = y′(s2) +x′(s2), we must have +y′(s1) < x′(s1)φ(x(s1)), y′(s2) > x′(s2)φ(x(s2)) +⇒ +y′(s2) − y′(s1) > x′(s2)φ(x(s2)) − x′(s1)φ(x(s1)) +• Therefore it suffices to show that for 0 < s1 < s2 < ǫ, +y′(s2) − y′(s1) ≤ x′(s2)φ(x(s2)) − x′(s1)φ(x(s1)). +On the one hand, +y′(s2) − y′(s1) = +� s2 +s1 +y′′(s)ds, +where y′′(s) = −z′′(s)(k + x(s)V2(s)) ≤ −z′′(s)(1 − β)k from (6). Thus +y′(s2) − y′(s1) ≤ +� s2 +s1 +−z′′(s)(1 − β)kds < 0. +On the other hand, +x′(s2)φ(x(s2)) − x′(s1)φ(x(s1)) += +� s2 +s1 +d +ds +� +x′(s)φ′(x(s)) +� += +� s2 +s1 +x′′(s)φ′(x(s)) + x′(s)2φ′′(x(s))ds. +Now let’s estimate x′′(s)φ′(x(s)) + x′(s)2φ′′(x(s)). Since γ(s) ∈ S1 for s ∈ +(s1, s2), one has x′′(s) = −z′′(s)x(s)V1(s) from (6). +Therefore |x′′(s)| ≤ +Ez′′(s)x(s) for some positive constant E. By hypothesis γ is parametrized +by arc length, so |x′(s)| ≤ 1. With (8) and (11) one differentiates z(s) = +g(x(s), y(s)) twice to obtain +z′′(s) += +gxx(x(s), y(s))x′(s)2 + 2gxy(x(s), y(s))x′(s)y′(s) + gyy(x(s), y(s))y′(s)2 ++gx(x(s), y(s))x′′(s) + gy(x(s), y(s))y′′(s) += +� +x(s)N−2[N(N − 1)a(x(s), y(s)) + x(s)p(x(s), y(s))] + y(s)q(x(s), y(s)) +� +x′(s)2 ++y′(s)[2gxy(x(s), y(s))x′(s) + gyy(x(s), y(s))y′(s)] ++[x(s)m(x(s), y(s)) + y(s)b(x(s), y(s))]x′′(s) ++[k + x(s)k(x(s), y(s)) + y(s)l(y(s))]y′′(s) +≤ +� +x(s)N−2C1 + |y(s)|C2 +� +· 1 + |y′(s)|C3 + C4|x′′(s)| + [k + βk]|y′′(s)| +≤ +x(s)N−2C1 + Bx(s)NC2 + Dx(s)N−1C3 + C4Ez′′(s)x(s) + (1 + β)kz′′(s)(k + |x(s)V2(x)|) +≤ +x(s)N−2(C1 + Bx(s)2C2 + Dx(s)C3) + z′′(s)(C4Ex(s) + (1 + β)k · (1 + β)k) +≤ +x(s)N−2C5 + z′′(s)C6 + (1 + β)2k2z′′(s). +15 + +By hypothesis (1 + β)k < 1, then we can choose ǫ small enough so that +C6 < 1 − (1 + β)2k2 implying that +z′′(s) ≤ Fx(s)N−2, and so |x′′(s)| ≤ EFx(s)N−1 = Gx(s)N−1. +Now let’s use (1) to approximate φ′′(x(s)) and φ′(x(s)): +0 < φ′(x) = MaMxM−1 + (M + 1)aM+1xM + · · · ≤ 2MaMxM−1 for x near 0. +So for ǫ sufficiently small, one has +φ′(x(s))x′′(s) ≥ φ′(x) · −Gx(s)N−1 +(19) +≥ +2MaMx(s)M−1 · −Gx(s)N−1 = −2MGaMx(s)M+N−2. +On the other hand, +φ′′(x) = M(M − 1)aMxM−2 + · · · ≥ 1 +2aMM(M − 1)xM−2 for x near 0. +So again by choosing ǫ small enough and assuming x′(s) ≥ 1 +2, one obtains +φ′′(x(s))x′(s)2 ≥ 1 +8aMM(M − 1)x(s)M−2. +(20) +Combing (19) and (20) we find that +φ′′(x(s))x′(s)2 + φ′(x(s))x′′(s) ≥ x(s)M−2aM(1 +8M(M − 1) − 2MGx(s)N). +Since N ≥ 2 the above difference can be made positive for every s ∈ [0, ǫ] if +ǫ is sufficiently small. Hence +x′(s2)φ(x(s2)) − x′(s1)φ(x(s1)) = +� s2 +s1 +x′′(s)φ′(x(s)) + x′(s)2φ′′(x(s))ds > 0. +We reach a contradiction. There γ(s) eventually stops bouncing between S1 +and S2 as s approaches 0, which reduces to the case of one obstacle. +11. If aM < 0, the curve y = φ(x) is concave downward for x > 0 nearby. +Replacing S1 by S2 and g by h in the previous argument gives a contradiction. +12. Trivial Case 1: φ(x) is identically zero, but g(x, 0) and h(x, 0) are not +identically zero. +Everything is fine until Step 9 by letting M = ∞. In Step 10, the proof +is as follows: +16 + +• If γ leaves a point in S2 and enters a point in S1, then (x(s), y(s)) crosses +φ from above to below at some s = s1. Since the curve y = φ(x) is the +x-axis, one must have dα +dx(x(s1)) < 0 and so y′(s1) < 0. +• Later (x(s), y(s)) crosses φ from below to above at some s = s2, thus +dα +dx(x(s2)) > 0 and so y′(s2) > 0. +• In between γ stays in S1 all the time where y′′(s) is always negative, so +y′(s2) < y′(s1), a contradiction. +13. Trivial Case 2: one of g(x, 0), h(x, 0) is identically zero, but not both. +Without loss of generality, let us assume that g(x, 0) is identically zero +but h(x, 0) is not. Then we can write g(x, y) as +g(x, y) = ky + xyb(x, y) + y2c(y). +If φ(x) is identically zero, then g(x, φ(x)) is also identically zero. However, +with equality (3) +h(x, φ(x)) = x +˜ +N˜a(x, 0) = x +˜ +N(˜a(0, 0) + . . . ) ̸= 0. +So φ(x) is nonzero whose power serious expansion is still (1). Everything is +fine until step 4. If γ(s) ∈ S1, the equality in (2) gives +z(s) += +g(x(s), y(s)) += +ky(s) + x(s)y(s)b(x(s), y(s)) + y(s)2c(y(s)) +≤ +k|y(s)| + |y(s)||x(s)b(x(s), y(s)) + y(s)c(y(s))| +≤ +k|y(s)| + C1|y(s)| +≤ +(k + C1)(1 + β)k1z(s) +By hypothesis (1+β)k < 1, for s close enough to 0 such that C1 < 1−(1+β)k2 +(1+β)k , +one obtains z(s) ≤ 0. On the other hand, z(s) > 0 for all s > 0. So the +geodesic does not touch S1 near the origin, which reduces to the case of one +obstacle. +14. g(x, 0) and h(x, 0) are both identically 0. One can show that φ(x) is +also identically zero. In this case the geodesic does not touch S1 and S2 near +the origin, so it must be a line segment at the beginning. +17 + +Theorem 2. Let M1 and M2 be 3-dimensional analytic manifolds with bound- +ary embedded in R3. Denote the boundary surfaces of M1 and M2 by S1 and +S2, respectively. Assume that S1 and S2 intersect transversally whose angle +is greater than or equal to 90◦. Let M be the intersection of M1 and M2. Let +γ be a geodesics in M parametrized by arc length s, with γ(0) = p ∈ S1 ∩ S2. +Then there is an ǫ > 0 such that γ has no switch point for 0 < s < ǫ. +Proof. 1. Set the coordinate system as in Theorem 1, except that the outward +normal vectors to S1 and S2 at p are no longer symmetrical with respect to +the z-axis. +Tilt the system appropriately so that the normal vectors are +(0, −k1, 1) and (0, k2, 1), respectively, where 0 < k1 < 1 < k2. Furthermore, +choose k1 and k2 so k1k2 < +1 +1+β < 1 for some β > 0. It follows that +gx(0, 0) = 0, +gy(0, 0) = k1. +And +hx(0, 0) = 0, +hy(0, 0) = −k2. +Since +∂ +∂y +�� +(0,0)[g(x, y) − h(x, y)] = k1 + k2 > 0, +again the implicit function theorem implies that the intersection of S1 and +S2 near p is a real analytic curve defined by the following equations: +y = φ(x), +z = g(x, φ(x)) = h(x, φ(x)), +where φ has the same power expansion as in (1) by assuming that φ is not +identically zero. Furthermore let us assume that g(x, 0) and h(x, 0) are not +identically zero. Therefore the equation defining S1 near p is of the form +g(x, y) = k1y + xNa(x, y) + xyb(x, y) + y2c(y), +(21) +where N ≥ 2, the functions a, b, c are analytic, and a(0, 0) ̸= 0. Moreover +the equation defining S2 near p is of the form +h(x, y) = −k2y + x +˜ +N˜a(x, y) + xy˜b(x, y) + y2˜c(y), +(22) +where ˜N ≥ 2, the functions ˜a,˜b, ˜c are analytic, and ˜a(0, 0) ̸= 0. +2. Same as in Theorem 1, the graph of φ(x) divides the (x, y)-plane into +two parts near 0: the part below the graph corresponding to the projection +18 + +of the surface S1 in M and the part above the graph corresponding to the +projection of the surface S2 in M. +3. Concavity of φ is determined by the sign of aM and there are two cases +to consider: aM > 0 and aM < 0. +4. Approximate y(s) and y′(s) using the normal vectors N1(s), N2(s) to +S1, S2. +Following the same procedure as in (6) and (7) one obtains if γ(s) ∈ S1, +x′′(s) = −z′′(s)x(s)V1(s), +y′′(s) = −z′′(s)(k + x(s)V2(s)), +where |x(s)V2(s)| ≤ βk1 for ǫ sufficiently small and therefore |y′′(s)| ≤ (1 + +β)k1|z′′(s)|; and if γ(s) ∈ S2, +x′′(s) = −z′′(s)x(s)W1(s), +y′′(s) = −z′′(s)(−k + x(s)W2(s)), +where |x(s)W2(s)| ≤ βk2 for ǫ small enough and hence |y′′(s)| ≤ (1 + +β)k2|z′′(s)|. +Case 1, if γ(s) ∈ S1, since k1(1 + β) < 1 one can use the same argument +as before to obtain +z(s) ≤ Ax(s)N ⇒ |y(s)| ≤ (1 + β)k1Ax(s)N = Bx(s)N. +(23) +Next differentiating z(s) = g(x(s), y(s)) once gives +z′(s) ≤ Cx(s)N−1 ⇒ |y′(s)| ≤ (1 + β)k1Cx(s)N−1 = Dx(s)N−1. +(24) +Case 2: if γ(s) ∈ S2, the arguments above fail because k2 > 1. Instead we +need to use a different approach. Since y′′(s) = −z′′(s)(−k2 + x(s)W2(s)), +z′′(s) + k2y′′(s) = z′′(s)[1 + k2 +2 − k2x(s)W2(s)] ≥ z′′(s)[1 + (1 − β)k2 +2]. (25) +On the other hand, if γ(s) ∈ S1, then y′′(s) = −z′′(s)(k1 + x(s)V2(s)) implies +that +z′′(s) + k2y′′(s) += +z′′(s)[1 − k1k2 − k2x(s)V2(s)] +(26) +≥ +z′′(s)[1 − k1k2 − k2k1β] += +z′′(s)[1 − (1 + β)k1k2], +Combining (25) with (26), together with the hypothesis (1 + β)k1k2, there +exists a constant H such that if γ(s) ∈ S1 or S2 +z′′(s) + k2y′′(s) ≥ Hz′′(s). +(27) +19 + +Note that the inequality (27) still holds if γ(s) does not touch any surface +since the acceleration is 0. It follows that for every s in the interval [0, ǫ], +z′(s) + k2y′(s) = +� s +0 +z′′(σ) + k2y′′(σ)dσ ≥ +� s +0 +Hz′′(σ)dσ = Hz′(s). +(28) +Moreover, +z(s) + k2y(s) ≥ Hz(s) for every s ∈ [0, ǫ]. +(29) +With (22) and (29) one deduces that +Hz(s) +≤ +x(s) +˜ +N˜a(x(s), y(s)) + y(s)[x(s)˜b(x(s), y(s)) + y(s)˜c(y(s))] +≤ +C1x(s) +˜ +N + |y(s)|C2 +≤ +C1x(s) +˜ +N + (1 + β)k2z(s)C2. +With ǫ sufficiently small C2 can be made as small as possible so that +H − (1 + β)k2C2 > 0, +implying +z(s) ≤ Ax(s) +˜ +N ⇒ |y(s)| ≤ (1 + β)k2Ax(s) +˜ +N = Bx(s) +˜ +N, +(30) +after enlarging A and B accordingly. +Differentiating z(s) = h(x(s), y(s)) once, together with (28) and (30), one +obtains +z′(s) += +−k2y′(s) + x(s) +˜ +N−1[ ˜Nx′(s)˜a(x(s), y(s)) ++x(s)(˜ax(x(s), y(s))x′(s) + ˜ay(x(s), y(s))y′(s))] ++y′(s)[x(s)˜b(x(s), y(s)) + x(s)y(s)˜by(x(s), y(s)) ++2y(s)˜c(y(s)) + y2(s)˜c′(y(s))] ++y(s)[x′(s)˜b(x(s), y(s)) + x(s)x′(s)˜bx(x(s), y(s))] +so z′(s) + k2y′(s) +≤ +C1x(s)N−1 + C2|y′(s)| + C3|y(s)| +≤ +C1x(s) +˜ +N−1 + C2(1 + β)k2z′(s) + C3Bx(s) +˜ +N +so Hz′(s) +≤ +C1x(s) +˜ +N−1 + C2(1 + β)k2z′(s) + C3Bx(s) +˜ +N +With ǫ sufficiently small C2 can be made as small as possible so that C2(1 + +β) < H and thus +z′(s) ≤ Cx(s) +˜ +N−1 ⇒ |y′(s)| ≤ (1 + β)k2z′(s) = Dx(s) +˜ +N−1, +(31) +20 + +by enlarging C and D accordingly. +Without loss of generality we may assume that N ≤ ˜N, then x(s) ˜ +N ≤ +x(s)N for x(s) ≤ 1. Thus (23) and (30) imply that if γ ∈ S1 or S2 +z(s) ≤ Ax(s)N, +|y(s)| ≤ Bx(s)N, +(32) +Furthermore (24) and (31) imply that if γ ∈ S1 or S2 +z′(s) ≤ Cx(s)N−1, +|y′(s)| ≤ Dx(s)N−1. +(33) +When γ(s) does not lie on S1 or S2, following the same argument as in +Theorem 1, one can show that +|T(x(s))| ≤ 2Dx(s)N−1. +(34) +|y(s)| ≤ Bx(s)N for every s ∈ [0, ǫ]. +(35) +From 5-10, one can copy the proof from Theorem 1 word by word to have: +M ≥ N, N = ˜N, a(0, 0) > 0, ˜a(0, 0) > 0, and γ is an alternating sequence +of boundaries segments on S1, S2 and line segments between S1 and S2. +11. If aM < 0, the argument is slightly different. +• If γ leaves a point in S1 and enters a point in S2, then (x(s), y(s)) crosses +φ from below to above at some s = s1. Since the curve y = φ(x) is +concave downward for x > 0, one must have dα +dx(x(s1)) > φ(x(s1)); +• Later (x(s), y(s)) crosses φ from above to below at some s = s2, then +dα +dx(x(s2)) < φ(x(s2)); +• In between γ stays in S2 all the time. +• Since dα +dx(x(s1)) = y′(s1) +x′(s1) and dα +dx(x(s2)) = y′(s2) +x′(s2), we must have +y′(s1) > x′(s1)φ(x(s1)), y′(s2) < x′(s2)φ(x(s2)) +⇒ y′(s2) − y′(s1) < x′(s2)φ(x(s2)) − x′(s1)φ(x(s1)). +• Therefore it suffices to show that for 0 < s1 < s2 < ǫ, +y′(s2) − y′(s1) ≥ x′(s2)φ(x(s2)) − x′(s1)φ(x(s1)). +21 + +On the one hand, +y′(s2) − y′(s1) = +� s2 +s1 +y′′(s)ds, +where y′′(s) = −z′′(s)(−k2 + x(s)W2(x)) ≥ z′′(s)(1 − β)k. Thus +y′(s2) − y′(s1) ≥ +� s2 +s1 +z′′(s)(1 − β)kds > 0. +On the other hand, +x′(s2)φ(x(s2)) − x′(s1)φ(x(s1)) += +� s2 +s1 +d +ds +� +x′(s)φ′(x(s)) +� += +� s2 +s1 +x′′(s)φ′(x(s)) + x′(s)2φ′′(x(s))ds. +Now let’s estimate x′′(s)φ′(x(s)) + x′(s)2φ′′(x(s)). Since γ(s) ∈ S2 for s ∈ +(s1, s2), one has x′′(s) = −z′′(s)x(s)W1(s) from (7). +Therefore |x′′(s)| ≤ +Ez′′(s)x(s) for some positive constant E. By hypothesis γ is parametrized +by arc length, so |x′(s)| ≤ 1. One differentiates z(s) = h(x(s), y(s)) twice to +get +z′′(s) += +hxx(x(s), y(s))x′(s)2 + 2hxy(x(s), y(s))x′(s)y′(s) + hyy(x(s), y(s))y′(s)2 ++hx(x(s), y(s))x′′(s) + hy(x(s), y(s))y′′(s) += +� +x(s)N−2[N(N − 1)˜a(x(s), y(s)) + x(s)˜p(x(s), y(s))] ++y(s)˜q(x(s), y(s)) +� +x′(s)2 ++y′(s)[2hxy(x(s), y(s))x′(s) + hyy(x(s), y(s))y′(s)] ++[x(s)W1(s)]x′′(s) + [−k2 + x(s)W2(s)]y′′(s) +One moves −k2y′′(s) to the other side of the inequality, together with (30) +and (31), to obtain +z′′(s) + k2y′′(s) +≤ +� +x(s)N−2C1 + |y(s)|C2 +� +· 1 + |y′(s)|C3 + C4|x′′(s)| + |x(s)W2(s)||y′′(s)| +≤ +x(s)N−2C1 + Bx(s)NC2 + Dx(s)N−1C3 + C4Ez′′(s)x(s) ++|x(s)W2(s)|z′′(s)(k2 + |x(s)W2(x)|) +≤ +x(s)N−2(C1 + Bx(s)2C2 + Dx(s)C3) ++z′′(s)(C4Ex(s) + βk2(1 + β)k2) +≤ +x(s)N−2C5 + β(1 + β)k2 +2z′′(s)C6. +22 + +Using (27) and choosing ǫ small enough so that β(1 + β)k2 +2C6 < H one gets +that +z′′(s) ≤ Fx(s)N−2, and so |x′′(s)| ≤ EFx(s)N−1 = Gx(s)N−1. +Now let’s use (1) to approximate φ′′(x(s)) and φ′(x(s)): +0 > φ′(x) = MaMxM−1 + (M + 1)aM+1xM + · · · ≥ 2MaMxM−1 for x near 0. +So for ǫ sufficiently small, one has +φ′(x(s))x′′(s) ≤ φ′(x(s)) · −Gx(s)N−1 +(36) +≤ +2MaMx(s)M−1 · −Gx(s)N−1 = −2MaMLx(s)M+N−2. +On the other hand, +φ′′(x) = M(M − 1)aMxM−2 + · · · ≤ 1 +2aMM(M − 1)xM−2 for x near 0. +So using ǫ small enough and assuming x′(s) ≥ 1 +2, one obtains +φ′′(x(s))x′(s)2 ≤ 1 +8aMM(M − 1)x(s)M−2. +(37) +Combining (36) and (37) we find that +φ′′(x(s))x′(s)2 + φ′(x(s))x′′(s) ≤ x(s)M−2aM(1 +8M(M − 1) − 2MGx(s)N). +Since N ≥ 2 the above difference can be made negative for every s ∈ [0, ǫ] if +ǫ is sufficiently small. Hence +x′(s2)φ(x(s2)) − x′(s1)φ(x(s1)) = +� s2 +s1 +x′′(s)φ′(x(s)) + x′(s)2φ′′(x(s))ds < 0. +We reach a contradiction. Thus γ eventually stops bouncing between S1 and +S2 as s approaches 0, which reduces to the case of one obstacle. +The trivial cases from 11-14 follow exactly the same proof in Theorem 1. +23 + +3 +Part Two +Theorem 3. Let M be an 3-dimensional manifold with boundary embedded +in R3. Denote the boundary surface of M by S and let γ(s) be a geodesic +on M parametrized by arc length s with γ(0) = p ∈ S. Then there exists +an ǫ > 0 such that the number of line segments in the image of γ within +the ǫ-ball of p is uniformly bounded, namely it is independent of the initial +velocity γ′(0). More precisely, there are at most two complete or partial line +segments. +The idea is to show that in each direction there exists a wedge and an ǫ +such that if γ′(0) is within this wedge, then γ has a uniform bound on the +number of switch points within the ǫ-ball of p. +1. Set up the coordinate system. +Choose an orientation of the coordinate system (x, y, z) so that p is the +origin, S near p can be parametrized by an analytic function z = g(x, y), and +the outward normal vector to S at p is in the positive z-axis. Notice that +γ′(0) is either tangent to S at p in which case γ′(0) is in the (x, y)-plane. (Or +γ′(0) has a negative z-component pointing towards the interior of M. We +will look at this case in the end.) +2. Without loss of generalizty, one may assume that the lowest degree in +the power series expansion of g(x, y) is 2. The idea for higher degrees is very +similar, which will be mentioned at the end. +Since the z-axis is normal to S at p and p is the origin, the Taylor series +expansion of g is +g(x, y) = 1 +2gxx(0, 0)x2 + gxy(0, 0)xy + 1 +2gyy(0, 0)y2 + higher-order terms, +where +1 +2gxx(0, 0)x2 + gxy(0, 0)xy + 1 +2gyy(0, 0)y2 +is not the zero polynomial. Let H be the Hessian matrix +H = +� +gxx(0, 0) +gxy(0, 0) +gxy(0, 0) +gyy(0, 0) +� +. +Since H is symmetric, the spectral theorem says that there exists a 2×2 real +orthogonal matrix P such that +PHP t = +�2a +0 +0 +2b +� +. +24 + +Rotating and/or reflecting the (x, y)-plane using P, the surface S near p can +be parametrized as follows: +g(x, y) = ax2 + by2 + higher-order terms, +where a and b are not identically zero. The signs of a, b tell us about the +shape of S near the origin, namely +• When a > 0, b ≥ 0 or a ≥ 0, b > 0, S is concave upward near the origin +and γ has at most one switch point near p. +• When a < 0, b ≤ 0 or a ≤ 0, b < 0, S is concave downward near the +origin and γ has no switch point near p. +• When one of a, b is positive and the other is negarive, S has a saddle +point at p and we are going to investigate this case in details. Without +loss of generality, we may assume that a > 0, b < 0. Moreover replacing +b by −b yields +g(x, y) = ax2 − by2 + higher-order terms, where a > 0, b > 0. +Notice that when ax2 − by2 = 0, y = ± +�a +b x, which gives rise to two +lines dividing the plane into four different regions. +There exists an +angle 0 < θ0 < π/2 such that +tan(θ0) = +�a +b. +Then we are going to prove the following: +1. In the positive direction of the x-axis: for any small positive δ, +there exists an ǫ > 0 such that if γ′(0) lies inside the wedge [−θ0 + +δ, θ0 − δ], then γ has at most one switch point within the ǫ-ball of +p. Similar statements hold in the negative direction of the x-axis +and positive and negative directions of the y-axis. +2. In the positive direction of y = +�a +bx : there exist an η > 0 and +an ǫ > 0 such that if γ′(0) is in the wedge [θ0 − η, θ0 + η], then +γ has at most two switch points within the ǫ-ball of p. Similar +25 + +statements also hold in the negative direction of y = +�a +bx and +the positive and negative directions of y = − +�a +bx. +3. Combining 1 and 2, the theorem follows. +Proposition 1. In the positive direction of the x-axis: for any 0 < δ < θ0, +there exists an ǫ > 0 such that if γ′(0) lies inside the wedge [−θ0 + δ, θ0 − δ], +then γ has at most one switch point within the ǫ-ball of p. +Proof. (1) Set up the frame. The first estimate of ǫ comes from that S is +parametrized by g(x, y) within the ǫ-ball of p. Since γ′(0) is a unit vector +tangent to S, one has x′(0) = cos θ and y′(0) = sin θ, where θ ∈ [−θ0+δ, θ0−δ] +by hypothesis. +(2) Show that x′(s) > 0 if ǫ is chosen small enough. This is the second +estimate of ǫ. Here we are assuming that if γ(s) is within the ǫ-ball of p, +then for every 0 ≤ σ ≤ s, γ(σ) also lies within the ǫ-neighbhorhood of p. +Let γ(s) = (x(s), y(s), z(s)) with |x(s)|,|y(s)|, |z(s)| less than or equal to +ǫ, so that γ(s) is within the ǫ-ball of p. If γ(s) ∈ S, then the normal vector +at γ(s) is +N(s) = (−gx(x(s), y(s)), −gy(x(s), y(s)), 1). +Let +g(x, y) = ax2 − by2 + x3c(x, y) + x2yd(x, y) + xy2e(x, y) + y3f(x, y), +(38) +then +gx(x, y) += +2ax + (3x2c + x3cx) + (2xyd + x2ydx) + (y2e + xy2ex) + y3fx. +gy(x, y) += +−2by + x3cy + (x2d + x2ydy) + (2xye + xy2ey) + (3y2f + y3fy). +There exists a positive constant A such that +gx(x, y) ≤ A and gy(x, y) ≤ A, if |x|, |y| ≤ ǫ, +(39) +where A → 0 as ǫ → 0. +Let s be such that γ′′(s) exists, then γ′′(s) = +z′′(s)N(s). This implies that +x′′(s) = −z′′(s)gx(x(s), y(s)) ⇒ |x′′(s)| ≤ Az′′(s). +26 + +Here z′′(s) ≥ 0, because within the ǫ-ball of p the outward normal vector to +S has a positive z-coordinate of 1 and γ′′(s) directs outward on a boundary +segment on S. Indeed γ(s) is a locally shortest path. If γ(s) lies on the +surface of M, its acceleration exists everywhere except at the switch points +and it outward normal to the surface. On the other hand, if γ(s) lies on a +line segment in the interior of M, then the acceleration is zero. So we obtain +x′(s) = x′(0) + +� s +0 +x′′(σ)dσ ≥ cos θ − A +� s +0 +z′′(σ)dσ = cos θ − Az′(s). (40) +Next let’s approximate z′(s). If γ(s) ∈ S, then +z(s) = g(x(s), y(s)) = ax(s)2−by(s)2+x(s)3c+x(s)2y(s)d+x(s)y(s)2e+y(s)3f, +so +z′(s) += +2ax(s)x′(s) − 2by(s)y′(s) + 3x(s)2x′(s)c(x(s), y(s)) + x(s)3cx(x(s), y(s))x′(s) ++x(s)3cy(x(s), y(s))y′(s) + 2x(s)x′(s)y(s)d(x(s), y(s)) + x(s)2y′(s)d(x(s), y(s)) ++x(s)2y(s)dx(x(s), y(s))x′(s) + x(s)2y(s)dy(x(s), y(s))y′(s)x′(s)y(s)2e(x(s), y(s)) ++2x(s)y(s)y′(s)e(x(s), y(s)) + x(s)y(s)2ex(x(s), y(s))x′(s) + x(s)y(s)2ey(x(s), y(s))y′(s) +3y(s)2y′(s)f(x(s), y(s)) + y(s)3fx(x(s), y(s))x′(s) + y(s)3fy(x(s), y(s))y′(s). +Since γ(s) is parametrized by arc length, |x′(s)| and |y′(s)| are no more than +1. Since each term in the above expression has either an x(s) or y(s), there +exists a positive constant B such that +|z′(s)| ≤ B, if |x(s)|, |y(s)| ≤ ǫ, +where B → 0 as ǫ → 0. On the other hand, if γ(s) is within an interior +line segment, then γ′(s) is constant and equal to the value at the endpoints. +Therefore |z′(s)| is still bounded by B. Thus we can choose ǫ small enough +so that B < cos |θ| +A , where |θ| ≤ θ0 − δ. It follows from (40) that +x′(s) > 0, +if |x(s)|, |y(s)| ≤ ǫ for ǫ sufficiently small. Therefore x(s) > 0. +(3) Approximate y′(s) and y(s). This is the third estimate of ǫ. +If γ(s) ∈ S, then the normal vector to S at γ(s) is +N(s) = (−gx(x(s), y(s)), −gy(x(s), y(s)), 1). +27 + +Let s be such that γ′′(s) exists, then γ′′(s) = z′′(s)N(s). From (39) one +obtains +y′′(s) = −z′′(s)gy(x(s), y(s)) ⇒ |y′′(s)| ≤ Az′′(s), +where A → 0 as ǫ → 0. Thus +y′(s) = y′(0) + +� s +0 +y′′(σ)dσ ≤ sin θ + A +� s +0 +z′′(σ)dσ = sin θ + Az′(s). +Let tan |θ| < c < tan θ0, where |θ| ≤ θ0 − δ. Choose ǫ sufficiently small so +that +sin |θ| + Az′(s) ≤ c [cos θ − Az′(s)] , i.e. A ≤ c cos θ − sin |θ| +1 + c +. +Therefore combining with (40) +|y′(s)| ≤ cx′(s). +(41) +With y(0) = x(0) = 0, one has +|y(s)| ≤ cx(s). +(42) +(4) Concavity. This is the last estimate of ǫ. +Suppose γ(s) leaves S at a switch point when s = s0 and dives into the +interior of M for increasing s until it enters S again at s = s1. +Since x′(s) > 0 within the ǫ-ball of p, then x(s) has a C1-inverse function +s(x) for s ∈ [0, s1]. Therefore we can express y(s) as +y(s) = y(s(x)) = α(x), +where α is a C1-function and α(0) = dα +dx = 0. +Consider the intersection of the two-dimensional plane y = y0 +T(x−x0) +with the surface z = g(x, y), where (x0, y0) = (x(s0), y(s0)), and T = dα +dx(x0). +With (41) +|T| = +���� +y′(s0) +x′(s0) +���� ≤ c. +Set +f(x) = g(x, y0 + T(x − x0)), +28 + +then +d2f +dx2 (x) = gxx + 2gxyT + gyyT 2, +where +gxx += +2a + (6xc + 6x2cx + x3cxx) + (2yd + 4xydx + x2ydxx) + (2y2ex + xy2exx) + y3fxx += +2a + x(. . .) + y(. . .) +gxy += +(3x2cy + x3cxy) + (2xd + 2xydy + x2dx + x2ydxy) + (2ye + y2ey + 2xyex + xy2exy) + x3fxy += +x(. . .) + y(. . .) +gyy += +−2b + x3cyy + (2x2dy + x2ydyy) + (2xe + 4xyey + xy2eyy) + (6yf + 6y2fy + y3fyy) += +−2b + x(. . .) + y(. . .) +So with y(s) = y0 + T(x − x0) for s ∈ [s0, s1] and (41), (42) +d2f +dx2(x) += +2a + x(. . .) + y(. . .) + 2Tx(. . .) + 2Ty(. . .) − 2bT 2 + T 2x(. . . ) + T 2y(. . .), so +d2f +dx2 (x(s)) +≥ +2a − x(s)C1 − cx(s)C2 − 2cx(s)C3 − 2c2x(s)C4 − 2bc2 − c2x(s)C5 − c3x(s)C6, +where C1, . . . , C6 are constants bounding the terms inside the corresponding +parentheses. By assumption c < tan θ0 = +�a +b, we can choose ǫ sufficiently +small such that the right side of the above inequality is positive, i.e., +x(s)C1 + cx(s)C2 + 2cx(s)C3 + 2c2x(s)C4 + c2x(s)C5 + c3x(s)C6 < 2a − 2bc2. +Therefore d2f +dx2(x(s)) > 0 for s ∈ [s0, s1]. It implies that f ′(x(s)) is increasing +as x(s) increases from x(s0) = x0 to x(s1) = x1. On the other hand, since +the interior line segment is tangent to S at the two endpoints, one must have +f ′(x0) = f ′(x1), +a contradiction. Therefore if γ leaves S at the switch point γ(s0), the geodesic +arc beyond this point is a line segment never returning to S again. So γ has +at most one switch point within the ǫ-ball of p, as desired. +Proposition 2. In the positive direction of the line y = +�a +b, there exist an +η > 0 and an ǫ > 0 such that if γ′(0) lies in the wedge [θ0 − η, θ0 + η], then +γ has at most two switch points within the ǫ-ball of p. +29 + +Proof. (1) Set up the frame. +Let’s rotate the (x, y)-plane so that the x-axis points in the positive di- +rection of the line y = +�a +b by the matrix +�cos θ0 +− sin θ0 +sin θ0 +cos θ0 +� +. +Thus with respect to this new coordinate system and in connection with (38) +the surface S near p can be parametrized by +g(cos θ0x − sin θ0y, sin θ0x + cos θ0y) += +a(cos θ0x − sin θ0y)2 − b(sin θ0x + cos θ0y)2 + +(cos θ0x − sin θ0y)3c + (cos θ0x − sin θ0y)2(sin θ0x + cos θ0y)d ++(cos θ0x − sin θ0y)(sin θ0x + cos θ0y)2e + (sin θ0x + cos θ0y)3f += +−2 +√ +abxy + (a − b)y2 ++x3 � +(cos θ0)3c + (cos θ0)2 sin θ0d + cos θ0(sin θ0)2e + (sin θ0)3f +� ++x2y +� +3(cos θ0)2 sin θ0c + cos θ0(1 − 3(sin θ0)2)d + sin θ0(3(cos θ0)2 − 1)e + 3(sin θ0)2 cos θ0f +� ++xy2 � +−3 cos θ0(sin θ0)2c + ((cos θ0)3 − 2(cos θ0)2 sin θ0(d + e) + 3 sin θ0(cos θ0)2f +� ++y3 � +−(sin θ0)3c + (sin θ0)2 cos θ0d − sin θ0(cos θ0)2e + (cos θ0)3f +� +where c, d, e, f are evaluated at (cos θ0x − sin θ0y, sin θ0x + cos θ0y). For con- +venience, let’s still use g(x, y) to denote the new parametrization +g(x, y) += +−2 +√ +abxy + (a − b)y2 +(43) ++x3c(x, y) + x2yd(x, y) + xy2e(x, y) + y3f(x, y), +where c, d, e, f are the (new) functions inside the corresponding brackets. +(2) Notice that within the ǫ-ball of p, we have |x|, |y| ≤ ǫ no matter how +we rotate the (x, y)-plane, because the distance to the origin is fixed. So if +γ(s) is within the ǫ-ball of p, then we have +|γ(s)| ≤ ǫ ⇒ |x(s)| ≤ ǫ, |y(s)| ≤ ǫ, |z(s)| ≤ ǫ. +(3) Concavity with respect to the new frame. If one moves slightly above +the x-axis, namely in the first quadrant, then the surface is concave down- +ward; if one moves slightly below the x-axis, namely in the fourth quadrant, +then the surface is concave upward. +30 + +(4) If γ′(0) points in the positive x-direction. +One can rewrite the g(x, y) again as follows: +g(x, y) = xNh(x, y) + xyi(x, y) + y2j(x, y), +(44) +where N ≥ 3, h(0, 0) ̸= 0, i(0, 0) = −2 +√ +ab, and j(0, 0) = a−b. According to +the theorem for a fixed direction (cite here), if γ′(0) = +∂ +∂x then there exists +an ǫ > 0 such that γ has at most one switch point before leaving the ǫ-ball +of p. +(5) γ′(0) is in the wedge [−η, η] where 0 < η < min(2θ0, π − 2θ0). This is +the first estimate of η. +Let δ ∈ [−η, η] and δ ̸= 0. So γ′(0) does NOT point in the positive x- +direction. If we rotate the (x, y)-plane according to the angle δ, the Taylor +expansion of g(cos δx − sin δy, sin δx + cos δy), denoted as gδ(x, y), has a +nonzero x2 term, because with (43) +gδ(x, y) += +−2 +√ +ab(cos δx − sin δy)(sin δx + cos δy) + (a − b)(sin δx + cos δy)2 +(45) ++(cos δx − sin δy)3c + (cos δx − sin δy)2(sin δx + cos δy)d ++(cos δx − sin δy)(sin δx + cos δy)2e + (sin δx + cos δy)3f += +x2 +� +bcos2(θ0 + δ) +cos2 θ0 +− b +� ++ xy +� +−2 +√ +ab cos 2δ + (a − b) sin 2δ +� ++y2 �√ +ab sin 2δ + (a − b) cos2 δ +� ++x3 � +(cos δ)3c + (cos δ)2 sin δd + cos δ(sin δ)2e + (sin δ)3f +� ++x2y +� +3(cos δ)2 sin δc + cos δ(1 − 3(sin δ)2)d + sin δ(3(cos δ)2 − 1)e + 3(sin δ)2 cos δf +� ++xy2 � +−3 cos δ(sin δ)2c + ((cos δ)3 − 2(cos δ)2 sin δ(d + e) + 3 sin δ(cos δ)2f +� ++y3 � +−(sin δ)3c + (sin δ)2 cos δd − sin δ(cos δ)2e + (cos δ)3f +� +where c, d, e, f are evaluated at (cos δx − sin δy, sin δx + cos δy). Notice that +if the coefficient of x2 is zero, then +bcos2(θ0 + δ) +cos2 θ0 +− b = 0 ⇒ cos(θ0 + δ) = ± cos θ0 ⇒ δ = 0, −2θ0, or π − 2θ0, +a contradiction. +Therefore we could rewrite gδ(x, y) to include the angle +δ = 0. +gδ(x, y) += +a2(δ)x2 + a3(δ)x3 + · · · + aN−1(δ)xN−1 +(46) ++xNhδ(x, y) + xyiδ(x, y) + y2jδ(x, y), +31 + +where a2(δ), . . . , aN−1(δ) are constant coefficients of x2, . . . , xN−1, respec- +tively. Moreover at δ = 0, these constants vanish, and h0(x, y) = h(x, y), +i0(x, y) = i(x, y), j0(x, y) = j(x, y). +(6) There exists an ǫ that works for all δ ∈ [−η, η]. This is the second +estimate of η (in step 4). There are four different cases depending on the +sign of a2(δ) and h(0, 0). +Case 1: a2(δ) > 0, h(0, 0) > 0. +When a2(δ) > 0, the angle δ < 0. There is actually a relationship between +a2(δ), . . . , aN−1(δ). +a2(δ) += +b +cos2 θ0 +[sin2(θ0) − sin2(θ0 + δ)] += +b +cos2 θ0 +[sin(θ0) + sin(θ0 + δ)][sin(θ0) − sin(θ0 + δ)] += +b +cos2 θ0 +[sin(θ0) + sin(θ0 + δ)][sin(θ0) − sin(θ0) cos δ − cos(θ0) sin δ] += +b +cos2 θ0 +[sin(θ0) + sin(θ0 + δ)][− cos(θ0) sin δ + sin(θ0)(1 − cos δ)] +≥ +b +cos2 θ0 +sin(θ0)[− cos(θ0) sin δ] = b tan(θ0)| sin δ|. +Furthermore, there are constants ci and M sufficiently large such that +a3(δ) += +c1 cos2 δ sin δ + c2 cos δ sin2 δ + c3 sin3 δ +a4(δ) += +c4 cos3 δ sin δ + c5 cos2 δ sin2 δ + c6 cos δ sin3 δ + c7 sin4 δ +... +aN−1(δ) += +c8 cosN−2 δ sin δ + · · · + c9 sinN−1 δ +⇒ +|a3(δ)|, . . . , |aN−1(δ)| ≤ M| sin δ| +(i) Let’s continue with the ǫ in (4). Then x′(s) ≥ 1 +2 if ǫ is chosen small +enough. Here we are assuming that if γ(s) is within the ǫ-ball of p, then for +every 0 ≤ σ ≤ s, γ(σ) also lies within the ǫ-ball of p. +(ii) Let γ(s) = (x(s), y(s), z(s)). If γ(s) ∈ S, then the normal vector to +S at γ(s) is +N(s) = (−(gδ)x(x(s), y(s)), −(gδ)y(x(s), y(s)), 1). +32 + +From (46) it follows that +(gδ)x(x, y) += +2a2(δ)x + · · · + (N − 1)aN−1(δ)xN−2 + NxN−1hδ + xN(hδ)x ++yiδ + xy(iδ)x + y2(jδ)x. +There exists a positive constant A such that +|(gδ)x(x, y)| ≤ A if |x|, |y| ≤ ǫ and δ ∈ [−η, η]. +Moreover A → 0 as ǫ → 0 for a fixed η. Let s be such that γ′′(s) exists, then +γ′′(s) = z′′(s)N(s). This implies that +x′′(s) = −z′′(s)(gδ)x(x(s), y(s)) ⇒ |x′′(s)| ≤ Az′′(s). +Here z′′(s) ≥ 0, because within the ǫ-ball of p the outward normal vector to +S has a positive z-coordinate of 1 and γ′′(s) directs outward on a boundary +segment on S. Indeed γ(s) is a locally shortest path. If γ(s) lies on the +surface of M, its acceleration exists everywhere except at the switch points +and is outward normal to the surface. +On the other hand, if γ(s) lies on a line segment in the interior of M, then +the acceleration γ′′(s) is zero. So the previous inequality still holds. Thus +x′(s) = x′(0) + +� s +0 +x′′(σ)dσ ≥ 1 − A +� s +0 +z′′(σ)dσ = 1 − Az′(s). +(47) +Next let’s approximate z′(s). If γ(s) ∈ S, using (46) one has +z′(s) += +2a2(δ)x(s)x′(s) + · · · + (N − 1)aN−1(δ)x(s)N−2x′(s) ++NxN−1x′(s)hδ + xN[(hδ)xx′(s) + (hδ)yy′(s)] ++x′(s)y(s)iδ + x(s)y′(s)iδ + x(s)y(s)[(iδ)xx′(s) + (iδ)yy′(s)] ++2y(s)y′(s)jδ + y(s)2[(jδ)xx′(s) + (jδ)yy′(s)]. +Since γ(s) is parametrized by arc length, |x′(s)| and |y′(s)| are no more than +1. So there exists a positive constant B such that +|z′(s)| ≤ B if |x(s)|, |y(s)| ≤ ǫ and δ ∈ [−η, η], +where B → 0 as ǫ → 0 for a fixed η. On the other hand, if γ(s) is within +an interior line segment, then γ′(s) is constant and equal to the value at the +33 + +endpoints. Therefore |z′(s)| is still bounded by B. Thus we can choose ǫ +small enough so that B ≤ +1 +2A. It follows from (47) that +x′(s) ≥ 1 +2, +(48) +if ǫ is chosen sufficiently small. Notice that this ǫ works for all δ because we +can bound hδ, iδ, jδ and their partial derivatives uniformly. +(iii) Approximate z(s), z′(s), y(s), y′(s). If γ(s) ∈ S, then +y′′(s) = −z′′(s)(gδ)y(x(s), y(s)) ⇒ |y′′(s)| ≤ Az′′(s), +where A → 0 if ǫ → 0 and A does not depend on δ. If γ(s) ̸∈ S, then γ(s) +is within an interior line segment, so the inequality still holds. With respect +to the new frame after rotating the (x, y)-plane by δ, γ′(0) = +∂ +∂x and so with +z(0) = z′(0) = y(0) = y′(0) = 0 one deduces that +|y′(s)| ≤ Az′(s), |y(s)| ≤ Az(s). +(49) +Now let’s estimate z(s). If γ(s) ∈ S, with (46) and (49) +z(s) +≤ +a2(δ)x(s)2 + |a3(δ)|x(s)3 + · · · + |aN−1(δ)|x(s)N−1 ++x(s)N|hδ(x(s), y(s))| + x(s)Az(s)|iδ(x(s), y(s))| + A2z(s)2|jδ(x(s), y(s))| +First, +|a3(δ)|x(s) + · · · + |aN−1(δ)|x(s)N−3 +≤ +M| sin δ|[x(s) + x(s)2 + · · · + x(s)N−3] +≤ +b tan(θ0)| sin δ| ≤ a2(δ), +if we choose ǫ sufficiently small. Second, +|hδ(x(s), y(s))| ≤ 2h(0, 0), +if η and ǫ are sufficiently small. Third, +x(s)Az(s)|iδ(x(s), y(s))|+A2z(s)2|jδ(x(s), y(s))| ≤ x(s)Az(s)C1+A2z(s)2C2, +where C1, C2 are some constants and we can choose ǫ small enough so that +Ax(s)C1 + A2z(s)C2 ≤ C3 < 1. +34 + +Therefore there exists a positive constant C such that +z(s) ≤ (2a2(δ)x(s)2 + 2h(0, 0)x(s)N)C +(50) +⇒ +|y(s)| ≤ (2a2(δ)x(s)2 + 2h(0, 0)x(s)N)AC +On the other hand, +z′(s) +≤ +2a2(δ)x(s) + · · · + (N − 1)|aN−1(δ)|x(s)N−2 ++Nx(s)N−1|hδ| + x(s)N[|(hδ)x| + |(hδ)y|] ++|y(s)||iδ| + x(s)Az′(s)|iδ| + x(s)|y(s)|[|(iδ)x| + |(iδ)y|] ++2y(s)Az′(s)|jδ| + |y(s)|2[|(jδ)x| + |(jδ)y|] +≤ +3a2(δ)x(s) + 2Nx(s)N−1h(0, 0) + (2a2(δ)x(s)2 + 2h(0, 0)x(s)N)ACC4 + C3z′(s) +≤ +4a2(δ)x(s) + 4Nx(s)N−1h(0, 0) + C3z′(s), +if η, ǫ are sufficiently small and C3 < 1, C4 are some constants. So there is +a constant C such that +z′(s) ≤ (4a2(δ)x(s) + 4Nx(s)N−1h(0, 0))C +(51) +⇒ +y′(s) ≤ (4a2(δ)x(s) + 4Nx(s)N−1h(0, 0))AC. +It follows that if y = α(x) and T(s) = α′(x(s)), then +|T(s)| = +���� +y′(s) +x′(s) +���� ≤ 2|y′(s)| ≤ (8a2(δ)x(s) + 8Nx(s)N−1h(0, 0))AC. +If γ(s) ̸∈ S, then γ(s) is in some line segment where y(s) = y(s0) + +T(s0)(x(s) − x(s0)) for some switch point at s0 < s. Since x(s) is increasing, +it follows that +|y(s)| +≤ +|y(s0)| + |T(s0)|[|x(s)| + |x(s0)|] +≤ +(2a2(δ)x(s0)2 + 2h(0, 0)x(s0)N)AC ++2(4a2(δ)x(s0) + 4Nx(s0)N−1h(0, 0))AC[|x(s)| + |x(s0)|] +≤ +(2a2(δ)x(s)2 + 2h(0, 0)x(s)N)AC + 2(4a2(δ)x(s) + 4Nx(s)N−1h(0, 0))AC · 2x(s) +≤ +(18a2(δ)x(s)2 + 18Nh(0, 0)x(s)N)AC. +The same inequality for T(s) still holds as before because T(s) = T(s0). +(iv) Now we are ready to show that there is at most one switch point. +Set f(x) = g(x, y0 + T(x − x0)) where (x0, y0) = (x(s0), y(s0)), T = T(s0), +and s0 is arbitrary. Then +f ′′(x0) += +gxx(x0, y0) + 2gxy(x0, y0)T + gyy(x0, y0)T 2, +35 + +where +gxx(x0, y0) += +2a2(δ) + 6a3(δ)x0 + · · · + (N − 1)(N − 2)aN−1(δ)xN−3 +0 ++xN−2 +0 +[N(N − 1)hδ + 2Nx0(hδ)x + x2 +0(hδ)xx] ++y0[2(iδ)x + x0(iδ)xx + y0(jδ)xx]. +First, for ǫ sufficiently small, +|6a3(δ)x0 + · · · + (N − 1)(N − 2)aN−1(δ)xN−3 +0 +| +≤ +M| sin δ|(6x0 + · · · + (N − 1)(N − 2)xN−3 +0 +) +≤ +b tan(θ0)| sin δ| ≤ a2(δ). +Second, for η and ǫ sufficiently small, +N(N − 1)hδ + 2Nx0(hδ)x + x2 +0(hδ)xx ≥ 1 +2N(N − 1)h(0, 0). +Third, there are constants C5, C6, C7 such that +|2(iδ)x + x0(iδ)xx + y0(jδ)xx| ≤ C5, |2gxy(x0, y0)| ≤ C6, |gyy(x0, y0)T| ≤ C7. +So +f ′′(x0) +≥ +2a2(δ) − a2(δ) + xN−2 +0 +1 +2N(N − 1)h(0, 0) +−(18a2(δ)x2 +0 + 18Nh(0, 0)xN +0 )ACC5 − (8a2(δ)x0 + 8NxN−1 +0 +h(0, 0))ACC6 +−(8a2(δ)x0 + 8NxN−1 +0 +h(0, 0))ACC7 += +a2(δ)[1 − 18ACC5x2 +0 − 8ACC6x0 − 8ACC7x0] ++xN−2 +0 +h(0, 0)[1 +2N(N − 1) − 18ACC5Nx2 +0 − 8ACC6Nx0 − 8ACC7Nx0]. +Therefore if ǫ is sufficiently small, for all |x0| ≤ ǫ, one has +1 − 18ACC5x2 +0 − 8ACC6x0 − 8ACC7x0 > 0, +1 +2N(N − 1) − 18ACC5Nx2 +0 − 8ACC6Nx0 − 8ACC7Nx0 > 0, +implying that γ can’t have a line segment within the ǫ-ball. Thus in the case +when a2(δ) > 0 and h(0, 0) > 0, γ has at most one switch point within the +ǫ-ball for any δ ∈ [−η, η]. +36 + +Case 2: a2(δ) < 0, h(0, 0) < 0. +Suppose for the sake of contradiction, γ′(0) is in the direction of +∂ +∂x. Then +we can approximate y(s), T(s) as in the paper (cite here) to show that γ has +no switch point close to the origin. It follows that γ has to be on the surface +initially. Given any two points γ(s1), γ(s2) on the geodesic close to the origin, +one can show that the line segment connecting them is actually in the interior +of M contradicting that γ is locally shortest. +Let (x1, y1) = (x(s1), y(s1)) and (x2, y2) = (x(s2), y(s2)). Then for t ∈ +[0, 1], set +f(t) = gδ(x1 + t(x2 − x1), y1 + t(y2 − y1)). +So +f ′′(t) = (gδ)xx(x2 − x1)2 + 2(gδ)xy(x2 − x1)(y2 − y1) + (gδ)yy(y2 − y1)2. +By the mean value theorem, +x2 − x1 = (s2 − s1)x′(˜s), y2 − y1 = (s2 − s1)y′(ˆs), +for some ˜s, ˆs in (s1, s2). Let +T = y2 − y1 +x2 − x1 += y′(ˆs) +x′(˜s) → 0 as s1, s2 → 0, +because γ′(0) = (1, 0, 0). Therefore +f ′′(t) = (x2 − x1)2[(gδ)xx + 2(gδ)xyT + (gδ)yyT 2]. +Since gxx(x, y) → 2a2(δ) as x, y → 0, then +f ′′(t) → (x2 − x1)2[2a2(δ)] +as s1, s2 → 0 and for every t ∈ [0, 1]. Thus the shortest path two points on γ +close to the orgin is the line segment in between which lies stricly below the +surface, a contradiction. So γ is a straight line initially. +The surface in the (x, z)-plane is the curve with equation +z(x) += +gδ(x, 0) += +a2(δ)x2 + · · · + aN−1(δ)xN−1 + xNhδ(x, 0), +which implies that +z′(0) = 0, z′′(0) = 2a2(δ) < 0. +37 + +So the slope of the line segment must be negative. If the line segment re- +enters the surface at some switch point, then the surface can’t be concave +downward there. Otherwise the line lies above the surface. +z′(x) += +2a2(δ)x + · · · + (N − 1)aN−1(δ)xN−2 + NxN−1hδ(x, 0) + xN(hδ)x(x, 0) +z′′(x) += +2a2(δ) + 6a3(δ)x + · · · + (N − 1)(N − 2)aN−1(δ)xN−3 ++N(N − 1)xN−2hδ(x, 0) + 2NxN−1(hδ)x(x, 0) + xN(hδ)xx(x, 0) +When a2(δ) < 0, the angle δ > 0. As before, one has +−a2(δ) += +b +cos2 θ0 +[sin(θ0) + sin(θ0 + δ)][sin(θ0 + δ) − sin(θ0)] += +b +cos2 θ0 +[sin(θ0) + sin(θ0 + δ)][δ cos(θ)], +where θ ∈ (θ0, θ0 + δ) by the mean value theorem. For η sufficiently small, +we have +δ = +δ +sin δ sin δ ≥ 1 +2 sin δ, +since +δ +sin δ → 1 as δ → 0. Thus +−a2(δ) ≥ +b +cos2 θ0 +sin(θ0)1 +2 sin δ cos(θ0 + η) ≥ 1 +4b tan(θ0) sin δ, +if η is sufficiently close to 0. Furthermore, there is M sufficiently large such +that +|a3(δ)|, . . . , |aN−1(δ)| ≤ M sin δ. +It follows that +|6a3(δ)x + · · · + (N − 1)(N − 2)aN−1(δ)xN−3| +≤ +M sin δ(6x + · · · + (N − 1)(N − 2)xN−3) +≤ +1 +4b tan(θ0) sin δ ≤ −a2(δ), +for all |x| ≤ ǫ if ǫ is sufficiently small. Moreover, if η and ǫ are sufficiently +small, then +N(N − 1)hδ(x, 0) + 2Nx(hδ)x(x, 0) + x2(hδ)xx(x, 0) ≤ 1 +2N(N − 1)h(0, 0), +38 + +Therefore +z′′(x) ≤ 2a2(δ) − a2(δ) + 1 +2N(N − 1)h(0, 0)xN−2 < 0. +So γ has no switch point unless it terminates at a point on the surface. As +a summary in the case when a2(δ) < 0 and h(0, 0) < 0, γ is either a straight +line exiting the ǫ-ball or a line segment terminating at some point on the +surface within the ǫ-ball. +Case 3: a2(δ) < 0, h(0, 0) > 0. +Since a2(δ) < 0, γ is initially a straight line just as shown in Case 2. +Suppose the angle between γ′(0) and the positive x-axis is −β. The line +either terminates at some point on the surface, or exits the ǫ-ball, or enters +the surface at some switch point at time s0. +Denote x(s0) as x0. +First, +the intersection of the line with the surface at γ(s0) satisfies − tan(β)x0 = +gδ(x0, 0) and so +− tan(β)x0 = a2(δ)x2 +0 + a3(δ)x3 +0 + · · · + aN−1(δ)xN−1 +0 ++ xN +0 hδ(x0, 0). +Next, the line is tangent to the surface at γ(s0), so − tan(β) = (gδ)x(x0, 0) +and +− tan(β) = 2a2(δ)x0+3a3(δ)x2 +0+· · ·+(N−1)aN−1(δ)xN−2 +0 ++NxN−1 +0 +hδ(x0, 0)+xN +0 (hδ)x(x0, 0). +Since x0 > 0, the above two equalities imply the following: +a2(δ)x0 + a3(δ)x2 +0 + · · · + aN−1(δ)xN−2 +0 ++ xN−1 +0 +hδ(x0, 0) += +2a2(δ)x0 + 3a3(δ)x2 +0 + · · · + (N − 1)aN−1(δ)xN−2 +0 ++ NxN−1 +0 +hδ(x0, 0) + xN +0 (hδ)x(x0, 0) +⇒ +a2(δ) + a3(δ)x0 + · · · + aN−1(δ)xN−3 +0 ++ xN−2 +0 +hδ(x0, 0) += +2a2(δ) + 3a3(δ)x0 + · · · + (N − 1)aN−1(δ)xN−3 +0 ++ NxN−2 +0 +hδ(x0, 0) + xN−1 +0 +(hδ)x(x0, 0) +⇒ +a2(δ) + 2a3(δ)x0 + · · · + (N − 2)aN−1(δ)xN−3 +0 ++ (N − 1)xN−2 +0 +hδ(x0, 0) + xN−1 +0 +(hδ)x(x0, 0) += +0. +Let 0 < c < 1 be a constant to be determined later. Then for ǫ sufficiently +small, +|2a3(δ)x0 + · · · + (N − 2)aN−1(δ)xN−3 +0 +| +≤ +M sin δ[2x0 + · · · + (N − 2)xN−3 +0 +] +≤ +c1 +4b tan(θ0) sin δ ≤ −ca2(δ). +39 + +Therefore +−a2(δ) − 2a3(δ)x0 − · · · − (N − 2)aN−1(δ)xN−3 +0 +≥ −a2(δ) + ca2(δ). +On the other hand, for η and ǫ sufficiently small, we have +|(N − 1)xN−2 +0 +hδ(x0, 0) + xN−1 +0 +(hδ)x(x0, 0)| ≤ (N − 1)xN−2 +0 +(1 + c)h(0, 0) +Thus +(1 − c)|a2(δ)| ≤ (N − 1)xN−2 +0 +(1 + c)h(0, 0) +(52) +⇒ +xN−2 +0 +≥ +(1 − c)|a2(δ)| +(N − 1)(1 + c)h(0, 0). +Now we are going to first shift our coordinates to have the origin at +γ(s0) = (x0, 0, z0 = − tan(β)x0) and then rotate the (x, z)-plane so that the +γ′(s0) points in the positive x-axis. Let’s use (u, v, w) for the new coordinates, +then with respect to the new frame +x = cos βu + sin βw + x0, y = v, z = − sin βu + cos βw + z0, +so the surface z = gδ(x, y) satisfies the equation +− sin βu + cos βw + z0 = gδ(cos βu + sin βw + x0, v). +Check that we can still solve for w analytically in terms of u, v within the +ǫ-ball. Taking the partial derivative of +sin βu − cos βw − z0 + gδ(cos βu + sin βw + x0, v) +with respect to w yields +− cos β + (gδ)x sin β = cos β[(gδ)x tan β − 1] = cos β[−(gδ)x(gδ)x(x0, 0) − 1], +where − tan β = (gδ)x(x0, 0) from before. Since (gδ)x(x, y) → 0 as x, y → 0, +for ǫ sufficiently small, +−(gδ)x(gδ)x(x0, 0) < 1. +Therefore there exists a real analytic function kδ such that w = kδ(u, v) with +kδ(0, 0) = 0, (kδ)u(0, 0) = 0, (kδ)v(0, 0) = 0. +40 + +Estimate γ(s) in the new frame starting from the point (x0, 0, z0). After +replacing s by s − s0, γ′(0) is equal to +∂ +∂u. +1. u′(s) ≥ 1 +2 if ǫ is chosen small enough. +By triangular inequality, |u(s)|, |v(s)|, |w(s)| are less than or equal to 2ǫ. +If γ(s) ∈ S, then the normal vector to S at γ(s) is +N(s) = (−(kδ)u(u(s), v(s)), −(kδ)v(u(s), v(s)), 1). +Since the lowest degree in kδ is at least two, there exists a positive constant +A such that +|(kδ)u(u, v)| ≤ A, |(kδ)v(u, v)| ≤ A, if |u|, |v| ≤ 2ǫ, +where A → 0 as ǫ → 0. +Let s be such that γ′′(s) exists, then γ′′(s) = +w′′(s)N(s). This implies that +u′′(s) = −w′′(s)(kδ)u(u(s), v(s)) ⇒ |u′′(s)| ≤ Aw′′(s). +Here w′′(s) ≥ 0 because within the ǫ-ball of p the surface S has the parametriza- +tion w = kδ(u, v) and thus the outward normal vector to S has a positive +w-coordinate of 1 and γ′′(s) is outward normal on a boundary segment in S. +If γ(s) ̸∈ S, γ′′(s) = 0 except at the switch points. Thus +u′(s) = u′(0) + +� s +0 +u′′(σ)dσ ≥ 1 − Aw′(s). +Next approximate w′(s). If γ(s) ∈ S, then +w(s) = kδ(u(s), v(s)) = u(s)2a(u(s), v(s))+u(s)v(s)b(u(s), v(s))+v(s)2c(u(s), v(s)), +for some analytic functions a, b, c. +Since γ is parametrized by arclength, +|u′(s)| ≤ 1 and |v′(s)| ≤ 1. Since each term in w′(s) has either u(s) or v(s) +and u′(s) or v′(s), there exists a positive constant B such that +|w′(s)| ≤ B, if |u(s)|, |v(s)| ≤ 2ǫ, +where B → 0 as ǫ → 0. On the other hand, if γ(s) ̸∈ S, γ′(s) is contant and +equal to the value at the endpoints. Therefore |w′(s)| is still bounded by B. +Thus one can choose ǫ small enough so that B < +1 +2A. It follows that +u′(s) ≥ 1 +2. +41 + +2. Approximate v′(s) and v(s). +If γ(s) ∈ S, then the normal vector to S at γ(s) is +N(s) = (−(kδ)u(u(s), v(s)), −(kδ)v(u(s), v(s)), 1). +Let s be such that γ′′(s) exists, then γ′′(s) = w′′(s)N(s). This implies that +v′′(s) = −w′′(s)(kδ)v(u(s), v(s)) ⇒ |v′′(s)| ≤ Aw′′(s). +Thus with v′(0) = w′(0) = 0, +|v′(s)| ≤ +� s +0 +|v′′(σ)|dσ ≤ A +� s +0 +w′′(σ)dσ = Aw′(s). +With v(0) = w(0) = 0, +|v(s)| ≤ Aw(s). +3. Coefficients of kδ(u, v). Denote kδ(u, v) as +kδ(u, v) = b2(δ)u2 + · · ·+ bN−1(δ)uN−1 + uNlδ(u, v) + uvmδ(u, v) + v2nδ(u, v), +where b2(δ), . . . , bN−1(δ) are constants and lδ, mδ, nδ are analytic functions of +u, v. Observe that for n between 2 and N − 1, +n!bn(δ) = ∂nkδ +∂un (0, 0). +The following lemma finds ∂nkδ +∂un (u, v) for n ≥ 2 by induction. +Lemma 2. Let A be cos β + sin β(kδ)u(u, v). Then for each n ≥ 2, +cos β∂nkδ +∂un (u, v) = +n−1 +� +p=0 +∂n−pgδ +∂xn−p +� +I +cIAn−p−|I|(∂A +∂u )i1(∂2A +∂u2 )i2 · · · (∂pA +∂up )ip, (53) +where I = (i1, i2, . . . , ip), i1 + 2i2 + · · · + pip = p, |I| = i1 + i2 + · · · + ip ≤ +n − p, cI ≥ 0, and the partial derivatives of gδ are evaluated at (cos βu + +sin βkδ(u, v) + x0, v). +Proof. When n = 2, differentiating the equation +− sin βu + cos βkδ(u, v) + z0 = gδ(cos βu + sin βkδ(u, v) + x0, v) +42 + +once with respect to u gives +− sin β + cos β(kδ)u = (gδ)x[cos β + sin β(kδ)u] = (gδ)xA. +Then taking the partial derivative with respect to u once more gives +cos β(kδ)uu = (gδ)xxA2 + (gδ)x∂uA. +In (53) when p = 0, there is no I so we have c0A2−0−0 = c0A2 where c0 = 1; +when p = 1, there is only one I = (1) so we have c1A2−1−1( ∂A +∂u )1 = c1∂uA +where c1 = 1. This coincides with the expression above. +When n ≥ 2, by inductive hypothesis we take the partial derivative of +(53) with respect to u. The left-hand side is cos β∂n+1 +u +kδ. The right-hand +side consists of three parts due to the product rule. +(1) +n−1 +� +p=0 +∂n+1−pgδ +∂xn+1−p +� +I +cIAn+1−p−|I|(∂A +∂u )i1(∂2A +∂u2 )i2 · · · (∂pA +∂up )ip, +where n becomes n + 1 and p stays the same. +(2) +n−1 +� +p=0 +∂n−pgδ +∂xn−p +� +I +cI(n − p − |I|)An−p−|I|−1(∂A +∂u )i1+1(∂2A +∂u2 )i2 · · · (∂pA +∂up )ip, +where n, p, i1 become n + 1, p + 1, i1 + 1, respectively. By letting ip+1 be +zero, one again has +i1+1+2i2+· · ·+pip+(p+1)ip+1 = p+1, n−p−|I|−1 = (n+1)−(p+1)−(|I|+1). +(3) +n−1 +� +p=0 +∂n−pgδ +∂xn−p +� +I +cIAn−p−|I| � +ij̸=0 +(∂A +∂u )i1 · · · ij(∂jA +∂uj )ij−1(∂j+1A +∂uj+1 )ij+1 · · ·(∂pA +∂up )ip, +where n, p become n + 1, p + 1, respectively. When j < p, ij and ij+1 are +replaced by ij + 1 and ij+1 + 1. By letting ip+1 = 0 one has +· · ·+j(ij−1)+(j+1)(ij+1+1)+· · ·+(p+1)ip+1 = p+1, · · ·+(ij−1)+(ij+1+1)+· · ·+ip+1 = |I|. +On the other hand, when j = p, ip = ip+1 = 1 and so +p(ip − 1) + (p + 1)ip+1 = p + 1, (ip − 1) + ip+1 = 1 = |I|. +It follows that cI are nonnegative integers and (53) is true. +43 + +Corollary 1. The coefficient of (gδ)x∂n−1 +u +A in (53) is always 1. +Proof. When n = 2, we’ve shown in the above lemma that the coefficient of +(gδ)x∂uA is 1. When n ≥ 2, if p = n − 1 then +i1 + 2i2 + · · · + (n − 1)in−1 = n − 1 and i1 + i2 + · · · + in−1 ≤ 1 +imply that +i1 = . . . = in−2 = 0 and in−1 = 1. +There is only one term of (gδ)x∂n−1 +u +A whose coefficient is 1 by induction. +Taking its derivative with respect to u yields +(gδ)xxA∂n−1 +u +A + (gδ)x∂n +uA, +so the coefficient of (gδ)x∂n +uA is still 1 completing the induction. +Corollary 2. The coefficient of ∂ngδ +∂xn An in (53) is always 1. +Proof. When n = 2, we’ve shown in the above lemma that the coefficient of +(gδ)xxA2 is 1. When n ≥ 2, if p = 0 there is no I since |I| = 0 and in (53) +we have only one term ∂ngδ +∂xn An whose coefficient is 1 by induction. Taking its +derivative with respect to u yields +∂n+1gδ +∂xn+1 An+1 + ∂ngδ +∂xn nAn−1∂A +∂u , +so the coefficient of ∂n+1gδ +∂xn+1 An+1 is still 1 completing the induction. +Since A = cos β +sin β(kδ)u(u, v) and (kδ)u(0, 0) = 0, A(0, 0) = cos β and +for 1 ≤ p ≤ N − 2 +∂pA +∂up (0, 0) = sin β∂p+1kδ +∂up+1 (0, 0) = sin β(p + 1)!bp+1(δ). +It follows that +cos βn!bn(δ) += +n−1 +� +p=0 +∂n−pgδ +∂xn−p (x0, 0) +� +I +cI(cos β)n−p−|I|(sin β)|I|b2(δ)i1b3(δ)i2 · · · bp+1(δ)ip +2!i13!i2 · · · (p + 1)!ip, +44 + +for 2 ≤ n ≤ N − 1. +Furthermore the Corollary (1) says that the term +corresponding to p = n − 1 in the above expression is +(gδ)x(x0, 0) sin βn!bn(δ) = − tan β sin βn!bn(δ). +So moving it to the other side yields +(cos β + tan β sin β)n!bn(δ) = +n−2 +� +p=0 +∂n−pgδ +∂xn−p (x0, 0) +� +I +cI(cos β)n−p−|I|(sin β)|I|b2(δ)i1b3(δ)i2 · · ·bp+1(δ)ip +2!i13!i2 · · · (p + 1)!ip +⇒ +n!bn(δ) = +n−2 +� +p=0 +∂n−pgδ +∂xn−p (x0, 0) +� +I +cI(cos β)n+1−p−|I|(sin β)|I|b2(δ)i1b3(δ)i2 · · · bp+1(δ)ip +2!i13!i2 · · · (p + 1)!ip, +using cos β + tan β sin β = sec β. So bn(δ) depends on the previous constants +for 3 ≤ n ≤ N − 1. +Lemma 3. bn(δ) > 0 for n between 2 and N − 1. +Proof. Before proceeding with the proof, we need to first estimate ∂pgδ +∂xp (x0, 0) +for 2 ≤ p ≤ N − 1. By induction one can show that +∂pgδ +∂xp (x0, 0) += +p!ap(δ) + · · · + (N − 1)(N − 2) · · ·(N − p)aN−1(δ)xN−1−p +0 ++ +p +� +q=0 +� +p +q +� +N(N − 1) · · ·(N − q + 1)xN−q +0 +∂p−q +x +hδ(x0, 0). +On the one hand, if ǫ is sufficiently small +|p!ap(δ) + · · · + (N − 1)(N − 2) · · ·(N − p)aN−1(δ)xN−1−p +0 +| +≤ +M sin δ[p! + · · · + (N − 1)(N − 2) · · ·(N − p)xN−1−p +0 +] +≤ +c1 +4b tan(θ0) sin δ ≤ c|a2(δ)|. +On the other hand, if η and ǫ are sufficiently small +p +� +q=0 +� +p +q +� +N(N − 1) · · ·(N − q + 1)xN−q +0 +∂p−q +x +hδ(x0, 0) +≥ +N(N − 1) · · ·(N − p + 1)xN−p +0 +h(0, 0)(1 − c). +45 + +Combining the two inequalities, together with (52), yields +∂pgδ +∂xp (x0, 0) +≥ +N(N − 1) · · · (N − p + 1)xN−p +0 +h(0, 0)(1 − c) − c|a2(δ)| +≥ +N(N − 1)(1 − c)2h(0, 0)|a2(δ)| +(N − 1)(1 + c)h(0, 0) +− c|a2(δ)| = +�N(1 − c)2 +1 + c +− c +� +|a2(δ)|, +which is positive if we choose c as follows. +N > c(1 + c) +(1 − c)2 ⇒ 2 > c(1 + c) +(1 − c)2 ⇒ 0 < c < 5 − +√ +17 +2 +< 1. +Denote the constant in the brackets as L = L(c, N), then for 2 ≤ p ≤ N − 1 +∂pgδ +∂xp (x0, 0) ≥ L|a2(δ)|. +Let’s determine the signs of bn(δ) for 2 ≤ n ≤ N − 1. When n = 2, +2!b2(δ) = cos3 β(gδ)xx(x0, 0) ≥ cos3 βL|a2(δ)| > 0 ⇒ b2(δ) > 0. +When n ≥ 3, by induction +n!bn(δ) +≥ +n−2 +� +p=0 +L|a2(δ)| +� +I +cI(cos β)n+1−p−|I|(sin β)|I|b2(δ)i1b3(δ)i2 · · · bp+1(δ)ip +2!i13!i2 · · · (p + 1)!ip > 0. +Indeed, when p = 0, the corresponding term in the above sum, together with +Corollary 2, is +∂ngδ +∂xn (x0, 0) cosn+1 β ≥ L|a2(δ)| cosn+1 β > 0. +So bn(δ) > 0, as desired. +The following lemma shows that the sign of lδ(0, 0) is also positive. Fur- +thermore, it gives a lower bound of lδ(0, 0). +Lemma 4. +lδ(0, 0) ≥ 1 − c +2N+1 h(0, 0) > 0. +46 + +Proof. By Lemma 2, +cos β∂Nkδ +∂uN (u, v) = +N−1 +� +p=0 +∂N−pgδ +∂xN−p +� +I +cIAN−p−|I|(∂A +∂u )i1(∂2A +∂u2 )i2 · · ·(∂pA +∂up )ip. +When p = N − 1, Corollary 1 suggests that we have +(gδ)x +∂N−1A +∂xN−1 (u, v). +Evaluating at (u, v) = (0, 0) gives us +cos βN!lδ(0, 0) − (gδ)x(x0, y0) sin βN!lδ(0, 0) += +N−2 +� +p=0 +∂N−pgδ +∂xN−p (x0, 0) +� +I +cI(cos β)N−p−|I|(sin β)|I|b2(δ)i1b3(δ)i2 · · · bp+1(δ)ip +2!i13!i2 · · · (p + 1)!ip. +Since (gδ)x(x0, y0) = − tan β, +N!lδ(0, 0) += +N−2 +� +p=0 +∂N−pgδ +∂xN−p (x0, 0) +� +I +cI(cos β)N+1−p−|I|(sin β)|I|b2(δ)i1b3(δ)i2 · · · bp+1(δ)ip +2!i13!i2 · · · (p + 1)!ip. +When p = 0, the corresponding term in the above summation by Corollary +2 is +∂Ngδ +∂xN (x0, 0) cosN+1 β. +Thus +N!lδ(0, 0) += +∂Ngδ +∂xN (x0, 0) cosN+1 β ++ +N−2 +� +p=1 +∂N−pgδ +∂xN−p (x0, 0) +� +I +cI(cos β)N+1−p−|I|(sin β)|I|b2(δ)i1b3(δ)i2 · · · bp+1(δ)ip +2!i13!i2 · · · (p + 1)!ip. +where the second term is positive by Lemma 3. Moreover, if η and ǫ are +sufficiently small +∂Ngδ +∂xN (x0, 0) += +N +� +q=0 +� +N +q +� +N(N − 1) · · ·(N − q + 1)xN−q +0 +∂q +xhδ(x0, 0) +≥ +N!h(0, 0)(1 − c) > 0. +47 + +Therefore +N!lδ(0, 0) ≥ N!h(0, 0)(1 − c) cosN+1 β ⇒ lδ(0, 0) ≥ h(0, 0)(1 − c) cosN+1 β. +Since − tan β = (gδ)x(x0, 0) and gx(0, 0) = 0, it follows that +β → 0, as δ, x0 → 0. +Therefore for η and ǫ sufficiently small, one can have +cos β ≥ 1 +2. +So +lδ(0, 0) ≥ 1 − c +2N+1 h(0, 0) > 0. +4. Approximate v(s) and v′(s) using the normal vector N(s) to S. We +denote γ(s) = (u(s), v(s), w(s)). If γ(s) ∈ S, the normal vector to S at γ(s) +is +N(s) = (−(kδ)u(v(s), v(s), −(kδ)v(u(s), v(s), 1). +Since u′(s) ≥ +1 +2, u(s) has a C1-inverse function s(u). +Therefore we can +express v(s) as +v(s) = v(s(u)) = α(u), +where α is a C1-function and α(0) = dα +du(0) = 0. Then one has v(s) = o(u(s)). +Hence +(kδ)u(u(s), v(s)) = u(s)V1(s); (kδ)v(u(s), v(s)) = u(s)V2(s), +where V1(s), V2(s) are bounded by some constant A. Let γ′′(s) exist, then +γ′′(s) = w′′(s)N(s), so +u′′(s) = −w′′(s)u(s)V1(s), v′′(s) = −w′′(s)u(s)V2(s). +When γ(s) does not touch the surface, the equalities still hold since γ′′(s) = 0. +With w(0) = w′(0) = v(0) = v′(0) = 0 one can approximate +|v′(s)| ≤ +� s +0 +|v′′(σ)|dσ ≤ Au(s)w′(s) ⇒ |v(s)| ≤ Au(s)w(s). +48 + +If γ(s) ∈ S, then +w(s) += +kδ(u(s), v(s)) += +b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)Nlδ + v(s)[u(s)mδ + v(s)nδ] +≤ +b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N|lδ| + Au(s)w(s)C1, +where C1 → 0 as ǫ → 0. Furthermore one can choose ǫ so small that +|lδ(u(s), v(s))−lδ(0, 0)| ≤ ch(0, 0)(1−c) ≤ clδ(0, 0) ⇒ |lδ(u(s), v(s))| ≤ (1+c)lδ(0, 0), +by uniform continuity. Therefore there is a constant B such that +w(s) ≤ B[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)]. +So +v(s) ≤ Au(s)B[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)]. +Next let’s pproximate v′(s). If γ(s) ∈ S, differentiating w(s) = kδ(u(s), v(s)) +gives +w′(s) += +2b2(δ)u(s)u′(s) + · · · + (N − 1)bN−1(δ)u(s)N−2u′(s) ++Nu(s)N−1u′(s)lδ + u(s)N[(lδ)uu′(s) + (lδ)vv′(s)] ++u′(s)v(s)mδ + u(s)v′(s)mδ + u(s)v(s)[(mδ)uu′(s) + (mδ)vv′(s)] ++2v(s)v′(s)nδ + v(s)2[(nδ)uu′(s) + (nδ)vv′(s)] +≤ +2b2(δ)u(s) + · · · + (N − 1)bN−1(δ)u(s)N−2 + u(s)N−1|Nlδ + u(s)[(lδ)u + (lδ)v]| ++|v(s)|C1 + u(s)|v′(s)|C2 + u(s)|v(s)|C3 + 2|v(s)||v′(s)|C4 + v(s)2C5 +≤ +2b2(δ)u(s) + · · · + (N − 1)bN−1(δ)u(s)N−2 + u(s)N−1N(1 + 2c)lδ(0, 0) ++(u(s)C2 + 2|v(s)|C4)Au(s)w′(s) + (C1 + u(s)C3 + |v(s)|C5)Au(s)B · +[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)], +≤ +3b2(δ)u(s) + · · · + NbN−1(δ)u(s)N−2 + u(s)N−1N(2 + 2c)lδ(0, 0) ++(u(s)C2 + 2|v(s)|C4)Au(s)w′(s), +where one can choose ǫ so small that +u(s)|(lδ)u + (lδ)v| ≤ Nc1 − c +2N+1 h(0, 0) ≤ Nclδ(0, 0). +(C1 + u(s)C3 + |v(s)|C5)Au(s)B ≤ 1, u(s) ≤ 1, 1 + c < N +49 + +By making (u(s)C2 + 2|v(s)|C4)Au(s) < 1, there exists a constant C such +that +w′(s) ≤ C[3b2(δ)u(s) + · · · + NbN−1(δ)u(s)N−2 + u(s)N−1N(2 + 2c)lδ(0, 0)]. +So +|v′(s)| ≤ Au(s)C[3b2(δ)u(s)+· · ·+NbN−1(δ)u(s)N−2+u(s)N−1N(2+2c)lδ(0, 0)]. +Now let’s look at the situation when γ(s) is on an interior line segment. +Consider a line segment in the image of γ with two endpoints γ(s1) and γ(s2), +we can parametrize v(s) for s ∈ [s1, s2] by +v(s) = v(s1)+T(u(s)−u(s1)), where T = dα +du(u(s1)) and |T| = +���� +v′(s1) +u′(s1) +���� ≤ 2|v′(s1)|. +|T| +≤ +2Au(s1)C[3b2(δ)u(s1) + · · · + NbN−1(δ)u(s1)N−2 + u(s1)N−1N(2 + 2c)lδ(0, 0)] +≤ +2Au(s)C[3b2(δ)u(s) + · · · + NbN−1(δ)u(s)N−2 + u(s)N−1N(2 + 2c)lδ(0, 0)], +where the last inequality holds because u(s) is increasing. Hence +|v(s)| +≤ +|v(s1)| + |T|(u(s) + u(s)) +≤ +Au(s)B[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)] ++4Au(s)2C[3b2(δ)u(s) + · · · + NbN−1(δ)u(s)N−2 + u(s)N−1N(2 + 2c)lδ(0, 0)] +≤ +(AB + 4AC · 2N)u(s)[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)] += +Du(s)[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)], +where D = (AB + 4AC · 2N). Furthermore, since v′(s) = v′(s1), +|v′(s)| +≤ +Au(s)C[3b2(δ)u(s) + · · · + NbN−1(δ)u(s)N−2 + u(s)N−1N(2 + 2c)lδ(0, 0)] +≤ +AC · 2Nu(s)[b2(δ)u(s) + · · · + bN−1(δ)u(s)N−2 + u(s)N−1(1 + c)lδ(0, 0)] +≤ +Du(s)[b2(δ)u(s) + · · · + bN−1(δ)u(s)N−2 + u(s)N−1(1 + c)lδ(0, 0)]. +5. Concavity. Suppose for the sake of contradiction that γ(s) leaves S at +a switch point when s = s0 and dives into the interior of M for increasing s +until it re-enters S again at s = s1. +50 + +Consider the intersection of the two-dimensional plane v = v0 +T(u−u0) +with the surface w = kδ(u, v), where (u0, v0) = (u(s0), v(s0)) and T = dα +du(u0). +Set f(u) = kδ(v, v0 + T(u − u0)), then with v(s) = v0 + T(u(s) − u0), +d2f +du2(u(s)) = (kδ)uu(u(s), v(s)) + 2(kδ)uv(u(s), v(s))T + (kδ)vv(u(s), v(s))T 2. +On the one hand, +(kδ)uu += +2b2(δ) + 6b3(δ)u(s) + · · · + (N − 1)(N − 2)bN−1(δ)u(s)N−3 ++N(N − 1)u(s)N−2lδ + 2Nu(s)N−1(lδ)u + u(s)N(lδ)uu ++v(s)[2(mδ)u + u(s)(mδ)uu] + v(s)2(nδ)uu, +where one can choose ǫ small enough so that +|lδ| ≥ (1−c)lδ(0, 0), |2Nu(s)(lδ)u + u(s)2(lδ)uu| +N(N − 1) +≤ clδ(0, 0), |2(mδ)u+u(s)(mδ)uu+v(s)(nδ)uu| ≤ C1. +Thus +(kδ)uu +≥ +2b2(δ) + 6b3(δ)u(s) + · · · + (N − 1)(N − 2)bN−1(δ)u(s)N−3 ++N(N − 1)u(s)N−2(1 − 2c)lδ(0, 0) +−C1Du(s)[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)] +≥ +2b2(δ) + 6b3(δ)u(s) + · · · + (N − 1)(N − 2)bN−1(δ)u(s)N−3 ++N(N − 1)u(s)N−2(1 − 2c)lδ(0, 0) +−1 +2[b2(δ) + · · · + bN−1(δ)u(s)N−3 + u(s)N−2(1 + c)lδ(0, 0)], +where we can choose ǫ small enough so that C1Du(s) ≤ +1 +2 and u(s) ≤ 1. +Note here that 1 − 2c > 0 because +0 < c < 5 − +√ +17 +2 +< 1 +2. +On the other hand, +|T| +≤ +2Au(s0)C[3b2(δ)u(s0) + · · · + NbN−1(δ)u(s0)N−2 + u(s0)N−1N(2 + 2c)lδ(0, 0)] +≤ +4ACNu(s0)[b2(δ)u(s0)2 + · · · + bN−1(δ)u(s0)N−1 + u(s0)N(1 + c)lδ(0, 0)] +≤ +Du(s)[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)]. +51 + +So for ǫ sufficiently small +|2(kδ)uvT + (kδ)vvT 2| ≤ C1|T| +≤ +C1Du(s)[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)] +≤ +1 +2[b2(δ) + · · · + bN−1(δ)u(s)N−3 + u(s)N−2(1 + c)lδ(0, 0)], +if C1Du(s) ≤ 1 +2 and u(s) ≤ 1. It follows that +d2f +du2(u(s)) +≥ +2b2(δ) + 6b3(δ)u(s) + · · · + (N − 1)(N − 2)bN−1(δ)u(s)N−3 ++N(N − 1)(1 − 2c)u(s)N−2lδ(0, 0) +−b2(δ) − b3(δ)u(s) − · · · − bN−1(δ)u(s)N−3 − u(s)N−2(1 + c)lδ(0, 0) += +b2(δ) + 5b3(δ)u(s) + · · · + [(N − 1)(N − 2) − 1]bN−1(δ)u(s)N−3 ++[N(N − 1)(1 − 2c) − (1 + c)]u(s)N−2lδ(0, 0), +where we want +N(N − 1)(1 − 2c) ≥ 2(1 − 2c) > 1 + c ⇒ 0 < c < 1 +5 < 5 − +√ +17 +2 +. +Therefore +d2f +du2(u(s)) > 0, +so f ′(u(s)) is increasing as u(s) increases from u(s0) = u0 to u(s1) = u1. On +the other hand, since the interior line segment is tangent to S at the two +endpoints, we must have +f ′(u0) = f ′(u1), +a contradiction. Therefore if γ leaves S at a switch point γ(s0), the geodesic +arc beyond this point is a line segment either exiting the ǫ-ball or terminating +at a point on S. So γ has at most two switch points in this case. +Case 4: a2(δ) > 0, h(0, 0) < 0. +This is a combination of Case 1, Case 2, and Case 3. When a2(δ) > 0, +the angle δ < 0. From Case 1 the constants a2(δ), a3(δ), . . ., aN−1(δ) satisfy +the following: +a2(δ) ≥ b tan(θ0)| sin θ|; |a3(δ)|, . . . , |aN−1(δ)| ≤ M| sin δ|. +52 + +Assume γ′(0) = +∂ +∂x. +1. x′(s) ≥ 1 +2. The argument is exactly the same as in Case 1 when we +proved for (48). +2. Approximate z(s), z′(s), y(s), y′(s). If γ(s) ∈ S, imitating the proof +for (50) in Case 1 one can show that there exists a positive constant C such +that +z(s) ≤ (2a2(δ)x(s)2 + 2|h(0, 0)|x(s)N)C +⇒ +|y(s)| ≤ (2a2(δ)x(s)2 + 2|h(0, 0)|x(s)N)AC +Simiarly, imitating the proof for (51) in Case 1 one can also show that +z′(s) ≤ (4a2(δ)x(s) + 4Nx(s)N−1|h(0, 0)|)C +⇒ +y′(s) ≤ (4a2(δ)x(s) + 4Nx(s)N−1|h(0, 0)|)AC. +It follows that if y = α(x) and T(s) = α′(x(s)), then +|T(s)| = +���� +y′(s) +x′(s) +���� ≤ 2|y′(s)| ≤ (8a2(δ)x(s) + 8Nx(s)N−1|h(0, 0)|)AC. +If γ(s) ̸∈ S, then γ(s) is in some line segment where y(s) = y(s0)+T(s0)(x(s)− +x(s0)) for some switch point at s0 < s. Since x(s) is increasing, it follows +that +|y(s)| +≤ +|y(s0)| + |T(s0)|[|x(s)| + |x(s0)|] +≤ +(2a2(δ)x(s0)2 + 2|h(0, 0)|x(s0)N)AC ++(8a2(δ)x(s0) + 8Nx(s0)N−1|h(0, 0)|)AC[|x(s)| + |x(s0)|] +≤ +(2a2(δ)x(s)2 + 2|h(0, 0)|x(s)N)AC + (8a2(δ)x(s) + 8Nx(s)N−1|h(0, 0)|)AC · 2x(s) +≤ +(18a2(δ)x(s)2 + 18N|h(0, 0)|x(s)N)AC. +Furthermore, +|z(s)| +≤ +|z(s0)| + 2|z′(s0)|[|x(s)| + |x(s0)|] +≤ +(2a2(δ)x(s0)2 + 2|h(0, 0)|x(s0)N)C + 2(4a2(δ)x(s0) + 4Nx(s0)N−1|h(0, 0)|)C · 2x(s) +≤ +(2a2(δ)x(s)2 + 2|h(0, 0)|x(s)N)C + 4x(s)(4a2(δ)x(s) + 4Nx(s)N−1|h(0, 0)|)C += +(18a2(δ)x(s)2 + 18N|h(0, 0)|x(s)N)C. +The same inequalities for z′(s), y′(s), and T(s) still hold as before because +z′(s) = z′(s0), y′(s) = y′(s0), T(s) = T(s0). +53 + +3. Now we are ready to estimate the location of the first switch point. Set +f(x) = gδ(x, y0 + T(x − x0)) where (x0, y0) = (x(s0), y(s0)) and T = T(s0). +Then +f ′′(x0) += +(gδ)xx(x0, y0) + 2(gδ)xy(x0, y0)T + (gδ)yy(x0, y0)T 2, +where +(gδ)xx(x0, y0) += +2a2(δ) + 6a3(δ)x0 + · · · + (N − 1)(N − 2)aN−1(δ)xN−3 +0 ++xN−2 +0 +[N(N − 1)hδ + 2Nx0(hδ)x + x2 +0(hδ)xx] ++y0[2(iδ)x + x0(iδ)xx + y0(jδ)xx]. +As x0 → 0, (gδ)xx(x0, y0) → 2a2(δ) and T(s0) → 0. Therefore f ′′(x0) > 0 +at the beginning and the geodesic initially stays on the surface if it is not a +straight line for which we will discuss later. +Suppose f ′′(x0) = 0. Let’s estimate x0. First, for ǫ sufficiently small, +|6a3(δ)x0 + · · · + (N − 1)(N − 2)aN−1(δ)xN−3 +0 +| +≤ +M| sin δ|(6x0 + · · · + (N − 1)(N − 2)xN−3 +0 +) +≤ +b tan(θ0)| sin δ| ≤ a2(δ). +Second, for η and ǫ sufficiently small, +N(N − 1)hδ + 2Nx0(hδ)x + x2 +0(hδ)xx ≥ (1 + c +2)N(N − 1)h(0, 0), +for some 0 < c < 1 to be determined later. Third, there are constants C5, C6 +such that +|2(iδ)x + x0(iδ)xx + y0(jδ)xx| ≤ C5, |2(gδ)xy(x0, y0) + (gδ)yy(x0, y0)T| ≤ C6. +If f ′′(x0) = 0, then +−xN−2 +0 +[N(N − 1)hδ + 2Nx0(hδ)x + x2 +0(hδ)xx] += +2a2(δ) + 6a3(δ)x0 + · · · + (N − 1)(N − 2)aN−1(δ)xN−3 +0 ++y0[2(iδ)x + x0(iδ)xx + y0(jδ)xx] + 2(gδ)xy(x0, y0)T + (gδ)yy(x0, y0)T 2. +On the one hand, +LHS ≤ −(1 + c +2)N(N − 1)h(0, 0)xN−2 +0 += (1 + c +2)N(N − 1)|h(0, 0)|xN−2 +0 +. +54 + +On the other hand, +RHS +≥ +2a2(δ) − a2(δ) − C5(18a2(δ)x2 +0 + 18N|h(0, 0)|xN +0 )AC +−C6(8a2(δ)x0 + 8NxN−1 +0 +|h(0, 0)|)AC. +Combining the two inequalities we get +N(N − 1)|h(0, 0)|xN−2 +0 +[(1 + c +2) + 18ACC5x2 +0 + 8ACC6x0 +N − 1 +] +≥ +a2(δ)[1 − 18ACC5x2 +0 − 8ACC6x0] +Choose ǫ small enough so that +18ACC5x2 +0 + 8ACC6x0 +N − 1 +≤ c +2 and 18ACC5x2 +0 + 8ACC6x0 ≤ c. +Therefore we obtain a lower bound for x0 +N(N − 1)|h(0, 0)|xN−2 +0 +(1 + c) ≥ a2(δ)(1 − c) +(54) +⇒ +xN−2 +0 +≥ +a2(δ)(1 − c) +N(N − 1)(1 + c)|h(0, 0)|. +Similarly, we can also get an upper bound for x0. Again suppose f ′′(x0) = +0. First, for ǫ sufficiently small, we still have +|6a3(δ)x0 + · · · + (N − 1)(N − 2)aN−1(δ)xN−3 +0 +| ≤ a2(δ). +Second, for η and ǫ sufficiently small, +N(N − 1)hδ + 2Nx0(hδ)x + x2 +0(hδ)xx ≤ (1 − c +2)N(N − 1)h(0, 0). +Third, there are still constants C5, C6 such that +|2(iδ)x + x0(iδ)xx + y0(jδ)xx| ≤ C5, |2(gδ)xy(x0, y0) + (gδ)yy(x0, y0)T| ≤ C6. +If f ′′(x0) = 0, then +−xN−2 +0 +[N(N − 1)hδ + 2Nx0(hδ)x + x2 +0(hδ)xx] += +2a2(δ) + 6a3(δ)x0 + · · · + (N − 1)(N − 2)aN−1(δ)xN−3 +0 ++y0[2(iδ)x + x0(iδ)xx + y0(jδ)xx] + 2(gδ)xy(x0, y0)T + (gδ)yy(x0, y0)T 2. +55 + +On the one hand, +LHS ≥ −(1 − c +2)N(N − 1)h(0, 0)xN−2 +0 += (1 − c +2)N(N − 1)|h(0, 0)|xN−2 +0 +. +On the other hand, +RHS +≤ +2a2(δ) + a2(δ) + C5(18a2(δ)x2 +0 + 18N|h(0, 0)|xN +0 )AC ++C6(8a2(δ)x0 + 8NxN−1 +0 +|h(0, 0)|)AC. +Combining the two yields +(1 − c +2)N(N − 1)|h(0, 0)|xN−2 +0 +≤ 3a2(δ) + (18ACC5x2 +0 + 8ACC6x0)a2(δ) ++xN−2 +0 +N|h(0, 0)|(18ACC5x2 +0 + 8ACC6x0) +⇒ +N(N − 1)|h(0, 0)|xN−2 +0 +[(1 − c +2) − 18ACC5x2 +0 + 8ACC6x0 +N − 1 +] +≤ +3a2(δ) + (18ACC5x2 +0 + 8ACC6x0)a2(δ). +So +N(N−1)|h(0, 0)|xN−2 +0 +(1−c) ≤ (3+c)a2(δ) ⇒ xN−2 +0 +≤ +(3 + c)a2(δ) +N(N − 1)|h(0, 0)|(1 − c). +4. +Now like in Case 3 we are going to shift our coordinates to have +the origin at γ(s0) = (x0, y0, z0) and then rotate the (x, y, z)-space so that +γ′(s0) = (x′(s0), y′(s0), z′(s0)) points in the positive x-axis. Let’s use (u, v, w) +for the new coordinates, then with respect to the new frame there is a rotation +matrix P ∈ SO3(R) such that + + +x − x0 +y − y0 +z − z0 + + = P + + +u +v +w + + , where P = + + +x′(s0) +p12 +p13 +y′(s0) +p22 +p23 +z′(s0) +p32 +p33 + + . +So +x += +x0 + x′(s0)u + p12v + p13w, +y += +y0 + y′(s0)u + p22v + p23w, +z += +z0 + z′(s0)u + p32v + p33w. +56 + +The second and third columns of P can be further specificed as below: + + +p12 +p22 +p32 + + = + + +−y′(s0) +x′(s0) +0 + + Y ; + + +p13 +p23 +p33 + + = + + +−x′(s0)2z′(s0) +−x′(s0)y′(s0)z′(s0) +x′(s0)[x′(s0)2 + y′(s0)2] + + Z, +(55) +where Y = 1/ +� +x′(s0)2 + y′(s0)2 and Z = 1/x′(s0) +� +x′(s0)2 + y′(s0)2. +Thus the surface z = gδ(x, y) satisfies the equation +z0 + z′(s0)u + p32v + p33w = gδ(x0 + x′(s0)u + p12v + p13w, y0 + y′(s0)u + p22v + p23w). +Check that we can still solve for w analytically in terms of u, v within the +ǫ-ball. Let’s take the partial derivate of +−z0−z′(s0)u−p32v−p33w+gδ(x0+x′(s0)u+p12v+p13w, y0+y′(s0)u+p22v+p23w) +with respect to w: +−p33 + (gδ)xp13 + (gδ)yp23. +Since Z > 0, it is equivalent to show that +−p33 + (gδ)xp13 + (gδ)yp23 +Z +̸= 0. +On the one hand, +(gδ)xp13 + (gδ)yp23 +Z += −x′(s0)2z′(s0)(gδ)x − x′(s0)y′(s0)z′(s0)(gδ)y ≤ |(gδ)x| + |(gδ)y|. +Since (gδ)x(x, y), (gδ)y(x, y) → 0 as x, y → 0, for ǫ sufficiently small +|(gδ)x| + |(gδ)y| ≤ 1 +16. +On the other hand, +p33 +Z = x′(s0)[x′(s0)2 + y′(s0)2] ≥ x′(s0)3 ≥ 1 +8. +so +−p33 + (gδ)xp13 + (gδ)yp23 +Z +≤ − 1 +16 < 0. +Therefore there exists a real analytic function kδ such that w = kδ(u, v) such +that kδ(0, 0) = 0, (kδ)u(0, 0) = 0, and (kδ)v(0, 0) = 0. +57 + +5. Estimate γ(s) in the new frame starting from (x0, y0, z0). After replac- +ing s by s − s0, we denote γ(s) as (u(s), v(s), w(s)). +(1). Coefficients of kδ(u, v). Denote kδ(u, v) as +kδ(u, v) = b2(δ)u2 + · · ·+ bN−1(δ)uN−1 + uNlδ(u, v) + uvmδ(u, v) + v2nδ(u, v), +where b2(δ), . . . , bN−1(δ) are constants and lδ, mδ, nδ are analytic functions. +Observe that for n between 2 and N − 1, +bn(δ) = 1 +n! +∂nkδ +∂un (0, 0). +Next let’s look for ∂nkδ +∂un (u, v) for n ≥ 2 by induction. +Lemma 5. Let A be x′(s0) + p13(kδ)u and B be y′(s0) + p23(kδ)u, then for +n ≥ 2, +p33 +∂nkδ +∂un (u, v) = +n +� +a+b=1 +∂a+bgδ +∂xa∂yb +� +I,J +cI,JAa−|I|Bb−|J|(∂A +∂u )i1 · · · (∂pA +∂up )ip(∂B +∂u )j1 · · ·(∂pB +∂up )jp, +where p = n − (a + b), I = (i1, i2, . . . , ip), J = (j1, j2, . . . , jp), (i1 + 2i2 + +· · · + pip) + (j1 + 2j2 + · · · + pjp) = p, |I| = i1 + i2 + · · · + ip ≤ a, |J| = +j1 + j2 + · · · + jp ≤ b, and the partial derivatives of gδ are evaluated at +(x0 + x′(s0)u + p12v + p13kδ(u, v), y0 + y′(s0)u + p22v + p23kδ(u, v)). +Proof. When n = 2, differentiating the following equation +z0+z′(s0)u+p32v+p33kδ(u, v) = gδ(x0+x′(s0)u+p12v+p13kδ(u, v), y0+y′(s0)u+p22v+p23kδ(u, v)) +with respect to u once gives us +z′(s0) + p33(kδ)u = (gδ)x[x′(s0) + p13(kδ)u] + (gδ)y[y′(s0) + p23(kδ)u]. +(56) +Let A = x′(s0) + p13(kδ)u and B = y′(s0) + p23(kδ)u, then +z′(s0) + p33(kδ)u = (gδ)xA + (gδ)yB. +Taking the partial derivative with respect to u once more gives +p33(kδ)uu = (gδ)xxA2 + (gδ)x∂uA + (gδ)yyB2 + (gδ)y∂uB + 2(gδ)xyAB. +58 + +When a + b = 2, p = 0 and so there are no I and J. There are three terms +corresponding to: a = 2, b = 0; a = 0, b = 2; a = 1, b = 1, respectively: +(gδ)xxA2, (gδ)yyB2, (gδ)xyAB. +When a + b = 1, either a = 1, b = 0 with I = (1), J = 0 or a = 0, b = 1 with +I = 0, J = (1). It follows that there are two terms +(gδ)x∂uA, (gδ)y∂uB. +When n ≥ 2, by inductive hypothesis we can take the partial derivative +of the expression in the lemma with respect to u. +The left-hand side is +p33 +∂n+1kδ +∂un+1 (u, v). The right-hand side consists of three parts due to the product +rule. +(1) +n +� +a+b=1 +∂a+b+1gδ +∂xa+1∂yb +� +I,J +cI,JAa+1−|I|Bb−|J|(∂A +∂u )i1 · · · (∂pA +∂up )ip(∂B +∂u )j1 · · · (∂pB +∂up )jp ++ +n +� +a+b=1 +∂a+b+1gδ +∂xa∂yb+1 +� +I,J +cI,JAa−|I|Bb+1−|J|(∂A +∂u )i1 · · ·(∂pA +∂up )ip(∂B +∂u )j1 · · · (∂pB +∂up )jp, +where a becomes a + 1 in the first term, b becomes b + 1 in the second term, +and p stays the same since (n + 1) − (a + 1 + b) = p. +(2) +n +� +a+b=1 +∂a+bgδ +∂xa∂yb +� +I,J +cI,J(a − |I|)Aa−|I|−1Bb−|J|(∂A +∂u )i1+1 · · · (∂pA +∂up )ip(∂B +∂u )j1 · · ·(∂pB +∂up )jp ++ +n +� +a+b=1 +∂a+bgδ +∂xa∂yb +� +I,J +cI,J(b − |J|)Aa−|I|Bb−|J|−1(∂A +∂u )i1+1 · · · (∂pA +∂up )ip(∂B +∂u )j1+1 · · · (∂pB +∂up )jp, +where a, b stay the same, so p becomes p + 1 = (n + 1) − (a + b). Moreover, +i1 becomes i1 + 1 in the first term and j1 becomes j1 + 1 in the second term. +So +(i1+1+2i2+· · ·+pip)+(j1+2j2+· · ·+pjp) = (i1+2i2+· · ·+pip)+(j1+1+2j2+· · ·+pjp) = p+1. +(3) +n +� +a+b=1 +∂a+bgδ +∂xa∂yb +� +I,J +cI,JAa−|I|Bb−|J| � +ik̸=0 +· · ·ik(∂A +∂u )ik−1(∂A +∂u )ik+1+1 · · · ++ +n +� +a+b=1 +∂a+bgδ +∂xa∂yb +� +I,J +cI,JAa−|I|Bb−|J| � +jk̸=0 +· · ·jk(∂B +∂u )jk−1(∂B +∂u )jk+1+1 · · · , +59 + +where a, b stay the same, so p becomes p + 1 = (n + 1) − (a + b). Moreover, +ik, ik+1 become ik − 1, ik+1 + 1 in the first term and jk, jk+1 become jk − 1, +jk+1 + 1 in the second term. So +· · ·+k(ik−1)+(k+1)(ik+1+1)+· · · = · · ·+k(jk−1)+(k+1)(jk+1+1)+· · · = p+1. +It follows that cI,J are nonnegative integers and the lemma is true. +The following two corollaries are analogous to Corollaries 1 and 2. +Corollary 3. The coefficients of (gδ)x∂n−1 +u +A and (gδ)y∂n−1 +u +B are always 1. +Corollary 4. The coefficients of ∂ngδ +∂xn An and ∂ngδ +∂yn Bn are always 1. +Now we let (u, v) = (0, 0), then (kδ)u(0, 0) = 0 implies that +A(0, 0) = x′(s0), B(0, 0) = y′(s0). +Furthermore for p ≥ 1, +∂pA +∂up (0, 0) = p13 +∂p+1kδ +∂up+1 (0, 0) = p13(p + 1)!bp+1(δ) +∂pB +∂up (0, 0) = p23 +∂p+1kδ +∂up+1 (0, 0) = p23(p + 1)!bp+1(δ). +It follows that for 2 ≤ n ≤ N − 1 +p33n!bn(δ) += +n +� +a+b=1 +∂a+bgδ +∂xa∂yb(x0, y0) +� +I,J +cI,Jx′(s0)a−|I|y′(s0)b−|J|(p13)|I|(p23)|J| +(2!)i1+j1(3!)i2+j2 · · · (p + 1)!ip+jpb2(δ)i1+j1 · · ·bp+1(δ)ip+jp. +Furthermore Corollary (3) suggests that the terms corresponding to a+b = 1 +or p = n − 1 in the expression are +(gδ)x(x0, y0)p13n!bn(δ) + (gδ)y(x0, y0)p23n!bn(δ). +Moving them to the other side of the expression yields +(p33 − (gδ)x(x0, y0)p13 − (gδ)y(x0, y0)p23)n!bn(δ) += +n +� +a+b=2 +∂a+bgδ +∂xa∂yb(x0, y0) +� +I,J +cI,Jx′(s0)a−|I|y′(s0)b−|J|(p13)|I|(p23)|J| +(2!)i1+j1(3!)i2+j2 · · · (p + 1)!ip+jpb2(δ)i1+j1 · · · bp+1(δ)ip+jp, +60 + +where p + 1 = n − (a + b) + 1 ≤ n − 1. So bn(δ) depends on the previous +coefficients b2(δ), . . ., bn−1(δ). +Let’s calculate the coefficient bn(δ). Evaluating (56) at (u, v) = (0, 0) +arrives +z′(s0) = (gδ)x(x0, y0)x′(s0) + (gδ)y(x0, y0)y′(s0), +together with (55), then +1 +Z [p33 − (gδ)x(x0, y0)p13 − (gδ)y(x0, y0)p23] += +x′(s0)[x′(s0)2 + y′(s0)2] + (gδ)x(x0, y0)x′(s0)2z′(s0) + (gδ)y(x0, y0)x′(s0)y′(s0)z′(s0) += +x′(s0)[x′(s0)2 + y′(s0)2] + x′(s0)z′(s0)2 += +x′(s0)[x′(s0)2 + y′(s0)2 + z′(s0)2] = x′(s0) · 1 = x′(s0). +So +n!bn(δ) +� +x′(s0)2 + y′(s0)2 += +n +� +a+b=2 +∂a+bgδ +∂xa∂yb(x0, y0) +� +I,J +cI,Jx′(s0)a−|I|y′(s0)b−|J|(p13)|I|(p23)|J| +(2!)i1+j1(3!)i2+j2 · · · (p + 1)!ip+jpb2(δ)i1+j1 · · · bp+1(δ)ip+jp += +n +� +a+b=2 +∂a+bgδ +∂xa∂yb(x0, y0) +� +I,J +cI,Jx′(s0)a−|I|y′(s0)b−|J| +� +−x′(s0)2z′(s0) +x′(s0) +� +x′(s0)2 + y′(s0)2 +�|I| � +−x′(s0)y′(s0)z′(s0) +x′(s0) +� +x′(s0)2 + y′(s0)2 +�|J| +(2!)i1+j1(3!)i2+j2 · · · (p + 1)!ip+jpb2(δ)i1+j1 · · · bp+1(δ)ip+jp += +n +� +a+b=2 +∂a+bgδ +∂xa∂yb(x0, y0) +� +I,J +cI,J(−1)|I|+|J|x′(s0)ay′(s0)b +� +z′(s0) +� +x′(s0)2 + y′(s0)2 +�|I|+|J| +(2!)i1+j1(3!)i2+j2 · · · (p + 1)!ip+jpb2(δ)i1+j1 · · · bp+1(δ)ip+jp. +Lemma 6. The signs of bn(δ) for n between 2 and N − 1 are all negative. +Proof. Before proving the lemma, one needs to estimate ∂pgδ +∂xp (x0, y0) for 2 ≤ +61 + +p ≤ N − 1. By induction one can show that +∂pgδ +∂xp (x0, y0) += +p!ap(δ) + · · · + (N − 1)(N − 2) · · ·(N − p)aN−1(δ)xN−1−p +0 ++ +p +� +q=0 +�p +q +� +N(N − 1) · · ·(N − q + 1)xN−q +0 +∂p−q +x +hδ(x0, y0) ++y0 +� +p +� +q=0 +�p +q +� dqx +dxq (x0)∂p−q +x +iδ(x0, y0) +� ++ y2 +0∂p +xjδ(x0, y0). +First if ǫ is sufficiently small, +|p!ap(δ) + · · · + (N − 1)(N − 2) · · ·(N − p)aN−1(δ)xN−1−p +0 +| +≤ +M| sin δ|[p! + · · · + (N − 1)(N − 2) · · ·(N − p)xN−1−p +0 +] +≤ +c +2b tan(θ0)| sin δ| ≤ c +2a2(δ). +Second if ǫ and η are sufficiently small, +p +� +q=0 +�p +q +� +N(N − 1) · · ·(N − q + 1)xN−q +0 +∂p−q +x +hδ(x0, y0) +≤ +N(N − 1) · · ·(N − p + 1)xN−p +0 +h(0, 0)(1 − c +2). +Third if ǫ is sufficiently small, +�����y0 +� +p +� +q=0 +� +p +q +� dqx +dxq (x0)∂p−q +x +iδ(x0, y0) +� ++ y2 +0∂p +xjδ(x0, y0) +����� +≤ +|y0|C1 ≤ (18a2(δ)x2 +0 + 18N|h(0, 0)|xN +0 )ACC1 +≤ +c +2a2(δ) + N(N − 1) · · ·(N − p + 1)xN−p +0 +|h(0, 0)|c +2. +Combining the above three inequalities, together with (54), yields +∂pgδ +∂xp (x0, y0) +≤ +ca2(δ) + N(N − 1) · · ·(N − p + 1)xN−p +0 +h(0, 0)(1 − c) +≤ +ca2(δ) − N(N − 1) · · ·(N − p + 1)|h(0, 0)|(1 − c)2a2(δ) +N(N − 1)(1 + c)|h(0, 0)| += +− +�(N − 2) · · ·(N − p + 1)(1 − c)2 +1 + c +− c +� +a2(δ) +≤ +− +�(1 − c)2 +1 + c +− c +� +a2(δ), +62 + +which is negative if we choose c as follows: +(1 − c)2 +1 + c +− c > 0 ⇒ (1 − c)2 > c(1 + c) ⇒ 0 < c < 1 +3. +Denote the constant in the brackets as L = L(c), then for 2 ≤ p ≤ N − 1 +∂pgδ +∂xp (x0, y0) ≤ −La2(δ). +Now we are ready to prove the lemma by induction. When n = 2, +2b2(δ) +� +x′(s0)2 + y′(s0)2 = (gδ)xx(x0, y0)x′(s0)2+2(gδ)xy(x0, y0)x′(s0)y′(s0)+(gδ)yy(x0, y0)y′(s0)2. +On the one hand, since x′(s0) ≥ 1 +2, +(gδ)xx(x0, y0)x′(s0)2 ≤ −L +4 a2(δ). +On the other hand, +y′(s0) [2(gδ)xy(x0, y0)x′(s0) + (gδ)yy(x0, y0)y′(s0)] ≤ C1|y′(s0)| +≤ +C1(4a2(δ)x0 + 4NxN−1 +0 +|h(0, 0)|)AC +≤ +4ACC1x0a2(δ) + 4ACC1x0 +(3 + c)a2(δ)N|h(0, 0)| +N(N − 1)|h(0, 0)|(1 − c) += +4ACC1x0a2(δ) +� +1 + +3 + c +(N − 1)(1 − c) +� +. +If we choose ǫ small enough so that +4ACC1x0 +� +1 + +3 + c +(N − 1)(1 − c) +� +≤ L +8 , +then +2(gδ)xy(x0, y0)x′(s0)y′(s0) + (gδ)yy(x0, y0)y′(s0)2 ≤ L +8 a2(δ). +It follows that +2b2(δ) +� +x′(s0)2 + y′(s0)2 ≤ −L +8 a2(δ) ⇒ b2(δ) < 0. +63 + +By inductive hypothesis, suppose b2(δ), . . . , bn−1(δ) are all negative, then it +suffices to show that +n!bn(δ) +� +x′(s0)2 + y′(s0)2 < 0. +There are two cases. Case 1: when b = 0, |J| = 0 since |J| ≤ b, then we have +n +� +a=2 +∂agδ +∂xa (x0, y0) +� +I,J +cI,J(−1)|I|x′(s0)a +� +z′(s0) +� +x′(s0)2 + y′(s0)2 +�|I| +2!i13!i2 · · · (p + 1)!ipb2(δ)i1b3(δ)i2 · · · bp+1(δ)ip. +Since z′(s0) > 0 and the sign of b2(δ)i1b3(δ)i2 · · ·bp+1(δ)ip is (−1)|I|, it follows +that for each I, J, +cI,J(−1)|I|x′(s0)a +� +z′(s0) +� +x′(s0)2 + y′(s0)2 +�|I| +2!i13!i2 · · · (p+1)!ipb2(δ)i1b3(δ)i2 · · ·bp+1(δ)ip ≥ 0. +Since ∂a +xgδ(x0, y0) < 0, the above sum is negative. Especially when a = n, +p = n − a − b = 0 and so there is only one term +∂ngδ +∂xn (x0, y0)x′(s0)n ≤ − 1 +2n La2(δ). +whose coefficient is 1 by Corollary 4. +Case 2: when b ̸= 0, there is at least one copy of y′(s0) in the summation, +so we can write the rest of the terms as +������ +y′(s0) + + +n +� +a+b=2,b≥1 +∂a+bgδ +∂xa∂yb(x0, y0) +� +I,J +cI,J(−1)|I|+|J|x′(s0)ay′(s0)b−1 +� +z′(s0) +� +x′(s0)2 + y′(s0)2 +�|I|+|J| +(2!)i1+j1(3!)i2+j2 · · · (p + 1)!ip+jpb2(δ)i1+j1 · · ·bp+1(δ)ip+jp��� +≤ +C1|y′(s0)| ≤ C1(4a2(δ)x0 + 4NxN−1 +0 +|h(0, 0)|)AC +≤ +4ACC1x0a2(δ) +� +1 + +3 + c +(N − 1)(1 − c) +� +. +where the first inequality is because everything inside the brackets is bounded. +If we choose ǫ small enough so that +4ACC1x0 +� +1 + +3 + c +(N − 1)(1 − c) +� +≤ +1 +2n+1L, +64 + +then +n!bn(δ) +� +x′(s0)2 + y′(s0)2 ≤ − 1 +2n+1La2(δ) < 0, +as desired. +The next lemma not only determines the sign of lδ(0, 0), but also gives +an upper bound of lδ(0, 0). +Lemma 7. +lδ(0, 0) ≤ 1 − c +2N−1 h(0, 0) < 0. +Proof. By Lemma 5, +p33 +∂Nkδ +∂uN (u, v) = +N +� +a+b=1 +∂a+bgδ +∂xa∂yb +� +I,J +cI,JAa−|I|Bb−|J|(∂A +∂u )i1 · · · (∂pA +∂up )ip(∂B +∂u )j1 · · · (∂pB +∂up )jp. +When a + b = 1 or p = N − 1 Corollary 3 suggests that we have in the above +sum +(gδ)x +∂N−1A +∂uN−1 + (gδ)y +∂N−1B +∂uN−1 . +Evaluating at (u, v) = (0, 0) gives us +p33N!lδ(0, 0) − (gδ)x(x0, y0)p13N!lδ(0, 0) − (gδ)y(x0, y0)p13N!lδ(0, 0) += +N +� +a+b=2 +∂a+bgδ +∂xa∂yb(x0, y0) +� +I,J +cI,Jx′(s0)a−|I|y′(s0)b−|J|(p13)|I|(p23)|J| +(2!)i1+j1(3!)i2+j2 · · · (p + 1)!ip+jpb2(δ)i1+j1 · · · bp+1(δ)ip+jp +Since p33 = Zx′(s0), p13 = −x′(s0)2z′(s0)Z, and p23 = −x′(s0)y′(s0)z′(s0)Z, +one has +N!lδ(0, 0) +� +x′(s0)2 + y′(s0)2 += +N +� +a+b=2 +∂a+bgδ +∂xa∂yb(x0, y0) +� +I,J +cI,J(−1)|I|+|J|x′(s0)ay′(s0)b +� +z′(s0) +� +x′(s0)2 + y′(s0)2 +�|I|+|J| +(2!)i1+j1(3!)i2+j2 · · · (p + 1)!ip+jpb2(δ)i1+j1 · · · bp+1(δ)ip+jp. +65 + +Next let’s use an analogous argument in Lemma 4. When b = 0, |J| = 0, +then we have +N +� +a=2 +∂agδ +∂xa (x0, y0) +� +I,J +cI,J(−1)|I|x′(s0)a +� +z′(s0) +� +x′(s0)2 + y′(s0)2 +�|I| +2!i13!i2 · · ·(p + 1)!ipb2(δ)i1b3(δ)i2 · · · bp+1(δ)ip +≤ +∂Ngδ +∂xN (x0, y0)x′(s0)N ≤ 1 +2N +∂Ngδ +∂xN (x0, y0), +Since +∂Ngδ +∂xN (x0, y0) += +N +� +q=0 +�N +q +� +N(N − 1) · · ·(N − q + 1)xN−q +0 +∂q +xhδ(x0, y0) ++y0 +� N +� +q=0 +� +N +q +� ∂N−qx +∂xN−q (x0)∂q +xiδ(x0, y0) + y0 +∂Njδ +∂xN (x0, y0) +� +. +On the one hand, if η and ǫ are sufficiently small, +N +� +q=0 +� +N +q +� +N(N − 1) · · ·(N − q + 1)xN−q +0 +∂q +xhδ(x0, y0) ≤ N!(1 − c +4)h(0, 0). +On the other hand, +�����y0 +� N +� +q=0 +�N +q +� ∂N−qx +∂xN−q (x0)∂q +xiδ(x0, y0) + y0 +∂Njδ +∂xN (x0, y0) +������ ≤ C1|y0| +≤ +(18a2(δ)x2 +0 + 18N|h(0, 0)|xN +0 )ACC1 ≤ c +4N!|h(0, 0)|, +for η and ǫ sufficiently small. Thus +∂Ngδ +∂xN (x0, y0) ≤ N!(1 − c +2)h(0, 0). +When b ̸= 0, there is always one copy of y′(s0) so we have +������ +y′(s0) + + +N +� +a+b=2 +∂a+bgδ +∂xa∂yb(x0, y0) +� +I,J +cI,J(−1)|I|+|J|x′(s0)ay′(s0)b−1 +� +z′(s0) +� +x′(s0)2 + y′(s0)2 +�|I|+|J| +(2!)i1+j1(3!)i2+j2 · · · (p + 1)!ip+jpb2(δ)i1+j1 · · ·bp+1(δ)ip+jp��� +≤ +C1|y′(s0)| ≤ C1(4a2(δ)x0 + 4NxN−1 +0 +|h(0, 0)|)AC +≤ +N! +2N +c +2|h(0, 0)|, +66 + +for η and ǫ sufficiently small. It follows that +N!lδ(0, 0) +� +x′(s0)2 + y′(s0)2 ≤ N! +2N (1 − c +2)h(0, 0) − N! +2N +c +2h(0, 0) = N! +2N (1 − c)h(0, 0) +⇒ +lδ(0, 0) ≤ +� +x′(s0)2 + y′(s0)2 +2N +(1 − c)h(0, 0) ≤ 2 +2N (1 − c)h(0, 0) < 0. +(2). Since b2(δ) < 0, we could show as in Case 2 that γ is initially a +straight line. The surface S in the (u, w)-plane is the curve +w = kδ(u, 0) = b2(δ)u2 + · · · + bN−1(δ)uN−1 + uNlδ(u, 0). +If the line segment re-enters the surface at some switch point, then the curve +can’t be concave downward there. Otherwise the line lies above the surface. +w′(u) += +2b2(δ)u + · · · + (N − 1)bN−1(δ)uN−2 + NuN−1lδ(u, 0), +w′′(u) += +2b2(δ) + 6b3(δ) + · · · + (N − 1)(N − 2)bN−1(δ)uN−3 ++N(N − 1)uN−2lδ(u, 0) + 2NuN−1(lδ)u(u, 0) + uN(lδ)uu(u, 0). +On the one hand, +2b2(δ) + 6b3(δ) + · · · + (N − 1)(N − 2)bN−1(δ)uN−3 < 0. +On the other hand, we can choose ǫ small enough so that for all u < ǫ, +lδ(u, 0) ≤ (1 − c +2)lδ(0, 0), |2Nu(lδ)u(u, 0) + u2(lδ)uu(u, 0)| +N(N − 1) +≤ c +2|lδ(0, 0)| +⇒ +N(N − 1)uN−2lδ(u, 0) + 2NuN−1(lδ)u(u, 0) + uN(lδ)uu(u, 0) +≤ N(N − 1)uN−2(1 − c)lδ(0, 0) < 0. +So w′′(u) < 0 and the graph is concave downward. Therefore γ never re- +enters the surface at a switch point. It follows that γ is a straight line that +either terminates at some point on the surface or exits the ǫ-ball. +Let’s summarize Case 4. In general, γ is initially a boundary segment +lying on the surface, then it leaves S in a straight line that exits the ǫ-ball. +So there is at most one interval in this case. +67 + +In the end, we are going to mention when the lowest degree in the Taylor +expansion of g(x, y) is greater than 2. Suppose the lowest degree is k ≥ 2, +the kth Taylor polynomial has the following form: +a0xk + a1xk−1y + · · · + ak−1xyk−1 + akyk +Consider the line y = mx, substituing it into the above polynomial gives us +xk(a0 + a1m + · · · + akmk). +Setting it to zero gives as at most k distinct solutions for m if ak ̸= 0, +otherwise adding the vertical line x = 0. Therefore the plane can be sliced +into at most 2k distinct pies using these slopes, such that within each slice, +the graph of g along a ray is either concave upward or downward near the +origin. The rest of the proof is analogous to the case when k = 2. +68 + +4 +Conclusion +It seems naturally that our theorems could be generalized to higher-dimensional +Euclidean space. However, the proof in Theorem 1 or 2 does not apply when +n > 3, because the intersection of M1 and M2 becomes a surface instead of a +curve. Furthermore, the proof in Theorem 3 involves dividing the plane into +finitely many slices using the lowest degree Taylor polynomial, which does +not make sense when we have more than two variables. Therefore we are +looking for new methods and we conjecture that all Theorems 1, 2, and 3 +do not generalize when n > 3, because of more free variables. That is there +exist counterexamples in higher-dimensionl Euclidean spaces. +69 + +References +[1] F. Albrecht and I.D. Berg, Geodescis in Euclidean Space with Analytic +Obstacle, Proceedings of the American Mathematical Society, Vol. 113, +No. 1 (Sep., 1991), pp. 201-207. +[2] S. B. Alexander, I.D. Berg, and R. L. Bishop, The Riemannian obstacle +problem, Illinois J. Math. 31 (1987), 167-184. +70 + diff --git a/N9E0T4oBgHgl3EQfTQCe/content/tmp_files/load_file.txt b/N9E0T4oBgHgl3EQfTQCe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3720970b44251c79b16aa34d74d7dc754a5e8564 --- /dev/null +++ b/N9E0T4oBgHgl3EQfTQCe/content/tmp_files/load_file.txt @@ -0,0 +1,1248 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf,len=1247 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='02234v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='DG] 5 Jan 2023 Geodesics in 3-dimensional Euclidean Space with One or Two Analytic Obstacles Chengcheng Yang (communited by Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Robert Hardt) January 6, 2023 1 Abstract F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Albrecht and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Berg proved that in an n-dimensional Eu- clidean space with an analytic obstacle a geodesic is locally an al- ternating finite union of a boundary segment on the surface of the obstacle and a line segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Their proof depends on the initial ve- locity of the geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' At the end of their paper, they conjected that there is a uniform bound on the number of line segments near a fixed initial position, so it should be independent of the initial velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In this paper we are going to show that this conjecture is true in the case when n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Furthermore, we are able to generalize the above local finiteness property to the union of two analytic obstacles which intersect transversally in a 3-dimensional Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' That is to say a geodesic does not bounce infinitely often between two obstacles near any point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 2 1 introduction In the paper [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Albrecht and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Berg proved that a geodesic cannot have an accumulation of interior line segments in Rn or Cn with an analytic obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' More precisely, let M be the closure of the complement of an obstacle in an Euclidean space and let S be its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' A geodesic in M (thought of as a string stretching over some obstacle) is a locally shortest path consisting of two types of segments: touching the boundary (which is known as a boundary segment and whose acceleration is outward normal to S) and not touching the boundary (which is a line segment known as an interior line segment or an interval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' We call the point connecting a boundary segment and an interior line segment a switch point following the tradition in the same paper above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' A geodesic is necessarily C1 at the switch points and can be parametrized by arc length;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' furthermore the acceleration exists everywhere except at the switch points [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When S is analytic, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Albrecht and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Berg showed that the switch points do not accumulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' That is to say, there exists an ǫ > 0 such that γ(s) has no switch point for 0 < s < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The result fails if S is just C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In our paper we are going to generalize the result by looking at the union of two obstacles in a 3-dimensional Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let M1, M2 be the clo- sure of the complement of two obstacles in an Euclidean space whose surfaces are S1, S2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Assume that S1 and S2 intersect transversally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let M be the intersection of M1 and M2 which is the closure of the complement of the union of the two obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Suppose γ is a geodesic in M, then the same conclusion holds so γ does not bounce back and forth between two surfaces infinitely often locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In other words, γ is eventually a boundary segment in one of the surfaces or a line segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The proof consists of two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The first part is concerned with the case of R3 and the angle between S1 and S2 is less than 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The argument uses symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The second part is still dealing with the case of R3 but the angle can be more than 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Here the symmetry arguemnt in the previous part fails, so we need to come up with an asymmetric argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Lastly R can be replaced by C but for visualization we stay with the real Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The second half of our paper concerns with the conjecture at end of the paper [1] by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Albrecht and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Berg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let M be the same as above, and fix one point p on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' They conjectured that there is a uniform bound on the number of intervals (or switch points) within a neighborhood of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' We are able to prove this conjecture if the dimension is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Namely, there exists 3 an ǫ such that for any geodesic γ initiating from p in M, there are at most two intervals within the ǫ-ball of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The proof shows that there are finitely many wedges covering the (x, y)- plane such that within each wedge such an ǫ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 4 2 Part One Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let M1 and M2 be 3-dimensional analytic manifolds with bound- ary embedded in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Denote the boundary surfaces of M1 and M2 by S1 and S2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Assume that S1 and S2 intersect transversally whose angle is less than 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let M be the intersection of M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ is a geodesics stretching over M parametrized by arc length s, with γ(0) = p ∈ S1 ∩ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then there exists an ǫ > 0 such that γ has no switch point for 0 < s < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Set the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Without loss of generality we may assume that p is the origin, the x-axis is tangent to the given geodesic γ at p, and the outward normal vectors to S1 and S2 at p are (0, −k, 1) and (0, k, 1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So the normal vec- tors at p are symmetric with respect to the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let 0 < k < 1 1+β for some β > 0 so k < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The two surfaces S1 and S2 are defined near the origin by analytic equations of the form z = g(x, y) and z = h(x, y), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that the outward normal vector to S1 at the origin is equal to (−gx(0, 0), −gy(0, 0), 1), which is also equal to (0, −k, 1) by hypothesis, therefore gx(0, 0) = 0, gy(0, 0) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Similarly, we have hx(0, 0) = 0, hy(0, 0) = −k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since ∂ ∂y �� (0,0)[g(x, y) − h(x, y)] = 2k > 0, the inverse function theorem implies that the intersection of S1 and S2 near p is a real analytic curve defined by the following equations: y = φ(x), z = g(x, φ(x)) = h(x, φ(x)), where φ is real analytic and φ(0) = 0, φ′(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In fact the tangent vector to S1 ∩ S2 at the origin is given by (1, φ′(x), gx(x, φ(x)) + gy(x, φ(x))φ′(x)) �� x=0 = (1, φ′(0), gx(0, 0) + gy(0, 0)φ′(0)) = (1, φ′(0), kφ′(0)), which is normal to (0, k, 1), thus (1, φ′(0), kφ′(0)) · (0, k, 1) = 2kφ′(0) = 0 =⇒ φ′(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 5 So φ(x) = aMxM + aM+1xM+1 + · · · , (1) where M ≥ 2 and aM ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Notice that we assume φ(x) is not identically zero and g(x, 0) is not identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Otherwise we will have trivial cases which will be included at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus the equation defining S1 near p is of the form g(x, y) = ky + xNa(x, y) + xyb(x, y) + y2c(y), (2) where N ≥ 2, the functions a, b, c are analytic, and a(0, 0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Likewise assume that h(x, 0) is not identically zero, the equation defining S2 near p is of the form h(x, y) = −ky + x ˜ N˜a(x, y) + xy˜b(x, y) + y2˜c(y), (3) where ˜N ≥ 2, the functions ˜a,˜b, ˜c are analytic, and ˜a(0, 0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Choose the orientation of the coordinate system so that γ′(0) = (1, 0, 0), M1 = {z ≤ g(x, y)}, and M2 = {z ≤ h(x, y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Project the S1 ∩ S2 onto the (x, y)-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The projection of the intersection of S1 and S2 onto the (x, y)-plane is a curve given by (x, φ(x)), where x ∈ (−δ, δ) for some δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then it divides the vertical strip (−δ, δ) × (−∞, ∞) into two disconnected regions: {x < φ(x)}, {x > φ(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Suppose y < 0 = φ(0), then using the linear approximation g(0, y) − h(0, y) ≈ gy(0, 0)y − hy(0, 0)y = 2ky < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus connectedness implies that {g(x, y) < h(x, y)} = {y < φ(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (4) Similarly, {g(x, y) > h(x, y)} = {y > φ(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (5) Given a point z = g(x, y) in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If the point lies in M2, then z ≤ h(x, y), thus g(x, y) ≤ h(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' With (4) it follows that the projection of S1 ∩ M2 near p is {g(x, y) ≤ h(x, y), −δ < x < δ} = {x ≤ φ(x), −δ < x < δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 6 Likewise with (5) the projection of S2 ∩ M1 near p is {g(x, y) ≥ h(x, y), −δ < x < δ} = {x ≥ φ(x), −δ < x < δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In conclusion the graph of φ(x) divides the (x, y)-plane into two parts near 0: the part below the graph corresponding to the projection of the surface S1 in M and the part above the graph corresponding to the projection of the surface S2 in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Concavity of φ, which will become crucial in proving the theorem later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since γ′(0) = (1, 0, 0), x′(s) > 0 for 0 ≤ s ≤ ǫ if we choose ǫ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore x(s) > 0 when 0 ≤ s ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' According to the equation (1), φ′′(x) = M(M − 1)aMxM−2 + (M + 1)MaM+1xM−1 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When aM > 0, φ(x) is concave upward over the interval (0, δ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' and when aM < 0, φ(x) is concave downward over the same interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Furthermore, we may also assume that x(s) < δ for 0 ≤ s ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So there are two cases to consider: aM > 0 and aM < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Approximate y(s) and y′(s) using the normal vectors N1(s), N2(s) to S1, S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' We denote γ(s) = (x(s), y(s), z(s)), for 0 ≤ s ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ∈ S1, the normal vector to S1 at γ(s) is N1(s) = (−gx(x(s), y(s)), −gy(x(s), y(s)), 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' From (2) it follows that gx(x(s), y(s)) = x(s)m(x(s), y(s)) + y(s)b(x(s), y(s));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' gy(x(s), y(s)) = k + x(s)k(x(s), y(s)) + y(s)l(y(s)), where the functions m, b, k, l are bounded near (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Moreover, since x′(s) > 0 for 0 ≤ s ≤ ǫ, x(s) has a C1-inverse function s(x) for s ∈ [0, ǫ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore we can express y(s) as y(s) = y(s(x)) = α(x), where α is a C1-function and α(0) = dα dx(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then one has y(s) = o(x(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Hence gx(x(s), y(s)) = x(s)[m(x(s), y(s)) + y(s) x(s)b(x(s), y(s))] = x(s)V1(s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' gy(x(s), y(s)) = k + x(s)[k(x(s), y(s)) + y(s) x(s)l(y(s))] = k + x(s)V2(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 7 Therefore N1(s) = (−x(s)V1(s), −k − x(s)V2(s), 1), where V1(s) and V2(s) are bounded for s near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let s be such that γ′′(s) exists, then γ′′(s) = z′′(s)N1(s) since the acceleration is outward normal to S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' This implies that x′′(s) = −z′′(s)x(s)V1(s), y′′(s) = −z′′(s)(k + x(s)V2(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (6) For ǫ sufficiently small, |x(s)V2(s)| ≤ βk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore |y′′(s)| ≤ (1 + β)k|z′′(s)| from the second equality in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Similarly, if γ(s) ∈ S2 and γ′′(s) exists, the equality in (3) deduces that the normal vector to S2 at γ(s) is N2(s) = (−x(s)W1(s), k − x(s)W2(s), 1), where W1(s) and W2(s) are bounded for s near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then it follows from γ′′(s) = z′′(s)N2(s) that x′′(s) = −z′′(s)x(s)W1(s), y′′(s) = −z′′(s)(−k + x(s)W2(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (7) Again for ǫ sufficiently small, one may assume that |x(s)W2(s)| ≤ βk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' There- fore |y′′(s)| ≤ (1 + β)k|z′′(s)| from the second equality in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When γ(s) does not lie on S1 and S2, γ is a line segment so γ′′(s) is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Combining with what we’ve found above, one gets |y′′(s)| ≤ (1+β)kz′′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Notice that z(0) = z′(0) = y(0) = y′(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Furthermore z′′(s) ≥ 0 (and hence z′(s) ≥ 0) on the interval [0, ǫ], because the outward normal vectors to S1 and S2 have a positive z-coordinate of 1 at the origin and γ′′(s) is directed outward on a boundary segment on S1 or S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Indeed, γ(s) is a locally shortest path and if γ(s) lies on the surface of M1 or M2, its acceleration exists everywhere except at the switch points and is outward normal to the surface [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So for s ∈ [0, ǫ], one can approximate |y′(s)| = | � s 0 y′′(σ)dσ| ≤ (1 + β)k � s 0 z′′(σ)dσ = (1 + β)kz′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Integrating again one obtains |y(s)| ≤ (1 + β)kz(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 8 If γ(s) ∈ S1, the equality in (2) gives z(s) = g(x(s), y(s)) = ky(s) + x(s)Na(x(s), y(s)) + x(s)y(s)b(x(s), y(s)) + y2(s)c(y(s)) ≤ k|y(s)| + x(s)N|a(x(s), y(s))| + |y(s)||x(s)b(x(s), y(s)) + y(s)c(y(s))| ≤ k|y(s)| + C1x(s)N + C2|y(s)| ≤ (k + C2)(1 + β)kz(s) + C1x(s)N, for some constants C1, C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since k < 1 1+β, one can choose ǫ small enough so that (k + C2)(1 + β)k < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore there exists a positive constant A such that z(s) ≤ Ax(s)N ⇒ |y(s)| ≤ (1 + β)kAx(s)N = Bx(s)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (8) Similarly, if γ(s) ∈ S2, the equality in (3) gives us z(s) ≤ Ax(s) ˜ N ⇒ |y(s)| ≤ Bx(s) ˜ N, (9) by enlarging A and B if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Choosing ǫ small enough so that x(s) < 1 and assuming without loss of generality that N ≤ ˜N, one has x(s) ˜ N ≤ x(s)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus with (8) and (9) z(s) ≤ Ax(s)N, |y(s)| ≤ Bx(s)N, (10) if γ(s) ∈ S1 or S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Next let’s approximate y′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ∈ S1, differentiating z(s) = g(x(s), y(s)) gives z′(s) = ky′(s) + x(s)N−1[Nx′(s)a(x(s), y(s)) + x(s)(ax(x(s), y(s))x′(s) + ay(x(s), y(s))y′(s))] +y′(s)[x(s)b(x(s), y(s)) + x(s)y(s)by(x(s), y(s)) + 2y(s)c(y(s)) + y2(s)c′(y(s))] +y(s)[x′(s)b(x(s), y(s)) + x(s)x′(s)bx(x(s), y(s))] ≤ k|y′(s)| + C1x(s)N−1 + C2|y′(s)| + C3|y(s)| ≤ (k + C2)(1 + β)kz′(s) + C1x(s)N−1 + C3Bx(s)N, for some constants C1, C2, C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Again since (1 + β)k < 1, for ǫ sufficiently small, one can make (k + C2)(1 + β)k < 1, so z′(s) ≤ Cx(s)N−1 ⇒ |y′(s)| ≤ (1 + β)kGx(s)N−1 = Dx(s)N−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (11) 9 Similarly, if γ(s) ∈ S2, differentiating z(s) = h(x(s), y(s)) gives z′(s) ≤ Cx(s) ˜ N−1 ⇒ |y′(s)| ≤ (1 + β)kGx(s) ˜ N−1 = Dx(s) ˜ N−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (12) by enlarging C and D if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Combining (11) and (12), together with ˜N ≥ N, one has z′(s) ≤ Cx(s)N−1, |y′(s)| ≤ (1 + β)kGx(s)N−1 = Dx(s)N−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (13) Now let’s look at the situation when γ(s) is in an interior line segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Considering a line segment in the image of γ with two endpoints γ(s1) and γ(s2), we can parametrize y(s) for s ∈ [s1, s2] by y(s) = y(s1) + T(x(s) − x(s1)), where T = dα dx(x(s1)), where with (13) ��dα dx(x(s)) �� = ��dα dx(x(s1)) �� = ��y′(s1) x′(s2) �� ≤ 2|y′(s1)| ≤ 2Dx(s1)N−1 ≤ 2Dx(s)N−1, (14) if x′ ≥ 1/2 by choosing ǫ small enough and the last inequality holds because x(s) is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Hence with (10) and (14) one obtains |y(s)| ≤ |y(s1)| + |T|(|x(s)| + |x(s1)|) ≤ Bx(s1)N + 2Dx(s)N−1(x(s) + x(s)) ≤ (B + 4D)x(s)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Replacing B by B + 4D, together with (10), yields that in general, |y(s)| ≤ Bx(s)N for every s ∈ [0, ǫ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (15) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Prove M ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If the geodesic γ moves from S1 to S2 or from S2 to S1, (x(s), y(s)) must cross the curve y = φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the one hand, |φ(x(s))| ≤ Bx(s)N according to (15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' on the other hand, |φ(x(s))| ≥ |aM | 2 x(s)M by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore one obtains the following relation: |aM| 2 x(s)M ≤ Bx(s)N =⇒ x(s)M−N ≤ 2B |aM|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 10 Suppose M < N the left-hand side converges to infinity as s approaches 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' This means that if M < N the geodesic γ eventually stops bouncing between S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore it reduces to the case of one obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Hence we proceed with M ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Next let’s prove that N = ˜N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Case 1: M > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the intersection of S1 and S2 we’ve shown that y = φ(x) for x ∈ (−δ, δ) and hence g(x, φ(x)) = h(x, φ(x)) over the interval (−δ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Using the equalities (1) and (2) one has g(x, φ(x)) = kφ(x) + xNa(x, φ(x)) + xφ(x)b(x, φ(x)) + φ2(x)c(φ(x)) = kxM(aM + aM+1x + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' ) + xN(a(0, 0) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' ) + xM+1(aMb(0, 0) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' ) + a2 Mx2M(c(0) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since M > N the first nonzero term in the power serious expansion of g(x, φ(x)) is a(0, 0)xN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Similarly the equalities (1) and (3) gives h(x, φ(x)) = −kφ(x) + x ˜ N˜a(x, φ(x)) + xφ(x)˜b(x, φ(x)) + φ2(x)˜c(φ(x)) = −kxM(aM + aM+1x + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' ) + x ˜ N(˜a(0, 0) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' ) + xM+1(aM˜b(0, 0) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' ) + a2 Mx2M(˜c(0) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' By the uniqueness of the power serious expansion one must have ˜N = N, otherwise the first nonzero term in the power serious expansion of h(x, φ(x)) has an order of at least N + 1 (we assumed ˜N ≥ N earlier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Case 2: M = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If there is an interior line segment in the image of γ with two endpoints γ(s1) and γ(s2) such that γ(s1) ∈ S2 and γ(s2) ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' We can parametrize y(s) for s ∈ [s1, s2] by y(s) = y(s1) + T(x(s) − x(s1)), where T = dα dx(x(s1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since γ(s1) ∈ S2, one can use the second inequality in (12) to estimate ��dα dx(x(s1)) �� = ��y′(s1) x′(s1) �� ≤ 2|y′(s1)| ≤ 2Dx(s1) ˜ N−1, if |x′(s)| ≥ 1 2 by choosing ǫ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' With (9) one gets |y(s)| ≤ |y(s1)| + |T|(|x(s)| + |x(s1)|) ≤ Bx(s1) ˜ N + 2Dx(s1) ˜ N−1(x(s) + x(s)) ≤ (B + 4D)x(s) ˜ N, 11 where the last inequality holds because x(s) is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' For some s ∈ (s1, s2), we have y(s) = φ(x(s)) and so |φ(x(s))| ≤ (B + 4D)x(s) ˜ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, |φ(x(s))| ≥ |aN| 2 x(s)N if s is sufficiently close to 0 by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus the following relation holds: |aN| 2 x(s)N ≤ (B + 4D)x(s) ˜ N =⇒ x(s)N− ˜ N ≤ 2(B + 4D) |aN| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Suppose ˜N > N then the left-hand side converges to infinity as s approaches 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So γ eventually stops going from S2 to S1 which reduces to the case of one obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Hence we proceed with ˜N = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Show a(0, 0) > 0, ˜a(0, 0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let’s prove by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Suppose that a(0, 0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Assume γ has a switch point inside S1 at s = s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' That is to say γ(s0) ∈ S1 and for either s > s0 nearby or s < s0 nearly, γ(s) is an interior line segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Denote (x(s0), y(s0)) by (x0, y0) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Consider the intersection of the two- dimensional plane y = y0 + T(x − x0) with the surface z = g(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Set f(x) = g(x, y0 + T(x − x0)), where T = dα dx(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that d2f dx2 (x0) = gxx(x0, y0) + 2gxy(x0, y0)T + gyy(x0, y0)T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (16) Using (2), (15) and choosing ǫ sufficiently small one can estimate gxx(x0, y0) = xN−2 0 [N(N − 1)a(x0, y0) + x0p(x0, y0)] + y0q(x0, y0) (17) ≤ xN−2 0 N(N − 1)1 2a(0, 0) + BxN 0 C1 ≤ xN−2 0 N(N − 1)1 2a(0, 0) − xN−2 0 N(N − 1)1 4a(0, 0) = xN−2 0 N(N − 1)1 4a(0, 0) Furthermore, using the inequality in (14) and letting ǫ be small enough one has |2gxy(x0, y0)T + gyy(x0, y0)T 2| (18) ≤ C2|T| ≤ C2 · 2DxN 0 ≤ −xN−2 0 N(N − 1)1 8a(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 12 So d2f dx2 (x0) ≤ xN−2 0 N(N − 1)1 8a(0, 0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore f is concave downward at x0 and the tangent line to the curve at x0 is above the graph, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In other words, γ has no switch point on S1 near the origin and so initially stays inside S2 or is a line segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Similarly γ initially stays inside S1 or is a line segment for ˜a(0, 0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Hence this reduces to the case of one obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus we proceed with assuming that a(0, 0) > 0 and ˜a(0, 0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Show that given ǫ small enough, if γ leaves S1 at a switch point, it will never enter S1 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Similarly, when γ leaves S2, it has to enter S1 at the next switch point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Indeed we can prove by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Suppose γ(s) leaves S1 at s = s0 and dives into the interior of M for increasing s until it enters S1 again at s = s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Again set f(x) = g(x, y0 + T(x − x0)), where (x0, y0) = (x(s0), y(s0)) and T = dα dx(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that for s ∈ [s0, s1] d2f dx2 (x(s)) = gxx(x(s), y(s)) + 2gxy(x(s), y(s))T + gyy(x(s), y(s))T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Note that when s = s0, this is just the expression in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Using an analogous argument as shown in (17) for the case a(0, 0) > 0 one yields gxx(x(s), y(s)) ≥ x(s)N−2N(N − 1)1 4a(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Moreover in analogy to (18) one has 2gxy(x(s), y(s))T + gyy(x(s), y(s))T 2 ≥ −xN−2 0 N(N − 1)1 8a(0, 0) ≥ −x(s)N−2N(N − 1)1 8a(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Hence d2f dx2 (x(s)) ≥ x(s)N−2N(N − 1)1 8a(0, 0) > 0 for all s ∈ [s0, s1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 13 Therefore f ′(x(s)) is increasing as x(s) increases from x(s0) = x0 to x(s1) = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, since the interior line segment is tangent to S1 at the two endpoints, we must have f ′(x0) = f ′(x1), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore if γ leaves S1 at the switch point γ(s0) for some s0 ∈ [0, ǫ], the geodesic arc beyond this point is a line segment never meeting S1 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Hence the next switch point (if there is one) lies on the surface S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The same argument holds for S2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The global behavior of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Near the origin the geodesic is an alternating sequence of a bound- ary segment on S1, an interval from S1 to S2, a boundary segment on S2, an interval from S2 to S1, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Each time the projection of γ crosses the graph of φ at time s, there is l(s) > 0 such that γ is an interior line segment over the interval [s − l(s), s + l(s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The set A of such s with 0 ≤ s ≤ ǫ is therefore countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Furthermore if A1 = sup A then A1 is actually the maximum of the set, because there is no s ∈ A within the l(A1)-distance of A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let A2 = sup(A − A1), A3 = sup(A − {A1, A2}), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that the set A can be linearly ordered as A = {A1 > A2 > A3 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' }, such that between An and An+1 the curve γ lies entirely in S1 or S2 for each n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If aM > 0, the curve y = φ(x) is concave upward for x > 0 nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' We can obtain a contradiction as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ leaves a point in S2 and enters a point in S1, then (x(s), y(s)) crosses φ from above to below at some s = s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' By concavity one must have dα dx(x(s1)) < φ(x(s1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Later (x(s), y(s)) crosses φ from below to above at some s = s2, then dα dx(x(s2)) > φ(x(s2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In between γ stays in S1 all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 14 Since dα dx(x(s1)) = y′(s1) x′(s1) and dα dx(x(s2)) = y′(s2) x′(s2), we must have y′(s1) < x′(s1)φ(x(s1)), y′(s2) > x′(s2)φ(x(s2)) ⇒ y′(s2) − y′(s1) > x′(s2)φ(x(s2)) − x′(s1)φ(x(s1)) Therefore it suffices to show that for 0 < s1 < s2 < ǫ, y′(s2) − y′(s1) ≤ x′(s2)φ(x(s2)) − x′(s1)φ(x(s1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the one hand, y′(s2) − y′(s1) = � s2 s1 y′′(s)ds, where y′′(s) = −z′′(s)(k + x(s)V2(s)) ≤ −z′′(s)(1 − β)k from (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus y′(s2) − y′(s1) ≤ � s2 s1 −z′′(s)(1 − β)kds < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, x′(s2)φ(x(s2)) − x′(s1)φ(x(s1)) = � s2 s1 d ds � x′(s)φ′(x(s)) � = � s2 s1 x′′(s)φ′(x(s)) + x′(s)2φ′′(x(s))ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Now let’s estimate x′′(s)φ′(x(s)) + x′(s)2φ′′(x(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since γ(s) ∈ S1 for s ∈ (s1, s2), one has x′′(s) = −z′′(s)x(s)V1(s) from (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore |x′′(s)| ≤ Ez′′(s)x(s) for some positive constant E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' By hypothesis γ is parametrized by arc length, so |x′(s)| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' With (8) and (11) one differentiates z(s) = g(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) twice to obtain z′′(s) = gxx(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))x′(s)2 + 2gxy(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))x′(s)y′(s) + gyy(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))y′(s)2 +gx(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))x′′(s) + gy(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))y′′(s) = � x(s)N−2[N(N − 1)a(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) + x(s)p(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))] + y(s)q(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) � x′(s)2 +y′(s)[2gxy(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))x′(s) + gyy(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))y′(s)] +[x(s)m(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) + y(s)b(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))]x′′(s) +[k + x(s)k(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) + y(s)l(y(s))]y′′(s) ≤ � x(s)N−2C1 + |y(s)|C2 � 1 + |y′(s)|C3 + C4|x′′(s)| + [k + βk]|y′′(s)| ≤ x(s)N−2C1 + Bx(s)NC2 + Dx(s)N−1C3 + C4Ez′′(s)x(s) + (1 + β)kz′′(s)(k + |x(s)V2(x)|) ≤ x(s)N−2(C1 + Bx(s)2C2 + Dx(s)C3) + z′′(s)(C4Ex(s) + (1 + β)k · (1 + β)k) ≤ x(s)N−2C5 + z′′(s)C6 + (1 + β)2k2z′′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 15 By hypothesis (1 + β)k < 1, then we can choose ǫ small enough so that C6 < 1 − (1 + β)2k2 implying that z′′(s) ≤ Fx(s)N−2, and so |x′′(s)| ≤ EFx(s)N−1 = Gx(s)N−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Now let’s use (1) to approximate φ′′(x(s)) and φ′(x(s)): 0 < φ′(x) = MaMxM−1 + (M + 1)aM+1xM + · · · ≤ 2MaMxM−1 for x near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So for ǫ sufficiently small, one has φ′(x(s))x′′(s) ≥ φ′(x) · −Gx(s)N−1 (19) ≥ 2MaMx(s)M−1 · −Gx(s)N−1 = −2MGaMx(s)M+N−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, φ′′(x) = M(M − 1)aMxM−2 + · · · ≥ 1 2aMM(M − 1)xM−2 for x near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So again by choosing ǫ small enough and assuming x′(s) ≥ 1 2, one obtains φ′′(x(s))x′(s)2 ≥ 1 8aMM(M − 1)x(s)M−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (20) Combing (19) and (20) we find that φ′′(x(s))x′(s)2 + φ′(x(s))x′′(s) ≥ x(s)M−2aM(1 8M(M − 1) − 2MGx(s)N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since N ≥ 2 the above difference can be made positive for every s ∈ [0, ǫ] if ǫ is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Hence x′(s2)φ(x(s2)) − x′(s1)φ(x(s1)) = � s2 s1 x′′(s)φ′(x(s)) + x′(s)2φ′′(x(s))ds > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' We reach a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' There γ(s) eventually stops bouncing between S1 and S2 as s approaches 0, which reduces to the case of one obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If aM < 0, the curve y = φ(x) is concave downward for x > 0 nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Replacing S1 by S2 and g by h in the previous argument gives a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Trivial Case 1: φ(x) is identically zero, but g(x, 0) and h(x, 0) are not identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Everything is fine until Step 9 by letting M = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In Step 10, the proof is as follows: 16 If γ leaves a point in S2 and enters a point in S1, then (x(s), y(s)) crosses φ from above to below at some s = s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since the curve y = φ(x) is the x-axis, one must have dα dx(x(s1)) < 0 and so y′(s1) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Later (x(s), y(s)) crosses φ from below to above at some s = s2, thus dα dx(x(s2)) > 0 and so y′(s2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In between γ stays in S1 all the time where y′′(s) is always negative, so y′(s2) < y′(s1), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Trivial Case 2: one of g(x, 0), h(x, 0) is identically zero, but not both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Without loss of generality, let us assume that g(x, 0) is identically zero but h(x, 0) is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then we can write g(x, y) as g(x, y) = ky + xyb(x, y) + y2c(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If φ(x) is identically zero, then g(x, φ(x)) is also identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' However, with equality (3) h(x, φ(x)) = x ˜ N˜a(x, 0) = x ˜ N(˜a(0, 0) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' ) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So φ(x) is nonzero whose power serious expansion is still (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Everything is fine until step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ∈ S1, the equality in (2) gives z(s) = g(x(s), y(s)) = ky(s) + x(s)y(s)b(x(s), y(s)) + y(s)2c(y(s)) ≤ k|y(s)| + |y(s)||x(s)b(x(s), y(s)) + y(s)c(y(s))| ≤ k|y(s)| + C1|y(s)| ≤ (k + C1)(1 + β)k1z(s) By hypothesis (1+β)k < 1, for s close enough to 0 such that C1 < 1−(1+β)k2 (1+β)k , one obtains z(s) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, z(s) > 0 for all s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So the geodesic does not touch S1 near the origin, which reduces to the case of one obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' g(x, 0) and h(x, 0) are both identically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' One can show that φ(x) is also identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In this case the geodesic does not touch S1 and S2 near the origin, so it must be a line segment at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 17 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let M1 and M2 be 3-dimensional analytic manifolds with bound- ary embedded in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Denote the boundary surfaces of M1 and M2 by S1 and S2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Assume that S1 and S2 intersect transversally whose angle is greater than or equal to 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let M be the intersection of M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let γ be a geodesics in M parametrized by arc length s, with γ(0) = p ∈ S1 ∩ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then there is an ǫ > 0 such that γ has no switch point for 0 < s < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Set the coordinate system as in Theorem 1, except that the outward normal vectors to S1 and S2 at p are no longer symmetrical with respect to the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Tilt the system appropriately so that the normal vectors are (0, −k1, 1) and (0, k2, 1), respectively, where 0 < k1 < 1 < k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Furthermore, choose k1 and k2 so k1k2 < 1 1+β < 1 for some β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that gx(0, 0) = 0, gy(0, 0) = k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' And hx(0, 0) = 0, hy(0, 0) = −k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since ∂ ∂y �� (0,0)[g(x, y) − h(x, y)] = k1 + k2 > 0, again the implicit function theorem implies that the intersection of S1 and S2 near p is a real analytic curve defined by the following equations: y = φ(x), z = g(x, φ(x)) = h(x, φ(x)), where φ has the same power expansion as in (1) by assuming that φ is not identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Furthermore let us assume that g(x, 0) and h(x, 0) are not identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore the equation defining S1 near p is of the form g(x, y) = k1y + xNa(x, y) + xyb(x, y) + y2c(y), (21) where N ≥ 2, the functions a, b, c are analytic, and a(0, 0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Moreover the equation defining S2 near p is of the form h(x, y) = −k2y + x ˜ N˜a(x, y) + xy˜b(x, y) + y2˜c(y), (22) where ˜N ≥ 2, the functions ˜a,˜b, ˜c are analytic, and ˜a(0, 0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Same as in Theorem 1, the graph of φ(x) divides the (x, y)-plane into two parts near 0: the part below the graph corresponding to the projection 18 of the surface S1 in M and the part above the graph corresponding to the projection of the surface S2 in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Concavity of φ is determined by the sign of aM and there are two cases to consider: aM > 0 and aM < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Approximate y(s) and y′(s) using the normal vectors N1(s), N2(s) to S1, S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Following the same procedure as in (6) and (7) one obtains if γ(s) ∈ S1, x′′(s) = −z′′(s)x(s)V1(s), y′′(s) = −z′′(s)(k + x(s)V2(s)), where |x(s)V2(s)| ≤ βk1 for ǫ sufficiently small and therefore |y′′(s)| ≤ (1 + β)k1|z′′(s)|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' and if γ(s) ∈ S2, x′′(s) = −z′′(s)x(s)W1(s), y′′(s) = −z′′(s)(−k + x(s)W2(s)), where |x(s)W2(s)| ≤ βk2 for ǫ small enough and hence |y′′(s)| ≤ (1 + β)k2|z′′(s)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Case 1, if γ(s) ∈ S1, since k1(1 + β) < 1 one can use the same argument as before to obtain z(s) ≤ Ax(s)N ⇒ |y(s)| ≤ (1 + β)k1Ax(s)N = Bx(s)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (23) Next differentiating z(s) = g(x(s), y(s)) once gives z′(s) ≤ Cx(s)N−1 ⇒ |y′(s)| ≤ (1 + β)k1Cx(s)N−1 = Dx(s)N−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (24) Case 2: if γ(s) ∈ S2, the arguments above fail because k2 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Instead we need to use a different approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since y′′(s) = −z′′(s)(−k2 + x(s)W2(s)), z′′(s) + k2y′′(s) = z′′(s)[1 + k2 2 − k2x(s)W2(s)] ≥ z′′(s)[1 + (1 − β)k2 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (25) On the other hand, if γ(s) ∈ S1, then y′′(s) = −z′′(s)(k1 + x(s)V2(s)) implies that z′′(s) + k2y′′(s) = z′′(s)[1 − k1k2 − k2x(s)V2(s)] (26) ≥ z′′(s)[1 − k1k2 − k2k1β] = z′′(s)[1 − (1 + β)k1k2], Combining (25) with (26), together with the hypothesis (1 + β)k1k2, there exists a constant H such that if γ(s) ∈ S1 or S2 z′′(s) + k2y′′(s) ≥ Hz′′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (27) 19 Note that the inequality (27) still holds if γ(s) does not touch any surface since the acceleration is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that for every s in the interval [0, ǫ], z′(s) + k2y′(s) = � s 0 z′′(σ) + k2y′′(σ)dσ ≥ � s 0 Hz′′(σ)dσ = Hz′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (28) Moreover, z(s) + k2y(s) ≥ Hz(s) for every s ∈ [0, ǫ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (29) With (22) and (29) one deduces that Hz(s) ≤ x(s) ˜ N˜a(x(s), y(s)) + y(s)[x(s)˜b(x(s), y(s)) + y(s)˜c(y(s))] ≤ C1x(s) ˜ N + |y(s)|C2 ≤ C1x(s) ˜ N + (1 + β)k2z(s)C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' With ǫ sufficiently small C2 can be made as small as possible so that H − (1 + β)k2C2 > 0, implying z(s) ≤ Ax(s) ˜ N ⇒ |y(s)| ≤ (1 + β)k2Ax(s) ˜ N = Bx(s) ˜ N, (30) after enlarging A and B accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Differentiating z(s) = h(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) once,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' together with (28) and (30),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' one obtains z′(s) = −k2y′(s) + x(s) ˜ N−1[ ˜Nx′(s)˜a(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) +x(s)(˜ax(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))x′(s) + ˜ay(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))y′(s))] +y′(s)[x(s)˜b(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) + x(s)y(s)˜by(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) +2y(s)˜c(y(s)) + y2(s)˜c′(y(s))] +y(s)[x′(s)˜b(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) + x(s)x′(s)˜bx(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))] so z′(s) + k2y′(s) ≤ C1x(s)N−1 + C2|y′(s)| + C3|y(s)| ≤ C1x(s) ˜ N−1 + C2(1 + β)k2z′(s) + C3Bx(s) ˜ N so Hz′(s) ≤ C1x(s) ˜ N−1 + C2(1 + β)k2z′(s) + C3Bx(s) ˜ N With ǫ sufficiently small C2 can be made as small as possible so that C2(1 + β) < H and thus z′(s) ≤ Cx(s) ˜ N−1 ⇒ |y′(s)| ≤ (1 + β)k2z′(s) = Dx(s) ˜ N−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (31) 20 by enlarging C and D accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Without loss of generality we may assume that N ≤ ˜N, then x(s) ˜ N ≤ x(s)N for x(s) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus (23) and (30) imply that if γ ∈ S1 or S2 z(s) ≤ Ax(s)N, |y(s)| ≤ Bx(s)N, (32) Furthermore (24) and (31) imply that if γ ∈ S1 or S2 z′(s) ≤ Cx(s)N−1, |y′(s)| ≤ Dx(s)N−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (33) When γ(s) does not lie on S1 or S2, following the same argument as in Theorem 1, one can show that |T(x(s))| ≤ 2Dx(s)N−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (34) |y(s)| ≤ Bx(s)N for every s ∈ [0, ǫ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (35) From 5-10, one can copy the proof from Theorem 1 word by word to have: M ≥ N, N = ˜N, a(0, 0) > 0, ˜a(0, 0) > 0, and γ is an alternating sequence of boundaries segments on S1, S2 and line segments between S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If aM < 0, the argument is slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ leaves a point in S1 and enters a point in S2, then (x(s), y(s)) crosses φ from below to above at some s = s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since the curve y = φ(x) is concave downward for x > 0, one must have dα dx(x(s1)) > φ(x(s1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Later (x(s), y(s)) crosses φ from above to below at some s = s2, then dα dx(x(s2)) < φ(x(s2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In between γ stays in S2 all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since dα dx(x(s1)) = y′(s1) x′(s1) and dα dx(x(s2)) = y′(s2) x′(s2), we must have y′(s1) > x′(s1)φ(x(s1)), y′(s2) < x′(s2)φ(x(s2)) ⇒ y′(s2) − y′(s1) < x′(s2)φ(x(s2)) − x′(s1)φ(x(s1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore it suffices to show that for 0 < s1 < s2 < ǫ, y′(s2) − y′(s1) ≥ x′(s2)φ(x(s2)) − x′(s1)φ(x(s1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 21 On the one hand, y′(s2) − y′(s1) = � s2 s1 y′′(s)ds, where y′′(s) = −z′′(s)(−k2 + x(s)W2(x)) ≥ z′′(s)(1 − β)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus y′(s2) − y′(s1) ≥ � s2 s1 z′′(s)(1 − β)kds > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, x′(s2)φ(x(s2)) − x′(s1)φ(x(s1)) = � s2 s1 d ds � x′(s)φ′(x(s)) � = � s2 s1 x′′(s)φ′(x(s)) + x′(s)2φ′′(x(s))ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Now let’s estimate x′′(s)φ′(x(s)) + x′(s)2φ′′(x(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since γ(s) ∈ S2 for s ∈ (s1, s2), one has x′′(s) = −z′′(s)x(s)W1(s) from (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore |x′′(s)| ≤ Ez′′(s)x(s) for some positive constant E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' By hypothesis γ is parametrized by arc length, so |x′(s)| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' One differentiates z(s) = h(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) twice to get z′′(s) = hxx(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))x′(s)2 + 2hxy(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))x′(s)y′(s) + hyy(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))y′(s)2 +hx(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))x′′(s) + hy(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))y′′(s) = � x(s)N−2[N(N − 1)˜a(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) + x(s)˜p(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))] +y(s)˜q(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) � x′(s)2 +y′(s)[2hxy(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))x′(s) + hyy(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))y′(s)] +[x(s)W1(s)]x′′(s) + [−k2 + x(s)W2(s)]y′′(s) One moves −k2y′′(s) to the other side of the inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' together with (30) and (31),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' to obtain z′′(s) + k2y′′(s) ≤ � x(s)N−2C1 + |y(s)|C2 � 1 + |y′(s)|C3 + C4|x′′(s)| + |x(s)W2(s)||y′′(s)| ≤ x(s)N−2C1 + Bx(s)NC2 + Dx(s)N−1C3 + C4Ez′′(s)x(s) +|x(s)W2(s)|z′′(s)(k2 + |x(s)W2(x)|) ≤ x(s)N−2(C1 + Bx(s)2C2 + Dx(s)C3) +z′′(s)(C4Ex(s) + βk2(1 + β)k2) ≤ x(s)N−2C5 + β(1 + β)k2 2z′′(s)C6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 22 Using (27) and choosing ǫ small enough so that β(1 + β)k2 2C6 < H one gets that z′′(s) ≤ Fx(s)N−2, and so |x′′(s)| ≤ EFx(s)N−1 = Gx(s)N−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Now let’s use (1) to approximate φ′′(x(s)) and φ′(x(s)): 0 > φ′(x) = MaMxM−1 + (M + 1)aM+1xM + · · · ≥ 2MaMxM−1 for x near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So for ǫ sufficiently small, one has φ′(x(s))x′′(s) ≤ φ′(x(s)) · −Gx(s)N−1 (36) ≤ 2MaMx(s)M−1 · −Gx(s)N−1 = −2MaMLx(s)M+N−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, φ′′(x) = M(M − 1)aMxM−2 + · · · ≤ 1 2aMM(M − 1)xM−2 for x near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So using ǫ small enough and assuming x′(s) ≥ 1 2, one obtains φ′′(x(s))x′(s)2 ≤ 1 8aMM(M − 1)x(s)M−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (37) Combining (36) and (37) we find that φ′′(x(s))x′(s)2 + φ′(x(s))x′′(s) ≤ x(s)M−2aM(1 8M(M − 1) − 2MGx(s)N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since N ≥ 2 the above difference can be made negative for every s ∈ [0, ǫ] if ǫ is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Hence x′(s2)φ(x(s2)) − x′(s1)φ(x(s1)) = � s2 s1 x′′(s)φ′(x(s)) + x′(s)2φ′′(x(s))ds < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' We reach a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus γ eventually stops bouncing between S1 and S2 as s approaches 0, which reduces to the case of one obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The trivial cases from 11-14 follow exactly the same proof in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 23 3 Part Two Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let M be an 3-dimensional manifold with boundary embedded in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Denote the boundary surface of M by S and let γ(s) be a geodesic on M parametrized by arc length s with γ(0) = p ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then there exists an ǫ > 0 such that the number of line segments in the image of γ within the ǫ-ball of p is uniformly bounded, namely it is independent of the initial velocity γ′(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' More precisely, there are at most two complete or partial line segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The idea is to show that in each direction there exists a wedge and an ǫ such that if γ′(0) is within this wedge, then γ has a uniform bound on the number of switch points within the ǫ-ball of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Set up the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Choose an orientation of the coordinate system (x, y, z) so that p is the origin, S near p can be parametrized by an analytic function z = g(x, y), and the outward normal vector to S at p is in the positive z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Notice that γ′(0) is either tangent to S at p in which case γ′(0) is in the (x, y)-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (Or γ′(0) has a negative z-component pointing towards the interior of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' We will look at this case in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Without loss of generalizty, one may assume that the lowest degree in the power series expansion of g(x, y) is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The idea for higher degrees is very similar, which will be mentioned at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since the z-axis is normal to S at p and p is the origin, the Taylor series expansion of g is g(x, y) = 1 2gxx(0, 0)x2 + gxy(0, 0)xy + 1 2gyy(0, 0)y2 + higher-order terms, where 1 2gxx(0, 0)x2 + gxy(0, 0)xy + 1 2gyy(0, 0)y2 is not the zero polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let H be the Hessian matrix H = � gxx(0, 0) gxy(0, 0) gxy(0, 0) gyy(0, 0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since H is symmetric, the spectral theorem says that there exists a 2×2 real orthogonal matrix P such that PHP t = �2a 0 0 2b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 24 Rotating and/or reflecting the (x, y)-plane using P, the surface S near p can be parametrized as follows: g(x, y) = ax2 + by2 + higher-order terms, where a and b are not identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The signs of a, b tell us about the shape of S near the origin, namely When a > 0, b ≥ 0 or a ≥ 0, b > 0, S is concave upward near the origin and γ has at most one switch point near p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When a < 0, b ≤ 0 or a ≤ 0, b < 0, S is concave downward near the origin and γ has no switch point near p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When one of a, b is positive and the other is negarive, S has a saddle point at p and we are going to investigate this case in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Without loss of generality, we may assume that a > 0, b < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Moreover replacing b by −b yields g(x, y) = ax2 − by2 + higher-order terms, where a > 0, b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Notice that when ax2 − by2 = 0, y = ± �a b x, which gives rise to two lines dividing the plane into four different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' There exists an angle 0 < θ0 < π/2 such that tan(θ0) = �a b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then we are going to prove the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In the positive direction of the x-axis: for any small positive δ, there exists an ǫ > 0 such that if γ′(0) lies inside the wedge [−θ0 + δ, θ0 − δ], then γ has at most one switch point within the ǫ-ball of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Similar statements hold in the negative direction of the x-axis and positive and negative directions of the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In the positive direction of y = �a bx : there exist an η > 0 and an ǫ > 0 such that if γ′(0) is in the wedge [θ0 − η, θ0 + η], then γ has at most two switch points within the ǫ-ball of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Similar 25 statements also hold in the negative direction of y = �a bx and the positive and negative directions of y = − �a bx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Combining 1 and 2, the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In the positive direction of the x-axis: for any 0 < δ < θ0, there exists an ǫ > 0 such that if γ′(0) lies inside the wedge [−θ0 + δ, θ0 − δ], then γ has at most one switch point within the ǫ-ball of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (1) Set up the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The first estimate of ǫ comes from that S is parametrized by g(x, y) within the ǫ-ball of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since γ′(0) is a unit vector tangent to S, one has x′(0) = cos θ and y′(0) = sin θ, where θ ∈ [−θ0+δ, θ0−δ] by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (2) Show that x′(s) > 0 if ǫ is chosen small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' This is the second estimate of ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Here we are assuming that if γ(s) is within the ǫ-ball of p, then for every 0 ≤ σ ≤ s, γ(σ) also lies within the ǫ-neighbhorhood of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let γ(s) = (x(s), y(s), z(s)) with |x(s)|,|y(s)|, |z(s)| less than or equal to ǫ, so that γ(s) is within the ǫ-ball of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ∈ S, then the normal vector at γ(s) is N(s) = (−gx(x(s), y(s)), −gy(x(s), y(s)), 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let g(x, y) = ax2 − by2 + x3c(x, y) + x2yd(x, y) + xy2e(x, y) + y3f(x, y), (38) then gx(x, y) = 2ax + (3x2c + x3cx) + (2xyd + x2ydx) + (y2e + xy2ex) + y3fx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' gy(x, y) = −2by + x3cy + (x2d + x2ydy) + (2xye + xy2ey) + (3y2f + y3fy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' There exists a positive constant A such that gx(x, y) ≤ A and gy(x, y) ≤ A, if |x|, |y| ≤ ǫ, (39) where A → 0 as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let s be such that γ′′(s) exists, then γ′′(s) = z′′(s)N(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' This implies that x′′(s) = −z′′(s)gx(x(s), y(s)) ⇒ |x′′(s)| ≤ Az′′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 26 Here z′′(s) ≥ 0, because within the ǫ-ball of p the outward normal vector to S has a positive z-coordinate of 1 and γ′′(s) directs outward on a boundary segment on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Indeed γ(s) is a locally shortest path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) lies on the surface of M, its acceleration exists everywhere except at the switch points and it outward normal to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, if γ(s) lies on a line segment in the interior of M, then the acceleration is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So we obtain x′(s) = x′(0) + � s 0 x′′(σ)dσ ≥ cos θ − A � s 0 z′′(σ)dσ = cos θ − Az′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (40) Next let’s approximate z′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ∈ S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' then z(s) = g(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) = ax(s)2−by(s)2+x(s)3c+x(s)2y(s)d+x(s)y(s)2e+y(s)3f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' so z′(s) = 2ax(s)x′(s) − 2by(s)y′(s) + 3x(s)2x′(s)c(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) + x(s)3cx(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))x′(s) +x(s)3cy(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))y′(s) + 2x(s)x′(s)y(s)d(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) + x(s)2y′(s)d(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) +x(s)2y(s)dx(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))x′(s) + x(s)2y(s)dy(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))y′(s)x′(s)y(s)2e(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) +2x(s)y(s)y′(s)e(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) + x(s)y(s)2ex(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))x′(s) + x(s)y(s)2ey(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))y′(s) 3y(s)2y′(s)f(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s)) + y(s)3fx(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))x′(s) + y(s)3fy(x(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y(s))y′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since γ(s) is parametrized by arc length, |x′(s)| and |y′(s)| are no more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since each term in the above expression has either an x(s) or y(s), there exists a positive constant B such that |z′(s)| ≤ B, if |x(s)|, |y(s)| ≤ ǫ, where B → 0 as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, if γ(s) is within an interior line segment, then γ′(s) is constant and equal to the value at the endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore |z′(s)| is still bounded by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus we can choose ǫ small enough so that B < cos |θ| A , where |θ| ≤ θ0 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows from (40) that x′(s) > 0, if |x(s)|, |y(s)| ≤ ǫ for ǫ sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore x(s) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (3) Approximate y′(s) and y(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' This is the third estimate of ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ∈ S, then the normal vector to S at γ(s) is N(s) = (−gx(x(s), y(s)), −gy(x(s), y(s)), 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 27 Let s be such that γ′′(s) exists, then γ′′(s) = z′′(s)N(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' From (39) one obtains y′′(s) = −z′′(s)gy(x(s), y(s)) ⇒ |y′′(s)| ≤ Az′′(s), where A → 0 as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus y′(s) = y′(0) + � s 0 y′′(σ)dσ ≤ sin θ + A � s 0 z′′(σ)dσ = sin θ + Az′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let tan |θ| < c < tan θ0, where |θ| ≤ θ0 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Choose ǫ sufficiently small so that sin |θ| + Az′(s) ≤ c [cos θ − Az′(s)] , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' A ≤ c cos θ − sin |θ| 1 + c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore combining with (40) |y′(s)| ≤ cx′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (41) With y(0) = x(0) = 0, one has |y(s)| ≤ cx(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (42) (4) Concavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' This is the last estimate of ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Suppose γ(s) leaves S at a switch point when s = s0 and dives into the interior of M for increasing s until it enters S again at s = s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since x′(s) > 0 within the ǫ-ball of p, then x(s) has a C1-inverse function s(x) for s ∈ [0, s1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore we can express y(s) as y(s) = y(s(x)) = α(x), where α is a C1-function and α(0) = dα dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Consider the intersection of the two-dimensional plane y = y0 +T(x−x0) with the surface z = g(x, y), where (x0, y0) = (x(s0), y(s0)), and T = dα dx(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' With (41) |T| = ���� y′(s0) x′(s0) ���� ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Set f(x) = g(x, y0 + T(x − x0)), 28 then d2f dx2 (x) = gxx + 2gxyT + gyyT 2, where gxx = 2a + (6xc + 6x2cx + x3cxx) + (2yd + 4xydx + x2ydxx) + (2y2ex + xy2exx) + y3fxx = 2a + x(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=') + y(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=') gxy = (3x2cy + x3cxy) + (2xd + 2xydy + x2dx + x2ydxy) + (2ye + y2ey + 2xyex + xy2exy) + x3fxy = x(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=') + y(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=') gyy = −2b + x3cyy + (2x2dy + x2ydyy) + (2xe + 4xyey + xy2eyy) + (6yf + 6y2fy + y3fyy) = −2b + x(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=') + y(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=') So with y(s) = y0 + T(x − x0) for s ∈ [s0, s1] and (41), (42) d2f dx2(x) = 2a + x(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=') + y(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=') + 2Tx(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=') + 2Ty(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=') − 2bT 2 + T 2x(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' ) + T 2y(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' ), so d2f dx2 (x(s)) ≥ 2a − x(s)C1 − cx(s)C2 − 2cx(s)C3 − 2c2x(s)C4 − 2bc2 − c2x(s)C5 − c3x(s)C6, where C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' , C6 are constants bounding the terms inside the corresponding parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' By assumption c < tan θ0 = �a b, we can choose ǫ sufficiently small such that the right side of the above inequality is positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=', x(s)C1 + cx(s)C2 + 2cx(s)C3 + 2c2x(s)C4 + c2x(s)C5 + c3x(s)C6 < 2a − 2bc2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore d2f dx2(x(s)) > 0 for s ∈ [s0, s1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It implies that f ′(x(s)) is increasing as x(s) increases from x(s0) = x0 to x(s1) = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, since the interior line segment is tangent to S at the two endpoints, one must have f ′(x0) = f ′(x1), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore if γ leaves S at the switch point γ(s0), the geodesic arc beyond this point is a line segment never returning to S again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So γ has at most one switch point within the ǫ-ball of p, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In the positive direction of the line y = �a b, there exist an η > 0 and an ǫ > 0 such that if γ′(0) lies in the wedge [θ0 − η, θ0 + η], then γ has at most two switch points within the ǫ-ball of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (1) Set up the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let’s rotate the (x, y)-plane so that the x-axis points in the positive di- rection of the line y = �a b by the matrix �cos θ0 − sin θ0 sin θ0 cos θ0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus with respect to this new coordinate system and in connection with (38) the surface S near p can be parametrized by g(cos θ0x − sin θ0y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' sin θ0x + cos θ0y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='a(cos θ0x − sin θ0y)2 − b(sin θ0x + cos θ0y)2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='(cos θ0x − sin θ0y)3c + (cos θ0x − sin θ0y)2(sin θ0x + cos θ0y)d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='+(cos θ0x − sin θ0y)(sin θ0x + cos θ0y)2e + (sin θ0x + cos θ0y)3f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='abxy + (a − b)y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='+x3 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='(cos θ0)3c + (cos θ0)2 sin θ0d + cos θ0(sin θ0)2e + (sin θ0)3f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='+x2y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='3(cos θ0)2 sin θ0c + cos θ0(1 − 3(sin θ0)2)d + sin θ0(3(cos θ0)2 − 1)e + 3(sin θ0)2 cos θ0f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='+xy2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='−3 cos θ0(sin θ0)2c + ((cos θ0)3 − 2(cos θ0)2 sin θ0(d + e) + 3 sin θ0(cos θ0)2f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='+y3 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='−(sin θ0)3c + (sin θ0)2 cos θ0d − sin θ0(cos θ0)2e + (cos θ0)3f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='where c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' f are evaluated at (cos θ0x − sin θ0y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' sin θ0x + cos θ0y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' For con- venience, let’s still use g(x, y) to denote the new parametrization g(x, y) = −2 √ abxy + (a − b)y2 (43) +x3c(x, y) + x2yd(x, y) + xy2e(x, y) + y3f(x, y), where c, d, e, f are the (new) functions inside the corresponding brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (2) Notice that within the ǫ-ball of p, we have |x|, |y| ≤ ǫ no matter how we rotate the (x, y)-plane, because the distance to the origin is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So if γ(s) is within the ǫ-ball of p, then we have |γ(s)| ≤ ǫ ⇒ |x(s)| ≤ ǫ, |y(s)| ≤ ǫ, |z(s)| ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (3) Concavity with respect to the new frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If one moves slightly above the x-axis, namely in the first quadrant, then the surface is concave down- ward;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' if one moves slightly below the x-axis, namely in the fourth quadrant, then the surface is concave upward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 30 (4) If γ′(0) points in the positive x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' One can rewrite the g(x, y) again as follows: g(x, y) = xNh(x, y) + xyi(x, y) + y2j(x, y), (44) where N ≥ 3, h(0, 0) ̸= 0, i(0, 0) = −2 √ ab, and j(0, 0) = a−b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' According to the theorem for a fixed direction (cite here), if γ′(0) = ∂ ∂x then there exists an ǫ > 0 such that γ has at most one switch point before leaving the ǫ-ball of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (5) γ′(0) is in the wedge [−η, η] where 0 < η < min(2θ0, π − 2θ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' This is the first estimate of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let δ ∈ [−η, η] and δ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So γ′(0) does NOT point in the positive x- direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If we rotate the (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y)-plane according to the angle δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' the Taylor expansion of g(cos δx − sin δy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' sin δx + cos δy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' denoted as gδ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' has a nonzero x2 term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' because with (43) gδ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ab(cos δx − sin δy)(sin δx + cos δy) + (a − b)(sin δx + cos δy)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='(45) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='+(cos δx − sin δy)3c + (cos δx − sin δy)2(sin δx + cos δy)d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='+(cos δx − sin δy)(sin δx + cos δy)2e + (sin δx + cos δy)3f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bcos2(θ0 + δ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='cos2 θ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='− b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='+ xy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ab cos 2δ + (a − b) sin 2δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='+y2 �√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ab sin 2δ + (a − b) cos2 δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='+x3 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='(cos δ)3c + (cos δ)2 sin δd + cos δ(sin δ)2e + (sin δ)3f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='+x2y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='3(cos δ)2 sin δc + cos δ(1 − 3(sin δ)2)d + sin δ(3(cos δ)2 − 1)e + 3(sin δ)2 cos δf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='+xy2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='−3 cos δ(sin δ)2c + ((cos δ)3 − 2(cos δ)2 sin δ(d + e) + 3 sin δ(cos δ)2f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='+y3 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='−(sin δ)3c + (sin δ)2 cos δd − sin δ(cos δ)2e + (cos δ)3f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='where c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' f are evaluated at (cos δx − sin δy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' sin δx + cos δy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Notice that if the coefficient of x2 is zero, then bcos2(θ0 + δ) cos2 θ0 − b = 0 ⇒ cos(θ0 + δ) = ± cos θ0 ⇒ δ = 0, −2θ0, or π − 2θ0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore we could rewrite gδ(x, y) to include the angle δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' gδ(x, y) = a2(δ)x2 + a3(δ)x3 + · · · + aN−1(δ)xN−1 (46) +xNhδ(x, y) + xyiδ(x, y) + y2jδ(x, y), 31 where a2(δ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' , aN−1(δ) are constant coefficients of x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' , xN−1, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Moreover at δ = 0, these constants vanish, and h0(x, y) = h(x, y), i0(x, y) = i(x, y), j0(x, y) = j(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (6) There exists an ǫ that works for all δ ∈ [−η, η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' This is the second estimate of η (in step 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' There are four different cases depending on the sign of a2(δ) and h(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Case 1: a2(δ) > 0, h(0, 0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When a2(δ) > 0, the angle δ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' There is actually a relationship between a2(δ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' , aN−1(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' a2(δ) = b cos2 θ0 [sin2(θ0) − sin2(θ0 + δ)] = b cos2 θ0 [sin(θ0) + sin(θ0 + δ)][sin(θ0) − sin(θ0 + δ)] = b cos2 θ0 [sin(θ0) + sin(θ0 + δ)][sin(θ0) − sin(θ0) cos δ − cos(θ0) sin δ] = b cos2 θ0 [sin(θ0) + sin(θ0 + δ)][− cos(θ0) sin δ + sin(θ0)(1 − cos δ)] ≥ b cos2 θ0 sin(θ0)[− cos(θ0) sin δ] = b tan(θ0)| sin δ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Furthermore, there are constants ci and M sufficiently large such that a3(δ) = c1 cos2 δ sin δ + c2 cos δ sin2 δ + c3 sin3 δ a4(δ) = c4 cos3 δ sin δ + c5 cos2 δ sin2 δ + c6 cos δ sin3 δ + c7 sin4 δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' aN−1(δ) = c8 cosN−2 δ sin δ + · · · + c9 sinN−1 δ ⇒ |a3(δ)|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' , |aN−1(δ)| ≤ M| sin δ| (i) Let’s continue with the ǫ in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then x′(s) ≥ 1 2 if ǫ is chosen small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Here we are assuming that if γ(s) is within the ǫ-ball of p, then for every 0 ≤ σ ≤ s, γ(σ) also lies within the ǫ-ball of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (ii) Let γ(s) = (x(s), y(s), z(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ∈ S, then the normal vector to S at γ(s) is N(s) = (−(gδ)x(x(s), y(s)), −(gδ)y(x(s), y(s)), 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 32 From (46) it follows that (gδ)x(x, y) = 2a2(δ)x + · · · + (N − 1)aN−1(δ)xN−2 + NxN−1hδ + xN(hδ)x +yiδ + xy(iδ)x + y2(jδ)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' There exists a positive constant A such that |(gδ)x(x, y)| ≤ A if |x|, |y| ≤ ǫ and δ ∈ [−η, η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Moreover A → 0 as ǫ → 0 for a fixed η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let s be such that γ′′(s) exists, then γ′′(s) = z′′(s)N(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' This implies that x′′(s) = −z′′(s)(gδ)x(x(s), y(s)) ⇒ |x′′(s)| ≤ Az′′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Here z′′(s) ≥ 0, because within the ǫ-ball of p the outward normal vector to S has a positive z-coordinate of 1 and γ′′(s) directs outward on a boundary segment on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Indeed γ(s) is a locally shortest path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) lies on the surface of M, its acceleration exists everywhere except at the switch points and is outward normal to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, if γ(s) lies on a line segment in the interior of M, then the acceleration γ′′(s) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So the previous inequality still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus x′(s) = x′(0) + � s 0 x′′(σ)dσ ≥ 1 − A � s 0 z′′(σ)dσ = 1 − Az′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (47) Next let’s approximate z′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ∈ S, using (46) one has z′(s) = 2a2(δ)x(s)x′(s) + · · · + (N − 1)aN−1(δ)x(s)N−2x′(s) +NxN−1x′(s)hδ + xN[(hδ)xx′(s) + (hδ)yy′(s)] +x′(s)y(s)iδ + x(s)y′(s)iδ + x(s)y(s)[(iδ)xx′(s) + (iδ)yy′(s)] +2y(s)y′(s)jδ + y(s)2[(jδ)xx′(s) + (jδ)yy′(s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since γ(s) is parametrized by arc length, |x′(s)| and |y′(s)| are no more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So there exists a positive constant B such that |z′(s)| ≤ B if |x(s)|, |y(s)| ≤ ǫ and δ ∈ [−η, η], where B → 0 as ǫ → 0 for a fixed η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, if γ(s) is within an interior line segment, then γ′(s) is constant and equal to the value at the 33 endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore |z′(s)| is still bounded by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus we can choose ǫ small enough so that B ≤ 1 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows from (47) that x′(s) ≥ 1 2, (48) if ǫ is chosen sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Notice that this ǫ works for all δ because we can bound hδ, iδ, jδ and their partial derivatives uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (iii) Approximate z(s), z′(s), y(s), y′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ∈ S, then y′′(s) = −z′′(s)(gδ)y(x(s), y(s)) ⇒ |y′′(s)| ≤ Az′′(s), where A → 0 if ǫ → 0 and A does not depend on δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ̸∈ S, then γ(s) is within an interior line segment, so the inequality still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' With respect to the new frame after rotating the (x, y)-plane by δ, γ′(0) = ∂ ∂x and so with z(0) = z′(0) = y(0) = y′(0) = 0 one deduces that |y′(s)| ≤ Az′(s), |y(s)| ≤ Az(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (49) Now let’s estimate z(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ∈ S, with (46) and (49) z(s) ≤ a2(δ)x(s)2 + |a3(δ)|x(s)3 + · · · + |aN−1(δ)|x(s)N−1 +x(s)N|hδ(x(s), y(s))| + x(s)Az(s)|iδ(x(s), y(s))| + A2z(s)2|jδ(x(s), y(s))| First, |a3(δ)|x(s) + · · · + |aN−1(δ)|x(s)N−3 ≤ M| sin δ|[x(s) + x(s)2 + · · · + x(s)N−3] ≤ b tan(θ0)| sin δ| ≤ a2(δ), if we choose ǫ sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Second, |hδ(x(s), y(s))| ≤ 2h(0, 0), if η and ǫ are sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Third, x(s)Az(s)|iδ(x(s), y(s))|+A2z(s)2|jδ(x(s), y(s))| ≤ x(s)Az(s)C1+A2z(s)2C2, where C1, C2 are some constants and we can choose ǫ small enough so that Ax(s)C1 + A2z(s)C2 ≤ C3 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 34 Therefore there exists a positive constant C such that z(s) ≤ (2a2(δ)x(s)2 + 2h(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0)x(s)N)C (50) ⇒ |y(s)| ≤ (2a2(δ)x(s)2 + 2h(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0)x(s)N)AC On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' z′(s) ≤ 2a2(δ)x(s) + · · · + (N − 1)|aN−1(δ)|x(s)N−2 +Nx(s)N−1|hδ| + x(s)N[|(hδ)x| + |(hδ)y|] +|y(s)||iδ| + x(s)Az′(s)|iδ| + x(s)|y(s)|[|(iδ)x| + |(iδ)y|] +2y(s)Az′(s)|jδ| + |y(s)|2[|(jδ)x| + |(jδ)y|] ≤ 3a2(δ)x(s) + 2Nx(s)N−1h(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0) + (2a2(δ)x(s)2 + 2h(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0)x(s)N)ACC4 + C3z′(s) ≤ 4a2(δ)x(s) + 4Nx(s)N−1h(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0) + C3z′(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' if η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' ǫ are sufficiently small and C3 < 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' C4 are some constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So there is a constant C such that z′(s) ≤ (4a2(δ)x(s) + 4Nx(s)N−1h(0, 0))C (51) ⇒ y′(s) ≤ (4a2(δ)x(s) + 4Nx(s)N−1h(0, 0))AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that if y = α(x) and T(s) = α′(x(s)), then |T(s)| = ���� y′(s) x′(s) ���� ≤ 2|y′(s)| ≤ (8a2(δ)x(s) + 8Nx(s)N−1h(0, 0))AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ̸∈ S, then γ(s) is in some line segment where y(s) = y(s0) + T(s0)(x(s) − x(s0)) for some switch point at s0 < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since x(s) is increasing, it follows that |y(s)| ≤ |y(s0)| + |T(s0)|[|x(s)| + |x(s0)|] ≤ (2a2(δ)x(s0)2 + 2h(0, 0)x(s0)N)AC +2(4a2(δ)x(s0) + 4Nx(s0)N−1h(0, 0))AC[|x(s)| + |x(s0)|] ≤ (2a2(δ)x(s)2 + 2h(0, 0)x(s)N)AC + 2(4a2(δ)x(s) + 4Nx(s)N−1h(0, 0))AC · 2x(s) ≤ (18a2(δ)x(s)2 + 18Nh(0, 0)x(s)N)AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The same inequality for T(s) still holds as before because T(s) = T(s0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (iv) Now we are ready to show that there is at most one switch point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Set f(x) = g(x, y0 + T(x − x0)) where (x0, y0) = (x(s0), y(s0)), T = T(s0), and s0 is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then f ′′(x0) = gxx(x0, y0) + 2gxy(x0, y0)T + gyy(x0, y0)T 2, 35 where gxx(x0, y0) = 2a2(δ) + 6a3(δ)x0 + · · · + (N − 1)(N − 2)aN−1(δ)xN−3 0 +xN−2 0 [N(N − 1)hδ + 2Nx0(hδ)x + x2 0(hδ)xx] +y0[2(iδ)x + x0(iδ)xx + y0(jδ)xx].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' First, for ǫ sufficiently small, |6a3(δ)x0 + · · · + (N − 1)(N − 2)aN−1(δ)xN−3 0 | ≤ M| sin δ|(6x0 + · · · + (N − 1)(N − 2)xN−3 0 ) ≤ b tan(θ0)| sin δ| ≤ a2(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Second, for η and ǫ sufficiently small, N(N − 1)hδ + 2Nx0(hδ)x + x2 0(hδ)xx ≥ 1 2N(N − 1)h(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Third, there are constants C5, C6, C7 such that |2(iδ)x + x0(iδ)xx + y0(jδ)xx| ≤ C5, |2gxy(x0, y0)| ≤ C6, |gyy(x0, y0)T| ≤ C7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So f ′′(x0) ≥ 2a2(δ) − a2(δ) + xN−2 0 1 2N(N − 1)h(0, 0) −(18a2(δ)x2 0 + 18Nh(0, 0)xN 0 )ACC5 − (8a2(δ)x0 + 8NxN−1 0 h(0, 0))ACC6 −(8a2(δ)x0 + 8NxN−1 0 h(0, 0))ACC7 = a2(δ)[1 − 18ACC5x2 0 − 8ACC6x0 − 8ACC7x0] +xN−2 0 h(0, 0)[1 2N(N − 1) − 18ACC5Nx2 0 − 8ACC6Nx0 − 8ACC7Nx0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore if ǫ is sufficiently small, for all |x0| ≤ ǫ, one has 1 − 18ACC5x2 0 − 8ACC6x0 − 8ACC7x0 > 0, 1 2N(N − 1) − 18ACC5Nx2 0 − 8ACC6Nx0 − 8ACC7Nx0 > 0, implying that γ can’t have a line segment within the ǫ-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus in the case when a2(δ) > 0 and h(0, 0) > 0, γ has at most one switch point within the ǫ-ball for any δ ∈ [−η, η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 36 Case 2: a2(δ) < 0, h(0, 0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Suppose for the sake of contradiction, γ′(0) is in the direction of ∂ ∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then we can approximate y(s), T(s) as in the paper (cite here) to show that γ has no switch point close to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that γ has to be on the surface initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Given any two points γ(s1), γ(s2) on the geodesic close to the origin, one can show that the line segment connecting them is actually in the interior of M contradicting that γ is locally shortest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let (x1, y1) = (x(s1), y(s1)) and (x2, y2) = (x(s2), y(s2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then for t ∈ [0, 1], set f(t) = gδ(x1 + t(x2 − x1), y1 + t(y2 − y1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So f ′′(t) = (gδ)xx(x2 − x1)2 + 2(gδ)xy(x2 − x1)(y2 − y1) + (gδ)yy(y2 − y1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' By the mean value theorem, x2 − x1 = (s2 − s1)x′(˜s), y2 − y1 = (s2 − s1)y′(ˆs), for some ˜s, ˆs in (s1, s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let T = y2 − y1 x2 − x1 = y′(ˆs) x′(˜s) → 0 as s1, s2 → 0, because γ′(0) = (1, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore f ′′(t) = (x2 − x1)2[(gδ)xx + 2(gδ)xyT + (gδ)yyT 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since gxx(x, y) → 2a2(δ) as x, y → 0, then f ′′(t) → (x2 − x1)2[2a2(δ)] as s1, s2 → 0 and for every t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus the shortest path two points on γ close to the orgin is the line segment in between which lies stricly below the surface, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So γ is a straight line initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The surface in the (x, z)-plane is the curve with equation z(x) = gδ(x, 0) = a2(δ)x2 + · · · + aN−1(δ)xN−1 + xNhδ(x, 0), which implies that z′(0) = 0, z′′(0) = 2a2(δ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 37 So the slope of the line segment must be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If the line segment re- enters the surface at some switch point, then the surface can’t be concave downward there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Otherwise the line lies above the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' z′(x) = 2a2(δ)x + · · · + (N − 1)aN−1(δ)xN−2 + NxN−1hδ(x, 0) + xN(hδ)x(x, 0) z′′(x) = 2a2(δ) + 6a3(δ)x + · · · + (N − 1)(N − 2)aN−1(δ)xN−3 +N(N − 1)xN−2hδ(x, 0) + 2NxN−1(hδ)x(x, 0) + xN(hδ)xx(x, 0) When a2(δ) < 0, the angle δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' As before, one has −a2(δ) = b cos2 θ0 [sin(θ0) + sin(θ0 + δ)][sin(θ0 + δ) − sin(θ0)] = b cos2 θ0 [sin(θ0) + sin(θ0 + δ)][δ cos(θ)], where θ ∈ (θ0, θ0 + δ) by the mean value theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' For η sufficiently small, we have δ = δ sin δ sin δ ≥ 1 2 sin δ, since δ sin δ → 1 as δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus −a2(δ) ≥ b cos2 θ0 sin(θ0)1 2 sin δ cos(θ0 + η) ≥ 1 4b tan(θ0) sin δ, if η is sufficiently close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Furthermore, there is M sufficiently large such that |a3(δ)|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' , |aN−1(δ)| ≤ M sin δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that |6a3(δ)x + · · · + (N − 1)(N − 2)aN−1(δ)xN−3| ≤ M sin δ(6x + · · · + (N − 1)(N − 2)xN−3) ≤ 1 4b tan(θ0) sin δ ≤ −a2(δ), for all |x| ≤ ǫ if ǫ is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Moreover, if η and ǫ are sufficiently small, then N(N − 1)hδ(x, 0) + 2Nx(hδ)x(x, 0) + x2(hδ)xx(x, 0) ≤ 1 2N(N − 1)h(0, 0), 38 Therefore z′′(x) ≤ 2a2(δ) − a2(δ) + 1 2N(N − 1)h(0, 0)xN−2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So γ has no switch point unless it terminates at a point on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' As a summary in the case when a2(δ) < 0 and h(0, 0) < 0, γ is either a straight line exiting the ǫ-ball or a line segment terminating at some point on the surface within the ǫ-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Case 3: a2(δ) < 0, h(0, 0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since a2(δ) < 0, γ is initially a straight line just as shown in Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Suppose the angle between γ′(0) and the positive x-axis is −β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The line either terminates at some point on the surface, or exits the ǫ-ball, or enters the surface at some switch point at time s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Denote x(s0) as x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' First, the intersection of the line with the surface at γ(s0) satisfies − tan(β)x0 = gδ(x0, 0) and so − tan(β)x0 = a2(δ)x2 0 + a3(δ)x3 0 + · · · + aN−1(δ)xN−1 0 + xN 0 hδ(x0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Next, the line is tangent to the surface at γ(s0), so − tan(β) = (gδ)x(x0, 0) and − tan(β) = 2a2(δ)x0+3a3(δ)x2 0+· · ·+(N−1)aN−1(δ)xN−2 0 +NxN−1 0 hδ(x0, 0)+xN 0 (hδ)x(x0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since x0 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' the above two equalities imply the following: a2(δ)x0 + a3(δ)x2 0 + · · · + aN−1(δ)xN−2 0 + xN−1 0 hδ(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0) = 2a2(δ)x0 + 3a3(δ)x2 0 + · · · + (N − 1)aN−1(δ)xN−2 0 + NxN−1 0 hδ(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0) + xN 0 (hδ)x(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0) ⇒ a2(δ) + a3(δ)x0 + · · · + aN−1(δ)xN−3 0 + xN−2 0 hδ(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0) = 2a2(δ) + 3a3(δ)x0 + · · · + (N − 1)aN−1(δ)xN−3 0 + NxN−2 0 hδ(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0) + xN−1 0 (hδ)x(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0) ⇒ a2(δ) + 2a3(δ)x0 + · · · + (N − 2)aN−1(δ)xN−3 0 + (N − 1)xN−2 0 hδ(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0) + xN−1 0 (hδ)x(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let 0 < c < 1 be a constant to be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then for ǫ sufficiently small, |2a3(δ)x0 + · · · + (N − 2)aN−1(δ)xN−3 0 | ≤ M sin δ[2x0 + · · · + (N − 2)xN−3 0 ] ≤ c1 4b tan(θ0) sin δ ≤ −ca2(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 39 Therefore −a2(δ) − 2a3(δ)x0 − · · · − (N − 2)aN−1(δ)xN−3 0 ≥ −a2(δ) + ca2(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, for η and ǫ sufficiently small, we have |(N − 1)xN−2 0 hδ(x0, 0) + xN−1 0 (hδ)x(x0, 0)| ≤ (N − 1)xN−2 0 (1 + c)h(0, 0) Thus (1 − c)|a2(δ)| ≤ (N − 1)xN−2 0 (1 + c)h(0, 0) (52) ⇒ xN−2 0 ≥ (1 − c)|a2(δ)| (N − 1)(1 + c)h(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Now we are going to first shift our coordinates to have the origin at γ(s0) = (x0, 0, z0 = − tan(β)x0) and then rotate the (x, z)-plane so that the γ′(s0) points in the positive x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let’s use (u, v, w) for the new coordinates, then with respect to the new frame x = cos βu + sin βw + x0, y = v, z = − sin βu + cos βw + z0, so the surface z = gδ(x, y) satisfies the equation − sin βu + cos βw + z0 = gδ(cos βu + sin βw + x0, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Check that we can still solve for w analytically in terms of u, v within the ǫ-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Taking the partial derivative of sin βu − cos βw − z0 + gδ(cos βu + sin βw + x0, v) with respect to w yields − cos β + (gδ)x sin β = cos β[(gδ)x tan β − 1] = cos β[−(gδ)x(gδ)x(x0, 0) − 1], where − tan β = (gδ)x(x0, 0) from before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since (gδ)x(x, y) → 0 as x, y → 0, for ǫ sufficiently small, −(gδ)x(gδ)x(x0, 0) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore there exists a real analytic function kδ such that w = kδ(u, v) with kδ(0, 0) = 0, (kδ)u(0, 0) = 0, (kδ)v(0, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 40 Estimate γ(s) in the new frame starting from the point (x0, 0, z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' After replacing s by s − s0, γ′(0) is equal to ∂ ∂u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' u′(s) ≥ 1 2 if ǫ is chosen small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' By triangular inequality, |u(s)|, |v(s)|, |w(s)| are less than or equal to 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ∈ S, then the normal vector to S at γ(s) is N(s) = (−(kδ)u(u(s), v(s)), −(kδ)v(u(s), v(s)), 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since the lowest degree in kδ is at least two, there exists a positive constant A such that |(kδ)u(u, v)| ≤ A, |(kδ)v(u, v)| ≤ A, if |u|, |v| ≤ 2ǫ, where A → 0 as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let s be such that γ′′(s) exists, then γ′′(s) = w′′(s)N(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' This implies that u′′(s) = −w′′(s)(kδ)u(u(s), v(s)) ⇒ |u′′(s)| ≤ Aw′′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Here w′′(s) ≥ 0 because within the ǫ-ball of p the surface S has the parametriza- tion w = kδ(u, v) and thus the outward normal vector to S has a positive w-coordinate of 1 and γ′′(s) is outward normal on a boundary segment in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ̸∈ S, γ′′(s) = 0 except at the switch points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus u′(s) = u′(0) + � s 0 u′′(σ)dσ ≥ 1 − Aw′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Next approximate w′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ∈ S, then w(s) = kδ(u(s), v(s)) = u(s)2a(u(s), v(s))+u(s)v(s)b(u(s), v(s))+v(s)2c(u(s), v(s)), for some analytic functions a, b, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since γ is parametrized by arclength, |u′(s)| ≤ 1 and |v′(s)| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since each term in w′(s) has either u(s) or v(s) and u′(s) or v′(s), there exists a positive constant B such that |w′(s)| ≤ B, if |u(s)|, |v(s)| ≤ 2ǫ, where B → 0 as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, if γ(s) ̸∈ S, γ′(s) is contant and equal to the value at the endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore |w′(s)| is still bounded by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus one can choose ǫ small enough so that B < 1 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that u′(s) ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 41 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Approximate v′(s) and v(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ∈ S, then the normal vector to S at γ(s) is N(s) = (−(kδ)u(u(s), v(s)), −(kδ)v(u(s), v(s)), 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let s be such that γ′′(s) exists, then γ′′(s) = w′′(s)N(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' This implies that v′′(s) = −w′′(s)(kδ)v(u(s), v(s)) ⇒ |v′′(s)| ≤ Aw′′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus with v′(0) = w′(0) = 0, |v′(s)| ≤ � s 0 |v′′(σ)|dσ ≤ A � s 0 w′′(σ)dσ = Aw′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' With v(0) = w(0) = 0, |v(s)| ≤ Aw(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Coefficients of kδ(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Denote kδ(u, v) as kδ(u, v) = b2(δ)u2 + · · ·+ bN−1(δ)uN−1 + uNlδ(u, v) + uvmδ(u, v) + v2nδ(u, v), where b2(δ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' , bN−1(δ) are constants and lδ, mδ, nδ are analytic functions of u, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Observe that for n between 2 and N − 1, n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bn(δ) = ∂nkδ ∂un (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The following lemma finds ∂nkδ ∂un (u, v) for n ≥ 2 by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let A be cos β + sin β(kδ)u(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then for each n ≥ 2, cos β∂nkδ ∂un (u, v) = n−1 � p=0 ∂n−pgδ ∂xn−p � I cIAn−p−|I|(∂A ∂u )i1(∂2A ∂u2 )i2 · · · (∂pA ∂up )ip, (53) where I = (i1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' , ip), i1 + 2i2 + · · · + pip = p, |I| = i1 + i2 + · · · + ip ≤ n − p, cI ≥ 0, and the partial derivatives of gδ are evaluated at (cos βu + sin βkδ(u, v) + x0, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When n = 2, differentiating the equation − sin βu + cos βkδ(u, v) + z0 = gδ(cos βu + sin βkδ(u, v) + x0, v) 42 once with respect to u gives − sin β + cos β(kδ)u = (gδ)x[cos β + sin β(kδ)u] = (gδ)xA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then taking the partial derivative with respect to u once more gives cos β(kδ)uu = (gδ)xxA2 + (gδ)x∂uA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In (53) when p = 0, there is no I so we have c0A2−0−0 = c0A2 where c0 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' when p = 1, there is only one I = (1) so we have c1A2−1−1( ∂A ∂u )1 = c1∂uA where c1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' This coincides with the expression above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When n ≥ 2, by inductive hypothesis we take the partial derivative of (53) with respect to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The left-hand side is cos β∂n+1 u kδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The right-hand side consists of three parts due to the product rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (1) n−1 � p=0 ∂n+1−pgδ ∂xn+1−p � I cIAn+1−p−|I|(∂A ∂u )i1(∂2A ∂u2 )i2 · · · (∂pA ∂up )ip, where n becomes n + 1 and p stays the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (2) n−1 � p=0 ∂n−pgδ ∂xn−p � I cI(n − p − |I|)An−p−|I|−1(∂A ∂u )i1+1(∂2A ∂u2 )i2 · · · (∂pA ∂up )ip, where n, p, i1 become n + 1, p + 1, i1 + 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' By letting ip+1 be zero, one again has i1+1+2i2+· · ·+pip+(p+1)ip+1 = p+1, n−p−|I|−1 = (n+1)−(p+1)−(|I|+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (3) n−1 � p=0 ∂n−pgδ ∂xn−p � I cIAn−p−|I| � ij̸=0 (∂A ∂u )i1 · · · ij(∂jA ∂uj )ij−1(∂j+1A ∂uj+1 )ij+1 · · ·(∂pA ∂up )ip, where n, p become n + 1, p + 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When j < p, ij and ij+1 are replaced by ij + 1 and ij+1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' By letting ip+1 = 0 one has · ·+j(ij−1)+(j+1)(ij+1+1)+· · ·+(p+1)ip+1 = p+1, · · ·+(ij−1)+(ij+1+1)+· · ·+ip+1 = |I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, when j = p, ip = ip+1 = 1 and so p(ip − 1) + (p + 1)ip+1 = p + 1, (ip − 1) + ip+1 = 1 = |I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that cI are nonnegative integers and (53) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 43 Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The coefficient of (gδ)x∂n−1 u A in (53) is always 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When n = 2, we’ve shown in the above lemma that the coefficient of (gδ)x∂uA is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When n ≥ 2, if p = n − 1 then i1 + 2i2 + · · · + (n − 1)in−1 = n − 1 and i1 + i2 + · · · + in−1 ≤ 1 imply that i1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' = in−2 = 0 and in−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' There is only one term of (gδ)x∂n−1 u A whose coefficient is 1 by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Taking its derivative with respect to u yields (gδ)xxA∂n−1 u A + (gδ)x∂n uA, so the coefficient of (gδ)x∂n uA is still 1 completing the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The coefficient of ∂ngδ ∂xn An in (53) is always 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When n = 2, we’ve shown in the above lemma that the coefficient of (gδ)xxA2 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When n ≥ 2, if p = 0 there is no I since |I| = 0 and in (53) we have only one term ∂ngδ ∂xn An whose coefficient is 1 by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Taking its derivative with respect to u yields ∂n+1gδ ∂xn+1 An+1 + ∂ngδ ∂xn nAn−1∂A ∂u , so the coefficient of ∂n+1gδ ∂xn+1 An+1 is still 1 completing the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since A = cos β +sin β(kδ)u(u, v) and (kδ)u(0, 0) = 0, A(0, 0) = cos β and for 1 ≤ p ≤ N − 2 ∂pA ∂up (0, 0) = sin β∂p+1kδ ∂up+1 (0, 0) = sin β(p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bp+1(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that cos βn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bn(δ) = n−1 � p=0 ∂n−pgδ ∂xn−p (x0, 0) � I cI(cos β)n−p−|I|(sin β)|I|b2(δ)i1b3(δ)i2 · · · bp+1(δ)ip 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i13!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ip, 44 for 2 ≤ n ≤ N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Furthermore the Corollary (1) says that the term corresponding to p = n − 1 in the above expression is (gδ)x(x0, 0) sin βn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bn(δ) = − tan β sin βn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bn(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So moving it to the other side yields (cos β + tan β sin β)n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bn(δ) = n−2 � p=0 ∂n−pgδ ∂xn−p (x0, 0) � I cI(cos β)n−p−|I|(sin β)|I|b2(δ)i1b3(δ)i2 · · ·bp+1(δ)ip 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i13!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ip ⇒ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bn(δ) = n−2 � p=0 ∂n−pgδ ∂xn−p (x0, 0) � I cI(cos β)n+1−p−|I|(sin β)|I|b2(δ)i1b3(δ)i2 · · · bp+1(δ)ip 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i13!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ip, using cos β + tan β sin β = sec β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So bn(δ) depends on the previous constants for 3 ≤ n ≤ N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' bn(δ) > 0 for n between 2 and N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Before proceeding with the proof, we need to first estimate ∂pgδ ∂xp (x0, 0) for 2 ≤ p ≤ N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' By induction one can show that ∂pgδ ∂xp (x0, 0) = p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ap(δ) + · · · + (N − 1)(N − 2) · · ·(N − p)aN−1(δ)xN−1−p 0 + p � q=0 � p q � N(N − 1) · · ·(N − q + 1)xN−q 0 ∂p−q x hδ(x0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the one hand, if ǫ is sufficiently small |p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ap(δ) + · · · + (N − 1)(N − 2) · · ·(N − p)aN−1(δ)xN−1−p 0 | ≤ M sin δ[p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' + · · · + (N − 1)(N − 2) · · ·(N − p)xN−1−p 0 ] ≤ c1 4b tan(θ0) sin δ ≤ c|a2(δ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, if η and ǫ are sufficiently small p � q=0 � p q � N(N − 1) · · ·(N − q + 1)xN−q 0 ∂p−q x hδ(x0, 0) ≥ N(N − 1) · · ·(N − p + 1)xN−p 0 h(0, 0)(1 − c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 45 Combining the two inequalities, together with (52), yields ∂pgδ ∂xp (x0, 0) ≥ N(N − 1) · · · (N − p + 1)xN−p 0 h(0, 0)(1 − c) − c|a2(δ)| ≥ N(N − 1)(1 − c)2h(0, 0)|a2(δ)| (N − 1)(1 + c)h(0, 0) − c|a2(δ)| = �N(1 − c)2 1 + c − c � |a2(δ)|, which is positive if we choose c as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' N > c(1 + c) (1 − c)2 ⇒ 2 > c(1 + c) (1 − c)2 ⇒ 0 < c < 5 − √ 17 2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Denote the constant in the brackets as L = L(c, N), then for 2 ≤ p ≤ N − 1 ∂pgδ ∂xp (x0, 0) ≥ L|a2(δ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let’s determine the signs of bn(δ) for 2 ≤ n ≤ N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When n = 2, 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='b2(δ) = cos3 β(gδ)xx(x0, 0) ≥ cos3 βL|a2(δ)| > 0 ⇒ b2(δ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When n ≥ 3, by induction n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bn(δ) ≥ n−2 � p=0 L|a2(δ)| � I cI(cos β)n+1−p−|I|(sin β)|I|b2(δ)i1b3(δ)i2 · · · bp+1(δ)ip 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i13!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ip > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Indeed, when p = 0, the corresponding term in the above sum, together with Corollary 2, is ∂ngδ ∂xn (x0, 0) cosn+1 β ≥ L|a2(δ)| cosn+1 β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So bn(δ) > 0, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The following lemma shows that the sign of lδ(0, 0) is also positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Fur- thermore, it gives a lower bound of lδ(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' lδ(0, 0) ≥ 1 − c 2N+1 h(0, 0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 46 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' By Lemma 2, cos β∂Nkδ ∂uN (u, v) = N−1 � p=0 ∂N−pgδ ∂xN−p � I cIAN−p−|I|(∂A ∂u )i1(∂2A ∂u2 )i2 · · ·(∂pA ∂up )ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When p = N − 1, Corollary 1 suggests that we have (gδ)x ∂N−1A ∂xN−1 (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Evaluating at (u, v) = (0, 0) gives us cos βN!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='lδ(0, 0) − (gδ)x(x0, y0) sin βN!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='lδ(0, 0) = N−2 � p=0 ∂N−pgδ ∂xN−p (x0, 0) � I cI(cos β)N−p−|I|(sin β)|I|b2(δ)i1b3(δ)i2 · · · bp+1(δ)ip 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i13!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since (gδ)x(x0, y0) = − tan β, N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='lδ(0, 0) = N−2 � p=0 ∂N−pgδ ∂xN−p (x0, 0) � I cI(cos β)N+1−p−|I|(sin β)|I|b2(δ)i1b3(δ)i2 · · · bp+1(δ)ip 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i13!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When p = 0, the corresponding term in the above summation by Corollary 2 is ∂Ngδ ∂xN (x0, 0) cosN+1 β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='lδ(0, 0) = ∂Ngδ ∂xN (x0, 0) cosN+1 β + N−2 � p=1 ∂N−pgδ ∂xN−p (x0, 0) � I cI(cos β)N+1−p−|I|(sin β)|I|b2(δ)i1b3(δ)i2 · · · bp+1(δ)ip 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i13!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' where the second term is positive by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Moreover, if η and ǫ are sufficiently small ∂Ngδ ∂xN (x0, 0) = N � q=0 � N q � N(N − 1) · · ·(N − q + 1)xN−q 0 ∂q xhδ(x0, 0) ≥ N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='h(0, 0)(1 − c) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 47 Therefore N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='lδ(0, 0) ≥ N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='h(0, 0)(1 − c) cosN+1 β ⇒ lδ(0, 0) ≥ h(0, 0)(1 − c) cosN+1 β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since − tan β = (gδ)x(x0, 0) and gx(0, 0) = 0, it follows that β → 0, as δ, x0 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore for η and ǫ sufficiently small, one can have cos β ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So lδ(0, 0) ≥ 1 − c 2N+1 h(0, 0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Approximate v(s) and v′(s) using the normal vector N(s) to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' We denote γ(s) = (u(s), v(s), w(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ∈ S, the normal vector to S at γ(s) is N(s) = (−(kδ)u(v(s), v(s), −(kδ)v(u(s), v(s), 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since u′(s) ≥ 1 2, u(s) has a C1-inverse function s(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore we can express v(s) as v(s) = v(s(u)) = α(u), where α is a C1-function and α(0) = dα du(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then one has v(s) = o(u(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Hence (kδ)u(u(s), v(s)) = u(s)V1(s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (kδ)v(u(s), v(s)) = u(s)V2(s), where V1(s), V2(s) are bounded by some constant A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let γ′′(s) exist, then γ′′(s) = w′′(s)N(s), so u′′(s) = −w′′(s)u(s)V1(s), v′′(s) = −w′′(s)u(s)V2(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When γ(s) does not touch the surface, the equalities still hold since γ′′(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' With w(0) = w′(0) = v(0) = v′(0) = 0 one can approximate |v′(s)| ≤ � s 0 |v′′(σ)|dσ ≤ Au(s)w′(s) ⇒ |v(s)| ≤ Au(s)w(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 48 If γ(s) ∈ S, then w(s) = kδ(u(s), v(s)) = b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)Nlδ + v(s)[u(s)mδ + v(s)nδ] ≤ b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N|lδ| + Au(s)w(s)C1, where C1 → 0 as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Furthermore one can choose ǫ so small that |lδ(u(s), v(s))−lδ(0, 0)| ≤ ch(0, 0)(1−c) ≤ clδ(0, 0) ⇒ |lδ(u(s), v(s))| ≤ (1+c)lδ(0, 0), by uniform continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore there is a constant B such that w(s) ≤ B[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So v(s) ≤ Au(s)B[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Next let’s pproximate v′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ∈ S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' differentiating w(s) = kδ(u(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' v(s)) gives w′(s) = 2b2(δ)u(s)u′(s) + · · · + (N − 1)bN−1(δ)u(s)N−2u′(s) +Nu(s)N−1u′(s)lδ + u(s)N[(lδ)uu′(s) + (lδ)vv′(s)] +u′(s)v(s)mδ + u(s)v′(s)mδ + u(s)v(s)[(mδ)uu′(s) + (mδ)vv′(s)] +2v(s)v′(s)nδ + v(s)2[(nδ)uu′(s) + (nδ)vv′(s)] ≤ 2b2(δ)u(s) + · · · + (N − 1)bN−1(δ)u(s)N−2 + u(s)N−1|Nlδ + u(s)[(lδ)u + (lδ)v]| +|v(s)|C1 + u(s)|v′(s)|C2 + u(s)|v(s)|C3 + 2|v(s)||v′(s)|C4 + v(s)2C5 ≤ 2b2(δ)u(s) + · · · + (N − 1)bN−1(δ)u(s)N−2 + u(s)N−1N(1 + 2c)lδ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0) +(u(s)C2 + 2|v(s)|C4)Au(s)w′(s) + (C1 + u(s)C3 + |v(s)|C5)Au(s)B · [b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' ≤ 3b2(δ)u(s) + · · · + NbN−1(δ)u(s)N−2 + u(s)N−1N(2 + 2c)lδ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0) +(u(s)C2 + 2|v(s)|C4)Au(s)w′(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' where one can choose ǫ so small that u(s)|(lδ)u + (lδ)v| ≤ Nc1 − c 2N+1 h(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0) ≤ Nclδ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (C1 + u(s)C3 + |v(s)|C5)Au(s)B ≤ 1, u(s) ≤ 1, 1 + c < N 49 By making (u(s)C2 + 2|v(s)|C4)Au(s) < 1, there exists a constant C such that w′(s) ≤ C[3b2(δ)u(s) + · · · + NbN−1(δ)u(s)N−2 + u(s)N−1N(2 + 2c)lδ(0, 0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So |v′(s)| ≤ Au(s)C[3b2(δ)u(s)+· · ·+NbN−1(δ)u(s)N−2+u(s)N−1N(2+2c)lδ(0, 0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Now let’s look at the situation when γ(s) is on an interior line segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Consider a line segment in the image of γ with two endpoints γ(s1) and γ(s2), we can parametrize v(s) for s ∈ [s1, s2] by v(s) = v(s1)+T(u(s)−u(s1)), where T = dα du(u(s1)) and |T| = ���� v′(s1) u′(s1) ���� ≤ 2|v′(s1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' |T| ≤ 2Au(s1)C[3b2(δ)u(s1) + · · · + NbN−1(δ)u(s1)N−2 + u(s1)N−1N(2 + 2c)lδ(0, 0)] ≤ 2Au(s)C[3b2(δ)u(s) + · · · + NbN−1(δ)u(s)N−2 + u(s)N−1N(2 + 2c)lδ(0, 0)], where the last inequality holds because u(s) is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Hence |v(s)| ≤ |v(s1)| + |T|(u(s) + u(s)) ≤ Au(s)B[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)] +4Au(s)2C[3b2(δ)u(s) + · · · + NbN−1(δ)u(s)N−2 + u(s)N−1N(2 + 2c)lδ(0, 0)] ≤ (AB + 4AC · 2N)u(s)[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)] = Du(s)[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)], where D = (AB + 4AC · 2N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Furthermore, since v′(s) = v′(s1), |v′(s)| ≤ Au(s)C[3b2(δ)u(s) + · · · + NbN−1(δ)u(s)N−2 + u(s)N−1N(2 + 2c)lδ(0, 0)] ≤ AC · 2Nu(s)[b2(δ)u(s) + · · · + bN−1(δ)u(s)N−2 + u(s)N−1(1 + c)lδ(0, 0)] ≤ Du(s)[b2(δ)u(s) + · · · + bN−1(δ)u(s)N−2 + u(s)N−1(1 + c)lδ(0, 0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Concavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Suppose for the sake of contradiction that γ(s) leaves S at a switch point when s = s0 and dives into the interior of M for increasing s until it re-enters S again at s = s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 50 Consider the intersection of the two-dimensional plane v = v0 +T(u−u0) with the surface w = kδ(u, v), where (u0, v0) = (u(s0), v(s0)) and T = dα du(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Set f(u) = kδ(v, v0 + T(u − u0)), then with v(s) = v0 + T(u(s) − u0), d2f du2(u(s)) = (kδ)uu(u(s), v(s)) + 2(kδ)uv(u(s), v(s))T + (kδ)vv(u(s), v(s))T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the one hand, (kδ)uu = 2b2(δ) + 6b3(δ)u(s) + · · · + (N − 1)(N − 2)bN−1(δ)u(s)N−3 +N(N − 1)u(s)N−2lδ + 2Nu(s)N−1(lδ)u + u(s)N(lδ)uu +v(s)[2(mδ)u + u(s)(mδ)uu] + v(s)2(nδ)uu, where one can choose ǫ small enough so that |lδ| ≥ (1−c)lδ(0, 0), |2Nu(s)(lδ)u + u(s)2(lδ)uu| N(N − 1) ≤ clδ(0, 0), |2(mδ)u+u(s)(mδ)uu+v(s)(nδ)uu| ≤ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus (kδ)uu ≥ 2b2(δ) + 6b3(δ)u(s) + · · · + (N − 1)(N − 2)bN−1(δ)u(s)N−3 +N(N − 1)u(s)N−2(1 − 2c)lδ(0, 0) −C1Du(s)[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)] ≥ 2b2(δ) + 6b3(δ)u(s) + · · · + (N − 1)(N − 2)bN−1(δ)u(s)N−3 +N(N − 1)u(s)N−2(1 − 2c)lδ(0, 0) −1 2[b2(δ) + · · · + bN−1(δ)u(s)N−3 + u(s)N−2(1 + c)lδ(0, 0)], where we can choose ǫ small enough so that C1Du(s) ≤ 1 2 and u(s) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Note here that 1 − 2c > 0 because 0 < c < 5 − √ 17 2 < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, |T| ≤ 2Au(s0)C[3b2(δ)u(s0) + · · · + NbN−1(δ)u(s0)N−2 + u(s0)N−1N(2 + 2c)lδ(0, 0)] ≤ 4ACNu(s0)[b2(δ)u(s0)2 + · · · + bN−1(δ)u(s0)N−1 + u(s0)N(1 + c)lδ(0, 0)] ≤ Du(s)[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 51 So for ǫ sufficiently small |2(kδ)uvT + (kδ)vvT 2| ≤ C1|T| ≤ C1Du(s)[b2(δ)u(s)2 + · · · + bN−1(δ)u(s)N−1 + u(s)N(1 + c)lδ(0, 0)] ≤ 1 2[b2(δ) + · · · + bN−1(δ)u(s)N−3 + u(s)N−2(1 + c)lδ(0, 0)], if C1Du(s) ≤ 1 2 and u(s) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that d2f du2(u(s)) ≥ 2b2(δ) + 6b3(δ)u(s) + · · · + (N − 1)(N − 2)bN−1(δ)u(s)N−3 +N(N − 1)(1 − 2c)u(s)N−2lδ(0, 0) −b2(δ) − b3(δ)u(s) − · · · − bN−1(δ)u(s)N−3 − u(s)N−2(1 + c)lδ(0, 0) = b2(δ) + 5b3(δ)u(s) + · · · + [(N − 1)(N − 2) − 1]bN−1(δ)u(s)N−3 +[N(N − 1)(1 − 2c) − (1 + c)]u(s)N−2lδ(0, 0), where we want N(N − 1)(1 − 2c) ≥ 2(1 − 2c) > 1 + c ⇒ 0 < c < 1 5 < 5 − √ 17 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore d2f du2(u(s)) > 0, so f ′(u(s)) is increasing as u(s) increases from u(s0) = u0 to u(s1) = u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, since the interior line segment is tangent to S at the two endpoints, we must have f ′(u0) = f ′(u1), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore if γ leaves S at a switch point γ(s0), the geodesic arc beyond this point is a line segment either exiting the ǫ-ball or terminating at a point on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So γ has at most two switch points in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Case 4: a2(δ) > 0, h(0, 0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' This is a combination of Case 1, Case 2, and Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When a2(δ) > 0, the angle δ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' From Case 1 the constants a2(δ), a3(δ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=', aN−1(δ) satisfy the following: a2(δ) ≥ b tan(θ0)| sin θ|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' |a3(δ)|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' , |aN−1(δ)| ≤ M| sin δ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 52 Assume γ′(0) = ∂ ∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' x′(s) ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The argument is exactly the same as in Case 1 when we proved for (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Approximate z(s), z′(s), y(s), y′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ∈ S, imitating the proof for (50) in Case 1 one can show that there exists a positive constant C such that z(s) ≤ (2a2(δ)x(s)2 + 2|h(0, 0)|x(s)N)C ⇒ |y(s)| ≤ (2a2(δ)x(s)2 + 2|h(0, 0)|x(s)N)AC Simiarly, imitating the proof for (51) in Case 1 one can also show that z′(s) ≤ (4a2(δ)x(s) + 4Nx(s)N−1|h(0, 0)|)C ⇒ y′(s) ≤ (4a2(δ)x(s) + 4Nx(s)N−1|h(0, 0)|)AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that if y = α(x) and T(s) = α′(x(s)), then |T(s)| = ���� y′(s) x′(s) ���� ≤ 2|y′(s)| ≤ (8a2(δ)x(s) + 8Nx(s)N−1|h(0, 0)|)AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If γ(s) ̸∈ S, then γ(s) is in some line segment where y(s) = y(s0)+T(s0)(x(s)− x(s0)) for some switch point at s0 < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since x(s) is increasing, it follows that |y(s)| ≤ |y(s0)| + |T(s0)|[|x(s)| + |x(s0)|] ≤ (2a2(δ)x(s0)2 + 2|h(0, 0)|x(s0)N)AC +(8a2(δ)x(s0) + 8Nx(s0)N−1|h(0, 0)|)AC[|x(s)| + |x(s0)|] ≤ (2a2(δ)x(s)2 + 2|h(0, 0)|x(s)N)AC + (8a2(δ)x(s) + 8Nx(s)N−1|h(0, 0)|)AC · 2x(s) ≤ (18a2(δ)x(s)2 + 18N|h(0, 0)|x(s)N)AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Furthermore, |z(s)| ≤ |z(s0)| + 2|z′(s0)|[|x(s)| + |x(s0)|] ≤ (2a2(δ)x(s0)2 + 2|h(0, 0)|x(s0)N)C + 2(4a2(δ)x(s0) + 4Nx(s0)N−1|h(0, 0)|)C · 2x(s) ≤ (2a2(δ)x(s)2 + 2|h(0, 0)|x(s)N)C + 4x(s)(4a2(δ)x(s) + 4Nx(s)N−1|h(0, 0)|)C = (18a2(δ)x(s)2 + 18N|h(0, 0)|x(s)N)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The same inequalities for z′(s), y′(s), and T(s) still hold as before because z′(s) = z′(s0), y′(s) = y′(s0), T(s) = T(s0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 53 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Now we are ready to estimate the location of the first switch point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Set f(x) = gδ(x, y0 + T(x − x0)) where (x0, y0) = (x(s0), y(s0)) and T = T(s0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Then f ′′(x0) = (gδ)xx(x0, y0) + 2(gδ)xy(x0, y0)T + (gδ)yy(x0, y0)T 2, where (gδ)xx(x0, y0) = 2a2(δ) + 6a3(δ)x0 + · · · + (N − 1)(N − 2)aN−1(δ)xN−3 0 +xN−2 0 [N(N − 1)hδ + 2Nx0(hδ)x + x2 0(hδ)xx] +y0[2(iδ)x + x0(iδ)xx + y0(jδ)xx].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' As x0 → 0, (gδ)xx(x0, y0) → 2a2(δ) and T(s0) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore f ′′(x0) > 0 at the beginning and the geodesic initially stays on the surface if it is not a straight line for which we will discuss later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Suppose f ′′(x0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let’s estimate x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' First, for ǫ sufficiently small, |6a3(δ)x0 + · · · + (N − 1)(N − 2)aN−1(δ)xN−3 0 | ≤ M| sin δ|(6x0 + · · · + (N − 1)(N − 2)xN−3 0 ) ≤ b tan(θ0)| sin δ| ≤ a2(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Second, for η and ǫ sufficiently small, N(N − 1)hδ + 2Nx0(hδ)x + x2 0(hδ)xx ≥ (1 + c 2)N(N − 1)h(0, 0), for some 0 < c < 1 to be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Third, there are constants C5, C6 such that |2(iδ)x + x0(iδ)xx + y0(jδ)xx| ≤ C5, |2(gδ)xy(x0, y0) + (gδ)yy(x0, y0)T| ≤ C6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If f ′′(x0) = 0, then −xN−2 0 [N(N − 1)hδ + 2Nx0(hδ)x + x2 0(hδ)xx] = 2a2(δ) + 6a3(δ)x0 + · · · + (N − 1)(N − 2)aN−1(δ)xN−3 0 +y0[2(iδ)x + x0(iδ)xx + y0(jδ)xx] + 2(gδ)xy(x0, y0)T + (gδ)yy(x0, y0)T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the one hand, LHS ≤ −(1 + c 2)N(N − 1)h(0, 0)xN−2 0 = (1 + c 2)N(N − 1)|h(0, 0)|xN−2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 54 On the other hand, RHS ≥ 2a2(δ) − a2(δ) − C5(18a2(δ)x2 0 + 18N|h(0, 0)|xN 0 )AC −C6(8a2(δ)x0 + 8NxN−1 0 |h(0, 0)|)AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Combining the two inequalities we get N(N − 1)|h(0, 0)|xN−2 0 [(1 + c 2) + 18ACC5x2 0 + 8ACC6x0 N − 1 ] ≥ a2(δ)[1 − 18ACC5x2 0 − 8ACC6x0] Choose ǫ small enough so that 18ACC5x2 0 + 8ACC6x0 N − 1 ≤ c 2 and 18ACC5x2 0 + 8ACC6x0 ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore we obtain a lower bound for x0 N(N − 1)|h(0, 0)|xN−2 0 (1 + c) ≥ a2(δ)(1 − c) (54) ⇒ xN−2 0 ≥ a2(δ)(1 − c) N(N − 1)(1 + c)|h(0, 0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Similarly, we can also get an upper bound for x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Again suppose f ′′(x0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' First, for ǫ sufficiently small, we still have |6a3(δ)x0 + · · · + (N − 1)(N − 2)aN−1(δ)xN−3 0 | ≤ a2(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Second, for η and ǫ sufficiently small, N(N − 1)hδ + 2Nx0(hδ)x + x2 0(hδ)xx ≤ (1 − c 2)N(N − 1)h(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Third, there are still constants C5, C6 such that |2(iδ)x + x0(iδ)xx + y0(jδ)xx| ≤ C5, |2(gδ)xy(x0, y0) + (gδ)yy(x0, y0)T| ≤ C6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If f ′′(x0) = 0, then −xN−2 0 [N(N − 1)hδ + 2Nx0(hδ)x + x2 0(hδ)xx] = 2a2(δ) + 6a3(δ)x0 + · · · + (N − 1)(N − 2)aN−1(δ)xN−3 0 +y0[2(iδ)x + x0(iδ)xx + y0(jδ)xx] + 2(gδ)xy(x0, y0)T + (gδ)yy(x0, y0)T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 55 On the one hand, LHS ≥ −(1 − c 2)N(N − 1)h(0, 0)xN−2 0 = (1 − c 2)N(N − 1)|h(0, 0)|xN−2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, RHS ≤ 2a2(δ) + a2(δ) + C5(18a2(δ)x2 0 + 18N|h(0, 0)|xN 0 )AC +C6(8a2(δ)x0 + 8NxN−1 0 |h(0, 0)|)AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Combining the two yields (1 − c 2)N(N − 1)|h(0, 0)|xN−2 0 ≤ 3a2(δ) + (18ACC5x2 0 + 8ACC6x0)a2(δ) +xN−2 0 N|h(0, 0)|(18ACC5x2 0 + 8ACC6x0) ⇒ N(N − 1)|h(0, 0)|xN−2 0 [(1 − c 2) − 18ACC5x2 0 + 8ACC6x0 N − 1 ] ≤ 3a2(δ) + (18ACC5x2 0 + 8ACC6x0)a2(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So N(N−1)|h(0, 0)|xN−2 0 (1−c) ≤ (3+c)a2(δ) ⇒ xN−2 0 ≤ (3 + c)a2(δ) N(N − 1)|h(0, 0)|(1 − c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Now like in Case 3 we are going to shift our coordinates to have the origin at γ(s0) = (x0, y0, z0) and then rotate the (x, y, z)-space so that γ′(s0) = (x′(s0), y′(s0), z′(s0)) points in the positive x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let’s use (u, v, w) for the new coordinates, then with respect to the new frame there is a rotation matrix P ∈ SO3(R) such that \uf8eb \uf8ed x − x0 y − y0 z − z0 \uf8f6 \uf8f8 = P \uf8eb \uf8ed u v w \uf8f6 \uf8f8 , where P = \uf8eb \uf8ed x′(s0) p12 p13 y′(s0) p22 p23 z′(s0) p32 p33 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So x = x0 + x′(s0)u + p12v + p13w, y = y0 + y′(s0)u + p22v + p23w, z = z0 + z′(s0)u + p32v + p33w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 56 The second and third columns of P can be further specificed as below: \uf8eb \uf8ed p12 p22 p32 \uf8f6 \uf8f8 = \uf8eb \uf8ed −y′(s0) x′(s0) 0 \uf8f6 \uf8f8 Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' \uf8eb \uf8ed p13 p23 p33 \uf8f6 \uf8f8 = \uf8eb \uf8ed −x′(s0)2z′(s0) −x′(s0)y′(s0)z′(s0) x′(s0)[x′(s0)2 + y′(s0)2] \uf8f6 \uf8f8 Z, (55) where Y = 1/ � x′(s0)2 + y′(s0)2 and Z = 1/x′(s0) � x′(s0)2 + y′(s0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus the surface z = gδ(x, y) satisfies the equation z0 + z′(s0)u + p32v + p33w = gδ(x0 + x′(s0)u + p12v + p13w, y0 + y′(s0)u + p22v + p23w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Check that we can still solve for w analytically in terms of u, v within the ǫ-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let’s take the partial derivate of −z0−z′(s0)u−p32v−p33w+gδ(x0+x′(s0)u+p12v+p13w, y0+y′(s0)u+p22v+p23w) with respect to w: −p33 + (gδ)xp13 + (gδ)yp23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since Z > 0, it is equivalent to show that −p33 + (gδ)xp13 + (gδ)yp23 Z ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the one hand, (gδ)xp13 + (gδ)yp23 Z = −x′(s0)2z′(s0)(gδ)x − x′(s0)y′(s0)z′(s0)(gδ)y ≤ |(gδ)x| + |(gδ)y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since (gδ)x(x, y), (gδ)y(x, y) → 0 as x, y → 0, for ǫ sufficiently small |(gδ)x| + |(gδ)y| ≤ 1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, p33 Z = x′(s0)[x′(s0)2 + y′(s0)2] ≥ x′(s0)3 ≥ 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' so −p33 + (gδ)xp13 + (gδ)yp23 Z ≤ − 1 16 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore there exists a real analytic function kδ such that w = kδ(u, v) such that kδ(0, 0) = 0, (kδ)u(0, 0) = 0, and (kδ)v(0, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 57 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Estimate γ(s) in the new frame starting from (x0, y0, z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' After replac- ing s by s − s0, we denote γ(s) as (u(s), v(s), w(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Coefficients of kδ(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Denote kδ(u, v) as kδ(u, v) = b2(δ)u2 + · · ·+ bN−1(δ)uN−1 + uNlδ(u, v) + uvmδ(u, v) + v2nδ(u, v), where b2(δ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' , bN−1(δ) are constants and lδ, mδ, nδ are analytic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Observe that for n between 2 and N − 1, bn(δ) = 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' ∂nkδ ∂un (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Next let’s look for ∂nkδ ∂un (u, v) for n ≥ 2 by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let A be x′(s0) + p13(kδ)u and B be y′(s0) + p23(kδ)u, then for n ≥ 2, p33 ∂nkδ ∂un (u, v) = n � a+b=1 ∂a+bgδ ∂xa∂yb � I,J cI,JAa−|I|Bb−|J|(∂A ∂u )i1 · · · (∂pA ∂up )ip(∂B ∂u )j1 · · ·(∂pB ∂up )jp, where p = n − (a + b), I = (i1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' , ip), J = (j1, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' , jp), (i1 + 2i2 + · · + pip) + (j1 + 2j2 + · · · + pjp) = p, |I| = i1 + i2 + · · · + ip ≤ a, |J| = j1 + j2 + · · · + jp ≤ b, and the partial derivatives of gδ are evaluated at (x0 + x′(s0)u + p12v + p13kδ(u, v), y0 + y′(s0)u + p22v + p23kδ(u, v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When n = 2, differentiating the following equation z0+z′(s0)u+p32v+p33kδ(u, v) = gδ(x0+x′(s0)u+p12v+p13kδ(u, v), y0+y′(s0)u+p22v+p23kδ(u, v)) with respect to u once gives us z′(s0) + p33(kδ)u = (gδ)x[x′(s0) + p13(kδ)u] + (gδ)y[y′(s0) + p23(kδ)u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (56) Let A = x′(s0) + p13(kδ)u and B = y′(s0) + p23(kδ)u, then z′(s0) + p33(kδ)u = (gδ)xA + (gδ)yB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Taking the partial derivative with respect to u once more gives p33(kδ)uu = (gδ)xxA2 + (gδ)x∂uA + (gδ)yyB2 + (gδ)y∂uB + 2(gδ)xyAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 58 When a + b = 2, p = 0 and so there are no I and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' There are three terms corresponding to: a = 2, b = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' a = 0, b = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' a = 1, b = 1, respectively: (gδ)xxA2, (gδ)yyB2, (gδ)xyAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When a + b = 1, either a = 1, b = 0 with I = (1), J = 0 or a = 0, b = 1 with I = 0, J = (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that there are two terms (gδ)x∂uA, (gδ)y∂uB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When n ≥ 2, by inductive hypothesis we can take the partial derivative of the expression in the lemma with respect to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The left-hand side is p33 ∂n+1kδ ∂un+1 (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The right-hand side consists of three parts due to the product rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (1) n � a+b=1 ∂a+b+1gδ ∂xa+1∂yb � I,J cI,JAa+1−|I|Bb−|J|(∂A ∂u )i1 · · · (∂pA ∂up )ip(∂B ∂u )j1 · · · (∂pB ∂up )jp + n � a+b=1 ∂a+b+1gδ ∂xa∂yb+1 � I,J cI,JAa−|I|Bb+1−|J|(∂A ∂u )i1 · · ·(∂pA ∂up )ip(∂B ∂u )j1 · · · (∂pB ∂up )jp, where a becomes a + 1 in the first term, b becomes b + 1 in the second term, and p stays the same since (n + 1) − (a + 1 + b) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (2) n � a+b=1 ∂a+bgδ ∂xa∂yb � I,J cI,J(a − |I|)Aa−|I|−1Bb−|J|(∂A ∂u )i1+1 · · · (∂pA ∂up )ip(∂B ∂u )j1 · · ·(∂pB ∂up )jp + n � a+b=1 ∂a+bgδ ∂xa∂yb � I,J cI,J(b − |J|)Aa−|I|Bb−|J|−1(∂A ∂u )i1+1 · · · (∂pA ∂up )ip(∂B ∂u )j1+1 · · · (∂pB ∂up )jp, where a, b stay the same, so p becomes p + 1 = (n + 1) − (a + b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Moreover, i1 becomes i1 + 1 in the first term and j1 becomes j1 + 1 in the second term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So (i1+1+2i2+· · ·+pip)+(j1+2j2+· · ·+pjp) = (i1+2i2+· · ·+pip)+(j1+1+2j2+· · ·+pjp) = p+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (3) n � a+b=1 ∂a+bgδ ∂xa∂yb � I,J cI,JAa−|I|Bb−|J| � ik̸=0 · ·ik(∂A ∂u )ik−1(∂A ∂u )ik+1+1 · · · + n � a+b=1 ∂a+bgδ ∂xa∂yb � I,J cI,JAa−|I|Bb−|J| � jk̸=0 · ·jk(∂B ∂u )jk−1(∂B ∂u )jk+1+1 · · · , 59 where a, b stay the same, so p becomes p + 1 = (n + 1) − (a + b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Moreover, ik, ik+1 become ik − 1, ik+1 + 1 in the first term and jk, jk+1 become jk − 1, jk+1 + 1 in the second term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So · ·+k(ik−1)+(k+1)(ik+1+1)+· · · = · · ·+k(jk−1)+(k+1)(jk+1+1)+· · · = p+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that cI,J are nonnegative integers and the lemma is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The following two corollaries are analogous to Corollaries 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The coefficients of (gδ)x∂n−1 u A and (gδ)y∂n−1 u B are always 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The coefficients of ∂ngδ ∂xn An and ∂ngδ ∂yn Bn are always 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Now we let (u, v) = (0, 0), then (kδ)u(0, 0) = 0 implies that A(0, 0) = x′(s0), B(0, 0) = y′(s0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Furthermore for p ≥ 1, ∂pA ∂up (0, 0) = p13 ∂p+1kδ ∂up+1 (0, 0) = p13(p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bp+1(δ) ∂pB ∂up (0, 0) = p23 ∂p+1kδ ∂up+1 (0, 0) = p23(p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bp+1(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that for 2 ≤ n ≤ N − 1 p33n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bn(δ) = n � a+b=1 ∂a+bgδ ∂xa∂yb(x0, y0) � I,J cI,Jx′(s0)a−|I|y′(s0)b−|J|(p13)|I|(p23)|J| (2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=')i1+j1(3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' )i2+j2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ip+jpb2(δ)i1+j1 · · ·bp+1(δ)ip+jp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Furthermore Corollary (3) suggests that the terms corresponding to a+b = 1 or p = n − 1 in the expression are (gδ)x(x0, y0)p13n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bn(δ) + (gδ)y(x0, y0)p23n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bn(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Moving them to the other side of the expression yields (p33 − (gδ)x(x0, y0)p13 − (gδ)y(x0, y0)p23)n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bn(δ) = n � a+b=2 ∂a+bgδ ∂xa∂yb(x0, y0) � I,J cI,Jx′(s0)a−|I|y′(s0)b−|J|(p13)|I|(p23)|J| (2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=')i1+j1(3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' )i2+j2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ip+jpb2(δ)i1+j1 · · · bp+1(δ)ip+jp, 60 where p + 1 = n − (a + b) + 1 ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So bn(δ) depends on the previous coefficients b2(δ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=', bn−1(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let’s calculate the coefficient bn(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Evaluating (56) at (u, v) = (0, 0) arrives z′(s0) = (gδ)x(x0, y0)x′(s0) + (gδ)y(x0, y0)y′(s0), together with (55), then 1 Z [p33 − (gδ)x(x0, y0)p13 − (gδ)y(x0, y0)p23] = x′(s0)[x′(s0)2 + y′(s0)2] + (gδ)x(x0, y0)x′(s0)2z′(s0) + (gδ)y(x0, y0)x′(s0)y′(s0)z′(s0) = x′(s0)[x′(s0)2 + y′(s0)2] + x′(s0)z′(s0)2 = x′(s0)[x′(s0)2 + y′(s0)2 + z′(s0)2] = x′(s0) · 1 = x′(s0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bn(δ) � x′(s0)2 + y′(s0)2 = n � a+b=2 ∂a+bgδ ∂xa∂yb(x0, y0) � I,J cI,Jx′(s0)a−|I|y′(s0)b−|J|(p13)|I|(p23)|J| (2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=')i1+j1(3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' )i2+j2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ip+jpb2(δ)i1+j1 · · · bp+1(δ)ip+jp = n � a+b=2 ∂a+bgδ ∂xa∂yb(x0, y0) � I,J cI,Jx′(s0)a−|I|y′(s0)b−|J| � −x′(s0)2z′(s0) x′(s0) � x′(s0)2 + y′(s0)2 �|I| � −x′(s0)y′(s0)z′(s0) x′(s0) � x′(s0)2 + y′(s0)2 �|J| (2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=')i1+j1(3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' )i2+j2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ip+jpb2(δ)i1+j1 · · · bp+1(δ)ip+jp = n � a+b=2 ∂a+bgδ ∂xa∂yb(x0, y0) � I,J cI,J(−1)|I|+|J|x′(s0)ay′(s0)b � z′(s0) � x′(s0)2 + y′(s0)2 �|I|+|J| (2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=')i1+j1(3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' )i2+j2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ip+jpb2(δ)i1+j1 · · · bp+1(δ)ip+jp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The signs of bn(δ) for n between 2 and N − 1 are all negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Before proving the lemma, one needs to estimate ∂pgδ ∂xp (x0, y0) for 2 ≤ 61 p ≤ N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' By induction one can show that ∂pgδ ∂xp (x0, y0) = p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ap(δ) + · · · + (N − 1)(N − 2) · · ·(N − p)aN−1(δ)xN−1−p 0 + p � q=0 �p q � N(N − 1) · · ·(N − q + 1)xN−q 0 ∂p−q x hδ(x0, y0) +y0 � p � q=0 �p q � dqx dxq (x0)∂p−q x iδ(x0, y0) � + y2 0∂p xjδ(x0, y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' First if ǫ is sufficiently small, |p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ap(δ) + · · · + (N − 1)(N − 2) · · ·(N − p)aN−1(δ)xN−1−p 0 | ≤ M| sin δ|[p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' + · · · + (N − 1)(N − 2) · · ·(N − p)xN−1−p 0 ] ≤ c 2b tan(θ0)| sin δ| ≤ c 2a2(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Second if ǫ and η are sufficiently small, p � q=0 �p q � N(N − 1) · · ·(N − q + 1)xN−q 0 ∂p−q x hδ(x0, y0) ≤ N(N − 1) · · ·(N − p + 1)xN−p 0 h(0, 0)(1 − c 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Third if ǫ is sufficiently small, �����y0 � p � q=0 � p q � dqx dxq (x0)∂p−q x iδ(x0, y0) � + y2 0∂p xjδ(x0, y0) ����� ≤ |y0|C1 ≤ (18a2(δ)x2 0 + 18N|h(0, 0)|xN 0 )ACC1 ≤ c 2a2(δ) + N(N − 1) · · ·(N − p + 1)xN−p 0 |h(0, 0)|c 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Combining the above three inequalities, together with (54), yields ∂pgδ ∂xp (x0, y0) ≤ ca2(δ) + N(N − 1) · · ·(N − p + 1)xN−p 0 h(0, 0)(1 − c) ≤ ca2(δ) − N(N − 1) · · ·(N − p + 1)|h(0, 0)|(1 − c)2a2(δ) N(N − 1)(1 + c)|h(0, 0)| = − �(N − 2) · · ·(N − p + 1)(1 − c)2 1 + c − c � a2(δ) ≤ − �(1 − c)2 1 + c − c � a2(δ), 62 which is negative if we choose c as follows: (1 − c)2 1 + c − c > 0 ⇒ (1 − c)2 > c(1 + c) ⇒ 0 < c < 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Denote the constant in the brackets as L = L(c), then for 2 ≤ p ≤ N − 1 ∂pgδ ∂xp (x0, y0) ≤ −La2(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Now we are ready to prove the lemma by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When n = 2, 2b2(δ) � x′(s0)2 + y′(s0)2 = (gδ)xx(x0, y0)x′(s0)2+2(gδ)xy(x0, y0)x′(s0)y′(s0)+(gδ)yy(x0, y0)y′(s0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the one hand, since x′(s0) ≥ 1 2, (gδ)xx(x0, y0)x′(s0)2 ≤ −L 4 a2(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, y′(s0) [2(gδ)xy(x0, y0)x′(s0) + (gδ)yy(x0, y0)y′(s0)] ≤ C1|y′(s0)| ≤ C1(4a2(δ)x0 + 4NxN−1 0 |h(0, 0)|)AC ≤ 4ACC1x0a2(δ) + 4ACC1x0 (3 + c)a2(δ)N|h(0, 0)| N(N − 1)|h(0, 0)|(1 − c) = 4ACC1x0a2(δ) � 1 + 3 + c (N − 1)(1 − c) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If we choose ǫ small enough so that 4ACC1x0 � 1 + 3 + c (N − 1)(1 − c) � ≤ L 8 , then 2(gδ)xy(x0, y0)x′(s0)y′(s0) + (gδ)yy(x0, y0)y′(s0)2 ≤ L 8 a2(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that 2b2(δ) � x′(s0)2 + y′(s0)2 ≤ −L 8 a2(δ) ⇒ b2(δ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 63 By inductive hypothesis, suppose b2(δ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' , bn−1(δ) are all negative, then it suffices to show that n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bn(δ) � x′(s0)2 + y′(s0)2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' There are two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Case 1: when b = 0, |J| = 0 since |J| ≤ b, then we have n � a=2 ∂agδ ∂xa (x0, y0) � I,J cI,J(−1)|I|x′(s0)a � z′(s0) � x′(s0)2 + y′(s0)2 �|I| 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i13!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ipb2(δ)i1b3(δ)i2 · · · bp+1(δ)ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since z′(s0) > 0 and the sign of b2(δ)i1b3(δ)i2 · · ·bp+1(δ)ip is (−1)|I|, it follows that for each I, J, cI,J(−1)|I|x′(s0)a � z′(s0) � x′(s0)2 + y′(s0)2 �|I| 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i13!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i2 · · · (p+1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ipb2(δ)i1b3(δ)i2 · · ·bp+1(δ)ip ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since ∂a xgδ(x0, y0) < 0, the above sum is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Especially when a = n, p = n − a − b = 0 and so there is only one term ∂ngδ ∂xn (x0, y0)x′(s0)n ≤ − 1 2n La2(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' whose coefficient is 1 by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Case 2: when b ̸= 0, there is at least one copy of y′(s0) in the summation, so we can write the rest of the terms as ������ y′(s0) \uf8ee \uf8f0 n � a+b=2,b≥1 ∂a+bgδ ∂xa∂yb(x0, y0) � I,J cI,J(−1)|I|+|J|x′(s0)ay′(s0)b−1 � z′(s0) � x′(s0)2 + y′(s0)2 �|I|+|J| (2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=')i1+j1(3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' )i2+j2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ip+jpb2(δ)i1+j1 · · ·bp+1(δ)ip+jp��� ≤ C1|y′(s0)| ≤ C1(4a2(δ)x0 + 4NxN−1 0 |h(0, 0)|)AC ≤ 4ACC1x0a2(δ) � 1 + 3 + c (N − 1)(1 − c) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' where the first inequality is because everything inside the brackets is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If we choose ǫ small enough so that 4ACC1x0 � 1 + 3 + c (N − 1)(1 − c) � ≤ 1 2n+1L, 64 then n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='bn(δ) � x′(s0)2 + y′(s0)2 ≤ − 1 2n+1La2(δ) < 0, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The next lemma not only determines the sign of lδ(0, 0), but also gives an upper bound of lδ(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' lδ(0, 0) ≤ 1 − c 2N−1 h(0, 0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' By Lemma 5, p33 ∂Nkδ ∂uN (u, v) = N � a+b=1 ∂a+bgδ ∂xa∂yb � I,J cI,JAa−|I|Bb−|J|(∂A ∂u )i1 · · · (∂pA ∂up )ip(∂B ∂u )j1 · · · (∂pB ∂up )jp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When a + b = 1 or p = N − 1 Corollary 3 suggests that we have in the above sum (gδ)x ∂N−1A ∂uN−1 + (gδ)y ∂N−1B ∂uN−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Evaluating at (u, v) = (0, 0) gives us p33N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='lδ(0, 0) − (gδ)x(x0, y0)p13N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='lδ(0, 0) − (gδ)y(x0, y0)p13N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='lδ(0, 0) = N � a+b=2 ∂a+bgδ ∂xa∂yb(x0, y0) � I,J cI,Jx′(s0)a−|I|y′(s0)b−|J|(p13)|I|(p23)|J| (2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=')i1+j1(3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' )i2+j2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ip+jpb2(δ)i1+j1 · · · bp+1(δ)ip+jp Since p33 = Zx′(s0), p13 = −x′(s0)2z′(s0)Z, and p23 = −x′(s0)y′(s0)z′(s0)Z, one has N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='lδ(0, 0) � x′(s0)2 + y′(s0)2 = N � a+b=2 ∂a+bgδ ∂xa∂yb(x0, y0) � I,J cI,J(−1)|I|+|J|x′(s0)ay′(s0)b � z′(s0) � x′(s0)2 + y′(s0)2 �|I|+|J| (2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=')i1+j1(3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' )i2+j2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ip+jpb2(δ)i1+j1 · · · bp+1(δ)ip+jp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 65 Next let’s use an analogous argument in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When b = 0, |J| = 0, then we have N � a=2 ∂agδ ∂xa (x0, y0) � I,J cI,J(−1)|I|x′(s0)a � z′(s0) � x′(s0)2 + y′(s0)2 �|I| 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i13!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='i2 · · ·(p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ipb2(δ)i1b3(δ)i2 · · · bp+1(δ)ip ≤ ∂Ngδ ∂xN (x0, y0)x′(s0)N ≤ 1 2N ∂Ngδ ∂xN (x0, y0), Since ∂Ngδ ∂xN (x0, y0) = N � q=0 �N q � N(N − 1) · · ·(N − q + 1)xN−q 0 ∂q xhδ(x0, y0) +y0 � N � q=0 � N q � ∂N−qx ∂xN−q (x0)∂q xiδ(x0, y0) + y0 ∂Njδ ∂xN (x0, y0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the one hand, if η and ǫ are sufficiently small, N � q=0 � N q � N(N − 1) · · ·(N − q + 1)xN−q 0 ∂q xhδ(x0, y0) ≤ N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (1 − c 4)h(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, �����y0 � N � q=0 �N q � ∂N−qx ∂xN−q (x0)∂q xiδ(x0, y0) + y0 ∂Njδ ∂xN (x0, y0) ������ ≤ C1|y0| ≤ (18a2(δ)x2 0 + 18N|h(0, 0)|xN 0 )ACC1 ≤ c 4N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='|h(0, 0)|, for η and ǫ sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Thus ∂Ngδ ∂xN (x0, y0) ≤ N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (1 − c 2)h(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' When b ̸= 0, there is always one copy of y′(s0) so we have ������ y′(s0) \uf8ee \uf8f0 N � a+b=2 ∂a+bgδ ∂xa∂yb(x0, y0) � I,J cI,J(−1)|I|+|J|x′(s0)ay′(s0)b−1 � z′(s0) � x′(s0)2 + y′(s0)2 �|I|+|J| (2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=')i1+j1(3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' )i2+j2 · · · (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='ip+jpb2(δ)i1+j1 · · ·bp+1(δ)ip+jp��� ≤ C1|y′(s0)| ≤ C1(4a2(δ)x0 + 4NxN−1 0 |h(0, 0)|)AC ≤ N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 2N c 2|h(0, 0)|, 66 for η and ǫ sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='lδ(0, 0) � x′(s0)2 + y′(s0)2 ≤ N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 2N (1 − c 2)h(0, 0) − N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 2N c 2h(0, 0) = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 2N (1 − c)h(0, 0) ⇒ lδ(0, 0) ≤ � x′(s0)2 + y′(s0)2 2N (1 − c)h(0, 0) ≤ 2 2N (1 − c)h(0, 0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Since b2(δ) < 0, we could show as in Case 2 that γ is initially a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The surface S in the (u, w)-plane is the curve w = kδ(u, 0) = b2(δ)u2 + · · · + bN−1(δ)uN−1 + uNlδ(u, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' If the line segment re-enters the surface at some switch point, then the curve can’t be concave downward there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Otherwise the line lies above the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' w′(u) = 2b2(δ)u + · · · + (N − 1)bN−1(δ)uN−2 + NuN−1lδ(u, 0), w′′(u) = 2b2(δ) + 6b3(δ) + · · · + (N − 1)(N − 2)bN−1(δ)uN−3 +N(N − 1)uN−2lδ(u, 0) + 2NuN−1(lδ)u(u, 0) + uN(lδ)uu(u, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the one hand, 2b2(δ) + 6b3(δ) + · · · + (N − 1)(N − 2)bN−1(δ)uN−3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' On the other hand, we can choose ǫ small enough so that for all u < ǫ, lδ(u, 0) ≤ (1 − c 2)lδ(0, 0), |2Nu(lδ)u(u, 0) + u2(lδ)uu(u, 0)| N(N − 1) ≤ c 2|lδ(0, 0)| ⇒ N(N − 1)uN−2lδ(u, 0) + 2NuN−1(lδ)u(u, 0) + uN(lδ)uu(u, 0) ≤ N(N − 1)uN−2(1 − c)lδ(0, 0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So w′′(u) < 0 and the graph is concave downward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore γ never re- enters the surface at a switch point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' It follows that γ is a straight line that either terminates at some point on the surface or exits the ǫ-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Let’s summarize Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' In general, γ is initially a boundary segment lying on the surface, then it leaves S in a straight line that exits the ǫ-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' So there is at most one interval in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 67 In the end, we are going to mention when the lowest degree in the Taylor expansion of g(x, y) is greater than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Suppose the lowest degree is k ≥ 2, the kth Taylor polynomial has the following form: a0xk + a1xk−1y + · · · + ak−1xyk−1 + akyk Consider the line y = mx, substituing it into the above polynomial gives us xk(a0 + a1m + · · · + akmk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Setting it to zero gives as at most k distinct solutions for m if ak ̸= 0, otherwise adding the vertical line x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore the plane can be sliced into at most 2k distinct pies using these slopes, such that within each slice, the graph of g along a ray is either concave upward or downward near the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' The rest of the proof is analogous to the case when k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 68 4 Conclusion It seems naturally that our theorems could be generalized to higher-dimensional Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' However, the proof in Theorem 1 or 2 does not apply when n > 3, because the intersection of M1 and M2 becomes a surface instead of a curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Furthermore, the proof in Theorem 3 involves dividing the plane into finitely many slices using the lowest degree Taylor polynomial, which does not make sense when we have more than two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Therefore we are looking for new methods and we conjecture that all Theorems 1, 2, and 3 do not generalize when n > 3, because of more free variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' That is there exist counterexamples in higher-dimensionl Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 69 References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Albrecht and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Berg, Geodescis in Euclidean Space with Analytic Obstacle, Proceedings of the American Mathematical Society, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 113, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 1 (Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=', 1991), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 201-207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Alexander, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Berg, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Bishop, The Riemannian obstacle problem, Illinois J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 31 (1987), 167-184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} +page_content=' 70' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfTQCe/content/2301.02234v1.pdf'} diff --git a/P9FQT4oBgHgl3EQfZDYV/content/tmp_files/2301.13314v1.pdf.txt b/P9FQT4oBgHgl3EQfZDYV/content/tmp_files/2301.13314v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a719b68d0a39cd878d3cbc1767c08ef29608c648 --- /dev/null +++ b/P9FQT4oBgHgl3EQfZDYV/content/tmp_files/2301.13314v1.pdf.txt @@ -0,0 +1,3586 @@ +arXiv:2301.13314v1 [math.OC] 30 Jan 2023 +Single-Loop Switching Subgradient Methods for Non-Smooth +Weakly Convex Optimization with Non-Smooth Convex +Constraints +Yankun Huang +Qihang Lin +Department of Business Analytics +University of Iowa, Iowa City, IA 52242. +yankun-huang@uiowa.edu, qihang-lin@uiowa.edu +Abstract +In this paper, we consider a general non-convex constrained optimization problem, +where the objective function is weakly convex and the constraint function is convex while +they can both be non-smooth. This class of problems arises from many applications in +machine learning such as fairness-aware supervised learning. To solve this problem, we +consider the classical switching subgradient method by [62], which is an intuitive and +easily implementable first-order method. Before this work, its iteration complexity was +only known for convex optimization. We prove its oracle complexity for finding a nearly +stationary point when the objective function is non-convex. +The analysis is derived +separately when the constraint function is deterministic and stochastic. Compared to +existing methods, especially the double-loop methods, the switching gradient method +can be applied to non-smooth problems and only has a single loop, which saves the effort +on tuning the number of inner iterations. +Keywords: Constrained optimization, First-order method, Non-smooth optimization, Non-convex +optimization +1 +Introduction +Continuous optimization with nonlinear constraints arises from many applications of machine learn- +ing and statistics. Examples include Neyman-Pearson classification [64] and learning with fairness +constraints [78]. In this paper, we consider the following general nonlinear constrained optimization +problem +f ∗ ≡ min +x∈X f(x) +s.t. +g(x) ≤ 0, +(1) +where X ⊂ Rd is a compact convex set that allows a computationally easy projection operator, f is +weakly-convex and g is convex. Functions f and g are not necessarily smooth and we assume their +function values and subgradients can be evaluated deterministically or through stochastic oracles. +When g(x) ≡ maxi=1,...,m gi(x) with convex gi’s, (1) is equivalent to an optimization problem with +multiple nonlinear constraints gi(x) ≤ 0 for i = 1, . . . , m. +A weakly convex function can be non-convex, so computing an optimal solution of (1) is chal- +lenging in general even without constraints. For this reason, theoretical analysis for gradient-based +1 + +algorithms for non-convex problems mostly focuses on an algorithm’s (oracle) complexity for finding +an ǫ-stationary solution for (1). When a problem is non-smooth, finding an ǫ-stationary solution is +generally difficult even if the problem is convex [48]. Hence, in this paper, we consider finding a nearly +ǫ-stationary solution for (1), whose definition will be stated later in Definition 3.2. +In the past decade, there have been many studies on non-convex constrained optimization. How- +ever, most of the existing algorithms and their theoretical complexity analysis are developed by as- +suming f and gi’s are all smooth or can be written as the sum of a smooth and a simple non-smooth +functions. A non-exhaustive list of the works under such an assumption includes [10, 33, 81, 82, 38, +41, 57, 36, 56, 46, 71, 45, 53, 65, 54, 52, 11, 14, 23, 8, 59, 22]. Their results cannot be applied to (1) +due to non-smoothness in the problem. +Non-smoothness occurs very often in optimization in machine learning, for example, when a +non-smooth loss function is applied, but there are much fewer studies on non-smooth non-convex +constrained optimization. Under the weakly-convexity assumption, an effective approach for solving a +non-smooth non-convex problem with theoretical guarantees is the (inexact) proximal point method, +where a quadratic proximal term is added to objective and constraint functions to construct a strongly +convex constrained subproblem and then a sequence of solutions can be generated by solving this +subproblem inexactly and updating the center of the proximal term. +Oracle complexity for this +method to find a nearly ǫ-stationary has been established by [12, 55, 40] under different constraint +qualification conditions. +The inexact proximal point method is a double-loop algorithm where the inner loop is typically +another optimization algorithm, e.g., the switching subgradient method [62], for solving the aforemen- +tioned strongly convex subproblems. The complexity results in [12, 55, 40] require the inner loop +solves each subproblem to achieve a targeted optimality gap. However, the optimality gap is hard to +evaluate, so we are unable to simply stop the inner loop based on the gap. Although the number of +inner iterations needed to achieve the targeted gap can be bounded theoretically, it usually involves +some constants that are unknown or can only be estimated conservatively. Hence, using the theoretical +iteration bound to stop the inner loop usually leads to significantly more inner iterations than what +is actually needed, which makes the whole algorithm inefficient. In practices, users often need to tune +the number of inner iterations to improve algorithm’s efficiency, which is a common inconvenience for +all double-loop methods. +On the contrary, a single-loop algorithm is usually easier to implement as it does not require tuning +the number of inner iterations. Therefore, the main contribution of this paper is showing that, when +the constraint function g is convex, a single-loop first-order algorithm is sufficient for finding a nearly +ǫ-stationary point of (1) with theoretically guaranteed oracle complexity. The algorithm we study is +the classical switching subgradient (SSG) method by [62], which is intuitive and easy to implement +but has only been analyzed before in the convex case. We show that the SSG method finds a nearly +ǫ-stationary point of (1) with complexity O(1/ǫ4) when f is either deterministic or stochastic but g +is deterministic. This complexity is optimal [3, 31] and matches the one achieved by the double-loop +methods in [12, 55, 40] under the same assumptions. To the best of our knowledge, this is also the +first complexity result on a single-loop algorithm for weakly convex non-smooth nonlinear constrained +optimization problem. When g is also stochastic, a large mini-batch is needed to estimate g in the +SSG method and we show that the complexity is still O(1/ǫ4) for the subgradient oracle but becomes +O(1/ǫ8) for the function value oracle. +2 +Related Work +Non-convex constrained optimization has a long history [34, 15, 29, 17, 30, 4, 16] and the interest on +this subject has been still growing in the machine learning community because of its new applications +such as learning with fairness constraints (see e.g., [78]). +In recent literature, the prevalent classes of algorithms for non-convex constrained optimization +are the augmented Lagrangian method (ALM) and the penalty method [35, 80, 81, 82, 37, 38, 41, 44, +2 + +57, 36, 56, 46, 71, 45, 53, 65, 54, 52]. Another actively studied class of algorithms is the sequential +quadratic programming method [10, 33, 5, 11, 23, 9, 8, 59, 7, 22]. An inexact projection gradient +method is developed by [14] and a level conditional gradient method is developed by [20]. However, +the papers above all focus on the case where g is smooth and f is either smooth or equals f1 + f2 +where f1 is smooth and non-convex while f2 is non-smooth and convex and has a simple structure that +allows a closed-form solution for the proximal mapping arg miny f2(y) + ρ +2∥y − x∥2 for any x. On the +contrary, we focus a non-smooth problem with no structure assumption other than weak convexity. +There are relatively fewer works on non-convex non-smooth constrained problems. An alternating +direction method of multipliers (ADMM) and an ALM are studied by [77] and [79], respectively, for +non-convex non-smooth problems with linear constraints while our study considers nonlinear non- +smooth constraints. The methods by [20] and [11] can be extended to a structured non-smooth case +where f = f1 + f2 with f1 being smooth non-convex and f2 = maxy y⊤Ax − φ(y) with a convex +φ, and g has a similar structure. The method by [13] can handle a specific non-smooth non-convex +constraint, i.e., g(x) = λ∥x∥1 − h(x) where h is a convex and smooth. However, our method does not +need these structure assumptions. +When f and g in (1) are weakly convex and non-smooth, the inexact proximal point method has +been studied by [12, 55, 40] under different constraint qualification conditions and different notions of +stationarity. Their complexity analysis utilizes the relationship between the gradient of the Moreau +envelope of (1) and the near stationarity of a solution, which is originally used to analyze complexity of +subgradient methods for weakly convex non-smooth unconstrained problems [25, 26, 27, 1, 28, 63, 84]. +Our analysis also follows a similar strategy. The methods mentioned above that can be applied to +(structured) non-smooth problems are all double-loop while our algorithm only requires a single loop. +When f is either deterministic or stochastic but g is convex and deterministic, our method has the +same complexity, i.e., O(1/ǫ4) as [12, 55, 40]. However, their methods allow g also to be non-convex +under additional assumptions. +The SSG algorithm is first proposed by [62]. It has been well-studied for convex problems [61, 6, +50, 67, 74, 75, 70, 69, 68, 2] and quasi-convex problems [67]. This paper provides the first complexity +analysis for the SSG method under weak convexity assumption. Non-smooth non-convex optimization +has also been studied without weak convexity assumption by [83, 47, 66, 48, 19, 72, 73]. These works +analyze the complexity of first-order methods for computing an (ǫ, δ)-Goldstein approximate stationary +point, which is a more general stationarity notation than what we consider here. However, these works +only focus on unconstrained problems. +3 +Preliminaries +Let ∥ · ∥ be the ℓ2-norm. For h : Rd → R ∪ {+∞}, the subdifferential of h at x is +∂h(x) = +� +ζ ∈ Rd �� h(x′) ≥ h(x) + ζ⊤(x′ − x) + o(∥x′ − x∥), x′ → x +� +, +and ζ ∈ ∂h(x) is a subgradient of h at x. We say h is µ-strongly convex (µ ≥ 0) on X if +h(x) ≥ h(x′) + ζ⊤(x − x′) + µ +2 ∥x − x′∥2 +for any (x, x′) ∈ X × X and any ζ ∈ ∂h(x′). We say h is ρ-weakly convex (ρ ≥ 0) on X if +h(x) ≥ h(x′) + ζ⊤(x − x′) − ρ +2∥x − x′∥2 +for any (x, x′) ∈ X ×X and any ζ ∈ ∂h(x′). We denote the normal cone of X at x by NX (x). +We say a point x is ǫ-feasible if x ∈ X and g(x) ≤ ǫ. Let I[A] be a zero-one indicator of event +A and let Dist(x, S) := miny∈S ∥x − y∥. +We make the following assumptions about (1) throughout the paper. +3 + +Assumption 3.1. The following statements hold: +A. f(x) is ρ-weakly convex with ∂f(x) ̸= ∅ on X. g(x) is convex with ∂g(x) ̸= ∅ on +X. Moreover, there exists M such that ∥ζf∥ ≤ M and ∥ζg∥ ≤ M for any x ∈ X, +ζf ∈ ∂f(x) and ζg ∈ ∂g(x). +B. There exists a strictly feasible solution xfeas with xfeas ∈ X and g(xfeas) < 0.1 +C. There exists D such that ∥x − x′∥ ≤ D for any x and x′ in X. +D. f ∗ > −∞. +Under Assumption 3.1, (1) is non-convex so finding an ǫ-optimal solution is intractable +in general. For a non-convex problem, the target is typically to find a stationary point of (1), +which is a point x∗ ∈ X that satisfies the following Karush-Kuhn-Tucker (KKT) conditions: +ζ∗ +f + λ∗ζ∗ +g ∈ −NX(x∗), +λ∗g(x∗) = 0, +g(x∗) ≤ 0, +λ∗ ≥ 0, +(2) +where λ∗ ∈ R is a Lagrangian multiplier, ζ∗ +f ∈ ∂f(x∗) and ζ∗ +g ∈ ∂g(x∗). Typically, an exact +stationary point can only be approached by an algorithm as full convergence, which may +require infinitely many iterations. Within a finite number of iterations, an algorithm can +only generate an ǫ-stationary point, which is a point �x ∈ X satisfying +Dist +� +�ζf + �λ�ζg, −NX (�x) +� +≤ ǫ, +|�λg(�x)| ≤ ǫ2, +g(�x) ≤ ǫ, +�λ ≥ 0, +(3) +where �λ ∈ R is a Lagrangian multiplier, �ζf ∈ ∂f(�x) and �ζg ∈ ∂g(�x). +However, when +f and g are non-smooth, computing an ǫ-stationary point with finite complexity is still +challenging even if there is no constraint. On the contrary, under weak convexity assumption +in Assumption 3.1A, it is possible to compute a nearly ǫ-stationary point defined later, which +is the goal of this paper. +Given any ˆρ > ρ and x ∈ X, the Moreau envelope and the proximal mapping of (1) are +defined as +ϕˆρ(x) ≡ min +y∈X +� +f(y) + ˆρ +2∥y − x∥2, s.t. g(y) ≤ 0 +� +, +(4) +�xˆρ(x) ≡ arg min +y∈X +� +f(y) + ˆρ +2∥y − x∥2, s.t. g(y) ≤ 0 +� +. +(5) +By Assumption 3.1A, the objective function in (4) and (5) is (ˆρ − ρ)-strongly convex and the +constraint function is convex, so �xˆρ(x) is uniquely defined. +Following the literature on weakly convex optimization [25, 27, 24, 12, 55, 40], we use the +value of ∥x − �xˆρ(x)∥ as a measure of the quality of a solution x because it can be interpreted +as a stationarity measure. To see this, consider the following KKT conditions of (4): +�ζf + ˆρ(�xˆρ(x) − x) + �λ�ζg ∈ −NX (�xˆρ(x)), +�λg(�xˆρ(x)) = 0, +g(�xˆρ(x)) ≤ 0, +�λ ≥ 0, +(6) +1We do not require that xfeas can be accessed by an algorithm. +4 + +where �ζf ∈ ∂f(�xˆρ(x)) and �ζg ∈ ∂g(�xˆρ(x)). These imply +Dist +� +�ζf + �λ�ζg, −NX (�xˆρ(x)) +� +≤ ˆρ∥�xˆρ(x) − x∥, +�λg(�xˆρ(x)) = 0, +g(�xˆρ(x)) ≤ 0, +�λ ≥ 0. +Therefore, as long as ∥�xˆρ(x) − x∥ ≤ ǫ, we have +Dist +� +�ζf + �λ�ζg, −NX (�xˆρ(x)) +� +≤ ˆρǫ, +�λg(�xˆρ(x)) = 0, and g(�xˆρ(x)) ≤ 0, which means �xˆρ(x) is an ˆρǫ-stationary point of the original +problem (1) by definition in (3). Since x is within an ǫ-distance from �xˆρ(x), we call such an +x a nearly ǫ-stationary point of (1). +Definition 3.2. Given ˆρ > ρ, a point x ∈ X is a nearly ǫ-stationary point of (1) if ∥�xˆρ(x) − +x∥ ≤ ǫ where �xˆρ(x) is defined in (5). +A random vector x ∈ X is a stochastic nearly ǫ- +stationary point of (1) if E∥�xˆρ(x) − x∥ ≤ ǫ. +Next, we will introduce a numerical method for finding a (stochastic) nearly ǫ-stationary +point of (1) with theoretically proved oracle complexity. Here, the oracle complexity is de- +fined as how many times the algorithm needs to query the subgradient or function value +of f or g through a deterministic or stochastic oracle in order to find a (stochastic) nearly +ǫ-stationary point. Our results are presented separately when the constraints are determin- +istic and stochastic. In both cases, the objective function f can be either deterministic or +stochastic, which does not affect the complexity. +Before showing the main results, we present the following lemma, which says any �λ in (6) +can be bounded from above by a constant independent of x. The proof is in Section A.1. +Lemma 3.3. Suppose Assumption 3.1 holds. Given any x ∈ X and ˆρ > ρ, let �xˆρ(x) defined +as in (5) and �λ is the associated Lagrangian multiplier satisfying (6). We have +�λ ≤ Λ := MD + ˆρD2 +−g(xfeas) +(7) +where M, D and xfeas are as in Assumption 3.1. +4 +Deterministic Constraints +In this section, we assume the subgradient and function value of g can be computed deter- +ministically. This includes the case when (1) has multiple convex deterministic constraints +gi(x) ≤ 0, i = 1, . . . , m, by having g(x) ≡ +max +i=1,...,m gi(x). This assumption is specified below. +Assumption 4.1. For any x ∈ X, g(x) and an ζg ∈ ∂g(x) can be evaluated exactly and a +stochastic subgradient ξf can be generated such that E ξf ∈ ∂f(x) and E exp +� +∥ξf∥2/M2� +≤ +exp(1) for any x ∈ X. +5 + +Algorithm 1 Switching Subgradient Method for Deterministic Constraints +1: Input: x(0) ∈ X, total number of iterations T, tolerance of infeasibility ǫt > 0, t = +0, 1, . . . , T − 1, and a starting index S for generating outputs. +2: Initialization: I ← ∅ and J ← ∅ +3: for iteration t = 0, 1, · · · , T − 1 do +4: +if g(x(t)) ≤ ǫt then +5: +Generate a stochastic subgradient ξ(t) +f +of f at x(t). +6: +x(t+1) ← projX (x(t) − ηtξ(t) +f ) +7: +I ← I ∪ {t} if t ≥ S +8: +else +9: +Generate a subgradient ζ(t) +g +of g at x(t). +10: +x(t+1) ← projX (x(t) − ηtζ(t) +g ) +11: +J ← J ∪ {t} if t ≥ S +12: +end if +13: end for +14: Output: x(τ) with τ randomly sampled from I with τ = t in a probability of ηt / � +t∈I ηt. +Assumption 4.1 means the distribution of ∥ξf∥2 has a light tail, which is a common +assumption for proving a stochastic first-order method converges in a high probability. See +(2.50) in [60] for an example. +We present the SSG method under Assumption 4.1 in Algorithm 1. This algorithm is +easy to implement and intuitive. At iteration t, we check if the current solution x(t) is nearly +feasible in the sense that g(x(t)) ≤ ǫt for a pre-determined tolerance of infeasibility ǫt. If yes, +the algorithm prioritizes reducing the objective value by taking a projected subgradient step +along the (stochastic) subgradient of f. If x(t) is not nearly feasible, the algorithm prioritizes +reducing the infeasibility by switching the updating direction to the subgradient of g. The +algorithm records the iteration indexes of the nearly feasible solutions in set I and other +indexes in set J. The final output is randomly sampled from the iterates whose indexes are +in I with a distribution weighted by the step sizes ηt’s. We also introduce an starting index +S so the algorithm only starts to record I and J when t ≥ S. +For simplicity of notation, we denote �xˆρ(x(t)) defined in (5) by �x(t). +Let Eτ be the +expectation taken only over the random index τ when the algorithms stop. We present the +convergence properties of Algorithm 1 separately when the subgradient of f is stochastic +and deterministic and when ǫt and ηt are static and diminishing. The complexity for finding +a stochastic nearly ǫ-stationary point T is the same in these four cases. The proof of the +following theorem is provided in Section A.3. +Theorem 4.2. Suppose Assumptions 3.1 and 4.1 hold and ˆρ > ρ, ǫ > 0 and Λ is defined in +(7). Moreover, suppose ξf is deterministic, namely, ξf = ζf ∈ ∂f(x) for any x ∈ X. Then +Algorithm 1 guarantees Eτ∥�x(τ) − x(τ)∥ ≤ ǫ and Eτg(x(τ)) ≤ ǫ2(ˆρ−ρ) +1+Λ +in either of the following +cases. +Case I: S = 0, ǫt = ǫ2(ˆρ−ρ) +1+Λ , ηt = +2ǫ2(ˆρ−ρ) +5(1+Λ)M2 and T ≥ 25M2D2(1+Λ)2 +4ǫ4(ˆρ−ρ)2 += O +� +1/ǫ4� +. +Case II: S = T +2 , ǫt = 5MD +√t+1, ηt = +D +M√t+1 and T ≥ 50M2D2(1+Λ)2 +ǫ4(ˆρ−ρ)2 += O +� +1/ǫ4� +. +6 + +Algorithm 1 is single-loop with O(1) oracle complexity per iteration, so its total complexity +is just T = O +� +1/ǫ4� +. +Theorem 4.3. Suppose Assumptions 3.1 and 4.1 hold and ˆρ > ρ, δ ∈ (0, 1), ǫ > 0 and Λ +is defined in (7). Then Algorithm 1 guarantees Eτ∥�x(τ) − x(τ)∥ ≤ ǫ and Eτg(x(τ)) ≤ ǫ2(ˆρ−ρ) +1+Λ +with probability at least 1 − δ in either of the following cases. +Case I: S, ǫt and ηt are chosen as Case I in Theorem 4.2 and +T ≥ + + + + + +25M2D2(1+Λ)2 +4ǫ4(ˆρ−ρ)2 +, +max{12 ln(8/δ), 16 +9 ln2(8/δ)}, +300 ln(4/δ)M2D2(1+Λ)2 +ǫ4(ˆρ−ρ)2 + + + + + += O +� 1 +ǫ4 +� +, +Case II: S and ηt are chosen as Case II in Theorem 4.2, ǫt = EMD +√t+1 where E is any +positive constant such that +E ≥ 4 + 2π +√ +6 max{ +� +12 ln(8/δ), 4 +3 ln(8/δ)} + 8 +� +3 ln(4/δ) +and T ≥ 2E2M2D2(1+Λ)2 +ǫ4(ˆρ−ρ)2 += O +� +1/ǫ4� +. +This theorem shows that, the complexity remains O(1/ǫ4) if the subgradient oracle of f +is stochastic. The only difference is that the result holds in a high probability. The proof can +be found in Section A.4. This complexity matches the lower-bound complexity for stochastic +smooth non-convex unconstrained optimization [3, 31], so it is optimal. In fact, we can also +obtain the same complexity if the subgradient oracles of both f and g are stochastic as long +as the function value oracle of g remains deterministic. Since the analysis and conclusion are +very similar to Theorem 4.3, we skip this result. +Remark 4.4. Property Eτg(x(τ)) ≤ ǫ2(ˆρ−ρ) +1+Λ +in the theorems above is not required by Defini- +tion 3.2. By Assumption 3.1, +Eτg(x(τ)) ≤ Eτg(�x(τ)) + MEτ∥�x(τ) − x(τ)∥ ≤ Mǫ, +which means a nearly ǫ-stationary point must be O(ǫ)-feasible by definition. +Property +Eτg(x(τ)) ≤ ǫ2(ˆρ−ρ) +1+Λ +implies O(ǫ2)-feasibility for the output, which is even better. +Remark 4.5. Strong convexity often leads to lower complexity for convex optimization, so an +interesting question is how the complexity of the SSG method changes if when g is µ-strongly +convex. We can show that, given strong convexity in g, the complexity is still O +� +1/ǫ4� +but +one can simply set ǫt = 0, which makes ηt the only tuning parameter. +This makes this +single-loop method even more attractive. We include this result in Section B. +5 +Stochastic Constraint +In this section, we consider the case where the oracles of the subgradient and function value +of g are stochastic. The assumption is formally stated below. +Assumption 5.1. For any x ∈ X, stochastic subgradients ξf and ξg and a stochastic value +ω can be generated independently such that E ω ∈ ∂g(x), E ξf ∈ ∂f(x) and E ξg ∈ +∂g(x). +Moreover, it holds that E exp +� +∥ξf∥2/M2� +≤ exp(1), E exp +� +∥ξg∥2/M2� +≤ exp(1), +and E exp +� +(ω − g(x))2/σ2� +≤ exp(1) for a constant σ for any x ∈ X. +7 + +Algorithm 2 Switching Subgradient Method for a Stochastic Constraint +1: Input: x(0) ∈ X, total number of iterations T, tolerance of infeasibility ǫt > 0, t = +0, 1, . . . , T − 1, mini-batch size B, and a starting index S for generating outputs. +2: Initialization: I ← ∅ and J ← ∅ +3: for iteration t = 0, 1, · · · , T − 1 do +4: +Generate a mini-batch of stochastic estimators of g at x(t), denoted by {ω(t) +i }B +i=1. +5: +¯ω(t) ← 1 +B +�B +i=1 ω(t) +i +6: +if ω(t) ≤ ǫt then +7: +Generate a stochastic subgradient ξ(t) +f +of f at x(t). +8: +x(t+1) ← projX (x(t) − ηtξ(t) +f ) +9: +I ← I ∪ {t} if t ≥ S +10: +else +11: +Generate a stochastic subgradient ξ(t) +g +of g at x(t). +12: +x(t+1) ← projX (x(t) − ηtξ(t) +g ) +13: +J ← J ∪ {t} if t ≥ S +14: +end if +15: end for +16: Output: x(τ) with τ randomly sampled from I with τ = t in a probability of ηt / � +t∈I ηt. +This case is fundamentally more challenging than the case with deterministic g. In fact, +the challenge comes only from the stochastic function value of g instead of its stochastic +subgradient. +In fact, in the fully deterministic case, Algorithm 1 essentially updates x(t) +along a hybrid subgradient +I(g(x(t)) ≤ ǫt)ζ(t) +f + I(g(x(t)) > ǫt)ζ(t) +g . +(8) +If only the subgradients are stochastic, the hybrid stochastic subgradient I(g(x(t)) ≤ ǫt)ξ(t) +f + +I(g(x(t)) > ǫt)ξ(t) +g +provides an unbiased estimation of (8), so we can still obtain complexity of +O(1/ǫ4) by a proof similar to Theorem 4.3. However, when g(x(t)) must be queried through +some unbiased estimator w(t), the naively constructed direction +I(w(t) ≤ ǫt)ξ(t) +f + I(w(t) > ǫt)ξ(t) +g +(9) +is not an unbiased estimator of (8). To tackle this issue, we have to query a mini-batch of +w(t) of size B to construct an estimator of g(x(t)), denoted by ¯w(t), with a high accuracy in a +high probability, so that (9) with w(t) replaced by ¯w(t) can be a nearly unbiased estimator of +(8). The SSG method with this modification is presented in Algorithm 2, where the condition +¯w(t) ≤ ǫt is used to determine if the stochastic subgradient should be switched. Its complexity +is characterized below when the control parameters are static and diminishing. The proof is +given in Section A.5. +Theorem 5.2. Suppose Assumptions 3.1 and 5.1 hold and ˆρ > ρ, δ ∈ (0, 1), ǫ > 0 and Λ is +defined in (7). Then Algorithm 2 guarantees Eτ∥�x(τ) − x(τ)∥ ≤ ǫ and Eτg(x(τ)) ≤ 2ǫ2(ˆρ−ρ) +1+Λ +with probability at least 1 − δ in either of the following cases. +8 + +Case I: If S, ǫt, ηt and T are chosen as Case I in Theorem 4.3 and B = 300σ2 ln(4T/δ)(Λ+1)2 +ǫ4(ˆρ−ρ)2 +. +Case II: If S, ǫt, ηt and T are chosen as Case II in Theorem 4.3 except that E is any +positive constant such that +E ≥ 8 + 2π +√ +6 +max{ +� +12 ln(8/δ), 4 +3 ln(8/δ)} + 8 +� +3 ln(4/δ) +and B = 3Tσ2 ln(2T/δ)(Λ+1)2 +2M2D2 +. +In each iteration of Algorithm 2, we query one stochastic subgradient of g or f but B +stochastic function values of g. In both Case I and Case II, we have T = O(1/ǫ4) and 2 +B = ˜O(1/ǫ4) so the subgradient oracle complexity is still O(1/ǫ4) but the function value +oracle complexity becomes O(1/ǫ8), which is higher than the O(1/ǫ6) complexity by the +double-loop methods in [12, 55]. It is our future work to reduce the complexity when g is +stochastic, for example, by a single-loop primal-dual method that uses a hybrid subgradient +like ξ(t) +f + λtξ(t) +g +with the dual variable λt updated by only one sample of w(t). +6 +Numerical Experiment +We demonstrate the performance of the SSG method on fairness-aware classification prob- +lems. It is implemented with both static and diminishing step sizes and compared with two +benchmarks. One is the constraint extrapolation (ConEx) method in Algorithm 1 in [12]. +It is a single-loop method for convex optimization, so it has no theoretical guarantee for +(1). The other is the inexact proximal point (IPP) method, which is a double-loop method +in [12, 55, 40]. It approximately solves a strongly convex constrained subproblem in each +outer iteration, and we use the SSG and ConEx methods as the solvers (inner loop) because +they both have the best theoretical complexity for that subproblem. We use IPP-SSG and +IPP-ConEx to denote these two implementations. +6.1 +Classification with ROC-based fairness +Given a feature vector a ∈ Rd and a class label b ∈ {1, −1}, the goal in a binary classification +problem is to learn a linear model w ∈ Rd to predict b based on the score w⊤a. +Let +D = {(ai, bi)}n +i=1 be a training set and ℓ(·) be a convex non-increasing loss function. Model +w can be learned by solving an empirical risk minimization problem +L∗ = min +w∈W +� +L(w) := 1 +n +n +� +i=1 +ℓ(biw⊤ai) +� +, +(10) +where W = {w ∈ Rd | ∥w∥ ≤ r}. Solving (10) may ensure good classification performance +of w but not its fairness. Suppose there exist two additional datasets containing the feature +vectors of a protected group Dp = {ap +i }np +i=1 and the feature vectors of an unprotected group +Du = {au +i }nu +i=1. We want to enhance the fairness of w between these two groups. There are +various fairness metrics in literature but we focus on the ROC-based fairness metric proposed +by [76]. Suppose we classify data a as positive if w⊤a ≥ θ and as negative otherwise, where θ +2 ˜O(·) suppresses logarithmic factors in the order of complexity. +9 + +Table 1: Information of the datasets. +Groups are males VS females in a9a and german, +users with age within [25, 60] VS outside [25, 60] in bank, and caucasian VS non-caucasian in +COMPAS. +Datasets +n +d +Label +Groups +a9a +48,842 +123 +Income +Gender +Bank +41,188 +54 +Subscription +Age +COMPAS +11,757 +14 +Recidivism +Race +german +1,000 +21 +Credit risk +Gender +is a threshold. The empirical ROC curve for these two groups is a curve in a 2D space whose +two coordinates are the predicted positive rates on the two groups, i.e., +1 +np +np +� +i=1 +I(w⊤ap +i ≥ θ) and 1 +nu +nu +� +i=1 +I(w⊤au +i ≥ θ), +as θ varies from −∞ to +∞. 3 The ROC-based fairness measure is defined as +max +θ∈Θ +��� 1 +np +np +� +i=1 +I(w⊤ap +i ≥ θ) − 1 +nu +nu +� +i=1 +I(w⊤au +i ≥ θ) +��� +or its continuous approximation +R(w) = max +θ∈Θ +��� 1 +np +np +� +i=1 +σ(w⊤ap +i ≥ θ) − 1 +nu +nu +� +i=1 +σ(w⊤au +i ≥ θ) +���, +(11) +where σ is the sigmoid function and Θ is a finite set of thresholds. If the value of this measure +is small, model w produces similar predicted positive rates for the protected and unprotected +groups on various θ’s, indicating the fairness of the model. To obtain a fair w, we balance +(10) and (11) by solving +min +w∈W R(w) s.t. L(w) ≤ L∗ + κ, +(12) +where κ is a slackness parameter indicating how much we are willing to increase the classifi- +cation loss in order to reduce R(w) to obtain a more fair model. Problem (12) is an instance +of (1) satisfying Assumption 3.1. +We solve (12) on three datasets: a9a [43], bank [58] and COMPAS [39]. The information +of these datasets is given in Table 1. We split each dataset into two subsets with a ratio of +2 : 1. The larger set is used as D in the constraint and the smaller set is further split into Dp +and Du based on the grouping variables in Table 1. +In our experiments, we first solve (10) using the subgradient method with a large enough +number of iterations to obtain L∗ and a solution wERM. Then we set κ = 0.001L∗, ℓ(z) = (1− +z)+ and r = 5∥wERM∥, and let Θ consist of 400 points equally spaced between mini w⊤ +ERMai− +0.5(maxi w⊤ +ERMai − mini w⊤ +ERMai) and maxi w⊤ +ERMai + 0.5(maxi w⊤ +ERMai − mini w⊤ +ERMai). +3It is different from the standard ROC curve whose coordinates are the predicted positive rates on positive +and negative classes. +10 + +ROC-Based Fair Classification +Wasserstein-Based Fair Classification +(Deterministic f) +(Stochastic f) +a9a +bank +COMPAS +german +COMPAS +Objective +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +Iteration t +104 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +0.22 +SSG-static +SSG-diminishing +ConEx +IPP-SSG +IPP-ConEx +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Iteration t +104 +0.005 +0.01 +0.015 +0.02 +0.025 +0.03 +0.035 +0.04 +0.045 +0.05 +0.055 +SSG-static +SSG-diminishing +ConEx +IPP-SSG +IPP-ConEx +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Iteration t +104 +0.008 +0.01 +0.012 +0.014 +0.016 +0.018 +0.02 +0.022 +0.024 +SSG-static +SSG-diminishing +ConEx +IPP-SSG +IPP-ConEx +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Iteration t +105 +0.6 +0.65 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +1.05 +SSG-static +SSG-diminishing +ConEx +IPP-SSG +IPP-ConEx +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Iteration t +105 +1.15 +1.2 +1.25 +1.3 +1.35 +1.4 +1.45 +1.5 +SSG-static +SSG-diminishing +ConEx +IPP-SSG +IPP-ConEx +Infeasibility +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +Iteration t +104 +-4 +-3 +-2 +-1 +0 +1 +2 +10-4 +SSG-static +SSG-diminishing +ConEx +IPP-SSG +IPP-ConEx +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Iteration t +104 +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +10-4 +SSG-static +SSG-diminishing +ConEx +IPP-SSG +IPP-ConEx +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Iteration t +104 +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +10-4 +SSG-static +SSG-diminishing +ConEx +IPP-SSG +IPP-ConEx +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Iteration t +105 +-4 +-3 +-2 +-1 +0 +1 +2 +10-3 +SSG-static +SSG-diminishing +ConEx +IPP-SSG +IPP-ConEx +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Iteration t +105 +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +2.5 +3 +10-3 +SSG-static +SSG-diminishing +ConEx +IPP-SSG +IPP-ConEx +Near Stationarity +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Iteration t +104 +0.005 +0.01 +0.015 +0.02 +0.025 +0.03 +0.035 +0.04 +SSG-static +SSG-diminishing +ConEx +IPP-SSG +IPP-ConEx +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Iteration t +104 +0 +0.005 +0.01 +0.015 +0.02 +0.025 +0.03 +0.035 +0.04 +0.045 +0.05 +SSG-static +SSG-diminishing +ConEx +IPP-SSG +IPP-ConEx +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Iteration t +104 +0 +0.005 +0.01 +0.015 +0.02 +0.025 +0.03 +0.035 +0.04 +SSG-static +SSG-diminishing +ConEx +IPP-SSG +IPP-ConEx +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Iteration t +105 +0.4 +0.45 +0.5 +0.55 +0.6 +0.65 +SSG-static +SSG-diminishing +ConEx +IPP-SSG +IPP-ConEx +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Iteration t +105 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +SSG-static +SSG-diminishing +ConEx +IPP-SSG +IPP-ConEx +Figure 1: Performances on fairness-aware classification. +We use a deterministic oracle in all methods. For SSG with a static step size, we select +ǫt from {5 × 10−6, 10−5, 2 × 10−5} and ηt from {5 × 10−4, 10−3, 2 × 10−3}. For SSG with a +diminishing step size, we set ǫt = +E1 +√t+1 and ηt = +E2 +√t+1 and select E1 from {0.01, 0.02, 0.05} +and E2 from {10−4, 2 × 10−4, 5 × 10−4}. For ConEx, we set the primal step size ηt = +c1 +√t+1 +and dual step size τt = c2 and select c1 from {0.01, 0.02, 0.05} and c2 from {20, 50, 100}. +We set θt = 1 in ConEx. We select the best set of parameters that produces the smallest +objective value after 5000 iterations. For IPP, we select ˆρ from max{ρ, 1} × {1, 1.5, 2} with +ρ = �n +i=1 ∥ai∥/n, and the proximal point subproblem is approximately solved by SSG and +ConEx both with 100 iterations indexed by k. For IPP-SSG, we apply a static step size and +the parameters are tuned the same way as SSG. For IPP-ConEx, we set the primal step size +ηk = c1(k + 1) and dual step size τk = +c2 +k+1 and select c1 from {20, 50, 100} and c2 from +{0.005, 0.01, 0.02}. We set θk = +k +k+1 in IPP-ConEx. +We report the performances of all methods on each dataset in the first three columns of +Figure 1. The x-axis represents the number of iterations. (For IPP, it represents the total +number of inner iterations across all outer iterations.) The y-axis represents the objective +value, infeasibility and near stationarity achieved at each iteration, respectively, in the three +rows. +To measure near stationarity, we solve (4) with x = x(t) using the SSG method +with sufficient iterations and use the last iterate as an approximation of �xˆρ(x(t)). Then we +plot ∥�xˆρ(x(t)) − x(t)∥ as near stationarity in Figure 1. Since computing �xˆρ(x(t)) with a high +precision is time-consuming, we only report near stationarity at 100 equally spaced iterations. +According to Figure 1, the ConEx method has the best performance overall, but it does +not have theoretical convergence guarantee. The IPP method and SSG method with static +11 + +step sizes have similar efficiency in reducing the objective value while keeping the solutions +nearly feasible on the instances we test on. The SSG method with diminishing step sizes +performs slightly better than these two on bank and COMPAS datasets. This is consistent +with our theory that the SSG and IPP methods have the same complexity. +6.2 +Classification with Wasserstein-based fairness +Wasserstein distance, which is a distance between two distributions, has also been considered +as a metric of fairness [42, 21]. Given a linear classifier w, the 1-Wasserstein distance between +the empirical distributions of the prediction scores on Du and Du is the optimal value of linear +program +min +P ∈P +� +H(P, w) := +np +� +i=1 +nu +� +j=1 +Pij|w⊤ap +i − w⊤au +i | +� +, +(13) +where P = {P ∈ Rnp×nu| �np +i=1 Pij = +1 +nu , �nu +j=1 Pij = +1 +np, Pij ≥ 0}. If this value is small, +the prediction score has similar distributions on the protected and unprotected groups. To +build a fair model, we just need to balance (10) and (13) by solving +min +P ∈P,w∈W H(P, w) s.t. L(w) ≤ L∗ + κ, +(14) +which is also an instance of (1) satisfying Assumption 3.1. +We solve (14) on datasets german [32, 18] and COMPAS [39] whose information is in +Table 1. We set κ and r and split each dataset into D, Dp and Du in the same way as in +the previous subsection. In each iteration of all methods in comparison, we need to project +P (t) to P, for which we use the algorithm and codes by [51]. Since their algorithm assume +np = nu, we further downsample the larger one of Dp and Du to the size of the smaller one. +We use a stochastic oracle for f. In fact, we generate the stochastic subgradient of H(P, w) +by sampling B = ⌊np +10 ⌋ = ⌊nu +10 ⌋ samples from Dp and Du, respectively, and forming B2 pairs +of protected and unprotected instances. Then, the stochastic subgradient is constructed only +using the summands of ∂wH(P, w) and the coordinates of ∂P H(P, w) that are associated to +the those pairs. +We use a deterministic oracle for g in all methods. +This is justified by the fact that +f has a quadratic computational cost, i.e., O(npnu), while g has only a linear cost O(n). +For SSG with a static step size, we select ǫt from {5 × 10−6, 10−5, 2 × 10−5} and ηt from +{0.01, 0.015, 0.02}. For SSG with a diminishing step size, we set ǫt = +E1 +√t+1 and ηt = +E2 +√t+1 +and select E1 from {0.05, 0.1, 0.15} and E2 from {10−4, 2 × 10−4, 5 × 10−4}. +For ConEx, +we set the primal step size ηt = +c1 +√t+1 and dual step size τt = +c2 +√t+1 and select c1 from +{0.01, 0.02, 0.05} and c2 from {20, 50, 100}. We set θt = 1 in ConEx. We select the best set +of parameters that produces the smallest objective value after 15000 iterations. For IPP, we +select ˆρ from max{ρ, 1} × {10−4, 5 × 10−4, 10−3} with ρ = ∥Ap∥F + ∥Au∥F where Ap is the +np × d matrix whose each row is a⊤ +i , Au is the nu × d matrix whose each row is a⊤ +i and ∥ · ∥F +is the Frobenius norm, and the proximal point subproblem is approximately solved by SSG +and ConEx both with 1000 iterations indexed by k. For IPP-SSG, we apply a static step +size and the parameters are tuned the same way as SSG. For IPP-ConEx, we set the primal +step size ηk = +c1 +√k+1 and dual step size τk = +c2 +k+1 and select c1 from {100, 200, 500} and c2 +12 + +from {0.005, 0.01, 0.02}. We set θk = +k +k+1 in IPP-ConEx. The performances of the methods +are given in the last two columns of Figure 1. This time ConEx does not perform as well as +it did for the previous application. 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On the iteration complexity of smoothed +proximal alm for nonconvex optimization problem with convex constraints. +arXiv +preprint arXiv:2207.06304, 2022. +[83] Jingzhao Zhang, Hongzhou Lin, Stefanie Jegelka, Ali Jadbabaie, and Suvrit Sra. Com- +plexity of finding stationary points of nonsmooth nonconvex functions. arXiv preprint +arXiv:2002.04130, 2020. +[84] Siqi Zhang and Niao He. +On the convergence rate of stochastic mirror descent for +nonsmooth nonconvex optimization. arXiv preprint arXiv:1806.04781, 2018. +19 + +A +Convergence Analysis +In this section, we present the proofs of all lemmas, propositions and theorems in the paper. +A.1 +Proof of Lemma 3.3 +Proof of Lemma 3.3. For simplicity of notation, we denote �xˆρ(x) in (5) by �x. +According to Assumption 3.1B, there exists a strictly feasible solution xfeas with xfeas ∈ X +and g(xfeas) < 0. As a result, the Lagrangian multiplier �λ ≥ 0 corresponding to �x is well- +defined and satisfies (6), which means �λg(�x) = 0 and +�ζf + ˆρ(�x − x) + �λ�ζg + �v = 0, +(15) +where �ζf ∈ ∂f(�x), �ζg ∈ ∂g(�x), �v ∈ NX (�x) and NX (�x) is the normal cone of X at �x. +If �λ = 0, the conclusion holds trivially. Hence, we focus on the case that �λ > 0. Note +that, in this case, we must have g(�x) = 0 since �λg(�x) = 0. Taking the inner product between +(15) and �x − xfeas gives +1 +�λ +⟨�ζf + ˆρ(�x − x), �x − xfeas⟩ = ⟨�ζg + �v/�λ, xfeas − �x⟩ ≤ g(xfeas) − g(�x) = g(xfeas), +(16) +where the inequality is because of convexity of g and the fact that �v/�λ ∈ NX(�x). Note that +Assumption 4.1 and Jensen’s inequality imply ∥�ζf∥ ≤ M. Moreover, we have ∥�x−xfeas∥ ≤ D +and ∥�x − x∥ ≤ D by Assumption 3.1C. Hence, from (16), we can obtain +�λ ≤ ⟨�ζf + ˆρ(�x − x), xfeas − �x⟩ +−g(xfeas) +≤ MD + ˆρD2 +−g(xfeas) . +A.2 +Technical Lemmas and Propositions +In this section, we first introduce additional notations and then present a few technical lemmas +and propositions which are needed to prove the convergence of the proposed algorithms. We +want to remind readers that we assume Assumption 3.1 for the entire paper so this assumption +will not be stated again in each lemma and proposition. Since the case under Assumption 5.1 +is more general than the case under Assumption 4.1, in the propositions below, the conclusions +under Assumption 5.1 will be presented before those under Assumption 5.1. +Let I[·] be an indicator of an event, which equals one if the event occurs and zero +otherwise. For each iterate x(t) in Algorithms 1 and +2, let ζ(t) +f +:= Eξ(t) +f +∈ ∂f(x(t)) and +ζ(t) +g +:= Eξ(t) +g +∈ ∂g(x(t)) and let �x(t) := �xˆρ(x(t)) defined in (5) and �λt ≥ 0 be the corresponding +Lagrangian multiplier satisfying (6), which exists by Assumption 3.1B. Under Assumption 5.1, +let Et[·] := E[·|ξ(0) +f , ξ(0) +g , ¯ω(0), ξ(1) +f , ξ(1) +g , , ¯ω(1), . . . , ξ(t−1) +f +, ξ(t−1) +g +, ¯ω(t−1)], i.e., the conditional ex- +pectation conditioning on all stochastic events before iteration t. Let Eτ be the expectation +taken only over the random index τ when the algorithm is terminated. +We first provide a proposition that characterizes the relationship between ∥x(t) − �x(t)∥ +and the control parameters T, S, B, ηt and ǫt. +20 + +Proposition A.1. Under Assumption 5.1 and assuming g is µ-strongly convex (µ can be +zero), Algorithm 2 guarantees +T−1 +� +t=S +� +ηtˆρ�λtI(¯ω(t) ≤ ǫt) + ηtˆρI(¯ω(t) > ǫt) +� µ +2∥�x(t) − x(t)∥2 ++ +T−1 +� +t=S +� +ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtg(x(t)) +� +I(¯ω(t) ≤ ǫt) + +T−1 +� +t=S +ηtˆρg(x(t))I(¯ω(t) > ǫt) +≤ +ˆρD2 +2 ++ ˆρ +2 +T−1 +� +t=S +� +η2 +t ∥ξ(t) +f ∥2I(¯ω(t) ≤ ǫt) + η2 +t ∥ξ(t) +g ∥2I(¯ω(t) > ǫt) +� ++ +T−1 +� +t=S +� +ˆρηt +� +ξ(t) +f − ζ(t) +f , �x(t) − x(t)� +I(¯ω(t) ≤ ǫt) + ˆρηt +� +ξ(t) +g − ζ(t) +g , �x(t) − x(t)� +I(¯ω(t) > ǫt) +� +. +As a special case of the result above, under Assumption 4.1 and assuming g is µ-strongly +convex (µ can be zero), Algorithm 1 guarantees +T−1 +� +t=S +� +ηtˆρ�λtI(g(x(t)) ≤ ǫt) + ηtˆρI(g(x(t)) > ǫt) +� µ +2 ∥�x(t) − x(t)∥2 ++ +T−1 +� +t=S +� +ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtg(x(t)) +� +I(g(x(t)) ≤ ǫt) + +T−1 +� +t=S +ηtˆρg(x(t))I(g(x(t)) > ǫt) +≤ +ˆρD2 +2 ++ ˆρ +2 +T−1 +� +t=S +� +η2 +t ∥ξ(t) +f ∥2I(g(x(t)) ≤ ǫt) + η2 +t ∥ζ(t) +g ∥2I(g(x(t)) > ǫt) +� ++ +T−1 +� +t=S +� +ˆρηt +� +ξ(t) +f − ζ(t) +f , �x(t) − x(t)� +I(g(x(t)) ≤ ǫt) +� +. +Proof. Let ξ = ξ(t) +f +if t ∈ I and ξ = ξ(t) +g +if t ∈ J. Similarly, let ζ = ζ(t) +f +∈ ∂f(x(t)) if t ∈ I +and ζ = ζ(t) +g +∈ ∂g(x(t)) if t ∈ J. +By the updating equation of x(t+1), we have +∥x(t+1) − �x(t)∥2 += +∥projX (x(t) − ηtξ(t)) − �x(t)∥2 = ∥projX (x(t) − ηtξ(t)) − projX (�x(t))∥2 +≤ +∥x(t) − ηtξ(t) − �x(t)∥2 = ∥x(t) − �x(t)∥2 − 2ηt +� +ξ(t), x(t) − �x(t)� ++ η2 +t ∥ξ(t)∥2. +Multiplying the inequality above by ˆρ/2 and adding f(�x(t)) to both sides, we obtain +f(�x(t)) + ˆρ +2∥x(t+1) − �x(t)∥2 +≤ f(�x(t)) + ˆρ +2∥x(t) − �x(t)∥2 − ˆρηt +� +ξ(t), x(t) − �x(t)� ++ ˆρη2 +t +2 ∥ξ(t)∥2 +(17) +≤ ϕˆρ(x(t)) − ˆρηt +� +ζ(t), x(t) − �x(t)� ++ ˆρη2 +t +2 ∥ξ(t)∥2 − ˆρηt +� +ξ(t) − ζ(t), x(t) − �x(t)� +, +(18) +21 + +where the second inequality is by the definition of ϕˆρ(x) in (4). Since �x(t) is a feasible solution +to problem (4) with x = x(t+1), we have +ϕˆρ(x(t+1)) ≤ f(�x(t)) + ˆρ +2∥x(t+1) − �x(t)∥2, +which, together with (18), implies +ˆρηt +� +ζ(t), x(t) − �x(t)� +≤ ϕˆρ(x(t)) − ϕˆρ(x(t+1)) + ˆρη2 +t +2 ∥ξ(t)∥2 − ˆρηt +� +ξ(t) − ζ(t), x(t) − �x(t)� +(19) +Next, we will bound +� +ζ(t), x(t) − �x(t)� +from below when t ∈ I and t ∈ J, separately. +Suppose t ∈ I so ¯ω(t) ≤ ǫt, ζ(t) = ζ(t) +f +and ξ(t) = ξ(t) +f . By ρ-weak convexity of f, we have +� +ζ(t), x(t) − �x(t)� +≥ +f(x(t)) − f(�x(t)) − ρ +2∥�x(t) − x(t)∥2 += +f(x(t)) − f(�x(t)) − ˆρ +2∥�x(t) − x(t)∥2 + ˆρ − ρ +2 +∥�x(t) − x(t)∥2 +(20) +Consider the convex optimization problem (4) with x = x(t). By Assumption 3.1B, there +exists a Lagrangian multiplier �λt ≥ 0 such that �λtg(�x(t)) = 0 (complementory slackness) and +�x(t) = arg min +x∈X +f(x) + ˆρ +2∥x − x(t)∥2 + �λtg(x). +Since the objective function above is (ˆρ − ρ + �λtµ)-strongly convex, we have +f(x(t)) + �λtg(x(t)) += +f(x(t)) + ˆρ +2∥x(t) − x(t)∥2 + �λtg(x(t)) +≥ +f(�x(t)) + ˆρ +2∥�x(t) − x(t)∥2 + �λtg(�x(t)) + ˆρ − ρ + �λtµ +2 +∥�x(t) − x(t)∥2, +which, by the facts that �λt ≥ 0 and �λtg(�x(t)) = 0, implies +f(x(t)) − f(�x(t)) − ˆρ +2∥�x(t) − x(t)∥2 ≥ −�λtg(x(t)) + ˆρ − ρ + �λtµ +2 +∥�x(t) − x(t)∥2. +Applying this inequality and inequality (20) to (19) leads to +ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtg(x(t)) + ηtˆρ�λtµ +2 +∥�x(t) − x(t)∥2 +≤ +ϕˆρ(x(t)) − ϕˆρ(x(t+1)) + ˆρη2 +t +2 ∥ξ(t) +f ∥2 − ˆρηt +� +ξ(t) +f − ζ(t) +f , x(t) − �x(t)� +. +(21) +Suppose t ∈ J so ¯ω(t) > ǫt, ζ(t) = ζ(t) +g +and ξ(t) = ξ(t) +g . By µ-convexity of g and the fact +that g(�x(t)) ≤ 0, we have +� +ζ(t), x(t) − �x(t)� +− µ +2 ∥�x(t) − x(t)∥2 ≥ g(x(t)) − g(�x(t)) ≥ g(x(t)). +22 + +Applying this inequality to (19) leads to +ηtˆρg(x(t)) + ηtˆρµ +2 +∥�x(t) − x(t)∥2 ≤ ϕˆρ(x(t)) − ϕˆρ(x(t+1)) + ˆρη2 +t +2 ∥ξ(t) +g ∥2 − ˆρηt +� +ξ(t) +g − ζ(t) +g , x(t) − �x(t)� +.(22) +Recall that I[t ∈ I] = I[¯ω(t) ≤ ǫt] and I[t ∈ J] = I[¯ω(t) > ǫt]. Summing up (21) and (22) +for t = S, S + 1, . . . , T − 1, we have +T−1 +� +t=S +� +ηtˆρ�λtI(¯ω(t) ≤ ǫt) + ηtˆρI(¯ω(t) > ǫt) +� µ +2∥�x(t) − x(t)∥2 ++ +T−1 +� +t=S +� +ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtg(x(t)) +� +I(¯ω(t) ≤ ǫt) + +T−1 +� +t=S +ηtˆρg(x(t))I(¯ω(t) > ǫt) +≤ +ϕˆρ(x(S)) − ϕˆρ(x(T)) + ˆρ +2 +T−1 +� +t=S +� +η2 +t ∥ξ(t) +f ∥2I(¯ω(t) ≤ ǫt) + η2 +t ∥ξ(t) +g ∥2I(¯ω(t) > ǫt) +� +− +T−1 +� +t=S +� +ˆρηt +� +ξ(t) +f − ζ(t) +f , x(t) − �x(t)� +I(¯ω(t) ≤ ǫt) + ˆρηt +� +ξ(t) +g − ζ(t) +g , x(t) − �x(t)� +I(¯ω(t) > ǫt) +� +. +Let x∗ be the optimal solution of (1). The first conclusion is then implied by the facts that +ϕˆρ(x(T)) ≥ f ∗ and ϕˆρ(x(S)) ≤ f(x∗) + ˆρ +2∥x∗ − x(S)∥2 ≤ f ∗ + ˆρD2 +2 , where f ∗ is finite by +Assumption 3.1D. The second conclusion is a special case of the first one when ¯ω(t) = g(x(t)) +and ξ(t) +g += ζ(t) +g . +Lemma A.2. Suppose Assumption 5.1 holds. Given any x ∈ X, let {ωi}B +i=1 be a mini-batch +of stochastic estimators of g at x and ¯ω = 1 +B +�B +i=1 ωi. It holds that, for any δ ∈ (0, 1), +Prob +� +¯ω > g(x) + +√ +3σ +� +ln(1/δ) +√ +B +� +≤ δ and Prob +� +¯ω < g(x) − +√ +3σ +� +ln(1/δ) +√ +B +� +≤ δ +Proof. The conclusion is guaranteed by Assumption 5.1 and Lemma 2 (Case A) in [49] by +choosing Ω = +� +3 ln(1/δ) in their bound. +Lemma A.3. Suppose Assumption 5.1 holds. For any δ ∈ (0, 1), Algorithm 2 guarantees +with probability at least 1 − δ that +g(x(t))I(¯ω(t) ≤ ǫt) +≤ +� +¯ω(t) + +√ +3σ +� +ln((T − S)/δ) +√ +B +� +I(¯ω(t) ≤ ǫt) +≤ +� +ǫt + +√ +3σ +� +ln((T − S)/δ) +√ +B +� +I(¯ω(t) ≤ ǫt) +(23) +for t = S, S + 1, . . . , T − 1 and, consequently, +T−1 +� +t=S +ηtˆρ�λtg(x(t))I(¯ω(t) ≤ ǫt) ≤ +T−1 +� +t=S +ηtˆρ�λt +� +ǫt + +√ +3σ +� +ln((T − S)/δ) +√ +B +� +I(¯ω(t) ≤ ǫt) +23 + +and Algorithm 2 guarantees with probability at least 1 − δ that +T−1 +� +t=S +ηtˆρg(x(t))I(¯ω(t) > ǫt) ≥ +T−1 +� +t=S +ηtˆρ +� +ǫt − +√ +3σ +� +ln((T − S)/δ) +√ +B +� +I(¯ω(t) > ǫt). +Proof. For any t, x(t) is determined by ξ(0) +f , ξ(0) +g , ¯ω(0), ξ(1) +f , ξ(1) +g , , ¯ω(1), . . . , ξ(t−1) +f +, ξ(t−1) +g +and +¯ω(t−1), while ¯ω(t) is generated after x(t). Hence, we have (23) for each t with a probability +of at least 1 − δ/(T − S) according to Lemma A.2. The first conclusion is then obtained by +taking the union bound for the events above for t = S, . . . , T − 1. The second conclusion can +be proved in a similar way. +Lemma A.4. Suppose Assumption 5.1 holds. For any δ ∈ (0, 1), Algorithm 2 guarantees +with probability at least 1 − δ that +T−1 +� +t=S +� +η2 +t ∥ξ(t) +f ∥2I(¯ω(t) ≤ ǫt) + η2 +t ∥ξ(t) +g ∥2I(¯ω(t) > ǫt) +� +≤ +T−1 +� +t=S +η2 +t M2 + max{ +� +12 ln(2/δ), 4 +3 ln(2/δ)} +� +� +� +� +T−1 +� +t=S +η4 +t M4. +Suppose Assumption 4.1 holds. For any δ ∈ (0, 1), Algorithm 1 guarantees +T−1 +� +t=S +� +η2 +t ∥ξ(t) +f ∥2I(g(x(t)) ≤ ǫt) + η2 +t ∥ζ(t) +g ∥2I(g(x(t)) > ǫt) +� +≤ +T−1 +� +t=S +η2 +t M2 + max{ +� +12 ln(2/δ), 4 +3 ln(2/δ)} +� +� +� +� +T−1 +� +t=S +η4 +t M4. +Proof. For any t, x(t) is determined by ξ(0) +f , ξ(0) +g , ¯ω(0), ξ(1) +f , ξ(1) +g , , ¯ω(1), . . . , ξ(t−1) +f +, ξ(t−1) +g +and +¯ω(t−1). Also, ξ(t) +f , ξ(t) +g +and ¯ω(t) are independent and generated after x(t). Hence, by Assump- +tion 5.1, we have +Et exp + +η2 +t ∥ξ(t) +f ∥2I(¯ω(t) ≤ ǫt) + η2 +t ∥ξ(t) +g ∥2I(¯ω(t) > ǫt) +η2 +t M2 + + += Et + +I(¯ω(t) ≤ ǫt) exp + +∥ξ(t) +f ∥2 +M2 + + + + + Et +� +I(¯ω(t) > ǫt) exp +� +∥ξ(t) +g ∥2 +M2 +�� += EtI(¯ω(t) ≤ ǫt)Et exp + +∥ξ(t) +f ∥2 +M2 + + + EtI(¯ω(t) > ǫt)Et exp +� +∥ξ(t) +g ∥2 +M2 +� +≤ EtI(¯ω(t) ≤ ǫt) exp (1) + EtI(¯ω(t) > ǫt) exp (1) = exp (1) , +where the second equality is by the conditional independence between ω(t), ξ(t) +f +and ξ(t) +g . +Then the first conclusion is guaranteed by Lemma 2 (Case B) in [49] by choosing Ω = +24 + +max{ +� +12 ln(2/δ), 4 +3 ln(2/δ)} in their bound. The second conclusion is a special case of the +first one when ¯ω(t) = g(x(t)) and ξ(t) +g += ζ(t) +g +and thus can be proved similarly. +Lemma A.5. Suppose Assumption 5.1 holds. For any δ ∈ (0, 1), Algorithm 2 guarantees +with probability at least 1 − δ that +T−1 +� +t=S +� +ˆρηt +� +ξ(t) +f − ζ(t) +f , �x(t) − x(t)� +I(¯ω(t) ≤ ǫt) + ˆρηt +� +ξ(t) +g − ζ(t) +g , �x(t) − x(t)� +I(¯ω(t) > ǫt) +� +≤ +� +3 ln(1/δ) +� +� +� +� +T−1 +� +t=S +4ˆρ2η2 +t M2D2. +Suppose Assumption 4.1 holds. For any δ ∈ (0, 1), Algorithm 1 guarantees +T−1 +� +t=S +ˆρηt +� +ξ(t) +f − ζ(t) +f , �x(t) − x(t)� +I(g(x(t)) ≤ ǫt) ≤ +� +3 ln(1/δ) +� +� +� +� +T−1 +� +t=S +4ˆρ2η2 +t M2D2. +Proof. By Assumption 5.1 and Jensen’s inequality, we have +exp + +∥ζ(t) +f ∥2 +M2 + + ≤ Et exp + +∥ξ(t) +f ∥2 +M2 + + ≤ exp(1) and exp +� +∥ζ(t) +g ∥2 +M2 +� +≤ Et exp +� +∥ξ(t) +g ∥2 +M2 +� +≤ exp(1) +which, by Jensen’s inequality again, implies +Et exp + +2∥ξ(t) +f ∥2 + 2∥ζ(t) +f ∥2 +4M2 + + = Et exp + +∥ξ(t) +f ∥2 +2M2 + + exp + +∥ζ(t) +f ∥2 +2M2 + + +≤ +� +� +� +� +�Et exp + +∥ξ(t) +f ∥2 +M2 + + exp +�1 +2 +� +≤ exp(1) +(24) +Et exp +� +2∥ξ(t) +g ∥2 + 2∥ζ(t) +g ∥2 +4M2 +� += Et exp +� +∥ξ(t) +g ∥2 +2M2 +� +exp +� +∥ζ(t) +g ∥2 +2M2 +� +≤ +� +� +� +�Et exp +� +∥ξ(t) +g ∥2 +M2 +� +exp +�1 +2 +� +≤ exp(1). +(25) +For any t, x(t) is determined by ξ(0) +f , ξ(0) +g , ¯ω(0), ξ(1) +f , ξ(1) +g , , ¯ω(1), . . . , ξ(t−1) +f +, ξ(t−1) +g +and ¯ω(t−1). +Also, ξ(t) +f , ξ(t) +g +and ¯ω(t) are independent and generated after x(t). Hence, by Assumption 5.1, +we have +Et +� +ˆρηt +� +ξ(t) +f − ζ(t) +f , �x(t) − x(t)� +I(¯ω(t) ≤ ǫt) + ˆρηt +� +ξ(t) +g − ζ(t) +g , �x(t) − x(t)� +I(¯ω(t) > ǫt) +� += 0. +25 + +and +Et exp + + + +� +ˆρηt +� +ξ(t) +f +− ζ(t) +f , �x(t) − x(t)� +I(¯ω(t) ≤ ǫt) + ˆρηt +� +ξ(t) +g +− ζ(t) +g , �x(t) − x(t)� +I(¯ω(t) > ǫt) +�2 +4ˆρ2η2 +t M 2D2 + + + += Et + +I(¯ω(t) ≤ ǫt) exp + + + +�� +ξ(t) +f +− ζ(t) +f , �x(t) − x(t)��2 +4M 2D2 + + + + + + Et + +I(¯ω(t) > ǫt) exp + + + +�� +ξ(t) +g +− ζ(t) +g , �x(t) − x(t)��2 +4M 2D2 + + + + + +≤ EtI(¯ω(t) ≤ ǫt)Et exp +� +∥ξ(t) +f +− ζ(t) +f ∥2∥�x(t) − x(t)∥2 +4M 2D2 +� ++ EtI(¯ω(t) > ǫt)Et exp +� +∥ξ(t) +g +− ζ(t) +g ∥2∥�x(t) − x(t)∥2 +4M 2D2 +� +≤ EtI(¯ω(t) ≤ ǫt)Et exp +� +2∥ξ(t) +f ∥2 + 2∥ζ(t) +f ∥2 +4M 2 +� ++ EtI(¯ω(t) > ǫt)Et exp +� +2∥ξ(t) +g ∥2 + 2∥ζ(t) +g ∥2 +4M 2 +� +≤ EtI(¯ω(t) ≤ ǫt) exp (1) + EtI(¯ω(t) > ǫt) exp (1) = exp (1) , +where the first inequality is by the Cauchy–Schwarz inequality and the conditional indepen- +dence between ω(t), ξ(t) +f +and ξ(t) +g , the second inequality is by Assumption 3.1C and the last +inequality is by (24) and (25). Then the first conclusion is guaranteed by Lemma 2 (Case A) +in [49] by choosing Ω = +� +3 ln(1/δ) in their bound. The second conclusion is a special case +of the first one when ¯ω(t) = g(x(t)) and ξ(t) +g += ζ(t) +g +and thus can be proved similarly. +Suppose g is convex but not strongly convex, i.e., µ = 0. Taking the union bound of +the four events in Lemmas A.3, A.4 and A.5 with δ replaced by δ +4 and applying the four +inequalities (two from Lemma A.3, one from Lemma A.4 and one from Lemma A.5) holding +in these events to Proposition A.1, we have the following bounds. +Proposition A.6. Under Assumption 5.1, Algorithm 2 guarantees with probability at least +1 − δ that +T−1 +� +t=S +� +ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtǫt +� +I(¯ω(t) ≤ ǫt) + +T−1 +� +t=S +ηtˆρǫtI(¯ω(t) > ǫt) +≤ +ˆρD2 +2 ++ ˆρ +2 +T−1 +� +t=S +η2 +t M2 + ˆρ +2 max{ +� +12 ln(8/δ), 4 +3 ln(8/δ)} +� +� +� +� +T−1 +� +t=S +η4 +t M4 +(26) ++ +� +3 ln(4/δ) +� +� +� +� +T−1 +� +t=S +4ˆρ2η2 +t M2D2 ++ +T−1 +� +t=S +ηtˆρ +√ +3σ +� +ln(4(T − S)/δ) +√ +B +� +�λtI(¯ω(t) ≤ ǫt) + I(¯ω(t) > ǫt) +� +. +26 + +Under Assumption 4.1, Algorithm 1 guarantees with probability at least 1 − δ that +T−1 +� +t=S +� +ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtǫt +� +I(g(x(t)) ≤ ǫt) + +T−1 +� +t=S +ηtˆρǫtI(g(x(t)) > ǫt) +≤ +ˆρD2 +2 ++ ˆρ +2 +T−1 +� +t=S +η2 +t M2 + ˆρ +2 max{ +� +12 ln(8/δ), 4 +3 ln(8/δ)} +� +� +� +� +T−1 +� +t=S +η4 +t M4 +(27) ++ +� +3 ln(4/δ) +� +� +� +� +T−1 +� +t=S +4ˆρ2η2 +t M2D2. +A.3 +Proof of Theorem 4.2 +Proof of Theorem 4.2. Since Algorithm 1 is fully deterministic under the assumptions and +we do not assume strong convexity in g so µ = 0, we can simplify the second inequality in +Proposition A.1 as follows +T −1 +� +t=S +� +ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtǫt +� +I(g(x(t)) ≤ ǫt) + +T −1 +� +t=S +ηtˆρǫtI(g(x(t)) > ǫt) +≤ +T −1 +� +t=S +� +ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtg(x(t)) +� +I(g(x(t)) ≤ ǫt) + +T −1 +� +t=S +ηtˆρg(x(t))I(g(x(t)) > ǫt) +≤ +ˆρD2 +2 ++ ˆρ +2 +T −1 +� +t=S +� +η2 +t ∥ξ(t) +f ∥2I(g(x(t)) ≤ ǫt) + η2 +t ∥ζ(t) +g ∥2I(g(x(t)) > ǫt) +� ++ +T −1 +� +t=S +� +ˆρηt +� +ξ(t) +f +− ζ(t) +f , �x(t) − x(t)� +I(g(x(t)) ≤ ǫt) +� +≤ +ˆρD2 +2 ++ ˆρ +2 +T −1 +� +t=S +η2 +t M 2, +(28) +where the second inequality is the second inequality in Proposition A.1 and the last inequality +is because ξ(t) +f += ζ(t) +f +and Assumption 4.1. +We first prove that, if S, T, ηt and ǫt are chosen such that +ǫt(1 + �λt) ≤ ǫ2(ˆρ − ρ) +(29) +and +T−1 +� +t=S +ηtˆρǫt > ˆρD2 +2 ++ ˆρ +2 +T−1 +� +t=S +η2 +t M2, +(30) +we must have g(x(t)) ≤ ǫt for at least one t in {S, . . . , T−1} (i.e., I ̸= ∅) and Eτ∥�x(τ)−x(τ)∥2 ≤ +ǫ2 (so Eτ∥�x(τ) − x(τ)∥ ≤ ǫ). +Suppose (30) holds and g(x(t)) > ǫt for t = S, . . . , T −1, i.e., I = ∅. (28) becomes exactly +the opposite of (30). This contradiction means g(x(t)) ≤ ǫt for at least one t in {S, . . . , T −1}. +27 + +Suppose (29) and (30) hold but Eτ∥�x(τ) − x(τ)∥2 > ǫ2. By the process of generating τ, we +have +ǫ2 < Eτ∥�x(τ) − x(τ)∥2 = +�T−1 +t=S ηtI(g(x(t)) ≤ ǫt)∥�x(t) − x(t)∥2 +�T−1 +t=S ηtI(g(x(t)) ≤ ǫt) +. +(31) +Note that the right-hand side of (31) is well-defined because we just proved I ̸= ∅. (31) and +(29) imply +T −1 +� +t=S +� +ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtǫt +� +I(g(x(t)) ≤ ǫt) + +T −1 +� +t=S +ηtˆρǫtI(g(x(t)) > ǫt) +> +T −1 +� +t=S +� +ηtˆρ(ˆρ − ρ)ǫ2 − ηtˆρ�λtǫt +� +I(g(x(t)) ≤ ǫt) + +T −1 +� +t=S +ηtˆρǫtI(g(x(t)) > ǫt) +≥ +T −1 +� +t=S +ηt ˆρǫtI(g(x(t)) ≤ ǫt) + +T −1 +� +t=S +ηtˆρǫtI(g(x(t)) > ǫt) ≥ +T −1 +� +t=S +ηt ˆρǫt, +(32) +where the second inequality is because of (29). Combining this inequality and (28) leads to +the opposite of (30). This contradiction means Eτ∥�x(τ) − x(τ)∥2 ≤ ǫ2. +Given the result above, we only need to show that the two choices of S, T, ηt and ǫt +ensure (29) and (30). +In Case I, (29) holds because of Lemma 3.3 and the choice of ǫt. Let η = ηt = +2ǫ2(ˆρ−ρ) +5(1+Λ)M2 +for any t. Using Lemma 3.3 and plugging the values of S, T, ηt and ǫt in (30), we can show +that (30) is equivalent to +Tηˆρǫ2(ˆρ − ρ) +1 + Λ +> ˆρD2 +2 ++ ˆρ +2Tη2M2, +which, after dividing both sides by Tηˆρ, can be equivalently written as +ǫ2(ˆρ − ρ) +1 + Λ +> D2 +2Tη + ηM2 +2 +. +By the values of η and T, each summand in the right-hand side of the inequality above is no +more than ǫ2(ˆρ−ρ) +5(1+Λ) so the right-hand side of the inequality above no more than 2ǫ2(ˆρ−ρ) +5(1+Λ) which +is strictly less than the left-hand side. This means (30) holds with this choice of parameters +and thus Eτ∥�x(τ) − x(τ)∥ ≤ ǫ. By the convexity of g and the choices of ηt and ǫt, we have +Eτg(x(τ)) ≤ ǫ2(ˆρ−ρ) +1+Λ . +In Case II, by the choices of ǫt and T, we have, for any t ∈ {S, . . . , T − 1}, +ǫt = 5MD +√t + 1 ≤ 5MD +√ +S + 1 = +5MD +� +T/2 + 1 +≤ ǫ2(ˆρ − ρ) +1 + Λ +. +(33) +This further implies (29) because of Lemma 3.3. Note that ηt and ǫt are decreasing in t. +Hence, the left-hand side of (30) satisfies +T−1 +� +t=S +ηtˆρǫt > T +2 ˆρηT ǫT = +5T +2T + 2 ˆρD2 ≥ 5ˆρD2 +4 +. +(34) +28 + +The right-hand side of (30) satisfies +ˆρD2 +2 ++ ˆρ +2 +T−1 +� +t=S +η2 +t M2 = ˆρD2 +2 ++ ˆρ +2D2 +T−1 +� +t=S +1 +t + 1 ≤ ˆρD2, +(35) +where the equality is obtained by plugging in the definition of ηt and the inequality is because +�T−1 +t=S +1 +t+1 ≤ +� T +S +1 +t dt = ln(T/S) = ln(2) ≤ 1. The right-hand side of (34) is strictly greater +than the right-hand side (35), which means (30) holds and thus Eτ∥�x(τ) − x(τ)∥ ≤ ǫ. By the +convexity of g and the choices of ηt and ǫt, we have Eτg(x(τ)) ≤ ǫS ≤ ǫ2(ˆρ−ρ) +1+λ +according to +(33). +A.4 +Proof of Theorem 4.3 +Proof of Theorem 4.3. By Proposition A.6, (27) holds with a probability of at least 1− δ. In +the rest of the proof, we always assume (27) holds. We first prove that, if S, T, ηt and ǫt are +chosen such that (29) holds and +T−1 +� +t=S +ηtˆρǫt +> +ˆρD2 +2 ++ ˆρ +2 +T−1 +� +t=S +η2 +t M2 + ˆρ +2 max{ +� +12 ln(8/δ), 4 +3 ln(8/δ)} +� +� +� +� +T−1 +� +t=S +η4 +t M4 +(36) ++ +� +3 ln(4/δ) +� +� +� +� +T−1 +� +t=S +4ˆρ2η2 +t M2D2, +we must have g(x(t)) ≤ ǫt for at least one t in {S, . . . , T−1} (i.e., I ̸= ∅) and Eτ∥�x(τ)−x(τ)∥2 ≤ +ǫ2 (so Eτ∥�x(τ) − x(τ)∥ ≤ ǫ). +Suppose (36) holds and g(x(t)) > ǫt for t = S, . . . , T − 1, i.e., I = ∅. (27) becomes +exactly the opposite of (36). +This contradiction means g(x(t)) ≤ ǫt for at least one t in +{S, . . . , T − 1}. Suppose (29) and (36) hold but Eτ∥�x(τ) − x(τ)∥2 > ǫ2. By the process of +generating τ, we have (31). Note that the right-hand side of (31) is well-defined because we +just proved I ̸= ∅. (31) and (29) imply (32). Combining (32) and (36) leads to the opposite +of (27). This contradiction means Eτ∥�x(τ) − x(τ)∥2 ≤ ǫ2. +Given the result above, we only need to show that the choices of S, T, ηt and ǫt ensure +(29) and (36). +In Case I, (29) holds because of Lemma 3.3 and the choice of ǫt. Let η = ηt = +2ǫ2(ˆρ−ρ) +5(1+Λ)M2 +for any t. Using Lemma 3.3 and plugging the values of S, T, ηt and ǫt in (36), we can show +that (36) is equivalent to +Tηˆρǫ2(ˆρ − ρ) +1 + Λ +> +ˆρD2 +2 ++ ˆρ +2Tη2M2 + ˆρ +2 max{ +� +12 ln(8/δ), 4 +3 ln(8/δ)} +√ +Tη2M2 ++2 +� +3 ln(4/δ)ˆρ +√ +TηMD, +which, after dividing both sides by Tηˆρ, can be equivalently written as +ǫ2(ˆρ − ρ) +1 + Λ +> D2 +2Tη + ηM2 +2 ++ 1 +2 max{ +� +12 ln(8/δ), 4 +3 ln(8/δ)}ηM2 +√ +T ++ 2 +� +3 ln(4/δ)MD +√ +T +. +29 + +By the values of η and T, each summand in the right-hand side of the inequality above is no +more than ǫ2(ˆρ−ρ) +5(1+Λ) so the right-hand side of the inequality above no more than 4ǫ2(ˆρ−ρ) +5(1+Λ) which +is strictly less than the left-hand side. This means (30) holds with this choice of parameters +and thus Eτ∥�x(τ) − x(τ)∥ ≤ ǫ. We can prove Eτg(x(τ)) ≤ ǫ2(ˆρ−ρ) +1+λ +in the same way as in +Theorem 4.2. +In Case II, by the choices of ǫt and T, we have, for any t ∈ {S, . . . , T − 1}, +ǫt = EMD +√t + 1 ≤ EMD +√ +S + 1 = +EMD +� +T/2 + 1 +≤ ǫ2(ˆρ − ρ) +1 + Λ +. +(37) +This further implies (29) because of Lemma 3.3. Note that ηt and ǫt are decreasing in t. +Hence, the left-hand side of (36) satisfies +T−1 +� +t=S +ηtˆρǫt > T +2 ˆρηT ǫT = +ET +2T + 2 ˆρD2 ≥ E ˆρD2 +4 +. +(38) +The right-hand side of (36) satisfies +ˆρD2 +2 ++ ˆρ +2 +T−1 +� +t=S +η2 +t M2 + ˆρ +2 max{ +� +12 ln(8/δ), 4 +3 ln(8/δ)} +� +� +� +� +T−1 +� +t=S +η4 +t M4 ++ +� +3 ln(4/δ) +� +� +� +� +T−1 +� +t=S +4ˆρ2η2 +t M2D2 += +ˆρD2 +2 ++ ˆρ +2D2 +T−1 +� +t=S +1 +t + 1 + ˆρ +2 max{ +� +12 ln(8/δ), 4 +3 ln(8/δ)}D2 +� +� +� +� +T−1 +� +t=S +1 +(t + 1)2 ++2 +� +3 ln(4/δ)ˆρD2 +� +� +� +� +T−1 +� +t=S +1 +t + 1 +≤ +ˆρD2 + ˆρπ +2 +√ +6 max{ +� +12 ln(8/δ), 4 +3 ln(8/δ)}D2 + 2 +� +3 ln(4/δ)ˆρD2, +(39) +where the equality is obtained by plugging in the definition of ηt and the inequality is because +�T−1 +t=S +1 +t+1 ≤ +� T +S +1 +t dt = ln(T/S) = ln(2) ≤ 1 and �T−1 +t=S +1 +(t+1)2 ≤ π2 +6 . By the condition satisfied +by E, the right-hand side of (38) is greater than or equal to the right-hand side (39). This +means (30) holds with this choice of parameters and thus Eτ∥�x(τ) − x(τ)∥ ≤ ǫ. We can prove +Eτg(x(τ)) ≤ ǫ2(ˆρ−ρ) +1+λ +in the same way as in Theorem 4.2. +A.5 +Proof of Theorem 5.2 +Proof of Theorem 5.2. By Lemma A.3 and Proposition A.6, (26) holds and, simultaneously, +(23) holds with δ replaced by δ/4 for t = S, S + 1, . . . , T − 1 with a probability of at least +1 − δ. In the rest of the proof, we always assume (26) holds and (23) holds with δ replaced +30 + +by δ/4 for t = S, S + 1, ˙,T − 1. We first prove that, if S, T, B, ηt and ǫt are chosen such that +(29) holds and +T−1 +� +t=S +ηtˆρǫt +> +ˆρD2 +2 ++ ˆρ +2 +T−1 +� +t=S +η2 +t M2 + ˆρ +2 max{ +� +12 ln(8/δ), 4 +3 ln(8/δ)} +� +� +� +� +T−1 +� +t=S +η4 +t M4 +(40) ++ +� +3 ln(4/δ) +� +� +� +� +T−1 +� +t=S +4ˆρ2η2 +t M2D2 + +T−1 +� +t=S +ηtˆρ +√ +3σ +� +ln(4(T − S)/δ) +√ +B +(Λ + 1), +we must have ¯ω(t) ≤ ǫt for at least one t in {S, S + 1, . . . , T − 1} (i.e., I ̸= ∅) and Eτ∥�x(τ) − +x(τ)∥2 ≤ ǫ2 (so Eτ∥�x(τ) − x(τ)∥ ≤ ǫ). Here, Λ is defined in (7) in Lemma 3.3. +Suppose (40) holds and ¯ω(t) > ǫt for t = S, S + 1, . . . , T − 1, i.e., I = ∅. (26) contradicts +with (40) as Λ ≥ �λt. This contradiction means ¯ω(t) ≤ ǫt for at least one t in {S, S+1, . . . , T−1} +so I ̸= ∅. Suppose (29) and (40) hold but Eτ∥�x(τ) −x(τ)∥2 > ǫ2. By the process of generating +τ, we have +ǫ2 < Eτ∥�x(τ) − x(τ)∥2 = +�T−1 +t=S ηtI(¯ω(t) ≤ ǫt)∥�x(t) − x(t)∥2 +�T−1 +t=S ηtI(¯ω(t) ≤ ǫt) +. +(41) +Note that the right-hand side of (41) is well-defined because we just proved I ̸= ∅. (41) and +(29) imply +T −1 +� +t=S +� +ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtǫt +� +I(¯ω(t) ≤ ǫt) + +T −1 +� +t=S +ηtˆρǫtI(¯ω(t) > ǫt) +> +T −1 +� +t=S +� +ηtˆρ(ˆρ − ρ)ǫ2 − ηtˆρ�λtǫt +� +I(¯ω(t) ≤ ǫt) + +T −1 +� +t=S +ηtˆρǫtI(¯ω(t) > ǫt) +≥ +T −1 +� +t=S +ηtˆρǫtI(¯ω(t) ≤ ǫt) + +T −1 +� +t=S +ηtˆρǫtI(¯ω(t) > ǫt) ≥ +T −1 +� +t=S +ηtˆρǫt, +(42) +where the second inequality is because of (29). Combining (42) and (40) leads to the opposite +of (26). This contradiction means Eτ∥�x(τ) − x(τ)∥2 ≤ ǫ2. +Given the result above, we only need to show that the choices of S, T, B, ηt and ǫt ensure +(29) and (40). +In Case I, (29) holds because of Lemma 3.3 and the choice of ǫt. Let η = ηt = +2ǫ2(ˆρ−ρ) +5(1+Λ)M2 +for any t. Using Lemma 3.3 and plugging the values of S, T, B, ηt and ǫt in (40), we can +show that (40) holds if +Tηˆρǫ2(ˆρ − ρ) +1 + Λ +> +ˆρD2 +2 ++ ˆρ +2Tη2M2 + ˆρ +2 max{ +� +12 ln(8/δ), 4 +3 ln(8/δ)} +√ +Tη2M2 ++2 +� +3 ln(4/δ)ˆρ +√ +TηMD + Tηˆρ +√ +3σ +� +ln(4T/δ) +√ +B +(Λ + 1), +which, after dividing both sides by Tηˆρ, can be equivalently written as +ǫ2(ˆρ − ρ) +1 + Λ +> +D2 +2Tη + ηM2 +2 ++ 1 +2 max{ +� +12 ln(8/δ), 4 +3 ln(8/δ)}ηM2 +√ +T ++2 +� +3 ln(4/δ)MD +√ +T ++ +√ +3σ +� +ln(4T/δ) +√ +B +(Λ + 1). +31 + +By the values of η, B and T, each of the first four summands in the right-hand side of the +inequality above is no more than ǫ2(ˆρ−ρ) +5(1+Λ) while the last summand is no more than ǫ2(ˆρ−ρ) +10(1+Λ), so +the right-hand side of the inequality above no more than 9ǫ2(ˆρ−ρ) +10(1+Λ) which is strictly less than +the left-hand side. This means (30) holds with this choice of parameters and thus Eτ∥�x(τ) − +x(τ)∥ ≤ ǫ. Moreover, since (23) holds with δ replaced by δ/4 for t = S, S + 1, . . . , T − 1, we +have +Eτg(x(τ)) += +�T−1 +t=0 ηtg(x(t))I(¯ω(t) ≤ ǫt) +�T−1 +t=0 ηtI(¯ω(t) ≤ ǫt) +≤ +�T−1 +t=0 ηt +� +ǫt + +√ +3σ√ +ln(4T/δ) +√ +B +� +I(¯ω(t) ≤ ǫt) +�T−1 +t=0 ηtI(¯ω(t) ≤ ǫt) +≤ +ǫ2(ˆρ − ρ) +1 + Λ ++ +√ +3σ +� +ln(4T/δ) +√ +B +≤ 2ǫ2(ˆρ − ρ) +1 + Λ +, +where the last inequality is because of the choice of B. +In Case II, by the choice of ǫt, we have (37) holds. This further implies (29) because of +Lemma 3.3. Note that ηt and ǫt are decreasing in t. Hence, we also have (38). The right-hand +side of (40) satisfies +ˆρD2 +2 ++ ˆρ +2 +T −1 +� +t=S +η2 +t M 2 + ˆρ +2 max{ +� +12 ln(8/δ), 4 +3 ln(8/δ)} +� +� +� +� +T −1 +� +t=S +η4 +t M 4 ++ +� +3 ln(4/δ) +� +� +� +� +T −1 +� +t=S +4ˆρ2η2 +t M 2D2 + +T −1 +� +t=S +ηtˆρ +√ +3σ +� +ln(4(T − S)/δ) +√ +B +(Λ + 1) += +ˆρD2 +2 ++ ˆρ +2D2 +T −1 +� +t=S +1 +t + 1 + ˆρ +2 max{ +� +12 ln(8/δ), 4 +3 ln(8/δ)}D2 +� +� +� +� +T −1 +� +t=S +1 +(t + 1)2 ++2 +� +3 ln(4/δ)ˆρD2 +� +� +� +� +T −1 +� +t=S +1 +t + 1 + Dˆρ +M +√ +3σ +� +ln(4(T − S)/δ) +√ +B +(Λ + 1) +T −1 +� +t=S +1 +√t + 1 +≤ +ˆρD2 + ˆρπ +2 +√ +6 max{ +� +12 ln(8/δ), 4 +3 ln(8/δ)}D2 + 2 +� +3 ln(4/δ)ˆρD2 + ˆρD2, +(43) +where the equality is obtained by plugging in the definition of ηt and the inequality is because +of the definition of B and the facts that �T−1 +t=S +1 +t+1 ≤ +� T +S +1 +t dt = ln(T/S) = ln(2) ≤ 1, +�T−1 +t=S +1 +(t+1)2 ≤ π2 +6 and �T−1 +t=S +1 +√t+1 ≤ +� T +S +1 +√ +tdt ≤ +� +T/2. By the condition of E, the right- +hand side of (38) is strictly greater than the right-hand side (43). This means (40) holds +with this choice of parameters and thus Eτ∥�x(τ) − x(τ)∥ ≤ ǫ. Moreover, since (23) holds with +δ replaced by δ/4 for t = S, S + 1, . . . , T − 1, we have +Eτg(x(τ)) += +�T−1 +t=S ηtg(x(t))I(¯ω(t) ≤ ǫt) +�T−1 +t=S ηtI(¯ω(t) ≤ ǫt) +≤ +�T−1 +t=S ηt +� +ǫt + +√ +3σ√ +ln(2T/δ) +√ +B +� +I(¯ω(t) ≤ ǫt) +�T−1 +t=S ηtI(¯ω(t) ≤ ǫt) +≤ +ǫT/2 + +√ +3σ +� +ln(2T/δ) +√ +B +≤ 2ǫ2(ˆρ − ρ) +1 + Λ +, +where the last inequality is because of (37) and the choices of B and T. +32 + +B +Complexity results when g is strongly convex +Suppose the constraint function g in (1) is deterministic and µ-strongly convex. We have the +following theorem. +Theorem B.1. Suppose Assumptions 3.1 and 4.1 hold , g is µ-strongly convex with µ > 0, +ˆρ > ρ and ǫ > 0. Moreover, suppose ξf is deterministic, namely, ξf = ζf ∈ ∂f(x) for any +x ∈ X. Suppose the output of Algorithm 1 is changed to x(τ) with τ randomly sampled from +S, S + 1, . . . , T − 1 with τ = t with a probability of ηt/ �T−1 +t=S ηt. +Then Algorithm 1 guarantees Eτ∥�x(τ) − x(τ)∥ ≤ ǫ in either of the following cases. +Case I: S = 0, ǫt = 0, ηt = ǫ2 min{ˆρ−ρ,µ/2} +M2 +and +T ≥ +M 2D2 +ǫ4 min{(ˆρ − ρ)2, µ2/4} = O +� 1 +ǫ4 +� +. +Case II: S = T +2 , ǫt = 0, ηt = +D +M√t+1 and +T ≥ +4M 2D2 +ǫ4 min{(ˆρ − ρ)2, µ2/4} = O +� 1 +ǫ4 +� +. +Before presenting its proof, we would like to make a few remarks. +Remark B.2. According to Theorem B.1, strong convexity in the constraint function g does +not reduce the O(1/ǫ4) complexity of the SSG method for finding a nearly ǫ-stationary point. +However, strong convexity brings benefit on other aspects. First, we can simply set ǫt = 0 +when g is strongly convex, which makes step size ηt the only tuning parameter. Second, the +theoretical complexity no longer depends on Λ, the upper bound of the dual variables, so it +can be strictly better than the one in Theorem 4.2 when Λ is very large. +Remark B.3. Theorem B.1 only shows the result when f and g are both deterministic. How- +ever, we can also establish O(1/ǫ4) oracle complexity for Algorithm 1 when f is stochastic +but g is deterministic, and establish O(1/ǫ4) subgradient oracle complexity and O(1/ǫ8) func- +tion value oracle complexity for Algorithm 2 when both f and g are stochastic. Similar to +Theorem B.1, we only need to set ǫt = 0 and define τ as in Theorem B.1. This indicates that +strong convexity in g brings the same convenience in these two cases but does not change +the complexity. The proofs for these two cases will be similar to that of Theorem B.1. Since +those results do not provide additional insights on complexity and analysis, we do not include +them in the paper. +Remark B.4. Theorem 4.2, 4.3 and 5.2 guarantee Eτg(x(τ)) ≤ O(ǫ2). This is no longer true +in Theorem B.1 because τ can take value in the index set J on which x(t) can be highly +infeasible. However, with Eτ∥�x(τ) − x(τ)∥ ≤ ǫ, we can at least derive Eτg(x(τ)) ≤ O(ǫ) from +Theorem B.1, which is good enough. See Remark 4.4 for the reason. +Proof of Theorem B.1. Since Algorithm 1 is fully deterministic with ǫt = 0 under the assump- +tions, we can bound the left-hand side of the second inequality in Proposition A.1 as follows +33 + +T−1 +� +t=S +� +ηtˆρ�λtI(g(x(t)) ≤ 0) + ηtˆρI(g(x(t)) > 0) +� µ +2 ∥�x(t) − x(t)∥2 ++ +T−1 +� +t=S +� +ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtg(x(t)) +� +I(g(x(t)) ≤ 0) + +T−1 +� +t=S +ηtˆρg(x(t))I(g(x(t)) > 0) +≥ +T−1 +� +t=S +ηtˆρI(g(x(t)) > 0)µ +2 ∥�x(t) − x(t)∥2 + +T−1 +� +t=S +ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2I(g(x(t)) ≤ 0) +≥ +ˆρ min{ˆρ − ρ, µ/2} +T−1 +� +t=S +ηt∥�x(t) − x(t)∥2, +where the first inequality is by dropping the non-negative terms and the fact that g(x(t))I(g(x(t)) ≤ +0) ≤ 0. Combining this lower bound with the second inequality in Proposition A.1 gives +ˆρ min{ˆρ − ρ, µ/2} +T−1 +� +t=S +ηt∥�x(t) − x(t)∥2 +≤ +ˆρD2 +2 ++ ˆρ +2 +T−1 +� +t=S +� +η2 +t ∥ξ(t) +f ∥2I(g(x(t)) ≤ 0) + η2 +t ∥ζ(t) +g ∥2I(g(x(t)) > 0) +� ++ +T−1 +� +t=S +� +ˆρηt +� +ξ(t) +f − ζ(t) +f , �x(t) − x(t)� +I(g(x(t)) ≤ 0) +� +≤ +ˆρD2 +2 ++ ˆρ +2 +T−1 +� +t=S +η2 +t M2, +(44) +where the last inequality is because ξ(t) +f += ζ(t) +f +and Assumption 4.1. By the (new) definition +of τ, after organizing terms, we have +Eτ∥�x(t) − x(t)∥2 ≤ +1 +min{ˆρ − ρ, µ/2} +� +D2 +2 + 1 +2 +T−1 +� +t=S +η2 +t M2 +� +/ +�T−1 +� +t=S +ηt +� +. +(45) +In Case I, let η = ηt = ǫ2 min{ˆρ−ρ,µ/2} +M2 +for any t. By the choices of ηt, S and T, (45) implies +Eτ∥�x(t) − x(t)∥2 ≤ +1 +min{ˆρ − ρ, µ/2} +� D2 +2Tη + ηM2 +2 +� +≤ ǫ2 +2 + ǫ2 +2 = ǫ2. +In Case II, by the choices of ηt, S and T, (45) implies +Eτ∥�x(t) − x(t)∥2 ≤ +1 +min{ˆρ − ρ, µ/2} +�DM +√ +T ++ DM +√ +T +� +≤ ǫ2 +2 + ǫ2 +2 = ǫ2. +where, in the first inequality, we use the facts that �T−1 +t=S ηt ≥ +T +2 ηT−1 = +√ +TD +2M +and that +�T−1 +t=S η2 +t = �T−1 +t=S +D2 +M2(t+1) ≤ +� T +S +D2 +M2tdt = +D2 +M2 ln(T/S) = +D2 +M2 ln(2) ≤ +D2 +M2 , and the second +inequality is by the choice of T. +34 + diff --git a/P9FQT4oBgHgl3EQfZDYV/content/tmp_files/load_file.txt b/P9FQT4oBgHgl3EQfZDYV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4ad2448b0d9b2be47c9b0ce0de47178e358a3ce --- /dev/null +++ b/P9FQT4oBgHgl3EQfZDYV/content/tmp_files/load_file.txt @@ -0,0 +1,1293 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf,len=1292 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='13314v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='OC] 30 Jan 2023 Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Optimization with Non-Smooth Convex Constraints Yankun Huang Qihang Lin Department of Business Analytics University of Iowa, Iowa City, IA 52242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' yankun-huang@uiowa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='edu, qihang-lin@uiowa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='edu Abstract In this paper, we consider a general non-convex constrained optimization problem, where the objective function is weakly convex and the constraint function is convex while they can both be non-smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This class of problems arises from many applications in machine learning such as fairness-aware supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' To solve this problem, we consider the classical switching subgradient method by [62], which is an intuitive and easily implementable first-order method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Before this work, its iteration complexity was only known for convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We prove its oracle complexity for finding a nearly stationary point when the objective function is non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The analysis is derived separately when the constraint function is deterministic and stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Compared to existing methods, especially the double-loop methods, the switching gradient method can be applied to non-smooth problems and only has a single loop, which saves the effort on tuning the number of inner iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Keywords: Constrained optimization, First-order method, Non-smooth optimization, Non-convex optimization 1 Introduction Continuous optimization with nonlinear constraints arises from many applications of machine learn- ing and statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Examples include Neyman-Pearson classification [64] and learning with fairness constraints [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In this paper, we consider the following general nonlinear constrained optimization problem f ∗ ≡ min x∈X f(x) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' g(x) ≤ 0, (1) where X ⊂ Rd is a compact convex set that allows a computationally easy projection operator, f is weakly-convex and g is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Functions f and g are not necessarily smooth and we assume their function values and subgradients can be evaluated deterministically or through stochastic oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' When g(x) ≡ maxi=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=',m gi(x) with convex gi’s, (1) is equivalent to an optimization problem with multiple nonlinear constraints gi(x) ≤ 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' A weakly convex function can be non-convex, so computing an optimal solution of (1) is chal- lenging in general even without constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For this reason, theoretical analysis for gradient-based 1 algorithms for non-convex problems mostly focuses on an algorithm’s (oracle) complexity for finding an ǫ-stationary solution for (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' When a problem is non-smooth, finding an ǫ-stationary solution is generally difficult even if the problem is convex [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Hence, in this paper, we consider finding a nearly ǫ-stationary solution for (1), whose definition will be stated later in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In the past decade, there have been many studies on non-convex constrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' How- ever, most of the existing algorithms and their theoretical complexity analysis are developed by as- suming f and gi’s are all smooth or can be written as the sum of a smooth and a simple non-smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' A non-exhaustive list of the works under such an assumption includes [10, 33, 81, 82, 38, 41, 57, 36, 56, 46, 71, 45, 53, 65, 54, 52, 11, 14, 23, 8, 59, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Their results cannot be applied to (1) due to non-smoothness in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Non-smoothness occurs very often in optimization in machine learning, for example, when a non-smooth loss function is applied, but there are much fewer studies on non-smooth non-convex constrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Under the weakly-convexity assumption, an effective approach for solving a non-smooth non-convex problem with theoretical guarantees is the (inexact) proximal point method, where a quadratic proximal term is added to objective and constraint functions to construct a strongly convex constrained subproblem and then a sequence of solutions can be generated by solving this subproblem inexactly and updating the center of the proximal term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Oracle complexity for this method to find a nearly ǫ-stationary has been established by [12, 55, 40] under different constraint qualification conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The inexact proximal point method is a double-loop algorithm where the inner loop is typically another optimization algorithm, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=', the switching subgradient method [62], for solving the aforemen- tioned strongly convex subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The complexity results in [12, 55, 40] require the inner loop solves each subproblem to achieve a targeted optimality gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' However, the optimality gap is hard to evaluate, so we are unable to simply stop the inner loop based on the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Although the number of inner iterations needed to achieve the targeted gap can be bounded theoretically, it usually involves some constants that are unknown or can only be estimated conservatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Hence, using the theoretical iteration bound to stop the inner loop usually leads to significantly more inner iterations than what is actually needed, which makes the whole algorithm inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In practices, users often need to tune the number of inner iterations to improve algorithm’s efficiency, which is a common inconvenience for all double-loop methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' On the contrary, a single-loop algorithm is usually easier to implement as it does not require tuning the number of inner iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Therefore, the main contribution of this paper is showing that, when the constraint function g is convex, a single-loop first-order algorithm is sufficient for finding a nearly ǫ-stationary point of (1) with theoretically guaranteed oracle complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The algorithm we study is the classical switching subgradient (SSG) method by [62], which is intuitive and easy to implement but has only been analyzed before in the convex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We show that the SSG method finds a nearly ǫ-stationary point of (1) with complexity O(1/ǫ4) when f is either deterministic or stochastic but g is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This complexity is optimal [3, 31] and matches the one achieved by the double-loop methods in [12, 55, 40] under the same assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' To the best of our knowledge, this is also the first complexity result on a single-loop algorithm for weakly convex non-smooth nonlinear constrained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' When g is also stochastic, a large mini-batch is needed to estimate g in the SSG method and we show that the complexity is still O(1/ǫ4) for the subgradient oracle but becomes O(1/ǫ8) for the function value oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 2 Related Work Non-convex constrained optimization has a long history [34, 15, 29, 17, 30, 4, 16] and the interest on this subject has been still growing in the machine learning community because of its new applications such as learning with fairness constraints (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=', [78]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In recent literature, the prevalent classes of algorithms for non-convex constrained optimization are the augmented Lagrangian method (ALM) and the penalty method [35, 80, 81, 82, 37, 38, 41, 44, 2 57, 36, 56, 46, 71, 45, 53, 65, 54, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Another actively studied class of algorithms is the sequential quadratic programming method [10, 33, 5, 11, 23, 9, 8, 59, 7, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' An inexact projection gradient method is developed by [14] and a level conditional gradient method is developed by [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' However, the papers above all focus on the case where g is smooth and f is either smooth or equals f1 + f2 where f1 is smooth and non-convex while f2 is non-smooth and convex and has a simple structure that allows a closed-form solution for the proximal mapping arg miny f2(y) + ρ 2∥y − x∥2 for any x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' On the contrary, we focus a non-smooth problem with no structure assumption other than weak convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' There are relatively fewer works on non-convex non-smooth constrained problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' An alternating direction method of multipliers (ADMM) and an ALM are studied by [77] and [79], respectively, for non-convex non-smooth problems with linear constraints while our study considers nonlinear non- smooth constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The methods by [20] and [11] can be extended to a structured non-smooth case where f = f1 + f2 with f1 being smooth non-convex and f2 = maxy y⊤Ax − φ(y) with a convex φ, and g has a similar structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The method by [13] can handle a specific non-smooth non-convex constraint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=', g(x) = λ∥x∥1 − h(x) where h is a convex and smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' However, our method does not need these structure assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' When f and g in (1) are weakly convex and non-smooth, the inexact proximal point method has been studied by [12, 55, 40] under different constraint qualification conditions and different notions of stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Their complexity analysis utilizes the relationship between the gradient of the Moreau envelope of (1) and the near stationarity of a solution, which is originally used to analyze complexity of subgradient methods for weakly convex non-smooth unconstrained problems [25, 26, 27, 1, 28, 63, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Our analysis also follows a similar strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The methods mentioned above that can be applied to (structured) non-smooth problems are all double-loop while our algorithm only requires a single loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' When f is either deterministic or stochastic but g is convex and deterministic, our method has the same complexity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=', O(1/ǫ4) as [12, 55, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' However, their methods allow g also to be non-convex under additional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The SSG algorithm is first proposed by [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' It has been well-studied for convex problems [61, 6, 50, 67, 74, 75, 70, 69, 68, 2] and quasi-convex problems [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This paper provides the first complexity analysis for the SSG method under weak convexity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Non-smooth non-convex optimization has also been studied without weak convexity assumption by [83, 47, 66, 48, 19, 72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' These works analyze the complexity of first-order methods for computing an (ǫ, δ)-Goldstein approximate stationary point, which is a more general stationarity notation than what we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' However, these works only focus on unconstrained problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 3 Preliminaries Let ∥ · ∥ be the ℓ2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For h : Rd → R ∪ {+∞}, the subdifferential of h at x is ∂h(x) = � ζ ∈ Rd �� h(x′) ≥ h(x) + ζ⊤(x′ − x) + o(∥x′ − x∥), x′ → x � , and ζ ∈ ∂h(x) is a subgradient of h at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We say h is µ-strongly convex (µ ≥ 0) on X if h(x) ≥ h(x′) + ζ⊤(x − x′) + µ 2 ∥x − x′∥2 for any (x, x′) ∈ X × X and any ζ ∈ ∂h(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We say h is ρ-weakly convex (ρ ≥ 0) on X if h(x) ≥ h(x′) + ζ⊤(x − x′) − ρ 2∥x − x′∥2 for any (x, x′) ∈ X ×X and any ζ ∈ ∂h(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We denote the normal cone of X at x by NX (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We say a point x is ǫ-feasible if x ∈ X and g(x) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Let I[A] be a zero-one indicator of event A and let Dist(x, S) := miny∈S ∥x − y∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We make the following assumptions about (1) throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 3 Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The following statements hold: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' f(x) is ρ-weakly convex with ∂f(x) ̸= ∅ on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' g(x) is convex with ∂g(x) ̸= ∅ on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Moreover, there exists M such that ∥ζf∥ ≤ M and ∥ζg∥ ≤ M for any x ∈ X, ζf ∈ ∂f(x) and ζg ∈ ∂g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' There exists a strictly feasible solution xfeas with xfeas ∈ X and g(xfeas) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' There exists D such that ∥x − x′∥ ≤ D for any x and x′ in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' f ∗ > −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1, (1) is non-convex so finding an ǫ-optimal solution is intractable in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For a non-convex problem, the target is typically to find a stationary point of (1), which is a point x∗ ∈ X that satisfies the following Karush-Kuhn-Tucker (KKT) conditions: ζ∗ f + λ∗ζ∗ g ∈ −NX(x∗), λ∗g(x∗) = 0, g(x∗) ≤ 0, λ∗ ≥ 0, (2) where λ∗ ∈ R is a Lagrangian multiplier, ζ∗ f ∈ ∂f(x∗) and ζ∗ g ∈ ∂g(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Typically, an exact stationary point can only be approached by an algorithm as full convergence, which may require infinitely many iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Within a finite number of iterations, an algorithm can only generate an ǫ-stationary point, which is a point �x ∈ X satisfying Dist � �ζf + �λ�ζg, −NX (�x) � ≤ ǫ, |�λg(�x)| ≤ ǫ2, g(�x) ≤ ǫ, �λ ≥ 0, (3) where �λ ∈ R is a Lagrangian multiplier, �ζf ∈ ∂f(�x) and �ζg ∈ ∂g(�x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' However, when f and g are non-smooth, computing an ǫ-stationary point with finite complexity is still challenging even if there is no constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' On the contrary, under weak convexity assumption in Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1A, it is possible to compute a nearly ǫ-stationary point defined later, which is the goal of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Given any ˆρ > ρ and x ∈ X, the Moreau envelope and the proximal mapping of (1) are defined as ϕˆρ(x) ≡ min y∈X � f(y) + ˆρ 2∥y − x∥2, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' g(y) ≤ 0 � , (4) �xˆρ(x) ≡ arg min y∈X � f(y) + ˆρ 2∥y − x∥2, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' g(y) ≤ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (5) By Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1A, the objective function in (4) and (5) is (ˆρ − ρ)-strongly convex and the constraint function is convex, so �xˆρ(x) is uniquely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Following the literature on weakly convex optimization [25, 27, 24, 12, 55, 40], we use the value of ∥x − �xˆρ(x)∥ as a measure of the quality of a solution x because it can be interpreted as a stationarity measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' To see this, consider the following KKT conditions of (4): �ζf + ˆρ(�xˆρ(x) − x) + �λ�ζg ∈ −NX (�xˆρ(x)), �λg(�xˆρ(x)) = 0, g(�xˆρ(x)) ≤ 0, �λ ≥ 0, (6) 1We do not require that xfeas can be accessed by an algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 4 where �ζf ∈ ∂f(�xˆρ(x)) and �ζg ∈ ∂g(�xˆρ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' These imply Dist � �ζf + �λ�ζg, −NX (�xˆρ(x)) � ≤ ˆρ∥�xˆρ(x) − x∥, �λg(�xˆρ(x)) = 0, g(�xˆρ(x)) ≤ 0, �λ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Therefore, as long as ∥�xˆρ(x) − x∥ ≤ ǫ, we have Dist � �ζf + �λ�ζg, −NX (�xˆρ(x)) � ≤ ˆρǫ, �λg(�xˆρ(x)) = 0, and g(�xˆρ(x)) ≤ 0, which means �xˆρ(x) is an ˆρǫ-stationary point of the original problem (1) by definition in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Since x is within an ǫ-distance from �xˆρ(x), we call such an x a nearly ǫ-stationary point of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Given ˆρ > ρ, a point x ∈ X is a nearly ǫ-stationary point of (1) if ∥�xˆρ(x) − x∥ ≤ ǫ where �xˆρ(x) is defined in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' A random vector x ∈ X is a stochastic nearly ǫ- stationary point of (1) if E∥�xˆρ(x) − x∥ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Next, we will introduce a numerical method for finding a (stochastic) nearly ǫ-stationary point of (1) with theoretically proved oracle complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Here, the oracle complexity is de- fined as how many times the algorithm needs to query the subgradient or function value of f or g through a deterministic or stochastic oracle in order to find a (stochastic) nearly ǫ-stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Our results are presented separately when the constraints are determin- istic and stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In both cases, the objective function f can be either deterministic or stochastic, which does not affect the complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Before showing the main results, we present the following lemma, which says any �λ in (6) can be bounded from above by a constant independent of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The proof is in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Given any x ∈ X and ˆρ > ρ, let �xˆρ(x) defined as in (5) and �λ is the associated Lagrangian multiplier satisfying (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We have �λ ≤ Λ := MD + ˆρD2 −g(xfeas) (7) where M, D and xfeas are as in Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 4 Deterministic Constraints In this section, we assume the subgradient and function value of g can be computed deter- ministically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This includes the case when (1) has multiple convex deterministic constraints gi(x) ≤ 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , m, by having g(x) ≡ max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=',m gi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This assumption is specified below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For any x ∈ X, g(x) and an ζg ∈ ∂g(x) can be evaluated exactly and a stochastic subgradient ξf can be generated such that E ξf ∈ ∂f(x) and E exp � ∥ξf∥2/M2� ≤ exp(1) for any x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 5 Algorithm 1 Switching Subgradient Method for Deterministic Constraints 1: Input: x(0) ∈ X, total number of iterations T, tolerance of infeasibility ǫt > 0, t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T − 1, and a starting index S for generating outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 2: Initialization: I ← ∅ and J ← ∅ 3: for iteration t = 0, 1, · · · , T − 1 do 4: if g(x(t)) ≤ ǫt then 5: Generate a stochastic subgradient ξ(t) f of f at x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 6: x(t+1) ← projX (x(t) − ηtξ(t) f ) 7: I ← I ∪ {t} if t ≥ S 8: else 9: Generate a subgradient ζ(t) g of g at x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 10: x(t+1) ← projX (x(t) − ηtζ(t) g ) 11: J ← J ∪ {t} if t ≥ S 12: end if 13: end for 14: Output: x(τ) with τ randomly sampled from I with τ = t in a probability of ηt / � t∈I ηt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 means the distribution of ∥ξf∥2 has a light tail, which is a common assumption for proving a stochastic first-order method converges in a high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' See (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='50) in [60] for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We present the SSG method under Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This algorithm is easy to implement and intuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' At iteration t, we check if the current solution x(t) is nearly feasible in the sense that g(x(t)) ≤ ǫt for a pre-determined tolerance of infeasibility ǫt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' If yes, the algorithm prioritizes reducing the objective value by taking a projected subgradient step along the (stochastic) subgradient of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' If x(t) is not nearly feasible, the algorithm prioritizes reducing the infeasibility by switching the updating direction to the subgradient of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The algorithm records the iteration indexes of the nearly feasible solutions in set I and other indexes in set J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The final output is randomly sampled from the iterates whose indexes are in I with a distribution weighted by the step sizes ηt’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We also introduce an starting index S so the algorithm only starts to record I and J when t ≥ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For simplicity of notation, we denote �xˆρ(x(t)) defined in (5) by �x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Let Eτ be the expectation taken only over the random index τ when the algorithms stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We present the convergence properties of Algorithm 1 separately when the subgradient of f is stochastic and deterministic and when ǫt and ηt are static and diminishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The complexity for finding a stochastic nearly ǫ-stationary point T is the same in these four cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The proof of the following theorem is provided in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 hold and ˆρ > ρ, ǫ > 0 and Λ is defined in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Moreover, suppose ξf is deterministic, namely, ξf = ζf ∈ ∂f(x) for any x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Then Algorithm 1 guarantees Eτ∥�x(τ) − x(τ)∥ ≤ ǫ and Eτg(x(τ)) ≤ ǫ2(ˆρ−ρ) 1+Λ in either of the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Case I: S = 0, ǫt = ǫ2(ˆρ−ρ) 1+Λ , ηt = 2ǫ2(ˆρ−ρ) 5(1+Λ)M2 and T ≥ 25M2D2(1+Λ)2 4ǫ4(ˆρ−ρ)2 = O � 1/ǫ4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Case II: S = T 2 , ǫt = 5MD √t+1, ηt = D M√t+1 and T ≥ 50M2D2(1+Λ)2 ǫ4(ˆρ−ρ)2 = O � 1/ǫ4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 6 Algorithm 1 is single-loop with O(1) oracle complexity per iteration, so its total complexity is just T = O � 1/ǫ4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 hold and ˆρ > ρ, δ ∈ (0, 1), ǫ > 0 and Λ is defined in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Then Algorithm 1 guarantees Eτ∥�x(τ) − x(τ)∥ ≤ ǫ and Eτg(x(τ)) ≤ ǫ2(ˆρ−ρ) 1+Λ with probability at least 1 − δ in either of the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Case I: S, ǫt and ηt are chosen as Case I in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2 and T ≥ \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 25M2D2(1+Λ)2 4ǫ4(ˆρ−ρ)2 , max{12 ln(8/δ), 16 9 ln2(8/δ)}, 300 ln(4/δ)M2D2(1+Λ)2 ǫ4(ˆρ−ρ)2 \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe = O � 1 ǫ4 � , Case II: S and ηt are chosen as Case II in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2, ǫt = EMD √t+1 where E is any positive constant such that E ≥ 4 + 2π √ 6 max{ � 12 ln(8/δ), 4 3 ln(8/δ)} + 8 � 3 ln(4/δ) and T ≥ 2E2M2D2(1+Λ)2 ǫ4(ˆρ−ρ)2 = O � 1/ǫ4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This theorem shows that, the complexity remains O(1/ǫ4) if the subgradient oracle of f is stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The only difference is that the result holds in a high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The proof can be found in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This complexity matches the lower-bound complexity for stochastic smooth non-convex unconstrained optimization [3, 31], so it is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In fact, we can also obtain the same complexity if the subgradient oracles of both f and g are stochastic as long as the function value oracle of g remains deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Since the analysis and conclusion are very similar to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3, we skip this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Property Eτg(x(τ)) ≤ ǫ2(ˆρ−ρ) 1+Λ in the theorems above is not required by Defini- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1, Eτg(x(τ)) ≤ Eτg(�x(τ)) + MEτ∥�x(τ) − x(τ)∥ ≤ Mǫ, which means a nearly ǫ-stationary point must be O(ǫ)-feasible by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Property Eτg(x(τ)) ≤ ǫ2(ˆρ−ρ) 1+Λ implies O(ǫ2)-feasibility for the output, which is even better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Strong convexity often leads to lower complexity for convex optimization, so an interesting question is how the complexity of the SSG method changes if when g is µ-strongly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We can show that, given strong convexity in g, the complexity is still O � 1/ǫ4� but one can simply set ǫt = 0, which makes ηt the only tuning parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This makes this single-loop method even more attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We include this result in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 5 Stochastic Constraint In this section, we consider the case where the oracles of the subgradient and function value of g are stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The assumption is formally stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For any x ∈ X, stochastic subgradients ξf and ξg and a stochastic value ω can be generated independently such that E ω ∈ ∂g(x), E ξf ∈ ∂f(x) and E ξg ∈ ∂g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Moreover, it holds that E exp � ∥ξf∥2/M2� ≤ exp(1), E exp � ∥ξg∥2/M2� ≤ exp(1), and E exp � (ω − g(x))2/σ2� ≤ exp(1) for a constant σ for any x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 7 Algorithm 2 Switching Subgradient Method for a Stochastic Constraint 1: Input: x(0) ∈ X, total number of iterations T, tolerance of infeasibility ǫt > 0, t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T − 1, mini-batch size B, and a starting index S for generating outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 2: Initialization: I ← ∅ and J ← ∅ 3: for iteration t = 0, 1, · · · , T − 1 do 4: Generate a mini-batch of stochastic estimators of g at x(t), denoted by {ω(t) i }B i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 5: ¯ω(t) ← 1 B �B i=1 ω(t) i 6: if ω(t) ≤ ǫt then 7: Generate a stochastic subgradient ξ(t) f of f at x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 8: x(t+1) ← projX (x(t) − ηtξ(t) f ) 9: I ← I ∪ {t} if t ≥ S 10: else 11: Generate a stochastic subgradient ξ(t) g of g at x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 12: x(t+1) ← projX (x(t) − ηtξ(t) g ) 13: J ← J ∪ {t} if t ≥ S 14: end if 15: end for 16: Output: x(τ) with τ randomly sampled from I with τ = t in a probability of ηt / � t∈I ηt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This case is fundamentally more challenging than the case with deterministic g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In fact, the challenge comes only from the stochastic function value of g instead of its stochastic subgradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In fact, in the fully deterministic case, Algorithm 1 essentially updates x(t) along a hybrid subgradient I(g(x(t)) ≤ ǫt)ζ(t) f + I(g(x(t)) > ǫt)ζ(t) g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (8) If only the subgradients are stochastic, the hybrid stochastic subgradient I(g(x(t)) ≤ ǫt)ξ(t) f + I(g(x(t)) > ǫt)ξ(t) g provides an unbiased estimation of (8), so we can still obtain complexity of O(1/ǫ4) by a proof similar to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' However, when g(x(t)) must be queried through some unbiased estimator w(t), the naively constructed direction I(w(t) ≤ ǫt)ξ(t) f + I(w(t) > ǫt)ξ(t) g (9) is not an unbiased estimator of (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' To tackle this issue, we have to query a mini-batch of w(t) of size B to construct an estimator of g(x(t)), denoted by ¯w(t), with a high accuracy in a high probability, so that (9) with w(t) replaced by ¯w(t) can be a nearly unbiased estimator of (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The SSG method with this modification is presented in Algorithm 2, where the condition ¯w(t) ≤ ǫt is used to determine if the stochastic subgradient should be switched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Its complexity is characterized below when the control parameters are static and diminishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The proof is given in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 hold and ˆρ > ρ, δ ∈ (0, 1), ǫ > 0 and Λ is defined in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Then Algorithm 2 guarantees Eτ∥�x(τ) − x(τ)∥ ≤ ǫ and Eτg(x(τ)) ≤ 2ǫ2(ˆρ−ρ) 1+Λ with probability at least 1 − δ in either of the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 8 Case I: If S, ǫt, ηt and T are chosen as Case I in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3 and B = 300σ2 ln(4T/δ)(Λ+1)2 ǫ4(ˆρ−ρ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Case II: If S, ǫt, ηt and T are chosen as Case II in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3 except that E is any positive constant such that E ≥ 8 + 2π √ 6 max{ � 12 ln(8/δ), 4 3 ln(8/δ)} + 8 � 3 ln(4/δ) and B = 3Tσ2 ln(2T/δ)(Λ+1)2 2M2D2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In each iteration of Algorithm 2, we query one stochastic subgradient of g or f but B stochastic function values of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In both Case I and Case II, we have T = O(1/ǫ4) and 2 B = ˜O(1/ǫ4) so the subgradient oracle complexity is still O(1/ǫ4) but the function value oracle complexity becomes O(1/ǫ8), which is higher than the O(1/ǫ6) complexity by the double-loop methods in [12, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' It is our future work to reduce the complexity when g is stochastic, for example, by a single-loop primal-dual method that uses a hybrid subgradient like ξ(t) f + λtξ(t) g with the dual variable λt updated by only one sample of w(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 6 Numerical Experiment We demonstrate the performance of the SSG method on fairness-aware classification prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' It is implemented with both static and diminishing step sizes and compared with two benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' One is the constraint extrapolation (ConEx) method in Algorithm 1 in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' It is a single-loop method for convex optimization, so it has no theoretical guarantee for (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The other is the inexact proximal point (IPP) method, which is a double-loop method in [12, 55, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' It approximately solves a strongly convex constrained subproblem in each outer iteration, and we use the SSG and ConEx methods as the solvers (inner loop) because they both have the best theoretical complexity for that subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We use IPP-SSG and IPP-ConEx to denote these two implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 Classification with ROC-based fairness Given a feature vector a ∈ Rd and a class label b ∈ {1, −1}, the goal in a binary classification problem is to learn a linear model w ∈ Rd to predict b based on the score w⊤a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Let D = {(ai, bi)}n i=1 be a training set and ℓ(·) be a convex non-increasing loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Model w can be learned by solving an empirical risk minimization problem L∗ = min w∈W � L(w) := 1 n n � i=1 ℓ(biw⊤ai) � , (10) where W = {w ∈ Rd | ∥w∥ ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Solving (10) may ensure good classification performance of w but not its fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose there exist two additional datasets containing the feature vectors of a protected group Dp = {ap i }np i=1 and the feature vectors of an unprotected group Du = {au i }nu i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We want to enhance the fairness of w between these two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' There are various fairness metrics in literature but we focus on the ROC-based fairness metric proposed by [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose we classify data a as positive if w⊤a ≥ θ and as negative otherwise, where θ 2 ˜O(·) suppresses logarithmic factors in the order of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 9 Table 1: Information of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Groups are males VS females in a9a and german, users with age within [25, 60] VS outside [25, 60] in bank, and caucasian VS non-caucasian in COMPAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Datasets n d Label Groups a9a 48,842 123 Income Gender Bank 41,188 54 Subscription Age COMPAS 11,757 14 Recidivism Race german 1,000 21 Credit risk Gender is a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The empirical ROC curve for these two groups is a curve in a 2D space whose two coordinates are the predicted positive rates on the two groups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=', 1 np np � i=1 I(w⊤ap i ≥ θ) and 1 nu nu � i=1 I(w⊤au i ≥ θ), as θ varies from −∞ to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 3 The ROC-based fairness measure is defined as max θ∈Θ ��� 1 np np � i=1 I(w⊤ap i ≥ θ) − 1 nu nu � i=1 I(w⊤au i ≥ θ) ��� or its continuous approximation R(w) = max θ∈Θ ��� 1 np np � i=1 σ(w⊤ap i ≥ θ) − 1 nu nu � i=1 σ(w⊤au i ≥ θ) ���, (11) where σ is the sigmoid function and Θ is a finite set of thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' If the value of this measure is small, model w produces similar predicted positive rates for the protected and unprotected groups on various θ’s, indicating the fairness of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' To obtain a fair w, we balance (10) and (11) by solving min w∈W R(w) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' L(w) ≤ L∗ + κ, (12) where κ is a slackness parameter indicating how much we are willing to increase the classifi- cation loss in order to reduce R(w) to obtain a more fair model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Problem (12) is an instance of (1) satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We solve (12) on three datasets: a9a [43], bank [58] and COMPAS [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The information of these datasets is given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We split each dataset into two subsets with a ratio of 2 : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The larger set is used as D in the constraint and the smaller set is further split into Dp and Du based on the grouping variables in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In our experiments, we first solve (10) using the subgradient method with a large enough number of iterations to obtain L∗ and a solution wERM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Then we set κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='001L∗, ℓ(z) = (1− z)+ and r = 5∥wERM∥, and let Θ consist of 400 points equally spaced between mini w⊤ ERMai− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='5(maxi w⊤ ERMai − mini w⊤ ERMai) and maxi w⊤ ERMai + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='5(maxi w⊤ ERMai − mini w⊤ ERMai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 3It is different from the standard ROC curve whose coordinates are the predicted positive rates on positive and negative classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 10 ROC-Based Fair Classification Wasserstein-Based Fair Classification (Deterministic f) (Stochastic f) a9a bank COMPAS 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='9 1 Iteration t 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='55 SSG-static SSG-diminishing ConEx IPP-SSG IPP-ConEx Figure 1: Performances on fairness-aware classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We use a deterministic oracle in all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For SSG with a static step size, we select ǫt from {5 × 10−6, 10−5, 2 × 10−5} and ηt from {5 × 10−4, 10−3, 2 × 10−3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For SSG with a diminishing step size, we set ǫt = E1 √t+1 and ηt = E2 √t+1 and select E1 from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='05} and E2 from {10−4, 2 × 10−4, 5 × 10−4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For ConEx, we set the primal step size ηt = c1 √t+1 and dual step size τt = c2 and select c1 from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='05} and c2 from {20, 50, 100}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We set θt = 1 in ConEx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We select the best set of parameters that produces the smallest objective value after 5000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For IPP, we select ˆρ from max{ρ, 1} × {1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='5, 2} with ρ = �n i=1 ∥ai∥/n, and the proximal point subproblem is approximately solved by SSG and ConEx both with 100 iterations indexed by k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For IPP-SSG, we apply a static step size and the parameters are tuned the same way as SSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For IPP-ConEx, we set the primal step size ηk = c1(k + 1) and dual step size τk = c2 k+1 and select c1 from {20, 50, 100} and c2 from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='02}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We set θk = k k+1 in IPP-ConEx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We report the performances of all methods on each dataset in the first three columns of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The x-axis represents the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (For IPP, it represents the total number of inner iterations across all outer iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=') The y-axis represents the objective value, infeasibility and near stationarity achieved at each iteration, respectively, in the three rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' To measure near stationarity, we solve (4) with x = x(t) using the SSG method with sufficient iterations and use the last iterate as an approximation of �xˆρ(x(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Then we plot ∥�xˆρ(x(t)) − x(t)∥ as near stationarity in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Since computing �xˆρ(x(t)) with a high precision is time-consuming, we only report near stationarity at 100 equally spaced iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' According to Figure 1, the ConEx method has the best performance overall, but it does not have theoretical convergence guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The IPP method and SSG method with static 11 step sizes have similar efficiency in reducing the objective value while keeping the solutions nearly feasible on the instances we test on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The SSG method with diminishing step sizes performs slightly better than these two on bank and COMPAS datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This is consistent with our theory that the SSG and IPP methods have the same complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2 Classification with Wasserstein-based fairness Wasserstein distance, which is a distance between two distributions, has also been considered as a metric of fairness [42, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Given a linear classifier w, the 1-Wasserstein distance between the empirical distributions of the prediction scores on Du and Du is the optimal value of linear program min P ∈P � H(P, w) := np � i=1 nu � j=1 Pij|w⊤ap i − w⊤au i | � , (13) where P = {P ∈ Rnp×nu| �np i=1 Pij = 1 nu , �nu j=1 Pij = 1 np, Pij ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' If this value is small, the prediction score has similar distributions on the protected and unprotected groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' To build a fair model, we just need to balance (10) and (13) by solving min P ∈P,w∈W H(P, w) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' L(w) ≤ L∗ + κ, (14) which is also an instance of (1) satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We solve (14) on datasets german [32, 18] and COMPAS [39] whose information is in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We set κ and r and split each dataset into D, Dp and Du in the same way as in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In each iteration of all methods in comparison, we need to project P (t) to P, for which we use the algorithm and codes by [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Since their algorithm assume np = nu, we further downsample the larger one of Dp and Du to the size of the smaller one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We use a stochastic oracle for f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In fact, we generate the stochastic subgradient of H(P, w) by sampling B = ⌊np 10 ⌋ = ⌊nu 10 ⌋ samples from Dp and Du, respectively, and forming B2 pairs of protected and unprotected instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Then, the stochastic subgradient is constructed only using the summands of ∂wH(P, w) and the coordinates of ∂P H(P, w) that are associated to the those pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We use a deterministic oracle for g in all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This is justified by the fact that f has a quadratic computational cost, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=', O(npnu), while g has only a linear cost O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For SSG with a static step size, we select ǫt from {5 × 10−6, 10−5, 2 × 10−5} and ηt from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='02}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For SSG with a diminishing step size, we set ǫt = E1 √t+1 and ηt = E2 √t+1 and select E1 from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='15} and E2 from {10−4, 2 × 10−4, 5 × 10−4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For ConEx, we set the primal step size ηt = c1 √t+1 and dual step size τt = c2 √t+1 and select c1 from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='05} and c2 from {20, 50, 100}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We set θt = 1 in ConEx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We select the best set of parameters that produces the smallest objective value after 15000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For IPP, we select ˆρ from max{ρ, 1} × {10−4, 5 × 10−4, 10−3} with ρ = ∥Ap∥F + ∥Au∥F where Ap is the np × d matrix whose each row is a⊤ i , Au is the nu × d matrix whose each row is a⊤ i and ∥ · ∥F is the Frobenius norm, and the proximal point subproblem is approximately solved by SSG and ConEx both with 1000 iterations indexed by k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For IPP-SSG, we apply a static step size and the parameters are tuned the same way as SSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For IPP-ConEx, we set the primal step size ηk = c1 √k+1 and dual step size τk = c2 k+1 and select c1 from {100, 200, 500} and c2 12 from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='02}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We set θk = k k+1 in IPP-ConEx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The performances of the methods are given in the last two columns of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This time ConEx does not perform as well as it did for the previous application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' On the 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' SIAM Journal on Optimization, 32(3):2319–2346, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' [82] Jiawei Zhang, Wenqiang Pu, and Zhi-Quan Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' On the iteration complexity of smoothed proximal alm for nonconvex optimization problem with convex constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='06304, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' [83] Jingzhao Zhang, Hongzhou Lin, Stefanie Jegelka, Ali Jadbabaie, and Suvrit Sra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Com- plexity of finding stationary points of nonsmooth nonconvex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' arXiv preprint arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='04130, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' [84] Siqi Zhang and Niao He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' On the convergence rate of stochastic mirror descent for nonsmooth nonconvex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' arXiv preprint arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='04781, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 19 A Convergence Analysis In this section, we present the proofs of all lemmas, propositions and theorems in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For simplicity of notation, we denote �xˆρ(x) in (5) by �x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' According to Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1B, there exists a strictly feasible solution xfeas with xfeas ∈ X and g(xfeas) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' As a result, the Lagrangian multiplier �λ ≥ 0 corresponding to �x is well- defined and satisfies (6), which means �λg(�x) = 0 and �ζf + ˆρ(�x − x) + �λ�ζg + �v = 0, (15) where �ζf ∈ ∂f(�x), �ζg ∈ ∂g(�x), �v ∈ NX (�x) and NX (�x) is the normal cone of X at �x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' If �λ = 0, the conclusion holds trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Hence, we focus on the case that �λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Note that, in this case, we must have g(�x) = 0 since �λg(�x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Taking the inner product between (15) and �x − xfeas gives 1 �λ ⟨�ζf + ˆρ(�x − x), �x − xfeas⟩ = ⟨�ζg + �v/�λ, xfeas − �x⟩ ≤ g(xfeas) − g(�x) = g(xfeas), (16) where the inequality is because of convexity of g and the fact that �v/�λ ∈ NX(�x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Note that Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 and Jensen’s inequality imply ∥�ζf∥ ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Moreover, we have ∥�x−xfeas∥ ≤ D and ∥�x − x∥ ≤ D by Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Hence, from (16), we can obtain �λ ≤ ⟨�ζf + ˆρ(�x − x), xfeas − �x⟩ −g(xfeas) ≤ MD + ˆρD2 −g(xfeas) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2 Technical Lemmas and Propositions In this section, we first introduce additional notations and then present a few technical lemmas and propositions which are needed to prove the convergence of the proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We want to remind readers that we assume Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 for the entire paper so this assumption will not be stated again in each lemma and proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Since the case under Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 is more general than the case under Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1, in the propositions below, the conclusions under Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 will be presented before those under Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Let I[·] be an indicator of an event, which equals one if the event occurs and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For each iterate x(t) in Algorithms 1 and 2, let ζ(t) f := Eξ(t) f ∈ ∂f(x(t)) and ζ(t) g := Eξ(t) g ∈ ∂g(x(t)) and let �x(t) := �xˆρ(x(t)) defined in (5) and �λt ≥ 0 be the corresponding Lagrangian multiplier satisfying (6), which exists by Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Under Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1, let Et[·] := E[·|ξ(0) f , ξ(0) g , ¯ω(0), ξ(1) f , ξ(1) g , , ¯ω(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , ξ(t−1) f , ξ(t−1) g , ¯ω(t−1)], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=', the conditional ex- pectation conditioning on all stochastic events before iteration t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Let Eτ be the expectation taken only over the random index τ when the algorithm is terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We first provide a proposition that characterizes the relationship between ∥x(t) − �x(t)∥ and the control parameters T, S, B, ηt and ǫt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 20 Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Under Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 and assuming g is µ-strongly convex (µ can be zero),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Algorithm 2 guarantees T−1 � t=S � ηtˆρ�λtI(¯ω(t) ≤ ǫt) + ηtˆρI(¯ω(t) > ǫt) � µ 2∥�x(t) − x(t)∥2 + T−1 � t=S � ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtg(x(t)) � I(¯ω(t) ≤ ǫt) + T−1 � t=S ηtˆρg(x(t))I(¯ω(t) > ǫt) ≤ ˆρD2 2 + ˆρ 2 T−1 � t=S � η2 t ∥ξ(t) f ∥2I(¯ω(t) ≤ ǫt) + η2 t ∥ξ(t) g ∥2I(¯ω(t) > ǫt) � + T−1 � t=S � ˆρηt � ξ(t) f − ζ(t) f ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' �x(t) − x(t)� I(¯ω(t) ≤ ǫt) + ˆρηt � ξ(t) g − ζ(t) g ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' �x(t) − x(t)� I(¯ω(t) > ǫt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' As a special case of the result above, under Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 and assuming g is µ-strongly convex (µ can be zero), Algorithm 1 guarantees T−1 � t=S � ηtˆρ�λtI(g(x(t)) ≤ ǫt) + ηtˆρI(g(x(t)) > ǫt) � µ 2 ∥�x(t) − x(t)∥2 + T−1 � t=S � ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtg(x(t)) � I(g(x(t)) ≤ ǫt) + T−1 � t=S ηtˆρg(x(t))I(g(x(t)) > ǫt) ≤ ˆρD2 2 + ˆρ 2 T−1 � t=S � η2 t ∥ξ(t) f ∥2I(g(x(t)) ≤ ǫt) + η2 t ∥ζ(t) g ∥2I(g(x(t)) > ǫt) � + T−1 � t=S � ˆρηt � ξ(t) f − ζ(t) f , �x(t) − x(t)� I(g(x(t)) ≤ ǫt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Let ξ = ξ(t) f if t ∈ I and ξ = ξ(t) g if t ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Similarly, let ζ = ζ(t) f ∈ ∂f(x(t)) if t ∈ I and ζ = ζ(t) g ∈ ∂g(x(t)) if t ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By the updating equation of x(t+1), we have ∥x(t+1) − �x(t)∥2 = ∥projX (x(t) − ηtξ(t)) − �x(t)∥2 = ∥projX (x(t) − ηtξ(t)) − projX (�x(t))∥2 ≤ ∥x(t) − ηtξ(t) − �x(t)∥2 = ∥x(t) − �x(t)∥2 − 2ηt � ξ(t), x(t) − �x(t)� + η2 t ∥ξ(t)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Multiplying the inequality above by ˆρ/2 and adding f(�x(t)) to both sides, we obtain f(�x(t)) + ˆρ 2∥x(t+1) − �x(t)∥2 ≤ f(�x(t)) + ˆρ 2∥x(t) − �x(t)∥2 − ˆρηt � ξ(t), x(t) − �x(t)� + ˆρη2 t 2 ∥ξ(t)∥2 (17) ≤ ϕˆρ(x(t)) − ˆρηt � ζ(t), x(t) − �x(t)� + ˆρη2 t 2 ∥ξ(t)∥2 − ˆρηt � ξ(t) − ζ(t), x(t) − �x(t)� , (18) 21 where the second inequality is by the definition of ϕˆρ(x) in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Since �x(t) is a feasible solution to problem (4) with x = x(t+1), we have ϕˆρ(x(t+1)) ≤ f(�x(t)) + ˆρ 2∥x(t+1) − �x(t)∥2, which, together with (18), implies ˆρηt � ζ(t), x(t) − �x(t)� ≤ ϕˆρ(x(t)) − ϕˆρ(x(t+1)) + ˆρη2 t 2 ∥ξ(t)∥2 − ˆρηt � ξ(t) − ζ(t), x(t) − �x(t)� (19) Next, we will bound � ζ(t), x(t) − �x(t)� from below when t ∈ I and t ∈ J, separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose t ∈ I so ¯ω(t) ≤ ǫt, ζ(t) = ζ(t) f and ξ(t) = ξ(t) f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By ρ-weak convexity of f, we have � ζ(t), x(t) − �x(t)� ≥ f(x(t)) − f(�x(t)) − ρ 2∥�x(t) − x(t)∥2 = f(x(t)) − f(�x(t)) − ˆρ 2∥�x(t) − x(t)∥2 + ˆρ − ρ 2 ∥�x(t) − x(t)∥2 (20) Consider the convex optimization problem (4) with x = x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1B, there exists a Lagrangian multiplier �λt ≥ 0 such that �λtg(�x(t)) = 0 (complementory slackness) and �x(t) = arg min x∈X f(x) + ˆρ 2∥x − x(t)∥2 + �λtg(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Since the objective function above is (ˆρ − ρ + �λtµ)-strongly convex, we have f(x(t)) + �λtg(x(t)) = f(x(t)) + ˆρ 2∥x(t) − x(t)∥2 + �λtg(x(t)) ≥ f(�x(t)) + ˆρ 2∥�x(t) − x(t)∥2 + �λtg(�x(t)) + ˆρ − ρ + �λtµ 2 ∥�x(t) − x(t)∥2, which, by the facts that �λt ≥ 0 and �λtg(�x(t)) = 0, implies f(x(t)) − f(�x(t)) − ˆρ 2∥�x(t) − x(t)∥2 ≥ −�λtg(x(t)) + ˆρ − ρ + �λtµ 2 ∥�x(t) − x(t)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Applying this inequality and inequality (20) to (19) leads to ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtg(x(t)) + ηtˆρ�λtµ 2 ∥�x(t) − x(t)∥2 ≤ ϕˆρ(x(t)) − ϕˆρ(x(t+1)) + ˆρη2 t 2 ∥ξ(t) f ∥2 − ˆρηt � ξ(t) f − ζ(t) f , x(t) − �x(t)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (21) Suppose t ∈ J so ¯ω(t) > ǫt, ζ(t) = ζ(t) g and ξ(t) = ξ(t) g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By µ-convexity of g and the fact that g(�x(t)) ≤ 0, we have � ζ(t), x(t) − �x(t)� − µ 2 ∥�x(t) − x(t)∥2 ≥ g(x(t)) − g(�x(t)) ≥ g(x(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 22 Applying this inequality to (19) leads to ηtˆρg(x(t)) + ηtˆρµ 2 ∥�x(t) − x(t)∥2 ≤ ϕˆρ(x(t)) − ϕˆρ(x(t+1)) + ˆρη2 t 2 ∥ξ(t) g ∥2 − ˆρηt � ξ(t) g − ζ(t) g , x(t) − �x(t)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (22) Recall that I[t ∈ I] = I[¯ω(t) ≤ ǫt] and I[t ∈ J] = I[¯ω(t) > ǫt].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Summing up (21) and (22) for t = S, S + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T − 1, we have T−1 � t=S � ηtˆρ�λtI(¯ω(t) ≤ ǫt) + ηtˆρI(¯ω(t) > ǫt) � µ 2∥�x(t) − x(t)∥2 + T−1 � t=S � ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtg(x(t)) � I(¯ω(t) ≤ ǫt) + T−1 � t=S ηtˆρg(x(t))I(¯ω(t) > ǫt) ≤ ϕˆρ(x(S)) − ϕˆρ(x(T)) + ˆρ 2 T−1 � t=S � η2 t ∥ξ(t) f ∥2I(¯ω(t) ≤ ǫt) + η2 t ∥ξ(t) g ∥2I(¯ω(t) > ǫt) � − T−1 � t=S � ˆρηt � ξ(t) f − ζ(t) f , x(t) − �x(t)� I(¯ω(t) ≤ ǫt) + ˆρηt � ξ(t) g − ζ(t) g , x(t) − �x(t)� I(¯ω(t) > ǫt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Let x∗ be the optimal solution of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The first conclusion is then implied by the facts that ϕˆρ(x(T)) ≥ f ∗ and ϕˆρ(x(S)) ≤ f(x∗) + ˆρ 2∥x∗ − x(S)∥2 ≤ f ∗ + ˆρD2 2 , where f ∗ is finite by Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The second conclusion is a special case of the first one when ¯ω(t) = g(x(t)) and ξ(t) g = ζ(t) g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Given any x ∈ X, let {ωi}B i=1 be a mini-batch of stochastic estimators of g at x and ¯ω = 1 B �B i=1 ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' It holds that, for any δ ∈ (0, 1), Prob � ¯ω > g(x) + √ 3σ � ln(1/δ) √ B � ≤ δ and Prob � ¯ω < g(x) − √ 3σ � ln(1/δ) √ B � ≤ δ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The conclusion is guaranteed by Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 and Lemma 2 (Case A) in [49] by choosing Ω = � 3 ln(1/δ) in their bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For any δ ∈ (0, 1), Algorithm 2 guarantees with probability at least 1 − δ that g(x(t))I(¯ω(t) ≤ ǫt) ≤ � ¯ω(t) + √ 3σ � ln((T − S)/δ) √ B � I(¯ω(t) ≤ ǫt) ≤ � ǫt + √ 3σ � ln((T − S)/δ) √ B � I(¯ω(t) ≤ ǫt) (23) for t = S, S + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T − 1 and, consequently, T−1 � t=S ηtˆρ�λtg(x(t))I(¯ω(t) ≤ ǫt) ≤ T−1 � t=S ηtˆρ�λt � ǫt + √ 3σ � ln((T − S)/δ) √ B � I(¯ω(t) ≤ ǫt) 23 and Algorithm 2 guarantees with probability at least 1 − δ that T−1 � t=S ηtˆρg(x(t))I(¯ω(t) > ǫt) ≥ T−1 � t=S ηtˆρ � ǫt − √ 3σ � ln((T − S)/δ) √ B � I(¯ω(t) > ǫt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For any t, x(t) is determined by ξ(0) f , ξ(0) g , ¯ω(0), ξ(1) f , ξ(1) g , , ¯ω(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , ξ(t−1) f , ξ(t−1) g and ¯ω(t−1), while ¯ω(t) is generated after x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Hence, we have (23) for each t with a probability of at least 1 − δ/(T − S) according to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The first conclusion is then obtained by taking the union bound for the events above for t = S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The second conclusion can be proved in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For any δ ∈ (0, 1), Algorithm 2 guarantees with probability at least 1 − δ that T−1 � t=S � η2 t ∥ξ(t) f ∥2I(¯ω(t) ≤ ǫt) + η2 t ∥ξ(t) g ∥2I(¯ω(t) > ǫt) � ≤ T−1 � t=S η2 t M2 + max{ � 12 ln(2/δ), 4 3 ln(2/δ)} � � � � T−1 � t=S η4 t M4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For any δ ∈ (0, 1), Algorithm 1 guarantees T−1 � t=S � η2 t ∥ξ(t) f ∥2I(g(x(t)) ≤ ǫt) + η2 t ∥ζ(t) g ∥2I(g(x(t)) > ǫt) � ≤ T−1 � t=S η2 t M2 + max{ � 12 ln(2/δ), 4 3 ln(2/δ)} � � � � T−1 � t=S η4 t M4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For any t, x(t) is determined by ξ(0) f , ξ(0) g , ¯ω(0), ξ(1) f , ξ(1) g , , ¯ω(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , ξ(t−1) f , ξ(t−1) g and ¯ω(t−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Also, ξ(t) f , ξ(t) g and ¯ω(t) are independent and generated after x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Hence, by Assump- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1, we have Et exp \uf8eb \uf8edη2 t ∥ξ(t) f ∥2I(¯ω(t) ≤ ǫt) + η2 t ∥ξ(t) g ∥2I(¯ω(t) > ǫt) η2 t M2 \uf8f6 \uf8f8 = Et \uf8ee \uf8f0I(¯ω(t) ≤ ǫt) exp \uf8eb \uf8ed∥ξ(t) f ∥2 M2 \uf8f6 \uf8f8 \uf8f9 \uf8fb + Et � I(¯ω(t) > ǫt) exp � ∥ξ(t) g ∥2 M2 �� = EtI(¯ω(t) ≤ ǫt)Et exp \uf8eb \uf8ed∥ξ(t) f ∥2 M2 \uf8f6 \uf8f8 + EtI(¯ω(t) > ǫt)Et exp � ∥ξ(t) g ∥2 M2 � ≤ EtI(¯ω(t) ≤ ǫt) exp (1) + EtI(¯ω(t) > ǫt) exp (1) = exp (1) , where the second equality is by the conditional independence between ω(t), ξ(t) f and ξ(t) g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Then the first conclusion is guaranteed by Lemma 2 (Case B) in [49] by choosing Ω = 24 max{ � 12 ln(2/δ), 4 3 ln(2/δ)} in their bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The second conclusion is a special case of the first one when ¯ω(t) = g(x(t)) and ξ(t) g = ζ(t) g and thus can be proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For any δ ∈ (0, 1), Algorithm 2 guarantees with probability at least 1 − δ that T−1 � t=S � ˆρηt � ξ(t) f − ζ(t) f , �x(t) − x(t)� I(¯ω(t) ≤ ǫt) + ˆρηt � ξ(t) g − ζ(t) g , �x(t) − x(t)� I(¯ω(t) > ǫt) � ≤ � 3 ln(1/δ) � � � � T−1 � t=S 4ˆρ2η2 t M2D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' For any δ ∈ (0, 1), Algorithm 1 guarantees T−1 � t=S ˆρηt � ξ(t) f − ζ(t) f , �x(t) − x(t)� I(g(x(t)) ≤ ǫt) ≤ � 3 ln(1/δ) � � � � T−1 � t=S 4ˆρ2η2 t M2D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 and Jensen’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' we have exp \uf8eb \uf8ed∥ζ(t) f ∥2 M2 \uf8f6 \uf8f8 ≤ Et exp \uf8eb \uf8ed∥ξ(t) f ∥2 M2 \uf8f6 \uf8f8 ≤ exp(1) and exp � ∥ζ(t) g ∥2 M2 � ≤ Et exp � ∥ξ(t) g ∥2 M2 � ≤ exp(1) which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' by Jensen’s inequality again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' implies Et exp \uf8eb \uf8ed2∥ξ(t) f ∥2 + 2∥ζ(t) f ∥2 4M2 \uf8f6 \uf8f8 = Et exp \uf8eb \uf8ed∥ξ(t) f ∥2 2M2 \uf8f6 \uf8f8 exp \uf8eb \uf8ed∥ζ(t) f ∥2 2M2 \uf8f6 \uf8f8 ≤ � � � � �Et exp \uf8eb \uf8ed∥ξ(t) f ∥2 M2 \uf8f6 \uf8f8 exp �1 2 � ≤ exp(1) (24) Et exp � 2∥ξ(t) g ∥2 + 2∥ζ(t) g ∥2 4M2 � = Et exp � ∥ξ(t) g ∥2 2M2 � exp � ∥ζ(t) g ∥2 2M2 � ≤ � � � �Et exp � ∥ξ(t) g ∥2 M2 � exp �1 2 � ≤ exp(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (25) For any t, x(t) is determined by ξ(0) f , ξ(0) g , ¯ω(0), ξ(1) f , ξ(1) g , , ¯ω(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , ξ(t−1) f , ξ(t−1) g and ¯ω(t−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Also, ξ(t) f , ξ(t) g and ¯ω(t) are independent and generated after x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Hence, by Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1, we have Et � ˆρηt � ξ(t) f − ζ(t) f , �x(t) − x(t)� I(¯ω(t) ≤ ǫt) + ˆρηt � ξ(t) g − ζ(t) g , �x(t) − x(t)� I(¯ω(t) > ǫt) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 25 and Et exp \uf8eb \uf8ec \uf8ed � ˆρηt � ξ(t) f − ζ(t) f ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' �x(t) − x(t)� I(¯ω(t) ≤ ǫt) + ˆρηt � ξ(t) g − ζ(t) g ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' �x(t) − x(t)� I(¯ω(t) > ǫt) �2 4ˆρ2η2 t M 2D2 \uf8f6 \uf8f7 \uf8f8 = Et \uf8ee \uf8ef\uf8f0I(¯ω(t) ≤ ǫt) exp \uf8eb \uf8ec \uf8ed �� ξ(t) f − ζ(t) f ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' �x(t) − x(t)��2 4M 2D2 \uf8f6 \uf8f7 \uf8f8 \uf8f9 \uf8fa\uf8fb + Et \uf8ee \uf8ef\uf8f0I(¯ω(t) > ǫt) exp \uf8eb \uf8ec \uf8ed �� ξ(t) g − ζ(t) g ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' �x(t) − x(t)��2 4M 2D2 \uf8f6 \uf8f7 \uf8f8 \uf8f9 \uf8fa\uf8fb ≤ EtI(¯ω(t) ≤ ǫt)Et exp � ∥ξ(t) f − ζ(t) f ∥2∥�x(t) − x(t)∥2 4M 2D2 � + EtI(¯ω(t) > ǫt)Et exp � ∥ξ(t) g − ζ(t) g ∥2∥�x(t) − x(t)∥2 4M 2D2 � ≤ EtI(¯ω(t) ≤ ǫt)Et exp � 2∥ξ(t) f ∥2 + 2∥ζ(t) f ∥2 4M 2 � + EtI(¯ω(t) > ǫt)Et exp � 2∥ξ(t) g ∥2 + 2∥ζ(t) g ∥2 4M 2 � ≤ EtI(¯ω(t) ≤ ǫt) exp (1) + EtI(¯ω(t) > ǫt) exp (1) = exp (1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' where the first inequality is by the Cauchy–Schwarz inequality and the conditional indepen- dence between ω(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' ξ(t) f and ξ(t) g ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' the second inequality is by Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1C and the last inequality is by (24) and (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Then the first conclusion is guaranteed by Lemma 2 (Case A) in [49] by choosing Ω = � 3 ln(1/δ) in their bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The second conclusion is a special case of the first one when ¯ω(t) = g(x(t)) and ξ(t) g = ζ(t) g and thus can be proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose g is convex but not strongly convex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=', µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Taking the union bound of the four events in Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='4 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='5 with δ replaced by δ 4 and applying the four inequalities (two from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3, one from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='4 and one from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='5) holding in these events to Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1, we have the following bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Under Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1, Algorithm 2 guarantees with probability at least 1 − δ that T−1 � t=S � ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtǫt � I(¯ω(t) ≤ ǫt) + T−1 � t=S ηtˆρǫtI(¯ω(t) > ǫt) ≤ ˆρD2 2 + ˆρ 2 T−1 � t=S η2 t M2 + ˆρ 2 max{ � 12 ln(8/δ), 4 3 ln(8/δ)} � � � � T−1 � t=S η4 t M4 (26) + � 3 ln(4/δ) � � � � T−1 � t=S 4ˆρ2η2 t M2D2 + T−1 � t=S ηtˆρ √ 3σ � ln(4(T − S)/δ) √ B � �λtI(¯ω(t) ≤ ǫt) + I(¯ω(t) > ǫt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 26 Under Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1, Algorithm 1 guarantees with probability at least 1 − δ that T−1 � t=S � ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtǫt � I(g(x(t)) ≤ ǫt) + T−1 � t=S ηtˆρǫtI(g(x(t)) > ǫt) ≤ ˆρD2 2 + ˆρ 2 T−1 � t=S η2 t M2 + ˆρ 2 max{ � 12 ln(8/δ), 4 3 ln(8/δ)} � � � � T−1 � t=S η4 t M4 (27) + � 3 ln(4/δ) � � � � T−1 � t=S 4ˆρ2η2 t M2D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Since Algorithm 1 is fully deterministic under the assumptions and we do not assume strong convexity in g so µ = 0, we can simplify the second inequality in Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 as follows T −1 � t=S � ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtǫt � I(g(x(t)) ≤ ǫt) + T −1 � t=S ηtˆρǫtI(g(x(t)) > ǫt) ≤ T −1 � t=S � ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtg(x(t)) � I(g(x(t)) ≤ ǫt) + T −1 � t=S ηtˆρg(x(t))I(g(x(t)) > ǫt) ≤ ˆρD2 2 + ˆρ 2 T −1 � t=S � η2 t ∥ξ(t) f ∥2I(g(x(t)) ≤ ǫt) + η2 t ∥ζ(t) g ∥2I(g(x(t)) > ǫt) � + T −1 � t=S � ˆρηt � ξ(t) f − ζ(t) f ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' �x(t) − x(t)� I(g(x(t)) ≤ ǫt) � ≤ ˆρD2 2 + ˆρ 2 T −1 � t=S η2 t M 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (28) where the second inequality is the second inequality in Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 and the last inequality is because ξ(t) f = ζ(t) f and Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We first prove that, if S, T, ηt and ǫt are chosen such that ǫt(1 + �λt) ≤ ǫ2(ˆρ − ρ) (29) and T−1 � t=S ηtˆρǫt > ˆρD2 2 + ˆρ 2 T−1 � t=S η2 t M2, (30) we must have g(x(t)) ≤ ǫt for at least one t in {S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T−1} (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=', I ̸= ∅) and Eτ∥�x(τ)−x(τ)∥2 ≤ ǫ2 (so Eτ∥�x(τ) − x(τ)∥ ≤ ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose (30) holds and g(x(t)) > ǫt for t = S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T −1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=', I = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (28) becomes exactly the opposite of (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This contradiction means g(x(t)) ≤ ǫt for at least one t in {S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 27 Suppose (29) and (30) hold but Eτ∥�x(τ) − x(τ)∥2 > ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By the process of generating τ, we have ǫ2 < Eτ∥�x(τ) − x(τ)∥2 = �T−1 t=S ηtI(g(x(t)) ≤ ǫt)∥�x(t) − x(t)∥2 �T−1 t=S ηtI(g(x(t)) ≤ ǫt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (31) Note that the right-hand side of (31) is well-defined because we just proved I ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (31) and (29) imply T −1 � t=S � ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtǫt � I(g(x(t)) ≤ ǫt) + T −1 � t=S ηtˆρǫtI(g(x(t)) > ǫt) > T −1 � t=S � ηtˆρ(ˆρ − ρ)ǫ2 − ηtˆρ�λtǫt � I(g(x(t)) ≤ ǫt) + T −1 � t=S ηtˆρǫtI(g(x(t)) > ǫt) ≥ T −1 � t=S ηt ˆρǫtI(g(x(t)) ≤ ǫt) + T −1 � t=S ηtˆρǫtI(g(x(t)) > ǫt) ≥ T −1 � t=S ηt ˆρǫt, (32) where the second inequality is because of (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Combining this inequality and (28) leads to the opposite of (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This contradiction means Eτ∥�x(τ) − x(τ)∥2 ≤ ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Given the result above, we only need to show that the two choices of S, T, ηt and ǫt ensure (29) and (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In Case I, (29) holds because of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3 and the choice of ǫt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Let η = ηt = 2ǫ2(ˆρ−ρ) 5(1+Λ)M2 for any t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3 and plugging the values of S, T, ηt and ǫt in (30), we can show that (30) is equivalent to Tηˆρǫ2(ˆρ − ρ) 1 + Λ > ˆρD2 2 + ˆρ 2Tη2M2, which, after dividing both sides by Tηˆρ, can be equivalently written as ǫ2(ˆρ − ρ) 1 + Λ > D2 2Tη + ηM2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By the values of η and T, each summand in the right-hand side of the inequality above is no more than ǫ2(ˆρ−ρ) 5(1+Λ) so the right-hand side of the inequality above no more than 2ǫ2(ˆρ−ρ) 5(1+Λ) which is strictly less than the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This means (30) holds with this choice of parameters and thus Eτ∥�x(τ) − x(τ)∥ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By the convexity of g and the choices of ηt and ǫt, we have Eτg(x(τ)) ≤ ǫ2(ˆρ−ρ) 1+Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In Case II, by the choices of ǫt and T, we have, for any t ∈ {S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T − 1}, ǫt = 5MD √t + 1 ≤ 5MD √ S + 1 = 5MD � T/2 + 1 ≤ ǫ2(ˆρ − ρ) 1 + Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (33) This further implies (29) because of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Note that ηt and ǫt are decreasing in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Hence, the left-hand side of (30) satisfies T−1 � t=S ηtˆρǫt > T 2 ˆρηT ǫT = 5T 2T + 2 ˆρD2 ≥ 5ˆρD2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (34) 28 The right-hand side of (30) satisfies ˆρD2 2 + ˆρ 2 T−1 � t=S η2 t M2 = ˆρD2 2 + ˆρ 2D2 T−1 � t=S 1 t + 1 ≤ ˆρD2, (35) where the equality is obtained by plugging in the definition of ηt and the inequality is because �T−1 t=S 1 t+1 ≤ � T S 1 t dt = ln(T/S) = ln(2) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The right-hand side of (34) is strictly greater than the right-hand side (35), which means (30) holds and thus Eτ∥�x(τ) − x(τ)∥ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By the convexity of g and the choices of ηt and ǫt, we have Eτg(x(τ)) ≤ ǫS ≤ ǫ2(ˆρ−ρ) 1+λ according to (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='4 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='6, (27) holds with a probability of at least 1− δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In the rest of the proof, we always assume (27) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We first prove that, if S, T, ηt and ǫt are chosen such that (29) holds and T−1 � t=S ηtˆρǫt > ˆρD2 2 + ˆρ 2 T−1 � t=S η2 t M2 + ˆρ 2 max{ � 12 ln(8/δ), 4 3 ln(8/δ)} � � � � T−1 � t=S η4 t M4 (36) + � 3 ln(4/δ) � � � � T−1 � t=S 4ˆρ2η2 t M2D2, we must have g(x(t)) ≤ ǫt for at least one t in {S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T−1} (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=', I ̸= ∅) and Eτ∥�x(τ)−x(τ)∥2 ≤ ǫ2 (so Eτ∥�x(τ) − x(τ)∥ ≤ ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose (36) holds and g(x(t)) > ǫt for t = S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=', I = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (27) becomes exactly the opposite of (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This contradiction means g(x(t)) ≤ ǫt for at least one t in {S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose (29) and (36) hold but Eτ∥�x(τ) − x(τ)∥2 > ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By the process of generating τ, we have (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Note that the right-hand side of (31) is well-defined because we just proved I ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (31) and (29) imply (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Combining (32) and (36) leads to the opposite of (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This contradiction means Eτ∥�x(τ) − x(τ)∥2 ≤ ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Given the result above, we only need to show that the choices of S, T, ηt and ǫt ensure (29) and (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In Case I, (29) holds because of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3 and the choice of ǫt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Let η = ηt = 2ǫ2(ˆρ−ρ) 5(1+Λ)M2 for any t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3 and plugging the values of S, T, ηt and ǫt in (36), we can show that (36) is equivalent to Tηˆρǫ2(ˆρ − ρ) 1 + Λ > ˆρD2 2 + ˆρ 2Tη2M2 + ˆρ 2 max{ � 12 ln(8/δ), 4 3 ln(8/δ)} √ Tη2M2 +2 � 3 ln(4/δ)ˆρ √ TηMD, which, after dividing both sides by Tηˆρ, can be equivalently written as ǫ2(ˆρ − ρ) 1 + Λ > D2 2Tη + ηM2 2 + 1 2 max{ � 12 ln(8/δ), 4 3 ln(8/δ)}ηM2 √ T + 2 � 3 ln(4/δ)MD √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 29 By the values of η and T, each summand in the right-hand side of the inequality above is no more than ǫ2(ˆρ−ρ) 5(1+Λ) so the right-hand side of the inequality above no more than 4ǫ2(ˆρ−ρ) 5(1+Λ) which is strictly less than the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This means (30) holds with this choice of parameters and thus Eτ∥�x(τ) − x(τ)∥ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We can prove Eτg(x(τ)) ≤ ǫ2(ˆρ−ρ) 1+λ in the same way as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In Case II, by the choices of ǫt and T, we have, for any t ∈ {S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T − 1}, ǫt = EMD √t + 1 ≤ EMD √ S + 1 = EMD � T/2 + 1 ≤ ǫ2(ˆρ − ρ) 1 + Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (37) This further implies (29) because of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Note that ηt and ǫt are decreasing in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Hence, the left-hand side of (36) satisfies T−1 � t=S ηtˆρǫt > T 2 ˆρηT ǫT = ET 2T + 2 ˆρD2 ≥ E ˆρD2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (38) The right-hand side of (36) satisfies ˆρD2 2 + ˆρ 2 T−1 � t=S η2 t M2 + ˆρ 2 max{ � 12 ln(8/δ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 4 3 ln(8/δ)} � � � � T−1 � t=S η4 t M4 + � 3 ln(4/δ) � � � � T−1 � t=S 4ˆρ2η2 t M2D2 = ˆρD2 2 + ˆρ 2D2 T−1 � t=S 1 t + 1 + ˆρ 2 max{ � 12 ln(8/δ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 4 3 ln(8/δ)}D2 � � � � T−1 � t=S 1 (t + 1)2 +2 � 3 ln(4/δ)ˆρD2 � � � � T−1 � t=S 1 t + 1 ≤ ˆρD2 + ˆρπ 2 √ 6 max{ � 12 ln(8/δ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 4 3 ln(8/δ)}D2 + 2 � 3 ln(4/δ)ˆρD2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (39) where the equality is obtained by plugging in the definition of ηt and the inequality is because �T−1 t=S 1 t+1 ≤ � T S 1 t dt = ln(T/S) = ln(2) ≤ 1 and �T−1 t=S 1 (t+1)2 ≤ π2 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By the condition satisfied by E, the right-hand side of (38) is greater than or equal to the right-hand side (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This means (30) holds with this choice of parameters and thus Eτ∥�x(τ) − x(τ)∥ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We can prove Eτg(x(τ)) ≤ ǫ2(ˆρ−ρ) 1+λ in the same way as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='5 Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2 Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3 and Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='6, (26) holds and, simultaneously, (23) holds with δ replaced by δ/4 for t = S, S + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T − 1 with a probability of at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In the rest of the proof, we always assume (26) holds and (23) holds with δ replaced 30 by δ/4 for t = S, S + 1, ˙,T − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We first prove that, if S, T, B, ηt and ǫt are chosen such that (29) holds and T−1 � t=S ηtˆρǫt > ˆρD2 2 + ˆρ 2 T−1 � t=S η2 t M2 + ˆρ 2 max{ � 12 ln(8/δ), 4 3 ln(8/δ)} � � � � T−1 � t=S η4 t M4 (40) + � 3 ln(4/δ) � � � � T−1 � t=S 4ˆρ2η2 t M2D2 + T−1 � t=S ηtˆρ √ 3σ � ln(4(T − S)/δ) √ B (Λ + 1), we must have ¯ω(t) ≤ ǫt for at least one t in {S, S + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T − 1} (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=', I ̸= ∅) and Eτ∥�x(τ) − x(τ)∥2 ≤ ǫ2 (so Eτ∥�x(τ) − x(τ)∥ ≤ ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Here, Λ is defined in (7) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose (40) holds and ¯ω(t) > ǫt for t = S, S + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=', I = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (26) contradicts with (40) as Λ ≥ �λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This contradiction means ¯ω(t) ≤ ǫt for at least one t in {S, S+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T−1} so I ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose (29) and (40) hold but Eτ∥�x(τ) −x(τ)∥2 > ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By the process of generating τ, we have ǫ2 < Eτ∥�x(τ) − x(τ)∥2 = �T−1 t=S ηtI(¯ω(t) ≤ ǫt)∥�x(t) − x(t)∥2 �T−1 t=S ηtI(¯ω(t) ≤ ǫt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (41) Note that the right-hand side of (41) is well-defined because we just proved I ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (41) and (29) imply T −1 � t=S � ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtǫt � I(¯ω(t) ≤ ǫt) + T −1 � t=S ηtˆρǫtI(¯ω(t) > ǫt) > T −1 � t=S � ηtˆρ(ˆρ − ρ)ǫ2 − ηtˆρ�λtǫt � I(¯ω(t) ≤ ǫt) + T −1 � t=S ηtˆρǫtI(¯ω(t) > ǫt) ≥ T −1 � t=S ηtˆρǫtI(¯ω(t) ≤ ǫt) + T −1 � t=S ηtˆρǫtI(¯ω(t) > ǫt) ≥ T −1 � t=S ηtˆρǫt, (42) where the second inequality is because of (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Combining (42) and (40) leads to the opposite of (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This contradiction means Eτ∥�x(τ) − x(τ)∥2 ≤ ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Given the result above, we only need to show that the choices of S, T, B, ηt and ǫt ensure (29) and (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In Case I, (29) holds because of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3 and the choice of ǫt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Let η = ηt = 2ǫ2(ˆρ−ρ) 5(1+Λ)M2 for any t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3 and plugging the values of S, T, B, ηt and ǫt in (40), we can show that (40) holds if Tηˆρǫ2(ˆρ − ρ) 1 + Λ > ˆρD2 2 + ˆρ 2Tη2M2 + ˆρ 2 max{ � 12 ln(8/δ), 4 3 ln(8/δ)} √ Tη2M2 +2 � 3 ln(4/δ)ˆρ √ TηMD + Tηˆρ √ 3σ � ln(4T/δ) √ B (Λ + 1), which, after dividing both sides by Tηˆρ, can be equivalently written as ǫ2(ˆρ − ρ) 1 + Λ > D2 2Tη + ηM2 2 + 1 2 max{ � 12 ln(8/δ), 4 3 ln(8/δ)}ηM2 √ T +2 � 3 ln(4/δ)MD √ T + √ 3σ � ln(4T/δ) √ B (Λ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 31 By the values of η, B and T, each of the first four summands in the right-hand side of the inequality above is no more than ǫ2(ˆρ−ρ) 5(1+Λ) while the last summand is no more than ǫ2(ˆρ−ρ) 10(1+Λ), so the right-hand side of the inequality above no more than 9ǫ2(ˆρ−ρ) 10(1+Λ) which is strictly less than the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This means (30) holds with this choice of parameters and thus Eτ∥�x(τ) − x(τ)∥ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Moreover, since (23) holds with δ replaced by δ/4 for t = S, S + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T − 1, we have Eτg(x(τ)) = �T−1 t=0 ηtg(x(t))I(¯ω(t) ≤ ǫt) �T−1 t=0 ηtI(¯ω(t) ≤ ǫt) ≤ �T−1 t=0 ηt � ǫt + √ 3σ√ ln(4T/δ) √ B � I(¯ω(t) ≤ ǫt) �T−1 t=0 ηtI(¯ω(t) ≤ ǫt) ≤ ǫ2(ˆρ − ρ) 1 + Λ + √ 3σ � ln(4T/δ) √ B ≤ 2ǫ2(ˆρ − ρ) 1 + Λ , where the last inequality is because of the choice of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In Case II, by the choice of ǫt, we have (37) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This further implies (29) because of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Note that ηt and ǫt are decreasing in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Hence, we also have (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The right-hand side of (40) satisfies ˆρD2 2 + ˆρ 2 T −1 � t=S η2 t M 2 + ˆρ 2 max{ � 12 ln(8/δ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 4 3 ln(8/δ)} � � � � T −1 � t=S η4 t M 4 + � 3 ln(4/δ) � � � � T −1 � t=S 4ˆρ2η2 t M 2D2 + T −1 � t=S ηtˆρ √ 3σ � ln(4(T − S)/δ) √ B (Λ + 1) = ˆρD2 2 + ˆρ 2D2 T −1 � t=S 1 t + 1 + ˆρ 2 max{ � 12 ln(8/δ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 4 3 ln(8/δ)}D2 � � � � T −1 � t=S 1 (t + 1)2 +2 � 3 ln(4/δ)ˆρD2 � � � � T −1 � t=S 1 t + 1 + Dˆρ M √ 3σ � ln(4(T − S)/δ) √ B (Λ + 1) T −1 � t=S 1 √t + 1 ≤ ˆρD2 + ˆρπ 2 √ 6 max{ � 12 ln(8/δ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 4 3 ln(8/δ)}D2 + 2 � 3 ln(4/δ)ˆρD2 + ˆρD2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (43) where the equality is obtained by plugging in the definition of ηt and the inequality is because of the definition of B and the facts that �T−1 t=S 1 t+1 ≤ � T S 1 t dt = ln(T/S) = ln(2) ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' �T−1 t=S 1 (t+1)2 ≤ π2 6 and �T−1 t=S 1 √t+1 ≤ � T S 1 √ tdt ≤ � T/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By the condition of E, the right- hand side of (38) is strictly greater than the right-hand side (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This means (40) holds with this choice of parameters and thus Eτ∥�x(τ) − x(τ)∥ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Moreover, since (23) holds with δ replaced by δ/4 for t = S, S + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T − 1, we have Eτg(x(τ)) = �T−1 t=S ηtg(x(t))I(¯ω(t) ≤ ǫt) �T−1 t=S ηtI(¯ω(t) ≤ ǫt) ≤ �T−1 t=S ηt � ǫt + √ 3σ√ ln(2T/δ) √ B � I(¯ω(t) ≤ ǫt) �T−1 t=S ηtI(¯ω(t) ≤ ǫt) ≤ ǫT/2 + √ 3σ � ln(2T/δ) √ B ≤ 2ǫ2(ˆρ − ρ) 1 + Λ , where the last inequality is because of (37) and the choices of B and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 32 B Complexity results when g is strongly convex Suppose the constraint function g in (1) is deterministic and µ-strongly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' We have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 hold , g is µ-strongly convex with µ > 0, ˆρ > ρ and ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Moreover, suppose ξf is deterministic, namely, ξf = ζf ∈ ∂f(x) for any x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Suppose the output of Algorithm 1 is changed to x(τ) with τ randomly sampled from S, S + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' , T − 1 with τ = t with a probability of ηt/ �T−1 t=S ηt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Then Algorithm 1 guarantees Eτ∥�x(τ) − x(τ)∥ ≤ ǫ in either of the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Case I: S = 0, ǫt = 0, ηt = ǫ2 min{ˆρ−ρ,µ/2} M2 and T ≥ M 2D2 ǫ4 min{(ˆρ − ρ)2, µ2/4} = O � 1 ǫ4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Case II: S = T 2 , ǫt = 0, ηt = D M√t+1 and T ≥ 4M 2D2 ǫ4 min{(ˆρ − ρ)2, µ2/4} = O � 1 ǫ4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Before presenting its proof, we would like to make a few remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' According to Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1, strong convexity in the constraint function g does not reduce the O(1/ǫ4) complexity of the SSG method for finding a nearly ǫ-stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' However, strong convexity brings benefit on other aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' First, we can simply set ǫt = 0 when g is strongly convex, which makes step size ηt the only tuning parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Second, the theoretical complexity no longer depends on Λ, the upper bound of the dual variables, so it can be strictly better than the one in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2 when Λ is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 only shows the result when f and g are both deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' How- ever, we can also establish O(1/ǫ4) oracle complexity for Algorithm 1 when f is stochastic but g is deterministic, and establish O(1/ǫ4) subgradient oracle complexity and O(1/ǫ8) func- tion value oracle complexity for Algorithm 2 when both f and g are stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Similar to Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1, we only need to set ǫt = 0 and define τ as in Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This indicates that strong convexity in g brings the same convenience in these two cases but does not change the complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' The proofs for these two cases will be similar to that of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Since those results do not provide additional insights on complexity and analysis, we do not include them in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='2 guarantee Eτg(x(τ)) ≤ O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' This is no longer true in Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 because τ can take value in the index set J on which x(t) can be highly infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' However, with Eτ∥�x(τ) − x(τ)∥ ≤ ǫ, we can at least derive Eτg(x(τ)) ≤ O(ǫ) from Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1, which is good enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' See Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='4 for the reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Since Algorithm 1 is fully deterministic with ǫt = 0 under the assump- tions, we can bound the left-hand side of the second inequality in Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 as follows 33 T−1 � t=S � ηtˆρ�λtI(g(x(t)) ≤ 0) + ηtˆρI(g(x(t)) > 0) � µ 2 ∥�x(t) − x(t)∥2 + T−1 � t=S � ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2 − ηtˆρ�λtg(x(t)) � I(g(x(t)) ≤ 0) + T−1 � t=S ηtˆρg(x(t))I(g(x(t)) > 0) ≥ T−1 � t=S ηtˆρI(g(x(t)) > 0)µ 2 ∥�x(t) − x(t)∥2 + T−1 � t=S ηtˆρ(ˆρ − ρ)∥�x(t) − x(t)∥2I(g(x(t)) ≤ 0) ≥ ˆρ min{ˆρ − ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' µ/2} T−1 � t=S ηt∥�x(t) − x(t)∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' where the first inequality is by dropping the non-negative terms and the fact that g(x(t))I(g(x(t)) ≤ 0) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' Combining this lower bound with the second inequality in Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1 gives ˆρ min{ˆρ − ρ, µ/2} T−1 � t=S ηt∥�x(t) − x(t)∥2 ≤ ˆρD2 2 + ˆρ 2 T−1 � t=S � η2 t ∥ξ(t) f ∥2I(g(x(t)) ≤ 0) + η2 t ∥ζ(t) g ∥2I(g(x(t)) > 0) � + T−1 � t=S � ˆρηt � ξ(t) f − ζ(t) f , �x(t) − x(t)� I(g(x(t)) ≤ 0) � ≤ ˆρD2 2 + ˆρ 2 T−1 � t=S η2 t M2, (44) where the last inequality is because ξ(t) f = ζ(t) f and Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By the (new) definition of τ, after organizing terms, we have Eτ∥�x(t) − x(t)∥2 ≤ 1 min{ˆρ − ρ, µ/2} � D2 2 + 1 2 T−1 � t=S η2 t M2 � / �T−1 � t=S ηt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' (45) In Case I, let η = ηt = ǫ2 min{ˆρ−ρ,µ/2} M2 for any t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' By the choices of ηt, S and T, (45) implies Eτ∥�x(t) − x(t)∥2 ≤ 1 min{ˆρ − ρ, µ/2} � D2 2Tη + ηM2 2 � ≤ ǫ2 2 + ǫ2 2 = ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' In Case II, by the choices of ηt, S and T, (45) implies Eτ∥�x(t) − x(t)∥2 ≤ 1 min{ˆρ − ρ, µ/2} �DM √ T + DM √ T � ≤ ǫ2 2 + ǫ2 2 = ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' where, in the first inequality, we use the facts that �T−1 t=S ηt ≥ T 2 ηT−1 = √ TD 2M and that �T−1 t=S η2 t = �T−1 t=S D2 M2(t+1) ≤ � T S D2 M2tdt = D2 M2 ln(T/S) = D2 M2 ln(2) ≤ D2 M2 , and the second inequality is by the choice of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} +page_content=' 34' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FQT4oBgHgl3EQfZDYV/content/2301.13314v1.pdf'} diff --git a/PNE2T4oBgHgl3EQfrggh/content/tmp_files/2301.04049v1.pdf.txt b/PNE2T4oBgHgl3EQfrggh/content/tmp_files/2301.04049v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c711e39b560c3f538558604f4bc7ffd9474972b6 --- /dev/null +++ b/PNE2T4oBgHgl3EQfrggh/content/tmp_files/2301.04049v1.pdf.txt @@ -0,0 +1,1089 @@ +Imbalanced Classication In Faulty Turbine +Data: New Proximal Policy Optimization +Mohammad Hossein Modirrousta, Mahdi Aliyari Shoorehdeli, Senior Member, IEEE, Mostafa Yari and +Arash Ghahremani +Abstract—There is growing importance to detecting +faults and implementing the best methods in industrial +and real-world systems. We are searching for the most +trustworthy and practical data-based fault detection meth- +ods proposed by articial intelligence applications. In this +paper, we propose a framework for fault detection based +on reinforcement learning and a policy known as proximal +policy optimization. As a result of the lack of fault data, one +of the signicant problems with the traditional policy is its +weakness in detecting fault classes, which was addressed +by changing the cost function. Using modied Proximal +Policy Optimization, we can increase performance, over- +come data imbalance, and better predict future faults. When +our modied policy is implemented, all evaluation metrics +will increase by 3% to 4% as compared to the traditional +policy in the rst benchmark, between 20% and 55% in the +second benchmark, and between 6% and 14% in the third +benchmark, as well as an improvement in performance and +prediction speed compared to previous methods. +Index Terms—Fault detection, Deep Learning, Reinforce- +ment learning, Proximal Policy Optimization. +I. INTRODUCTION +P +RODUCTION safety and product quality are key factors +to ensure economic benets in the industrial process [1], +[2]. Monitoring and diagnosing faults play an important role +and have received considerable attention from academia and +industry [3], [4]. +Operational failures are mainly traced to external environ- +mental factors during the in-service period. Structures can +fail due to ice and sand accumulations, erosion, corrosion, +lightning damage, dust, and insect contamination [5]. Cracks +and delaminations can also occur during the operational phase. +When the incipient damage is not detected early, it may +result in severe malfunctions, fatal injuries, or, worst case, the +collapse of the entire turbine. +Condition Monitoring (CM) systems for turbines rely on +fault detection and prediction algorithms, but there are two +major challenges. Sensor data generation is the rst of these. +Mohammad +Hossein +Modirrousta +(e-mail: +moham- +madbc@email.kntu.ac.ir) is with the Faculty of Electrical Engineering, +K. N. Toosi University of Technology, Tehran, Iran. +Mahdi Aliyari Shoorehdeli (e-mail: aliyari@kntu.ac.ir) is with the Fac- +ulty of Mechatronics Engineering, K. N. Toosi University of Technology, +Tehran, Iran. +Mostafa Yari (e-mail: yari.mostafa@mapnaec.com) is with the Faculty +of Mechatronics Engineering, K. N. Toosi University of Technology, +Tehran, Iran. +Arash Ghahremani (e-mail: ghahremani.arash@mapnaec.com) is +with the Faculty of Mechanical Engineering, K. N. Toosi University of +Technology, Tehran, Iran. +CM systems and their interpretation are complicated by the +need to store and process these data. Sensor reliability is +the second issue. [6], [7] have pointed out issues regarding +the accuracy and accountability of sensors used for pattern +recognition and fault identication. The accuracy of sensor +data strongly inuences a CM system’s performance. Addi- +tionally, the use of a large number of sensors, and hence +monitoring variables, may reduce the overall reliability of the +sensor system. Consequently, fault detection methods are a +necessary component of ensuring the operational safety and +reliability of mechanical systems and reducing maintenance +costs. +A. Related work +According to [8], there are three types of fault detection +methods for turbine systems, including signal-based methods +[9], [10], model-based methods [11], and knowledge-based +methods [12], [13]. A robust fault estimation and fault-tolerant +control approach for Takagi-Sugeno fuzzy systems is proposed +in [14]. Furthermore, multivariate statistical methods have also +successfully detected faults [15], [16]. +The use of SCADA data for monitoring wind turbine +conditions has been proposed in several ways. There has been +extensive use of articial neural networks (ANNs), Gaussian +processes (GP), support vector machines (SVMs), and random +forests (RFs) in order to improve the performance of turbines. +In [17], [18], [19], [20], [21], general overviews of these +techniques are provided. +The development of turbine models and fault detection +methods based on articial intelligence has been impressive +[22], [23]. As part of the analysis for turbine condition +monitoring [24], a method based on articial neural networks +(ANNs) has been proposed. [25] presents an algorithm for +ANN pattern recognition and its application to controls for +turbines. In [26], support vector machines (SVMs) were com- +bined with a residual-based method to detect and isolate faults +in wind turbines. Using Shannon wavelet SVMs and manifold +learning to diagnose faults in turbine transmission systems +were proposed in [27]. Detecting turbine faults can also be +accomplished using AI-based methods based on fuzzy logic +or expert systems. According to [28], adaptive neuro-fuzzy +inference systems can be used to monitor the condition of +turbines. +Deep learning (DL) has achieved enormous success in +many elds in recent years. Researchers are also interested in +the eld of fault detection [29], [30]. An unsupervised deep + +learning method (denoising autoencoder) has been proposed +for detecting turbine faults [31]. In addition to the domain +adaptation using the maximum mean discrepancy, DL models +such as sparse autoencoder [32] and convolutional neural +networks (CNN) [33] are used for condition recognition. An +ImageNet pre-trained network is used in the paper [34] to train +a deep-learning network to classify faults. In order to obtain a +time-frequency distribution to ne-tune the high-level network +layers, sensor data are transformed into image data by plotting +or using wavelet transformation [35]. +An imbalance of class samples occurs in real-world ap- +plications when one class’s samples exceed those of other +classes [36]. Turbine fault detection, for instance, exhibits class +imbalance because these machines generally operate under +normal conditions and occasionally fail; these conditions result +in many normal operations and few faulty ones. Data and +algorithm-level research has been conducted to alleviate the +problem of class imbalance [37]. This technique is a data-level +method that involves random under- and oversampling. Over- +sampling techniques have been studied extensively [38]. A +focus on cost-sensitive algorithms [39] and ensemble learning +[40] is discussed. Misclassied positive and negative samples +incur high and low costs, respectively. Costs are difcult to +determine. An ensemble learning algorithm such as Adaboost +[14] uses an iterative boosting algorithm to increase the weight +of misclassied samples and decrease the weight of correctly +classied samples after each iteration. The performance of +boosting depends strongly on the base classier. +Articial intelligence (AI) also includes reinforcement +learning (RL). Intelligent computing techniques are used +to automate and understand problem-oriented learning and +decision-making [41]. In contrast to other intelligent methods, +it emphasizes that the agent learns through direct interaction +with the environment without requiring imitation of supervi- +sion signals. Deep reinforcement learning (DRL) combines +RL with deep learning (DL). DRL has achieved great success +in games [42], recommendation systems [43], and robotics +control [44]. Nevertheless, DRL is rarely mentioned in fault +identication, which DL dominates. We propose a new method +for identifying faults in real turbines through DRL. Intelligent +methods will be more universal with fault parameterization and +DRL implementation. Several faults are parameterized here, +which enables DRL to transform a classication problem into +a sequential decision problem. +A Markov decision process for classication was proposed +by Wiering et al. This framework dened a standard clas- +sication problem as a sequential decision-making problem, +and an MLP model trained in it outperformed a regular +MLP model trained by backpropagation [45]. Using DRL to +identify bearing health states, Ding et al. proposed an approach +[46]; Huang et al. adopted DRL to implement a preventive +maintenance policy for serial production lines [47]. Based on +time-frequency representations (TFR) and dynamic response +mappings (DRLs), Wang et al. developed a new fault diagnosis +methodology [48]. As described in [49], DDQN improves +the detection performance of cyberattacks by adopting a ne- +grained trafc ow monitoring mechanism. +In contrast to traditional policy gradient algorithms, PPO is +an advanced algorithm capable of overcoming the problem of +low learning efciency caused by the inuence of the step size +on learning efciency. PPO has several primary advantages +for training control policies. First of all, in [50], the hyper- +parameters of PPO were proved to be robust when training +various tasks, and PPO can balance the complexity and accu- +racy of control policies. Second, in [51], the training control +policy of PPO was found to be superior to that of other RL +algorithms on all metrics compared to the performance indica- +tors. Based on the full six-degree-of-freedom system dynamics +of the UAV, in [52], PPO is used to train quadrotor control +policies, achieving stable hovering. In the context of Software- +Dened Networking (SDN), Zolotukhin et al. proposed an +interesting approach [53]. The authors investigate Deep Q- +Network (DQN) and PPO in response to an attack. DQN +and PPO show promising results, which further motivates +this study. PPO also simplies implementation and improves +performance in IoT applications [54]. The authors of [55] +propose an agent-based reinforcement learning scheme using +PPO to allow multiple agents to control their own devices. +An intrusion detection hyperparameter control procedure is +built in [56], which controls and trains a deep neural network +feature extractor based on proximal policy optimization (PPO). +A new automated lane change strategy using proximal policy +optimization is proposed [57], which shows excellent benets +while maintaining performance stability. +B. Our Contributions +The ndings of the above studies led us to propose a new +fault detection method based on DRL. It is based on Classi- +cation Markov Decision Process (CMDP), which denes the +fault detection problem as a guessing game. The diagnosis +agent rst learns an optimal recognition policy within the +framework of DRL. By using experience replay (ER), the +agent automatically interacts with the environment, creating +experiences, and updating the model based on those expe- +riences. Furthermore, the proposed method has been tested +on detection tasks and is compared to existing detection +methods. In addition to exhibiting better generalization and +speed compiling, this method also performs well when dealing +with imbalanced problems. +The following are the main contributions of our framework +as described in this paper: +1) We are using reinforcement learning to build a recom- +mender label system. In order to discover a new algorithm +for fault detection from a DRL perspective, we consider +fault detection as a guessing game and describe it as a +sequential decision-making problem based on CMDP. +2) With the help of experience replay (ER) and reward-based +learning tools, an optimal model for fault detection can be +developed based on Proximal Policy Optimization (PPO). +3) It is necessary to make some changes to the regular cost +function of PPO. As a result of these changes, imbalances +in data will be addressed. Using our approach, we can make +decisions without relying on feature engineering. +4) We examined and compared the performance of this +method with and without changes in the cost function, as + +well as with other methods on multiple datasets. We found +that this method enhanced fault detection abilities. +II. PROBLEM DEFINITION +The diagnosis of faults is often considered to be a +classication problem. The primary purpose of DRL is to +solve the sequential decision-making problem. A guessing +game is used in this work to diagnose turbine faults. We +also create a game simulation that converts fault diagno- +sis into a sequential decision-making problem using the +DRL. This illustration illustrates how a training dataset that +might be regarded as a guessing question set is Xtrain += +(s1, l1) , (s2l2) , . . . , (sn, ln) where si is the i − th sample +and li is the i − th label corresponding to sample si. As part +of this game, each round consists of T questions matching +training data generated from the training dataset Xtrain . Agents +guess these questions sequentially by the order in which the +samples in D are arranged. +This game involves the agent observing a sample each time +and assuming the class of the sample. Following the guessing +question, the environment provides an immediate reward to +the agent and the next guessing question (i.e., the following +sample). A positive reward is awarded to the agent if the +agent correctly identies the sample’s category; otherwise, a +negative reward is given. The agent’s objective in this game is +to maximize accumulated rewards within the constraints of an +optimal behavior policy that has been learned due to constant +interaction with the environment, as shown in Fig. 1. +Fig. 1: Overview of the Agent-User interaction in the Classi- +cation MDP (CMDP). +III. METHODOLOGY +A. Proximal Policy Optimization +Using reinforcement learning, an agent can learn how to +interact with its environment to maximize its expected cu- +mulative rewards. An RL algorithm can be divided into two +general categories: value-based and policy-based. Even though +value-based methods can approximate the value function us- +ing neural networks in an off-policy manner, policy-based +methods, such as the REINFORCE algorithm [20], offer the +primary advantage of optimizing the quantity of advantage +directly while maintaining stability during the approximation +of functions. As a result, our study focuses on RL methods +based on policy. +During a general policy gradient reinforcement learning, the +objective function is as follows: +LP (θ) = ˆEt + +log πθ (at  st) ˆAt + +(1) +In this case, ˆEt represents the expectation operator, πθ repre- +sents a stochastic RL policy, and ˆAt represents the estimated +advantage function at time step t. +Using the discount factor γ  [0, 1], we can calculate ˆAt +using the generalized advantage estimator [30]. Generally, the +generalized advantage estimator can be described as follows: +ˆAt = δt + (γλ)δt+1 + · · · + · · · + (γλ)T −t+1δT −1 +(2) +Where δt = rt + γVϕ (st+1) − Vϕ (st), and T is the sampled +mini-batch size. The parameter λ  [0, 1] represents the +generalized advantage estimator. +According to LV , the objective function is as follows: +LV (ϕ) = E + +LV +t (ϕ) + += E + ˆV target +ϕ +(st) − Vϕ (st) + + +(3) +In this case, the target value of the time-difference error (TD- +Error) is +ˆV target +ϕ +(st) = rt+1 + γVϕ (st+1) +(4) +A gradient descent algorithm is used to update the parameters +of Vϕ, with the gradient LV : +ϕ = ϕ − ηϕLV (ϕ) +(5) +In the critic model optimization, ηϕ is the learning rate. +Using the actor model, the PPO uses the importance sam- +pling method in place of the objective function presented in +Equation (1) to estimate the expectation of samples collected +from the old policy πθold under the new policy πθ. Using LCP I +as a surrogate objective function, the algorithm maximizes: +LCP I(θ) = Et + πθ (at  st) +πθold (at  st) +At + +(6) +The PPO optimizes LCP I with a small value δ based on +the amount of the policy update as a constraint, according to +the equation below: +Et [KL [πθold (·  st) , πθ (·  st)]] ≤ δ +(7) +Kullback-Leibler divergence is indicated by KL [32]. The +actor-critic structure is used in advanced policy-based algo- +rithms, including TRPO [18] and PPO [17], which combine +the advantages of traditional value-based and policy-based ap- +proaches. As well as being more straightforward to implement +and allowing multiple optimization iterations, the PPO algo- +rithm also has a higher sample complexity. In particular, PPO +proposes a modied surrogate loss function, which combines +the policy surrogate and the error term associated with the +value function, dened as follows [17] +LCLIP +V +S +t +(θ) = ˆEt + +LCLIP +t +(θ) − c1LV +t (θ) + c2S [πθ] (st) + +(8) +In this equation, LCLIP +t +stands for the clipped surrogate +objective, c1 and c2 represent the coefcients, LV +t stands for +the squared-error loss of the value function, and S stands for +the entropy loss. In order to ensure that our agents have enough + +Store Expenence +Reward Y +Environment +Agent +S1 +Action +St +ER Buffer +Lata +State s,exploration, we use an entropy term. By using this term, the +policy will be pushed to behave more spontaneously until the +other objective overtakes it. +In more detail, the clipped surrogate objective LCLIP +t +is as +follows: +LCLIP (θ) = Et + +LCLIP +t +(ϕ) + += Et + +min (Rt(θ), clip (Rt(θ), 1 − ϵ, 1 + ϵ)) At + +(9) +Where ϵ indicates a clipping parameter and Rt(θ) += +πθ(at|st) +πθold(at|st) indicates a probability ratio. This procedure results +in a clipping of the probability ratio Rt(θ) at time 1 − ϵ or +1 + ϵ, depending upon whether the advantage is positive or +negative, which forms the clipped objective after multiplying +the advantage approximator ˆAt. In the end, LCLIP +t +is calcu- +lated by taking the minimum of this clipped objective and the +unclipped objective Rt(θ) ˆAt, thereby reducing the need for a +substantial policy update compared to the unclipped version +[17]. This loss function is known as the conservative policy +iteration algorithm loss function [21]. +Using gradient descent, the parameters of πθ are updated +according to the gradient LCLIP of the negative of the +clipped objective function: +θ = θ − ηθLCLIP (θ) +(10) +Assume that ηθ is the learning rate for the actor model +optimization. +IV. PROPOSED DRL BASED FAULT DIAGNOSIS +A. Data Pre-processing +Turbine Databases Some rows with a ”drop-it” title are +available in the datasets available. As such data is considered +outlier data, it should be deleted. Following that, the data must +be normalized using a Min-Max scaler as follows: +DNormalize = +D − Dmin +Dmax − Dmin +. +(11) +D and DNormalize represent features before and after +normalization, respectively, and Dmin and Dmax represent +minimum and maximum features based on data before nor- +malization. +CICIDS2017 Database. This dataset is also normalized +using the same technique as the previous dataset. In addition, +some values had inf values, which posed a problem in the +normalization and classication procedure. These values have +been dropped from the list. +B. Agent Architecture Design +We have faced many challenges in dealing with the im- +balanced classication problem. In industrial settings, it is +essential to detect faulty data, and many researchers have +attempted to reduce the number of false positives and false +negatives. The loss functions were modied to emphasize the +few data that address the imbalanced data problem. Consider +the objective function in the actor model. As follows, we dene +a new surrogate objective function: +LNCP I(θ) = ˆEt + πθ (at  st) +πθold (at  st) +ˆAt + β log (πθ (at  st)) + +(12) +New Conservative Policy Iteration is indicated by the super- +script N C P I. The term entropy into a new policy was added +to the cost function compared to equation 6. The purpose of +this work is to place a greater emphasis on new policies rather +than old ones and to improve and accelerate the convergence +of actor losses and total losses. In the results section, we will +compare the results with and without this change. +The large class imbalance observed during the training of +dense detectors overwhelms the cross-entropy loss, as demon- +strated in [58]. The authors propose adapting the loss function +to down-weight easy examples and adding a modulating factor +to the cross-entropy loss. Therefore, the following is the +denition of focal loss: +FL (pt) = −αt (1 − pt)γ log (pt) +(13) +p  [0, 1] is the estimated probability according to the +model. Specically, α  [0, 1] is a weighting factor, and +γ  [0, 1] is a tunable focusing parameter. A modulating factor +reduces the loss contribution from easy examples and extends +the range of examples subject to low loss contributions. +For the above reasons, we replace the entropy loss with a +focal loss in equation 8. Results show a signicant improve- +ment in performance and a rapid convergence in total loss. This +change will also be revealed in the results section. Below is a +denition of the new total loss: +LNCLIP +V +S +t +(θ) = Et + +LNCLIP +t +(θ) − c1LV +t (θ) + c2FL [πθ] (st) + +(14) +In Algorithm 1 and Fig. 2, the learning phase is described +in detail, and a simulated environment model with new cost +functions is presented. The evaluation part is described in +Algorithm 2. +V. EXPERIMENTAL VERIFICATION AND ANALYSIS +A. Data Description +Turbine Database 1. This study used data from a real +working steam turbine. The turbine’s data was collected over +124 days at one minute per day sampling rate. The total +number of data collected was 207,361. After preprocessing +and removing outlier data from the non-dominant operating +mode, 190,635 data could be used. A total of 121,279 data +points are included in the normal category, and 69,356 data +points are included in the fault category. There are 31 features +included in this program, such as condenser pressure, pressure +on inlet valves, steam ow value, active and reactive power, +and others. The problem was caused by a leak in one of +the pressure valves, causing the entire system to fail. This +study examines the system’s behavior in its dominant operating +mode, the high-pressure mode, and the data are sufciently +comprehensive to understand the system’s behavior. + +Fig. 2: Process of our actor-critic proximal policy optimization. +Algorithm 1 Proposed Proximal Policy Optimization +Require: +1: States S = s1, . . . , sn +2: Actions A = a1, . . . , an, +A : S ⇒ A +3: Reward function R : S × A → R +4: Probabilistic transition function P : S × A → S +5: Initialize ER buffer B with experience replay memory M +6: procedure PROXIMAL POLICY OPTIMIZATION +7: +for each step of an episode do +8: +Run the actor model πθ(st, at) +9: +Store (st, at, rt, st+1) in B +10: +if Learning is not nished then +11: +πθold ← πθ +12: +Random sample (si, ai, ri, si+1) from M +13: +Compute  +At (using the critic model) +14: +Get the value Vϕ (st), ˆV target +ϕ +(st) +15: +Compute LV +t (ϕ) +16: +Compute LNCLIP +t +(θ) (using the actor model) +17: +Update critic model Vϕ using LV (ϕ) +18: +Update actor model πθ using LNCLIP (θ) +19: +Compute FL [πθ] (st) +20: +Update policy π +21: +if Learning is nished then +22: +Store policy πθ (at  st) +23: +Evaluate π on the test database +Turbine Database 2. Another set of data with a different +turbine is collected. The data was collected over 16 days at +one minute per day sampling rate. Following preprocessing +and removing outliers from the non-dominant operating mode, +240763 data in 8 classes could be used. The normal category +contains 188,974 data points, while the fault category named +”rpm low” contains 5456 data points. There has been a de- +Algorithm 2 Environment Evaluation +Require: +1: Labels L = l1, . . . , ln +2: Batch Samples X = (s1, l1) , (s2, l2) , . . . , (sT , lB) +3: procedure EVALUATION +4: +Our new policy π(S) = A +5: +for sampled skB +k=1 do +6: +if at = lt then +7: +rt = positive +8: +if at ̸= lt then +9: +rt = negative +10: +Receive next state st+1 +11: +if t == B then +12: +Store variables and go to another batch. +crease in turbine frequency, which has resulted in a decrease in +turbine rotation speed, which is the cause of this fault. Another +class has been labeled ”unknown,” and we have six different +unknown labels. It was unclear to the expert man what the +cause of the fault was. We have 32841, 6590, 5865, 593, and +285 records in each unknown class. This document, 65 features +are included, such as compressor discharge pressure, active +and reactive power of the generator, exhaust temperature, +interstage fuel gas pressure, and others. This study examines +the system’s behavior in its dominant operating mode, the +L30 mode. In this operating mode, known as the temperature +working mode, the turbine produces power to the maximum +capacity allowed by the system. The absolute limit of the +allowed value is the inlet temperature of the turbine (after +combustion), which should be at most, a specic value to +prevent overheating of the blades. The fuel valve controls the +inlet turbine’s end temperature. This turbine has other working +modes (start-up mode, coast-down mode, acceleration mode), + +but it works almost in this mode. +CICIDS2017 Database. The third dataset we used was the +CICIDS2017 dataset [59], which represents external network +data. This dataset contains the most up-to-date attack patterns +in terms of network security. In order to speed up the evalu- +ation process, we used one document from the database. As +shown in Table I, the dataset is summarized. +TABLE I: Dataset summary for CICDS-2017. +Filename +Classes +Samples +Wednesday working hours +Benign +440031 +DoS Hulk +231073 +DoS GoldenEye +10293 +DoS Slowloris +5796 +DoS Slowhttptest +5499 +Heartbleed +11 +B. Evaluation Conguration +In order to implement the Actor–Critic PPO algorithm, +Python 3.8 and PyTorch 1.11 are used. Actor and critic +models are constructed using multilayer perceptrons of four +hidden layers each and two separate output layers. For turbine +database 1 and the CICIDS2017 database, each hidden layer +consists of 32 neurons. For turbine database 2, we also use +48 neurons. The rst output layer of the actor model results +in several values that sum to one, for instance, eight actions +for turbine database 2. As a result of the second output layer +for the critic model, a single value can be obtained as an +evaluation of the action selected by the actor model. Table +II lists the other hyper-parameters of the Actor–Critic PPO +algorithm. +The β value in equation 12 is equal to 0.01. Equation 13 +uses the α value of 0.25 and the γ value of 2. Equation 14 +uses 0.5 and 0.2 for the coefcients c1 and c2. +TABLE II: PPO training hyperparameters. +The hyperparameter +Value +Optimization algorithm +Adam +GAE parameter +0.95 +Clipping parameter +0.2 +Learning rate (actor, critic) 0.001 +Batch size +256 +Discount factor +0.99 +The number of steps +256 +epoch +100 +C. Results and Analysis +Based on the data descriptions in this section, the algo- +rithm’s productivity is evaluated on the above data. According +to these results, the agent has learned a series of recognition +strategies well. The agent can realize action learning quickly, +mainly when performing diagnostic tasks related to turbine +faults. +We considered three models to evaluate the PPO algorithm +on our benchmarks, besides evaluating the changes in cost +functions on the result. Firstly, we named traditional PPO +”Model 1”. As a second step, we changed entropy in equation +8 to focal loss, and we named our new model ”Model 2”. +Lastly, after changing the cost function in the actor to equation +12 in addition to the use of focal loss and reach to equation +14, we named the nal model ”Model 3”. +Fig. 3 illustrates that all evaluation criteria increased by +approximately 3% with Model 2 compared to Model 1. In +particular, Model 2 has an accuracy of 93.86%, the precision +of 93.70%, recall of 92.94%, and f1 score of 93.32%. the +values of evaluation criteria for Model 3 have increased to +94.84%, 94.88%, 94.84%, and 94.86%, respectively. +The change in the cost functions led to an increase in per- +formance for the second dataset, as seen in Fig. 4. According +to the evaluation criteria, Model 1 does not perform well. As a +result of using Model 2, the accuracy, precision, recall, and F1 +score are 98%, 78%, 88%, and 81%, respectively. Model 3’s +criteria values are 98%, 90%, 91%, and 90%, with an increase +of 3%-12% in precision, recall, and F1 score over Model 2. +According to Fig. 5, traditional PPO performs at 85%, 93%, +85%, and 88% in criteria values based on the CICIDS2017 +database. Model 2 increased all criteria values to 96%. As a +nal result in Model 3, all the criteria values were increased +by about 3% rather than in Model 2 to 99%. +There is a signicant increase in performance between +Model 2 and Model 1, as can be seen in almost all gures. The +performance of Model 3 has also been improved in comparison +to Model 2 in terms of evaluation criteria. Due to the reasons +we mentioned at the beginning of the article, fault detection is +critical in industrial systems. The lack of labeled data in the +rst and second datasets, which relate to a real turbine, and the +third dataset, which relates to an attack on a real system, make +fault detection challenging. By enhancing performance in all +benchmarks studied, the third model has solved the challenge +of an imbalanced problem. +Table III compares our results on Turbine Database 1 with +our previous work on this data [60]. Values of evaluation +criteria were collected using weighted averages. The result +we obtained with modied proximal policy is higher than +that obtained with another RL framework named DDQN with +Update Policy. In DDQN with Update Policy, we use double +deep Q networks in a classication Markov decision process, +and we periodically update the policy based on the data we +receive. In this mode, we must update the initial model 124 +times (because our data was gathered over 124 days), and +each time, the model is run in 1000 iterations. Using our new +method based on PPO, we achieve better performance in less +time, and we only train our model on 100 epochs. As a result, +with our new algorithm, we have maintained the performance +while reducing computational time and cost. +To verify the validity of the proposed method and see the +results and compare it with previous works, we considered one +of the reliable datasets in the eld of cyberattacks. Now we +can see the advantages of the cost function changes compared +to the earlier methods. It can be seen from Table III that +our method performs better than contrastive learning-based +methods [61], [62] and is comparable to the LSTM approach +used in [63]. + +Fig. 3: Performance comparison of Models on Turbine +Database 1. +Fig. 4: Performance comparison of Models on Turbine +Database 2. +Fig. 5: Performance comparison of Models on CICIDS2017 +Database. +TABLE III: Comparison of our PPO policy with prior work. +Data +method +Accuracy Precision Recall F1 Score +Turbine Database 1 DDQN without Update Policy [60] +0.91 +0.91 +0.91 +0.91 +DDQN with Update Policy [60] +0.94 +0.94 +0.94 +0.94 +Our Proximal Policy Optimization +0.95 +0.95 +0.95 +0.95 +CIC IDS2017 +BoTNet [61] +0.97 +0.95 +0.96 +0.96 +Contrastive learning [62] +0.98 +0.98 +0.98 +0.98 +Rened LSTM [63] +0.99 +0.99 +0.99 +0.99 +Our Proximal Policy Optimization +0.99 +0.99 +0.99 +0.99 +VI. CONCLUSIONS +In this paper, we present a method for fault detection based +on neural networks and reinforcement learning. Our approach +is based on a famous policy based on actor and critic networks, +known as Proximal Policy Optimization. A disadvantage of +the traditional policy is its inability to detect fault classes due +to the lack of fault data, which was addressed by changing +the cost function in actor loss and total loss. We observe a +signicant improvement in all evaluation criteria, as shown in +the gures, and results are comparable to previous works in +the rst and third benchmarks. In addition, despite the second +benchmark having eight imbalance classes, the new policy +classies it well, while the old policy does not. In addition, +our new policy is more reasonable in terms of time and +computational cost than our previous policy with the Double +Deep Q Network in the rst benchmark. +REFERENCES +[1] X. Cheng and J. 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Alimi, “Rened lstm based intrusion detection for denial- +of-service attack in internet of things,” Journal of Sensor and Actuator +Networks, vol. 11, no. 3, p. 32, 2022. +Mahdi Aliyari Shoorehdeli received his B.Sc. +degree in electronics engineering from KNTU +in 2001. He then pursued his studies in control +engineering and, therefore, obtained his M.Eng. +and Ph.D. degree from the same university in +2003 and 2008, respectively. He is currently +appointed to the Department of Mechatronics +Engineering of KNTU as an Assistant Professor. +Dr. Aliyari is the author of more than 200 papers +in international journals and conferences. His +research interests include fault detection and +isolation and system identication and Machine Learning. +Mostafa Yari received his M.Eng. degree in +mechatronics engineering from KNTU in 2013. +His research interests include deep learning and +fault detection and condition monitoring and im- +age processing. +Arash Ghahremani received his M.Eng. degree +in mechanical engineering (Energy Conversion) +from KNTU in 2017. As a Ph.D. student, he +is pursuing his studies in the same eld and +at the same university. His research interests +include mechanics of uids and solids, energy +conversion and condition monitoring. + diff --git a/PNE2T4oBgHgl3EQfrggh/content/tmp_files/load_file.txt b/PNE2T4oBgHgl3EQfrggh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..be7b04ce0fe195035d425b7a1776c63ffce04ff8 --- /dev/null +++ b/PNE2T4oBgHgl3EQfrggh/content/tmp_files/load_file.txt @@ -0,0 +1,915 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf,len=914 +page_content='Imbalanced Classi\ue000cation In Faulty Turbine Data: New Proximal Policy Optimization Mohammad Hossein Modirrousta, Mahdi Aliyari Shoorehdeli, Senior Member, IEEE, Mostafa Yari and Arash Ghahremani Abstract—There is growing importance to detecting faults and implementing the best methods in industrial and real-world systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' We are searching for the most trustworthy and practical data-based fault detection meth- ods proposed by arti\ue000cial intelligence applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In this paper, we propose a framework for fault detection based on reinforcement learning and a policy known as proximal policy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' As a result of the lack of fault data, one of the signi\ue000cant problems with the traditional policy is its weakness in detecting fault classes, which was addressed by changing the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Using modi\ue000ed Proximal Policy Optimization, we can increase performance, over- come data imbalance, and better predict future faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' When our modi\ue000ed policy is implemented, all evaluation metrics will increase by 3% to 4% as compared to the traditional policy in the \ue000rst benchmark, between 20% and 55% in the second benchmark, and between 6% and 14% in the third benchmark, as well as an improvement in performance and prediction speed compared to previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Index Terms—Fault detection, Deep Learning, Reinforce- ment learning, Proximal Policy Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' INTRODUCTION P RODUCTION safety and product quality are key factors to ensure economic bene\ue000ts in the industrial process [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Monitoring and diagnosing faults play an important role and have received considerable attention from academia and industry [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Operational failures are mainly traced to external environ- mental factors during the in-service period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Structures can fail due to ice and sand accumulations, erosion, corrosion, lightning damage, dust, and insect contamination [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Cracks and delaminations can also occur during the operational phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' When the incipient damage is not detected early, it may result in severe malfunctions, fatal injuries, or, worst case, the collapse of the entire turbine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Condition Monitoring (CM) systems for turbines rely on fault detection and prediction algorithms, but there are two major challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Sensor data generation is the \ue000rst of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Mohammad Hossein Modirrousta (e-mail: moham- madbc@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='kntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='ir) is with the Faculty of Electrical Engineering, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Toosi University of Technology, Tehran, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Mahdi Aliyari Shoorehdeli (e-mail: aliyari@kntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='ir) is with the Fac- ulty of Mechatronics Engineering, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Toosi University of Technology, Tehran, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Mostafa Yari (e-mail: yari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='mostafa@mapnaec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='com) is with the Faculty of Mechatronics Engineering, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Toosi University of Technology, Tehran, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Arash Ghahremani (e-mail: ghahremani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='arash@mapnaec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='com) is with the Faculty of Mechanical Engineering, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Toosi University of Technology, Tehran, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' CM systems and their interpretation are complicated by the need to store and process these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Sensor reliability is the second issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' [6], [7] have pointed out issues regarding the accuracy and accountability of sensors used for pattern recognition and fault identi\ue000cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The accuracy of sensor data strongly in\ue001uences a CM system’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Addi- tionally, the use of a large number of sensors, and hence monitoring variables, may reduce the overall reliability of the sensor system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Consequently, fault detection methods are a necessary component of ensuring the operational safety and reliability of mechanical systems and reducing maintenance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Related work According to [8], there are three types of fault detection methods for turbine systems, including signal-based methods [9], [10], model-based methods [11], and knowledge-based methods [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' A robust fault estimation and fault-tolerant control approach for Takagi-Sugeno fuzzy systems is proposed in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Furthermore, multivariate statistical methods have also successfully detected faults [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The use of SCADA data for monitoring wind turbine conditions has been proposed in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' There has been extensive use of arti\ue000cial neural networks (ANNs), Gaussian processes (GP), support vector machines (SVMs), and random forests (RFs) in order to improve the performance of turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In [17], [18], [19], [20], [21], general overviews of these techniques are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The development of turbine models and fault detection methods based on arti\ue000cial intelligence has been impressive [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' As part of the analysis for turbine condition monitoring [24], a method based on arti\ue000cial neural networks (ANNs) has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' [25] presents an algorithm for ANN pattern recognition and its application to controls for turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In [26], support vector machines (SVMs) were com- bined with a residual-based method to detect and isolate faults in wind turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Using Shannon wavelet SVMs and manifold learning to diagnose faults in turbine transmission systems were proposed in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Detecting turbine faults can also be accomplished using AI-based methods based on fuzzy logic or expert systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' According to [28], adaptive neuro-fuzzy inference systems can be used to monitor the condition of turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Deep learning (DL) has achieved enormous success in many \ue000elds in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Researchers are also interested in the \ue000eld of fault detection [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' An unsupervised deep learning method (denoising autoencoder) has been proposed for detecting turbine faults [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In addition to the domain adaptation using the maximum mean discrepancy, DL models such as sparse autoencoder [32] and convolutional neural networks (CNN) [33] are used for condition recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' An ImageNet pre-trained network is used in the paper [34] to train a deep-learning network to classify faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In order to obtain a time-frequency distribution to \ue000ne-tune the high-level network layers, sensor data are transformed into image data by plotting or using wavelet transformation [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' An imbalance of class samples occurs in real-world ap- plications when one class’s samples exceed those of other classes [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Turbine fault detection, for instance, exhibits class imbalance because these machines generally operate under normal conditions and occasionally fail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' these conditions result in many normal operations and few faulty ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Data and algorithm-level research has been conducted to alleviate the problem of class imbalance [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' This technique is a data-level method that involves random under- and oversampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Over- sampling techniques have been studied extensively [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' A focus on cost-sensitive algorithms [39] and ensemble learning [40] is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Misclassi\ue000ed positive and negative samples incur high and low costs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Costs are dif\ue000cult to determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' An ensemble learning algorithm such as Adaboost [14] uses an iterative boosting algorithm to increase the weight of misclassi\ue000ed samples and decrease the weight of correctly classi\ue000ed samples after each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The performance of boosting depends strongly on the base classi\ue000er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Arti\ue000cial intelligence (AI) also includes reinforcement learning (RL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Intelligent computing techniques are used to automate and understand problem-oriented learning and decision-making [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In contrast to other intelligent methods, it emphasizes that the agent learns through direct interaction with the environment without requiring imitation of supervi- sion signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Deep reinforcement learning (DRL) combines RL with deep learning (DL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' DRL has achieved great success in games [42], recommendation systems [43], and robotics control [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Nevertheless, DRL is rarely mentioned in fault identi\ue000cation, which DL dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' We propose a new method for identifying faults in real turbines through DRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Intelligent methods will be more universal with fault parameterization and DRL implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Several faults are parameterized here, which enables DRL to transform a classi\ue000cation problem into a sequential decision problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' A Markov decision process for classi\ue000cation was proposed by Wiering et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' This framework de\ue000ned a standard clas- si\ue000cation problem as a sequential decision-making problem, and an MLP model trained in it outperformed a regular MLP model trained by backpropagation [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Using DRL to identify bearing health states, Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' proposed an approach [46];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' adopted DRL to implement a preventive maintenance policy for serial production lines [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Based on time-frequency representations (TFR) and dynamic response mappings (DRLs), Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' developed a new fault diagnosis methodology [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' As described in [49], DDQN improves the detection performance of cyberattacks by adopting a \ue000ne- grained traf\ue000c \ue001ow monitoring mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In contrast to traditional policy gradient algorithms, PPO is an advanced algorithm capable of overcoming the problem of low learning ef\ue000ciency caused by the in\ue001uence of the step size on learning ef\ue000ciency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' PPO has several primary advantages for training control policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' First of all, in [50], the hyper- parameters of PPO were proved to be robust when training various tasks, and PPO can balance the complexity and accu- racy of control policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Second, in [51], the training control policy of PPO was found to be superior to that of other RL algorithms on all metrics compared to the performance indica- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Based on the full six-degree-of-freedom system dynamics of the UAV, in [52], PPO is used to train quadrotor control policies, achieving stable hovering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In the context of Software- De\ue000ned Networking (SDN), Zolotukhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' proposed an interesting approach [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The authors investigate Deep Q- Network (DQN) and PPO in response to an attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' DQN and PPO show promising results, which further motivates this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' PPO also simpli\ue000es implementation and improves performance in IoT applications [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The authors of [55] propose an agent-based reinforcement learning scheme using PPO to allow multiple agents to control their own devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' An intrusion detection hyperparameter control procedure is built in [56], which controls and trains a deep neural network feature extractor based on proximal policy optimization (PPO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' A new automated lane change strategy using proximal policy optimization is proposed [57], which shows excellent bene\ue000ts while maintaining performance stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Our Contributions The \ue000ndings of the above studies led us to propose a new fault detection method based on DRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' It is based on Classi\ue000- cation Markov Decision Process (CMDP), which de\ue000nes the fault detection problem as a guessing game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The diagnosis agent \ue000rst learns an optimal recognition policy within the framework of DRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' By using experience replay (ER), the agent automatically interacts with the environment, creating experiences, and updating the model based on those expe- riences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Furthermore, the proposed method has been tested on detection tasks and is compared to existing detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In addition to exhibiting better generalization and speed compiling, this method also performs well when dealing with imbalanced problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The following are the main contributions of our framework as described in this paper: 1) We are using reinforcement learning to build a recom- mender label system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In order to discover a new algorithm for fault detection from a DRL perspective, we consider fault detection as a guessing game and describe it as a sequential decision-making problem based on CMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 2) With the help of experience replay (ER) and reward-based learning tools, an optimal model for fault detection can be developed based on Proximal Policy Optimization (PPO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 3) It is necessary to make some changes to the regular cost function of PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' As a result of these changes, imbalances in data will be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Using our approach, we can make decisions without relying on feature engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 4) We examined and compared the performance of this method with and without changes in the cost function, as well as with other methods on multiple datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' We found that this method enhanced fault detection abilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' PROBLEM DEFINITION The diagnosis of faults is often considered to be a classi\ue000cation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The primary purpose of DRL is to solve the sequential decision-making problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' A guessing game is used in this work to diagnose turbine faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' We also create a game simulation that converts fault diagno- sis into a sequential decision-making problem using the DRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' This illustration illustrates how a training dataset that might be regarded as a guessing question set is Xtrain = \ue00d(s1, l1) , (s2l2) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' , (sn, ln)\ue00e where si is the i − th sample and li is the i − th label corresponding to sample si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' As part of this game, each round consists of T questions matching training data generated from the training dataset Xtrain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Agents guess these questions sequentially by the order in which the samples in D are arranged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' This game involves the agent observing a sample each time and assuming the class of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Following the guessing question, the environment provides an immediate reward to the agent and the next guessing question (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=', the following sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' A positive reward is awarded to the agent if the agent correctly identi\ue000es the sample’s category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' otherwise, a negative reward is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The agent’s objective in this game is to maximize accumulated rewards within the constraints of an optimal behavior policy that has been learned due to constant interaction with the environment, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 1: Overview of the Agent-User interaction in the Classi- \ue000cation MDP (CMDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Proximal Policy Optimization Using reinforcement learning, an agent can learn how to interact with its environment to maximize its expected cu- mulative rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' An RL algorithm can be divided into two general categories: value-based and policy-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Even though value-based methods can approximate the value function us- ing neural networks in an off-policy manner, policy-based methods, such as the REINFORCE algorithm [20], offer the primary advantage of optimizing the quantity of advantage directly while maintaining stability during the approximation of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' As a result, our study focuses on RL methods based on policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' During a general policy gradient reinforcement learning, the objective function is as follows: LP (θ) = ˆEt \ue007 log πθ (at \ue00f st) ˆAt \ue008 (1) In this case, ˆEt represents the expectation operator, πθ repre- sents a stochastic RL policy, and ˆAt represents the estimated advantage function at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Using the discount factor γ \ue007 [0, 1], we can calculate ˆAt using the generalized advantage estimator [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Generally, the generalized advantage estimator can be described as follows: ˆAt = δt + (γλ)δt+1 + · · · + · · · + (γλ)T −t+1δT −1 (2) Where δt = rt + γVϕ (st+1) − Vϕ (st), and T is the sampled mini-batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The parameter λ \ue007 [0, 1] represents the generalized advantage estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' According to LV , the objective function is as follows: LV (ϕ) = \ue005E \ue000 LV t (ϕ) \ue001 = \ue005E \ue007\ue002\ue002\ue002 ˆV target ϕ (st) − Vϕ (st) \ue002\ue002\ue002 \ue008 (3) In this case, the target value of the time-difference error (TD- Error) is ˆV target ϕ (st) = rt+1 + γVϕ (st+1) (4) A gradient descent algorithm is used to update the parameters of Vϕ, with the gradient \ue010LV : ϕ = ϕ − ηϕ\ue010LV (ϕ) (5) In the critic model optimization, ηϕ is the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Using the actor model, the PPO uses the importance sam- pling method in place of the objective function presented in Equation (1) to estimate the expectation of samples collected from the old policy πθold under the new policy πθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Using LCP I as a surrogate objective function, the algorithm maximizes: LCP I(θ) = \ue005Et \ue003 πθ (at \ue00f st) πθold (at \ue00f st) \ue005At \ue004 (6) The PPO optimizes LCP I with a small value δ based on the amount of the policy update as a constraint, according to the equation below: \ue005Et [KL [πθold (· \ue00f st) , πθ (· \ue00f st)]] ≤ δ (7) Kullback-Leibler divergence is indicated by KL [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The actor-critic structure is used in advanced policy-based algo- rithms, including TRPO [18] and PPO [17], which combine the advantages of traditional value-based and policy-based ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' As well as being more straightforward to implement and allowing multiple optimization iterations, the PPO algo- rithm also has a higher sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In particular, PPO proposes a modi\ue000ed surrogate loss function, which combines the policy surrogate and the error term associated with the value function, de\ue000ned as follows [17] LCLIP +V +S t (θ) = ˆEt \ue000 LCLIP t (θ) − c1LV t (θ) + c2S [πθ] (st) \ue001 (8) In this equation, LCLIP t stands for the clipped surrogate objective, c1 and c2 represent the coef\ue000cients, LV t stands for the squared-error loss of the value function, and S stands for the entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In order to ensure that our agents have enough Store Expenence Reward Y Environment Agent S1 Action St ER Buffer Lata State s,exploration, we use an entropy term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' By using this term, the policy will be pushed to behave more spontaneously until the other objective overtakes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In more detail, the clipped surrogate objective LCLIP t is as follows: LCLIP (θ) = \ue005Et \ue000 LCLIP t (ϕ) \ue001 = \ue005Et \ue007 min (Rt(θ), clip (Rt(θ), 1 − ϵ, 1 + ϵ)) \ue005At \ue008 (9) Where ϵ indicates a clipping parameter and Rt(θ) = πθ(at|st) πθold(at|st) indicates a probability ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' This procedure results in a clipping of the probability ratio Rt(θ) at time 1 − ϵ or 1 + ϵ, depending upon whether the advantage is positive or negative, which forms the clipped objective after multiplying the advantage approximator ˆAt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In the end, LCLIP t is calcu- lated by taking the minimum of this clipped objective and the unclipped objective Rt(θ) ˆAt, thereby reducing the need for a substantial policy update compared to the unclipped version [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' This loss function is known as the conservative policy iteration algorithm loss function [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Using gradient descent, the parameters of πθ are updated according to the gradient \ue010LCLIP of the negative of the clipped objective function: θ = θ − ηθ\ue010LCLIP (θ) (10) Assume that ηθ is the learning rate for the actor model optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' PROPOSED DRL BASED FAULT DIAGNOSIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Data Pre-processing Turbine Databases Some rows with a ”drop-it” title are available in the datasets available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' As such data is considered outlier data, it should be deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Following that, the data must be normalized using a Min-Max scaler as follows: DNormalize = D − Dmin Dmax − Dmin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' (11) D and DNormalize represent features before and after normalization, respectively, and Dmin and Dmax represent minimum and maximum features based on data before nor- malization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' CICIDS2017 Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' This dataset is also normalized using the same technique as the previous dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In addition, some values had inf values, which posed a problem in the normalization and classi\ue000cation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' These values have been dropped from the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Agent Architecture Design We have faced many challenges in dealing with the im- balanced classi\ue000cation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In industrial settings, it is essential to detect faulty data, and many researchers have attempted to reduce the number of false positives and false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The loss functions were modi\ue000ed to emphasize the few data that address the imbalanced data problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Consider the objective function in the actor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' As follows, we de\ue000ne a new surrogate objective function: LNCP I(θ) = ˆEt \ue003 πθ (at \ue00f st) πθold (at \ue00f st) ˆAt + β log (πθ (at \ue00f st)) \ue004 (12) New Conservative Policy Iteration is indicated by the super- script N C P I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The term entropy into a new policy was added to the cost function compared to equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The purpose of this work is to place a greater emphasis on new policies rather than old ones and to improve and accelerate the convergence of actor losses and total losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In the results section, we will compare the results with and without this change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The large class imbalance observed during the training of dense detectors overwhelms the cross-entropy loss, as demon- strated in [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The authors propose adapting the loss function to down-weight easy examples and adding a modulating factor to the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Therefore, the following is the de\ue000nition of focal loss: FL (pt) = −αt (1 − pt)γ log (pt) (13) p \ue007 [0, 1] is the estimated probability according to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Speci\ue000cally, α \ue007 [0, 1] is a weighting factor, and γ \ue007 [0, 1] is a tunable focusing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' A modulating factor reduces the loss contribution from easy examples and extends the range of examples subject to low loss contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' For the above reasons, we replace the entropy loss with a focal loss in equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Results show a signi\ue000cant improve- ment in performance and a rapid convergence in total loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' This change will also be revealed in the results section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Below is a de\ue000nition of the new total loss: LNCLIP +V +S t (θ) = \ue005Et \ue000 LNCLIP t (θ) − c1LV t (θ) + c2FL [πθ] (st) \ue001 (14) In Algorithm 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 2, the learning phase is described in detail, and a simulated environment model with new cost functions is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The evaluation part is described in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' EXPERIMENTAL VERIFICATION AND ANALYSIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Data Description Turbine Database 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' This study used data from a real working steam turbine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The turbine’s data was collected over 124 days at one minute per day sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The total number of data collected was 207,361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' After preprocessing and removing outlier data from the non-dominant operating mode, 190,635 data could be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' A total of 121,279 data points are included in the normal category, and 69,356 data points are included in the fault category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' There are 31 features included in this program, such as condenser pressure, pressure on inlet valves, steam \ue001ow value, active and reactive power, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The problem was caused by a leak in one of the pressure valves, causing the entire system to fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' This study examines the system’s behavior in its dominant operating mode, the high-pressure mode, and the data are suf\ue000ciently comprehensive to understand the system’s behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 2: Process of our actor-critic proximal policy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Algorithm 1 Proposed Proximal Policy Optimization Require: 1: States S = \ue00ds1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' , sn\ue00e 2: Actions A = \ue00da1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' an\ue00e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' A : S ⇒ A 3: Reward function R : S × A → R 4: Probabilistic transition function P : S × A → S 5: Initialize ER buffer B with experience replay memory M 6: procedure PROXIMAL POLICY OPTIMIZATION 7: for each step of an episode do 8: Run the actor model πθ(st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' at) 9: Store (st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' at,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' rt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' st+1) in B 10: if Learning is not \ue000nished then 11: πθold ← πθ 12: Random sample (si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' si+1) from M 13: Compute \ue006 At (using the critic model) 14: Get the value Vϕ (st),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' ˆV target ϕ (st) 15: Compute LV t (ϕ) 16: Compute LNCLIP t (θ) (using the actor model) 17: Update critic model Vϕ using \ue010LV (ϕ) 18: Update actor model πθ using \ue010LNCLIP (θ) 19: Compute FL [πθ] (st) 20: Update policy π 21: if Learning is \ue000nished then 22: Store policy πθ (at \ue00f st) 23: Evaluate π on the test database Turbine Database 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Another set of data with a different turbine is collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The data was collected over 16 days at one minute per day sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Following preprocessing and removing outliers from the non-dominant operating mode, 240763 data in 8 classes could be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The normal category contains 188,974 data points, while the fault category named ”rpm low” contains 5456 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' There has been a de- Algorithm 2 Environment Evaluation Require: 1: Labels L = \ue00dl1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' , ln\ue00e 2: Batch Samples X = \ue00d(s1, l1) , (s2, l2) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' , (sT , lB)\ue00e 3: procedure EVALUATION 4: Our new policy π(S) = A 5: for sampled \ue00dsk\ue00eB k=1 do 6: if at = lt then 7: rt = positive 8: if at ̸= lt then 9: rt = negative 10: Receive next state st+1 11: if t == B then 12: Store variables and go to another batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' crease in turbine frequency, which has resulted in a decrease in turbine rotation speed, which is the cause of this fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Another class has been labeled ”unknown,” and we have six different unknown labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' It was unclear to the expert man what the cause of the fault was.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' We have 32841, 6590, 5865, 593, and 285 records in each unknown class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' This document, 65 features are included, such as compressor discharge pressure, active and reactive power of the generator, exhaust temperature, interstage fuel gas pressure, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' This study examines the system’s behavior in its dominant operating mode, the L30 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In this operating mode, known as the temperature working mode, the turbine produces power to the maximum capacity allowed by the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The absolute limit of the allowed value is the inlet temperature of the turbine (after combustion), which should be at most, a speci\ue000c value to prevent overheating of the blades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The fuel valve controls the inlet turbine’s end temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' This turbine has other working modes (start-up mode, coast-down mode, acceleration mode), but it works almost in this mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' CICIDS2017 Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The third dataset we used was the CICIDS2017 dataset [59], which represents external network data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' This dataset contains the most up-to-date attack patterns in terms of network security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In order to speed up the evalu- ation process, we used one document from the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' As shown in Table I, the dataset is summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' TABLE I: Dataset summary for CICDS-2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Filename Classes Samples Wednesday working hours Benign 440031 DoS Hulk 231073 DoS GoldenEye 10293 DoS Slowloris 5796 DoS Slowhttptest 5499 Heartbleed 11 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Evaluation Con\ue000guration In order to implement the Actor–Critic PPO algorithm, Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='8 and PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='11 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Actor and critic models are constructed using multilayer perceptrons of four hidden layers each and two separate output layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' For turbine database 1 and the CICIDS2017 database, each hidden layer consists of 32 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' For turbine database 2, we also use 48 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The \ue000rst output layer of the actor model results in several values that sum to one, for instance, eight actions for turbine database 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' As a result of the second output layer for the critic model, a single value can be obtained as an evaluation of the action selected by the actor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Table II lists the other hyper-parameters of the Actor–Critic PPO algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The β value in equation 12 is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Equation 13 uses the α value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='25 and the γ value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Equation 14 uses 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='2 for the coef\ue000cients c1 and c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' TABLE II: PPO training hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The hyperparameter Value Optimization algorithm Adam GAE parameter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='95 Clipping parameter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='2 Learning rate (actor, critic) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='001 Batch size 256 Discount factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='99 The number of steps 256 epoch 100 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Results and Analysis Based on the data descriptions in this section, the algo- rithm’s productivity is evaluated on the above data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' According to these results, the agent has learned a series of recognition strategies well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The agent can realize action learning quickly, mainly when performing diagnostic tasks related to turbine faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' We considered three models to evaluate the PPO algorithm on our benchmarks, besides evaluating the changes in cost functions on the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Firstly, we named traditional PPO ”Model 1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' As a second step, we changed entropy in equation 8 to focal loss, and we named our new model ”Model 2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Lastly, after changing the cost function in the actor to equation 12 in addition to the use of focal loss and reach to equation 14, we named the \ue000nal model ”Model 3”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 3 illustrates that all evaluation criteria increased by approximately 3% with Model 2 compared to Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In particular, Model 2 has an accuracy of 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='86%, the precision of 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='70%, recall of 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='94%, and f1 score of 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='32%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' the values of evaluation criteria for Model 3 have increased to 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='84%, 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='88%, 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='84%, and 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='86%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The change in the cost functions led to an increase in per- formance for the second dataset, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' According to the evaluation criteria, Model 1 does not perform well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' As a result of using Model 2, the accuracy, precision, recall, and F1 score are 98%, 78%, 88%, and 81%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Model 3’s criteria values are 98%, 90%, 91%, and 90%, with an increase of 3%-12% in precision, recall, and F1 score over Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 5, traditional PPO performs at 85%, 93%, 85%, and 88% in criteria values based on the CICIDS2017 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Model 2 increased all criteria values to 96%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' As a \ue000nal result in Model 3, all the criteria values were increased by about 3% rather than in Model 2 to 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' There is a signi\ue000cant increase in performance between Model 2 and Model 1, as can be seen in almost all \ue000gures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The performance of Model 3 has also been improved in comparison to Model 2 in terms of evaluation criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Due to the reasons we mentioned at the beginning of the article, fault detection is critical in industrial systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The lack of labeled data in the \ue000rst and second datasets, which relate to a real turbine, and the third dataset, which relates to an attack on a real system, make fault detection challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' By enhancing performance in all benchmarks studied, the third model has solved the challenge of an imbalanced problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Table III compares our results on Turbine Database 1 with our previous work on this data [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Values of evaluation criteria were collected using weighted averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' The result we obtained with modi\ue000ed proximal policy is higher than that obtained with another RL framework named DDQN with Update Policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In DDQN with Update Policy, we use double deep Q networks in a classi\ue000cation Markov decision process, and we periodically update the policy based on the data we receive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In this mode, we must update the initial model 124 times (because our data was gathered over 124 days), and each time, the model is run in 1000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Using our new method based on PPO, we achieve better performance in less time, and we only train our model on 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' As a result, with our new algorithm, we have maintained the performance while reducing computational time and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' To verify the validity of the proposed method and see the results and compare it with previous works, we considered one of the reliable datasets in the \ue000eld of cyberattacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Now we can see the advantages of the cost function changes compared to the earlier methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' It can be seen from Table III that our method performs better than contrastive learning-based methods [61], [62] and is comparable to the LSTM approach used in [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 3: Performance comparison of Models on Turbine Database 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 4: Performance comparison of Models on Turbine Database 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 5: Performance comparison of Models on CICIDS2017 Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' TABLE III: Comparison of our PPO policy with prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Data method Accuracy Precision Recall F1 Score Turbine Database 1 DDQN without Update Policy [60] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='91 DDQN with Update Policy [60] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='94 Our Proximal Policy Optimization 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='95 CIC IDS2017 BoTNet [61] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='96 Contrastive learning [62] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='98 Re\ue000ned LSTM [63] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='99 Our Proximal Policy Optimization 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='99 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' CONCLUSIONS In this paper, we present a method for fault detection based on neural networks and reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Our approach is based on a famous policy based on actor and critic networks, known as Proximal Policy Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' A disadvantage of the traditional policy is its inability to detect fault classes due to the lack of fault data, which was addressed by changing the cost function in actor loss and total loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' We observe a signi\ue000cant improvement in all evaluation criteria, as shown in the \ue000gures, and results are comparable to previous works in the \ue000rst and third benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In addition, despite the second benchmark having eight imbalance classes, the new policy classi\ue000es it well, while the old policy does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' In addition, our new policy is more reasonable in terms of time and computational cost than our previous policy with the Double Deep Q Network in the \ue000rst benchmark.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Ghorbani, “Toward generating a new intrusion detection dataset and intrusion traf\ue000c characterization.” ICISSp, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 108–116, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' [60] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Modirrousta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Shoorehdeli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Yari, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Ghahremani, “Dqlap: Deep q-learning recommender algorithm with update policy for a real steam turbine system,” arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='06399, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' [61] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Wang, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Li, “Network intrusion detection model based on improved byol self-supervised learning,” Security and Mohammad Hossein Modirrousta received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' degree in control engineering from KNTU in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' He then pursued his studies in control engineering and, therefore, obtained his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' degree from the same university in 2022, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' His research interests include deep learning and reinforcement learning and fault detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Communication Networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 2021, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' [62] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Lot\ue000, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Modirrousta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Shashaani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Amini, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Shoore- hdeli, “Network intrusion detection with limited labeled data,” arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='03147, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' [63] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Adefemi Alimi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Ouahada, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Abu-Mahfouz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Rimer, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Alimi, “Re\ue000ned lstm based intrusion detection for denial- of-service attack in internet of things,” Journal of Sensor and Actuator Networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' 32, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Mahdi Aliyari Shoorehdeli received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' degree in electronics engineering from KNTU in 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' He then pursued his studies in control engineering and, therefore, obtained his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' degree from the same university in 2003 and 2008, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' He is currently appointed to the Department of Mechatronics Engineering of KNTU as an Assistant Professor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Aliyari is the author of more than 200 papers in international journals and conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' His research interests include fault detection and isolation and system identi\ue000cation and Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Mostafa Yari received his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' degree in mechatronics engineering from KNTU in 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' His research interests include deep learning and fault detection and condition monitoring and im- age processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' Arash Ghahremani received his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' degree in mechanical engineering (Energy Conversion) from KNTU in 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' As a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' student, he is pursuing his studies in the same \ue000eld and at the same university.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} +page_content=' His research interests include mechanics of \ue001uids and solids, energy conversion and condition monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfrggh/content/2301.04049v1.pdf'} diff --git a/PdFJT4oBgHgl3EQfIywm/vector_store/index.faiss b/PdFJT4oBgHgl3EQfIywm/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..e284d72e1cb38e0d5c46bbd2b7a9a5bed4c8757f --- /dev/null +++ b/PdFJT4oBgHgl3EQfIywm/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c673115d51d18ab508946a948f3a389d9ca94845491c74f1fa8d38a35ebf1131 +size 5373997 diff --git a/QtAyT4oBgHgl3EQftvnR/content/tmp_files/2301.00602v1.pdf.txt b/QtAyT4oBgHgl3EQftvnR/content/tmp_files/2301.00602v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a93e94255ab5f8844fd2342040614394a09eb09f --- /dev/null +++ b/QtAyT4oBgHgl3EQftvnR/content/tmp_files/2301.00602v1.pdf.txt @@ -0,0 +1,1103 @@ +Software engineering for mobile applications, a survey on +challenges and solutions + +Shehab Eldeen Ayman Mounir +Shehab.ayman@bue.edu.eg +The British University in Egypt + +Abstract +Mobile app development has become the front +line in software engineering. With the recent +years many smartphone platforms have grew +including but not limited to webOS, blackberry +os, Tizen, android, and iOS. The coexistence of +these platforms results in a challenging situation +where apps must be developed and maintained +to the same level. The mobile app development +scene has recently seen a noticeable rise in the +number of applications that adapt web elements +like HTML5 to produce native like applications +that are essentially web views wrapped into +containers to appear as any normal application. +This means that the application behavior can +vary drastically from one user to another +meaning that the app behavior can be changed +drastically. Therefore, application developers +rely on an agile or an ad-hoc approach to +development that is mostly autonomous. In this +paper, we describe the current state of the art of +context +awareness +in +mobile +application +development. +introduction +No one can deny that the astronomical rise in +smartphone popularity in the last decade has +reshaped the software engineering industry. +With billions in app downloads, mobile app +development has become the front line in +software engineering. Through the recent years +many smartphone platforms have grew include +but not limited to webOS, blackberry os, Tizen, +android and iOS. With IOS and android taking +most of the share of the smartphone platform. +some of the common challenges of building a +multifunctional software has moved from the +desktop to mobile but developing software for +smartphones has come with its own set of new +challenges that needs to be overcome. +Recent estimations done in 2017 state that the +apple app store contains more than 2.2 million +apps while the google play store contains 2.8 +million and they are in a constant growth. Even +though both windows app store and blackberry +world are both discontinued, they contained 600 +thousand and 200 thousand apps respectively +[1]. +Developing for these mobile platforms is not as +flexible as developing for the desktop. The +programming languages, tools, frameworks and +APIs required for the development process are +platform specific, for instance, native android +applications +are +built +using +the +java +programming language on the android studio + +2 + +integrated development environment, whereas +apple’s IOS applications are developed using the +swift language with XCode tool [2]. The +coexistence of these platforms results in a +challenging situation where apps have to be +developed and maintained to the same level. +Traditional software engineering approaches +have been somewhat phased out in the context +of mobile development as these approaches +cannot be directly applied in the mobile context. +Mobile graphical user interfaces are considered +as a whole new paradigm for human computer +interaction study. Mobile phones provide a +whole new way of interaction that is considered +untraditional comparing +it +the +computer +situation which involves sitting down and +focusing attention and resources towards the +interaction. And even the interaction method +has drastically changed from the typical mouse +and keyboard to touch screens and gestures. +And the way of interaction seems to be evolving +constantly as voice commands and augment +reality/mixed reality are becoming more of a +possibility [3]. Nowadays the variety of hardware +and software platform have forced developers to +make a group of different applications that on +the outside might look the same but on the +inside are completely different just to suite every +platform [4]. +Challenges with mobile development +1. Create a universal graphical user +interface +Research conducted by Tarasewich and Gong +suggest that at least four of Shneiderman’s +design principals -which will be looked at +further- are applicable -without modification- to +mobile +phones. +These +include +“enabling +frequent users to use shortcuts”, “designing +dialogue to yield closure”, “offering informative +feedback”, “supporting internal locus of control” +[5]. As technologies tend to evolve, challenges +tend to evolve as well that’s why recently efforts +have been into researching ways of streamlining +application development regardless of the +hardware or the software platform. Focusing on +streamlining the process will free up a lot for +resources which could then be directed towards +making better user interfaces. +2. Reusing +software +across +different +platforms +Companies take the hard decision whether to +focus on one or two platforms and enrich the +user experience or go all out and support +natively every single platform that is currently on +the market compromising their software quality +[6]. +There is no other way of saying it but sometimes +user interface cannot be unified across IOS and +android not due to some technical limitations in +any of the two platforms but due to the way each + +3 + +platform users have grown accustomed to +certain design aspects. An application with an +insert circular icon in the bottom right corner +and three vertical dashes in the top left corner - +that when clicked bring a charms menu- is easily +identified by mobile developers as an android +application. While applications that have their +back/previous button in the top right corner +tend to be IOS applications. Android applications +tend not to have a back button as android users +are accustomed to using the back button +implemented in the operating system itself. +While IOS users are accustomed to look for a +back button in the application itself and when it +is not present, a swipe to the right would +function exactly like a back button. +Many software companies have separate teams +for development where each time only focuses +on one single platform. This means that if an app +is to be developed for IOS and android -for +example- the software engineering effort +needed to build the app are doubled to provide +the same functionalities on the two operating +systems. +The +basic +software engineering +concepts of reusing and refactoring parts of +software are not applicable here as they cannot +be transferred from one software to another. +This leads to a very limited coordination +between development teams. It mostly relies on +ad-hoc basis without any real effort in reducing +the resources -especially time and cost- +allocated for the project. +The mobile development scene has recently +seen a noticeable rise in the number of +applications that adapt web elements like +HTML5 to produce native like applications which +are essentially web views wrapped into +containers to appear as any normal application. +These kind of applications -which will be +discussed further- do not have the rich +capabilities of native apps as they cannot access +the +platform’s +APIs +which +limits +their +functionalities, but on the other side allow for +the reuse of almost an entire application +interface on more than one platform [7]. +3. Context aware applications +Mobile devices are far in contrast to traditional +stationary computer platforms. Mobile phones +are highly customizable and adaptive to user’s +needs which means that they must constantly +monitor the environment. This means that +applications on mobile need to be constantly +aware of time, weather, location, orientation, +proximity, etc. [8] Mobile applications are now +able to use all of this contextual awareness to +make a very specific, very specialized experience +that is very special to its own user meaning that +the application behavior can vary drastically +from one user to another. This idea of context +awareness is no new feature, web applications +have been providing similar kind of user tailored +experience for years but it has never got to the +same extent as mobile applications due to the + +4 + +sensors and capabilities of mobile phones. This +kind of sensory overload has never been present +or dealt with in traditional software engineering, +it was always associated with robotics and smart +objects which are dealt with using agent +oriented +software +engineering +[9]. +The +availability of these sensors has put their +utilization in the forefront of application +development. So, developers pour careful +attention +into +analyzing +application +requirements and utilizing context awareness to +much improve the quality of their application +resulting in a better user experience. +Agent oriented software engineering (AOSE) is +concerned with building software agents that +are mostly autonomous. This approach provides +all the necessary models, abstractions and +“pure” software engineering approaches to build +a contextually aware application of a multi-agent +system (MAS) [10]. For simplification lets +imagine that the multiagent system is a physical +robot. Using its sensors, the robot must use all of +its sensors to sense and understand its +surrounding environment and react to it +accordingly in order to achieve its goals. +4. Balancing agility and uncertainty +There is no denying that the existence of mobile +applications +has +changed +the +way +a +development process is looked at. Gone are the +times +where +almost +all +the +application +requirements are clearly specified and fully +implemented on the first run. It is now +completely normal to regularly receive updates +for applications that add new features or focus +on stability improvement and bug fixes. This +means that the traditional waterfall model +cannot be realistically applied in mobile +development that’s why application developers +rely on an agile or an ad-hoc approach to +development. The constant growth in demand +for more context aware applications, tailored +user experience, very heavy competition and low +tolerance by users for unresponsive applications +have derived more of a semi-formal approach +development that does not necessarily go by the +books. This approach has to be somehow +integrated within the agile method. The very +dynamic very user specific experience provided +my mobile applications allows for scenarios and +situations that may not be fully specified within +the functional and nonfunctional requirements. +If mobile applications were strictly to follow their +functional / nonfunctional requirements, this +will result in a less quality experience. This +means +that +applications +need +to +run +continuously and autonomously adapt and +modify the behavior and provide more +functionality than strictly specified. +It is hard to imagine the idea of how applications +can provide an improved service than originally +specified within the design documents and how +this constantly adaptive service can improve the +quality and engagement with applications. A + +5 + +great example of adaptive service is user tailored +ad experience. This has become an essential way +of digital advertising where the products listed in +the ad spots depending on the user interests. +Gathering user interests has lately been a +relatively easy task. For example, google ad +services track each user’s search history to +determine what are this specific user’s interest +and then display ads of objects that the user may +find tempting to buy. Another great example is +Facebook application, it uses location services +and profile analysis to recommend friends. And +this context awareness feature keeps updating +frequently. For example, if a person left his +workplace, university or even the country, the +application will keep providing relevant and new +recommendations based on the updated +location. Another area where Facebook has +ventured into and has made a big impact is with +the Facebook marketplace service on which +people can buy and sell almost anything from +used phones to used cars. The application here +utilizes context awareness to link users with the +nearest buyers and promote products to users +who are interested in as it will most likely end up +in a successful buy / sell operation. +Cross platform development +An ongoing challenge in mobile development +especially for startups and relatively small +companies is to choose the platforms that they +are going to develop for. Companies tend to +focus their development on IOS and android to +cover the largest amount of smartphone users +possible. Having the application accessible to the +most possible number of users brings more +profit and makes more of an impact on the +market. So, it is time and money consuming for +companies to hire skilled developers that can +develop applications up to the same standard on +each platform. +Challenges +with +creating +cross +platform +application development +1. Creating a universal graphical user interface +Regardless from the actual design itself, every +mobile platform provides its own way for +developers +to +address +user +interface +requirements and manipulating in to developer’s +needs. A good start would be to cope with the +variety screen sizes and resolutions of different +smartphones and tablets. Android for instance is +very flexible with this aspect as it gives +developers more flexibility when dealing with +screens with different sizes and resolutions. +Unlike Apple which seems to be very rigid in this +aspect. IOS applications are restricted in size and +resolution based on the specific iPhone/iPad +models that are targeted. Thus, a unified user +interface design is a bit of a challenge for +developers to implement across different +platforms. + +6 + +a survey published in the international journal of +advanced computer science and applications +interviewed a diverse group of smartphone +application developers with different years of +experience to see how they approach UI design. +It was noticed in the survey that IOS developers +tend to follow Apple’s own UI guidelines which +apply constraints on sizes and resolutions but +also give some flexibility in how to approach this +design whether to rely on apple’s own XIB with +storyboards or MVC. On the other hand, android +developers seemed to follow Google’s own +guidelines on material design as it provides +better user experience [11]. +2. Issues within a single platform +Within a single platform, internal challenges may +arise that could make development even more +difficult. Android is a perfect example for that. A +wide variety of android operating system flavors +co-exist in the market. Samsung, Xaiomi, Oppo +and Huawei each have their own skin on top of +the android operating system which sometimes +leads to incompatibilities with certain parts of +applications. A new dilemma has unfolded in +early 2020 with the trade wars between the +united states of America and china has affected +software +development. +Due +to +recent +developments, +Chinese +telecommunication +company Huawei can no longer take advantage +of American made software. Although Huawei +would still use the android operating system, +they get to lose out on Google’s complimentary +services like the Google play store and Google +play services. This issue has caused a lot of +problems for companies as now decision makers +have to choose whether to integrate google play +services which are arguably essential to develop +an application that is up to the highest standards +or to ditch the services in favor of making an app +that can be published independent from +Google’s own play store. This is a tough decision +to take as decision makers have to choose +whether to integrate the services and appeal to +north American and European users and lose a +potential of a billion smartphone users in south +east Asia or develop an app that is available in +the Asian market and risk providing an app that +is +substandard +in +quality +compared +to +competitors applications. +There exist some cross-platform frameworks +that aim to ease the development process for +making different versions of the same app to run +on completely different platforms. The original +goal of any cross-platform framework is to “write +once compile more” but the success of a +framework is not always guaranteed. One of +most famous cross-platform frameworks is +Xamarin. It is developed my Microsoft and it +supports +both +IOS +and +Android. +The +programming language used to write for both +platforms is C#. developers write the apps as if +they are developing for windows phone devices. +Using compatibility libraries, Xamarin then takes + +7 + +Windows SDK API calls and translates them to +their equivalent API calls for IOS and android. +Another new framework that has become a hot +topic is React-Native which is developed by +Facebook and also aims to provide a cross- +platform development environment which +supports both IOS and android. + +Figure 1 shows how source code is transformed to IOS and +android code [12] +Developers often face a lot of challenges when +trying to use these cross-platform frameworks. +These frameworks often tend to lack proper +support for their own libraries so when +developers get stuck with an issue, they find it +hard to get the proper community support. And +some of the libraries themselves tend to have +limited power when trying to access mobile- +specific functionality and proper hardware +utilization which leads to memory leakage issue +and less CPU utilization. +It is hard to find good cross-platform developers +and especially those needed to write to a specific +framework and have a background in writing, +integrating and testing in previous projects. +Projects that involve great risks or constraints - +whether time or money constraints- tend to stick +to the more stable and secure native +development +approach +and +avoid +cross- +platform. +In the survey some of the developers stated that +sticking to native applications makes for an +easier development process as there is a better +community support and it takes a lot of time and +effort to make a cross-platform app imitate the +look and feel of a native app. Some other +developers said that they were open to learn +new ways to develop applications but the +environment they are in is not really helping. +Companies with big projects tend to be +constrained on time and money when it comes +to delivering big projects, so it puts the project at +risk if it was to be developed in a relatively new - +new to the developers- environment. One +developer said that a good developer if given the +proper time and resources would not find it +difficult to learn any of the new cross-platform +frameworks. Another developer said that +learning +a +framework +for +cross-platform +development should not be hard at all as all of +them share the same basic object-oriented +programming concepts that lay the foundation +for programming. +3. Changes in requirement + +Windows +Native +App +VisualStudio +Windows +WindowsSDK +App +App +NativeiOS +iOS App +source +App +code +Comoabibility +X +Xamarir +lbrary +iOSSDK +Witten inCs +withcallsto +Native +Android +WindowsAP +Android +App +App +Compebiblity +Android SDK +Compilation +Runtine8 + +Whether it is a desktop application or a mobile +application, if the client changes or modifies +some requirements during the initial stage of +development, these modifications can be made. +If the requirements are constantly changing +while enough time is given for development and +proper testing then there is still no problem. But +of requirements change in a late stage of the +development life cycle then the associated cost +of fixing them would be quite high. Frequent +changes in development whether due to vague +requirements from the first place or the sudden +change in client needs is very costly because +these +changes +have +to +be +taken +into +consideration +and +sometimes +the +whole +development process would need to be +restarted from the beginning. Unclear changes +waste time and money that’s why agile +processes have slots for coping with these kinds +of occasions [13]. +4. The +importance +of +testing +during +development +Testing is a very important part of the +development process; its aim is to find and +correct the errors that are present in the +application to give more effect. Testing in +general aims to increase the quality of the +application. But small organizations cannot +really afford to allocate resources to testing. And +sometimes small organizations cannot really +afford to hire testers so developers double as +testers as well [14]. And it is generally hard for a +developer to find bugs in a code that he wrote +himself. While organizations may not find the +resources to test at all, other organizations test +their applications only on emulators. Emulators +by nature are very limited in features, they lack +in terms of mobility, dynamic resolutions and +aspect ratios and mobile oriented features like +compass, gyroscope and GPS. +Through the survey it has been concluded that +the ways of testing vary from one company to +another. Some companies preferred automated +testing while others preferred manual testing. +Other companies follow load, alpha, beta and +smoke testing. +One developer stated that their company follows +a scrum methodology even though they +understand that it is not a good practice for their +relatively small apps. While another developer +said that their company has a dedicated QA +department which manually tests the apps on +actual devices. In general, almost all the +developers agreed on the idea that testing is an +absolute necessity to deliver a bug free app. +5. Adapting old APIs to support new features +Another big dilemma that faces developers +when trying to make an app -whether native or +cross platform- is whether or not to utilize +bleeding edge technologies. These technologies +may be appealing to implement in the app but + +9 + +the problem always lies with supporting old +software and old hardware because most of the +app users will not have access to the latest +resources so this has to be put in mind. A way of +getting around the problem is using support +libraries which help bring new features from new +APIs to old APIs. +6. Maintenance +Another challenge in developing any kind of +software whether for desktop or mobile Is to +maintain the app and analyze crashes. Due to the +almost infinite combination between hardware +and software it is hard to replicate the crashes +for the purpose of analysis and fixing. +Research efforts towards better software +engineering for mobile devices +1. Designing special user interface for +people with accessibility issues +According to a report published by the US +Census, about 15 percent of people living in the +united states suffer from at least one disability. +This disability can be physical, sensory or other. +Very few advancements in research has been +conducted in this area with regards to human +mobile interaction due to the limited studies that +show communities’ relations with their mobile +phones [15]. +There exist some general guidelines on how to +modify existing applications to assist people with +visual impairment. Those guidelines mostly +suggest using “text to speech” services to assist +with low vision / blind users. A text to speech +service acts as a screen reader, it reads whatever +is on the screen with a clear and loud voice which +eases the developers work as no significant +modification needs to be made to the design of +an application to make it work. For example, +Apple has built a “voice over” service inside IOS +platform so it can work on almost all applications +[16]. +2. Software product line engineering +One of the development approaches that are +geared towards bringing down development +cost is software product line engineering. The +aim of this approach is to determine and group +applications that are similar in functionality and +reuse existing code / logic across them. This +approach can be considered the most successful +approach yet for efficient development in the +era of multiplatform applications [17]. +The book “Software product line engineering: a +family-based software engineering process” +highlights two main phases for software product +line engineering which are domain engineering +phase and application engineering phase. +Assuming that a software company has defined +its product line, the domain phase then defines +both the common and application specific +requirements for the whole product line. Then +comes the application phase where actual +coding takes place, the common functionality + +10 + +logic is shared and used across the entire +product line. The product line might not consist +of entirely different applications it may be the +same application but on different platforms. This +approach really encourages developers to focus +more on the common elements in the product +line such as requirements and design. It helps +developers +understand +the +requirements +regardless which specific platform the software +is going to be written to. This approach also shifts +the +development +process +by +putting +requirement gathering and understand upfront +rather than starting with design approaches +which -in theory- should yield a faster +development process. +3. Self-adaptive requirements +By nature, functional requirements tend to take +all the attention and focus of developers and +clients from non-functional requirements which +are as critical as their counterparts [4]. +Applications need to constantly “self-adapt” to +provide users with the functionality they need. +In some situations, applications need to self- +adapt +to +provide +users +with +reduced +functionality to provide a better support for +dynamism. +A good approach is to use already existing self- +adaptive system requirement specification +languages such as “RELAX” [18]. The language +“Relax” was designed as a middleware language +that aims to explicitly express the behavioral and +environmental uncertainties that are come +attached to self-adaptive systems. Relax takes a +simple approach to dividing up requirements +into those that must be satisfied completely +(invariant) and those whose partial satisfaction / +completion is not a necessity (variant). All the +requirements are then expressed using a +combination of natural language and fuzzy logic. +The RELAX documentation then lists out the +variant requirements and how the environment +can affect them. +Integrating +a +self-adaptive +specification +language like RELAX into an existing agile +software engineering approach would provide a +better structure for requirement gathering and +specification and would overall improve the +quality of non-functional system requirements in +the context of environment changes. Adapting +this approach will allow developers to better +consider and adapt their application’s behavior +in a non-optimal environment. +Shneiderman’s Eight Golden Rules to design +better interfaces +About Shneiderman’s +Ben Shneiderman is an American computer +science professor how specializes in human +computer interaction in the University of +Maryland. He made an infamous book about +human computer interaction called “Designing +the User Interface: Strategies for effective + +11 + +human-computer Interaction” in which he +outlines eight golden principals for designing an +interface. Four of these principles can be directly +translated into application design while the +other four -while still relevant- would need some +modifications to translate to mobile app +development [19]. +1. Strive for consistency +Consistency in design relies on using same design +elements through the entire application. Design +elements are the icons, menu hierarchy, colors +and actions. Users need to easily understand the +flow of the application and how they can do a +sequence of actions to achieve their goals. The +experience gained from interaction with a page +needs to be transferrable and applied to the +other pages within the same application without +needing to learn a new skill. Consistency in +design helps with the feeling of familiarity with +the software product. +A better way to implement a consistent design is +not only to have consistency within all the pages +of the application but also to have consistency +with the design philosophy of the platform. One +example that has been adapted by many android +developers is the charms menu. It is very often +now on android applications is to find on the top +left corner 3 parallel horizontal dashes that when +pressed bring up a sliding charms menu. This +menu contains all the necessary options from +settings to account management. The main +reason android developers use the charms menu +is because it is recommended by google as part +of the “material design” interface. Utilizing +material design elements means that users can +translate their experience from one application +to another making the interaction easier. +2. Enable frequent users to use shortcuts +An important demographic that must not be +ignored when designing an application is the +power users. Power users do not look or behave +similar to casual users. For power users the +entire phone is just a utility to achieve multiple +goals and sometimes multiple goals at the same +time. So, for power users, there has to exist +shortcuts that would enable them to achieve +their goals in a shorter time and with less steps. +A great implementation on shortcuts is Apple’s +built in shortcut app on IOS. The application +enables users to automate almost any kind of +tasks on IOS. These tasks can be combined in any +order and be are contextually aware so they can +be triggered by time or location. These tasks can +range from morning chores to congratulating +someone on a special occasion. A great example +is to setup a morning routine that triggers every +morning on an exact time. It would start with an +alarm to wake up the user then inform him / her +with set day’s temperature and the probability of +rain. It could go on to then read the important +news of the day, turn on the coffee machine and +order an uber for work. After the user leaves for + +12 + +work, the phone then proceeds to lock the house +doors using the smart lock. This kind of +automation was in recent years considered a +future dream that has become a reality due to +the advancements in mobile development. +3. Offer informative feedback +A proper engaging feedback should inform the +user of what actions they made and also it +should come in a proper timely manner. A good +example for a good feedback is a processing +status bar that would indicate to the user how +long a certain process will take. A very bad +example, is to have an application that produces +error codes whenever it encounters errors. Error +codes are not understood by users which makes +the experience frustrating. +4. Design dialogue to yield closure +Communication is key, it is how humans +understand +each +other +and +exchange +information. The same principals should be +applied when developing an application. For +example, a form of reassurance has to be +present after completing a transaction process. +It would be better if the application could display +a message confirming that the transaction has +been completed successfully or if it has failed. +Even if it has failed the application should display +a message indicating that it had failed and +explaining where exactly an error has happened. +5. Offer simple error handling +Developers should never expect that their +applications are used by power users and +experienced testers. An application should be +fool-proof and even in a rare case of an +application breaking error, the user should be +guided by an easy step by step solution that he / +she could follow. +6. Permit easy reversal actions +Continuing on with the idea of not expecting +application users to be experts, there should +always exist a way to reverse an action or a +sequence of actions in the cases of accidental +presses. The undo button is frequently used in +photo editing applications where users might +accidently apply the wrong filter on a picture. +7. Support internal locus of control +A good way of interaction between users and an +application is to give users sense of control and +that they always initiate the actions. +8. Reduce short term memory load +Application design interfaces should be simply +organized with a proper layout and hierarchy. +The design should encourage the user to rely on +recognition not recall as it is much easier. Using +the application should not feel like a puzzle or a +brain exercise. An application should always give +users visual or auditory clues and provide +relevant information at all times. +conclusion + +13 + +Mobile platforms are moving towards more and +more fragmentation whether its due to the +variety of software platforms or even to the +different flavors within each platform. An +application can have the same name, design, +look and feel on two different platforms but still +be treated as two different applications due to +the nature of development for each platform. +So, to ensure consistency, functionalities of each +app have to be manually checked against similar +versions of it on the other platforms. Creating a +general-purpose +reusable +graphical +user +interface depends on balancing compromises +whether to be consistent or to follow the design +language of each platform. Testing is still +considered as a huge challenge among the +mobile app development industry. Unit testing is +still +not +common +among +the +mobile +development community and the current testing +structures are not mature enough. But not all the +blame has to be put on developers as the current +mobile testing tools look very powerless as they +tend to lack the fundamental features of mobile +phones like sensors, gesture navigation support +and location support. + + + + +References + +[ +1 +] +I. Malavolta, "Web-based hybrid mobile +apps: state of the practice and research +opportunities," 2016. +[ +2 +] +Microsoft, "building cross platform +applications," [Online]. Available: +https://developer.xamarin.com/guides/cross +- +platform/application_fundamentals/building +_cross_platform_applications/part_0_- +_overview/. +[ +3 +] +J. D. a. J. Dixon, "Mobile Application +Software Engineering: Challenges and +Research Directions," Department of +Computer and Information Sciences Towson +University. +[ +4 +] +A. I. Wasserman, "Software engineering +issues for mobile application development," +in FSE/SDP workshop on Future of software +engineering research, 2010. +[ +5 +] +J. G. a. P. Tarasewich, "Guidelines for +handheld mobile device interface design," in +Proceedings of DSI 2004 Annual Meeting,, +2004. +[ +6 +] +B. Fling, Mobile design and development, +O’Reilly, 2009. +[ +7 +] +S. Allen, Pro Smartphone Cross-Platform +Development: iPhone, Blackberry, Windows +Mobile, and Android Development and +Distribution, 2010: Apress. +[ +8 +] +W. S. M. P. G. L. J. A. a. W. R. T. Hofer, +"Context-awareness on mobile devices - the +hydrogen approach," in 36th Annual Hawaii + +14 + +International Conference on System Sciences, +2003, Hawaii , 2003. +[ +9 +] +N. Jennings, "Agent-oriented software +engineering," in Multi-Agent System +Engineering, 1999. +[ +1 +0 +] +N. Jennings, "Multi-Agent System +Engineering," in Agent-oriented software +engineering, 1999. +[ +1 +1 +] +M. S. M. A. S. B. M.-u.-d. A. S. Naila Kousar, +"Software Engineering: Challenges and their +Solution in Mobile App Development," +(IJACSA) International Journal of Advanced +Computer Science and Applications, vol. 9, +no. 1, 2018. +[ +1 +2 +] +N. G. V. N. S. a. I. L. Boushehrinejadmoradi, +Testing Cross-Platform Mobile App +Development Frameworks, 2015. +[ +1 +3 +] +A. Wasserman, "Software engineering issues +for mobile application development," in +FSE/SDP workshop on Future of software +engineering research, New Mexico, 2010. +[ +1 +4 +] +S. a. K. A. Amatya, "Cross-Platform Mobile +Development Challenges and Opportunities," +„ICT Innovations 2013: ICT Innovations and +Education, 2013. +[ +1 +5 +] +M. Brault, "Disability status and the +characteristics of people in group quarters: a +brief analysis of disability prevalence among +the civilian noninstitutionalized and total +populations in the American community +survey," US Census Bureau, 2008. +[ +1 +6 +] +Apple Inc., "iOS Human Interface Guidelines: +Introduction.," [Online]. Available: +http://developer.apple.com/library/ios/#DO +CUMENTATION/UserExperience/Conceptual/ +MobileHIG/Introduction/Introduction.html. +[ +1 +7 +] +D. W. a. C. Lai, Software product line +engineering: a family based software +engineering process, Addison-Wesley, 1999. +[ +1 +8 +] +P. S. N. B. B. H. C. C. a. J.-M. B. J. Whittle, +"RELAX: a language to address uncertainty in +self-adaptive systems requirement,," +Requirements Engineering,, vol. vol. 15, no. +no. 2, pp. pp. 177-196, 2010.. +[ +1 +9 +] +Interaction design foundation, +"Shneiderman’s Eight Golden Rules Will Help +You Design Better Interfaces," 2020. +[Online]. Available: https://www.interaction- +design.org/literature/article/shneiderman-s- +eight-golden-rules-will-help-you-design- +better-interfaces. + + + + diff --git a/QtAyT4oBgHgl3EQftvnR/content/tmp_files/load_file.txt b/QtAyT4oBgHgl3EQftvnR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..330411683d4bb9a8b85bf33fb762a861f03ec079 --- /dev/null +++ b/QtAyT4oBgHgl3EQftvnR/content/tmp_files/load_file.txt @@ -0,0 +1,350 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf,len=349 +page_content='Software engineering for mobile applications, a survey on challenges and solutions Shehab Eldeen Ayman Mounir Shehab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content='ayman@bue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content='eg The British University in Egypt Abstract Mobile app development has become the front line in software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' With the recent years many smartphone platforms have grew including but not limited to webOS, blackberry os, Tizen, android, and iOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The coexistence of these platforms results in a challenging situation where apps must be developed and maintained to the same level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The mobile app development scene has recently seen a noticeable rise in the number of applications that adapt web elements like HTML5 to produce native like applications that are essentially web views wrapped into containers to appear as any normal application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This means that the application behavior can vary drastically from one user to another meaning that the app behavior can be changed drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Therefore, application developers rely on an agile or an ad-hoc approach to development that is mostly autonomous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' In this paper, we describe the current state of the art of context awareness in mobile application development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' introduction No one can deny that the astronomical rise in smartphone popularity in the last decade has reshaped the software engineering industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' With billions in app downloads, mobile app development has become the front line in software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Through the recent years many smartphone platforms have grew include but not limited to webOS, blackberry os, Tizen, android and iOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' With IOS and android taking most of the share of the smartphone platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' some of the common challenges of building a multifunctional software has moved from the desktop to mobile but developing software for smartphones has come with its own set of new challenges that needs to be overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Recent estimations done in 2017 state that the apple app store contains more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content='2 million apps while the google play store contains 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content='8 million and they are in a constant growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Even though both windows app store and blackberry world are both discontinued, they contained 600 thousand and 200 thousand apps respectively [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Developing for these mobile platforms is not as flexible as developing for the desktop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The programming languages, tools, frameworks and APIs required for the development process are platform specific, for instance, native android applications are built using the java programming language on the android studio 2 integrated development environment, whereas apple’s IOS applications are developed using the swift language with XCode tool [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The coexistence of these platforms results in a challenging situation where apps have to be developed and maintained to the same level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Traditional software engineering approaches have been somewhat phased out in the context of mobile development as these approaches cannot be directly applied in the mobile context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Mobile graphical user interfaces are considered as a whole new paradigm for human computer interaction study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Mobile phones provide a whole new way of interaction that is considered untraditional comparing it the computer situation which involves sitting down and focusing attention and resources towards the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' And even the interaction method has drastically changed from the typical mouse and keyboard to touch screens and gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' And the way of interaction seems to be evolving constantly as voice commands and augment reality/mixed reality are becoming more of a possibility [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Nowadays the variety of hardware and software platform have forced developers to make a group of different applications that on the outside might look the same but on the inside are completely different just to suite every platform [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Challenges with mobile development 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Create a universal graphical user interface Research conducted by Tarasewich and Gong suggest that at least four of Shneiderman’s design principals -which will be looked at further- are applicable -without modification- to mobile phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' These include “enabling frequent users to use shortcuts”, “designing dialogue to yield closure”, “offering informative feedback”, “supporting internal locus of control” [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' As technologies tend to evolve, challenges tend to evolve as well that’s why recently efforts have been into researching ways of streamlining application development regardless of the hardware or the software platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Focusing on streamlining the process will free up a lot for resources which could then be directed towards making better user interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Reusing software across different platforms Companies take the hard decision whether to focus on one or two platforms and enrich the user experience or go all out and support natively every single platform that is currently on the market compromising their software quality [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' There is no other way of saying it but sometimes user interface cannot be unified across IOS and android not due to some technical limitations in any of the two platforms but due to the way each 3 platform users have grown accustomed to certain design aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' An application with an insert circular icon in the bottom right corner and three vertical dashes in the top left corner - that when clicked bring a charms menu- is easily identified by mobile developers as an android application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' While applications that have their back/previous button in the top right corner tend to be IOS applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Android applications tend not to have a back button as android users are accustomed to using the back button implemented in the operating system itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' While IOS users are accustomed to look for a back button in the application itself and when it is not present, a swipe to the right would function exactly like a back button.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Many software companies have separate teams for development where each time only focuses on one single platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This means that if an app is to be developed for IOS and android -for example- the software engineering effort needed to build the app are doubled to provide the same functionalities on the two operating systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The basic software engineering concepts of reusing and refactoring parts of software are not applicable here as they cannot be transferred from one software to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This leads to a very limited coordination between development teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' It mostly relies on ad-hoc basis without any real effort in reducing the resources -especially time and cost- allocated for the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The mobile development scene has recently seen a noticeable rise in the number of applications that adapt web elements like HTML5 to produce native like applications which are essentially web views wrapped into containers to appear as any normal application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' These kind of applications -which will be discussed further- do not have the rich capabilities of native apps as they cannot access the platform’s APIs which limits their functionalities, but on the other side allow for the reuse of almost an entire application interface on more than one platform [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Context aware applications Mobile devices are far in contrast to traditional stationary computer platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Mobile phones are highly customizable and adaptive to user’s needs which means that they must constantly monitor the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This means that applications on mobile need to be constantly aware of time, weather, location, orientation, proximity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' [8] Mobile applications are now able to use all of this contextual awareness to make a very specific, very specialized experience that is very special to its own user meaning that the application behavior can vary drastically from one user to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This idea of context awareness is no new feature, web applications have been providing similar kind of user tailored experience for years but it has never got to the same extent as mobile applications due to the 4 sensors and capabilities of mobile phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This kind of sensory overload has never been present or dealt with in traditional software engineering, it was always associated with robotics and smart objects which are dealt with using agent oriented software engineering [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The availability of these sensors has put their utilization in the forefront of application development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' So, developers pour careful attention into analyzing application requirements and utilizing context awareness to much improve the quality of their application resulting in a better user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Agent oriented software engineering (AOSE) is concerned with building software agents that are mostly autonomous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This approach provides all the necessary models, abstractions and “pure” software engineering approaches to build a contextually aware application of a multi-agent system (MAS) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' For simplification lets imagine that the multiagent system is a physical robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Using its sensors, the robot must use all of its sensors to sense and understand its surrounding environment and react to it accordingly in order to achieve its goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Balancing agility and uncertainty There is no denying that the existence of mobile applications has changed the way a development process is looked at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Gone are the times where almost all the application requirements are clearly specified and fully implemented on the first run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' It is now completely normal to regularly receive updates for applications that add new features or focus on stability improvement and bug fixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This means that the traditional waterfall model cannot be realistically applied in mobile development that’s why application developers rely on an agile or an ad-hoc approach to development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The constant growth in demand for more context aware applications, tailored user experience, very heavy competition and low tolerance by users for unresponsive applications have derived more of a semi-formal approach development that does not necessarily go by the books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This approach has to be somehow integrated within the agile method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The very dynamic very user specific experience provided my mobile applications allows for scenarios and situations that may not be fully specified within the functional and nonfunctional requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' If mobile applications were strictly to follow their functional / nonfunctional requirements, this will result in a less quality experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This means that applications need to run continuously and autonomously adapt and modify the behavior and provide more functionality than strictly specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' It is hard to imagine the idea of how applications can provide an improved service than originally specified within the design documents and how this constantly adaptive service can improve the quality and engagement with applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' A 5 great example of adaptive service is user tailored ad experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This has become an essential way of digital advertising where the products listed in the ad spots depending on the user interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Gathering user interests has lately been a relatively easy task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' For example, google ad services track each user’s search history to determine what are this specific user’s interest and then display ads of objects that the user may find tempting to buy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Another great example is Facebook application, it uses location services and profile analysis to recommend friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' And this context awareness feature keeps updating frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' For example, if a person left his workplace, university or even the country, the application will keep providing relevant and new recommendations based on the updated location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Another area where Facebook has ventured into and has made a big impact is with the Facebook marketplace service on which people can buy and sell almost anything from used phones to used cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The application here utilizes context awareness to link users with the nearest buyers and promote products to users who are interested in as it will most likely end up in a successful buy / sell operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Cross platform development An ongoing challenge in mobile development especially for startups and relatively small companies is to choose the platforms that they are going to develop for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Companies tend to focus their development on IOS and android to cover the largest amount of smartphone users possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Having the application accessible to the most possible number of users brings more profit and makes more of an impact on the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' So, it is time and money consuming for companies to hire skilled developers that can develop applications up to the same standard on each platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Challenges with creating cross platform application development 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Creating a universal graphical user interface Regardless from the actual design itself, every mobile platform provides its own way for developers to address user interface requirements and manipulating in to developer’s needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' A good start would be to cope with the variety screen sizes and resolutions of different smartphones and tablets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Android for instance is very flexible with this aspect as it gives developers more flexibility when dealing with screens with different sizes and resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Unlike Apple which seems to be very rigid in this aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' IOS applications are restricted in size and resolution based on the specific iPhone/iPad models that are targeted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Thus, a unified user interface design is a bit of a challenge for developers to implement across different platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 6 a survey published in the international journal of advanced computer science and applications interviewed a diverse group of smartphone application developers with different years of experience to see how they approach UI design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' It was noticed in the survey that IOS developers tend to follow Apple’s own UI guidelines which apply constraints on sizes and resolutions but also give some flexibility in how to approach this design whether to rely on apple’s own XIB with storyboards or MVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' On the other hand, android developers seemed to follow Google’s own guidelines on material design as it provides better user experience [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Issues within a single platform Within a single platform, internal challenges may arise that could make development even more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Android is a perfect example for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' A wide variety of android operating system flavors co-exist in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Samsung, Xaiomi, Oppo and Huawei each have their own skin on top of the android operating system which sometimes leads to incompatibilities with certain parts of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' A new dilemma has unfolded in early 2020 with the trade wars between the united states of America and china has affected software development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Due to recent developments, Chinese telecommunication company Huawei can no longer take advantage of American made software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Although Huawei would still use the android operating system, they get to lose out on Google’s complimentary services like the Google play store and Google play services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This issue has caused a lot of problems for companies as now decision makers have to choose whether to integrate google play services which are arguably essential to develop an application that is up to the highest standards or to ditch the services in favor of making an app that can be published independent from Google’s own play store.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This is a tough decision to take as decision makers have to choose whether to integrate the services and appeal to north American and European users and lose a potential of a billion smartphone users in south east Asia or develop an app that is available in the Asian market and risk providing an app that is substandard in quality compared to competitors applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' There exist some cross-platform frameworks that aim to ease the development process for making different versions of the same app to run on completely different platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The original goal of any cross-platform framework is to “write once compile more” but the success of a framework is not always guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' One of most famous cross-platform frameworks is Xamarin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' It is developed my Microsoft and it supports both IOS and Android.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The programming language used to write for both platforms is C#.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' developers write the apps as if they are developing for windows phone devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Using compatibility libraries, Xamarin then takes 7 Windows SDK API calls and translates them to their equivalent API calls for IOS and android.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Another new framework that has become a hot topic is React-Native which is developed by Facebook and also aims to provide a cross- platform development environment which supports both IOS and android.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Figure 1 shows how source code is transformed to IOS and android code [12] Developers often face a lot of challenges when trying to use these cross-platform frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' These frameworks often tend to lack proper support for their own libraries so when developers get stuck with an issue, they find it hard to get the proper community support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' And some of the libraries themselves tend to have limited power when trying to access mobile- specific functionality and proper hardware utilization which leads to memory leakage issue and less CPU utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' It is hard to find good cross-platform developers and especially those needed to write to a specific framework and have a background in writing, integrating and testing in previous projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Projects that involve great risks or constraints - whether time or money constraints- tend to stick to the more stable and secure native development approach and avoid cross- platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' In the survey some of the developers stated that sticking to native applications makes for an easier development process as there is a better community support and it takes a lot of time and effort to make a cross-platform app imitate the look and feel of a native app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Some other developers said that they were open to learn new ways to develop applications but the environment they are in is not really helping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Companies with big projects tend to be constrained on time and money when it comes to delivering big projects, so it puts the project at risk if it was to be developed in a relatively new - new to the developers- environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' One developer said that a good developer if given the proper time and resources would not find it difficult to learn any of the new cross-platform frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Another developer said that learning a framework for cross-platform development should not be hard at all as all of them share the same basic object-oriented programming concepts that lay the foundation for programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Changes in requirement Windows Native App VisualStudio Windows WindowsSDK App App NativeiOS iOS App source App code Comoabibility X Xamarir lbrary iOSSDK Witten inCs withcallsto Native Android WindowsAP Android App App Compebiblity Android SDK Compilation Runtine8 Whether it is a desktop application or a mobile application, if the client changes or modifies some requirements during the initial stage of development, these modifications can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' If the requirements are constantly changing while enough time is given for development and proper testing then there is still no problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' But of requirements change in a late stage of the development life cycle then the associated cost of fixing them would be quite high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Frequent changes in development whether due to vague requirements from the first place or the sudden change in client needs is very costly because these changes have to be taken into consideration and sometimes the whole development process would need to be restarted from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Unclear changes waste time and money that’s why agile processes have slots for coping with these kinds of occasions [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The importance of testing during development Testing is a very important part of the development process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' its aim is to find and correct the errors that are present in the application to give more effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Testing in general aims to increase the quality of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' But small organizations cannot really afford to allocate resources to testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' And sometimes small organizations cannot really afford to hire testers so developers double as testers as well [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' And it is generally hard for a developer to find bugs in a code that he wrote himself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' While organizations may not find the resources to test at all, other organizations test their applications only on emulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Emulators by nature are very limited in features, they lack in terms of mobility, dynamic resolutions and aspect ratios and mobile oriented features like compass, gyroscope and GPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Through the survey it has been concluded that the ways of testing vary from one company to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Some companies preferred automated testing while others preferred manual testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Other companies follow load, alpha, beta and smoke testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' One developer stated that their company follows a scrum methodology even though they understand that it is not a good practice for their relatively small apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' While another developer said that their company has a dedicated QA department which manually tests the apps on actual devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' In general, almost all the developers agreed on the idea that testing is an absolute necessity to deliver a bug free app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Adapting old APIs to support new features Another big dilemma that faces developers when trying to make an app -whether native or cross platform- is whether or not to utilize bleeding edge technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' These technologies may be appealing to implement in the app but 9 the problem always lies with supporting old software and old hardware because most of the app users will not have access to the latest resources so this has to be put in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' A way of getting around the problem is using support libraries which help bring new features from new APIs to old APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Maintenance Another challenge in developing any kind of software whether for desktop or mobile Is to maintain the app and analyze crashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Due to the almost infinite combination between hardware and software it is hard to replicate the crashes for the purpose of analysis and fixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Research efforts towards better software engineering for mobile devices 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Designing special user interface for people with accessibility issues According to a report published by the US Census, about 15 percent of people living in the united states suffer from at least one disability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This disability can be physical, sensory or other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Very few advancements in research has been conducted in this area with regards to human mobile interaction due to the limited studies that show communities’ relations with their mobile phones [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' There exist some general guidelines on how to modify existing applications to assist people with visual impairment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Those guidelines mostly suggest using “text to speech” services to assist with low vision / blind users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' A text to speech service acts as a screen reader, it reads whatever is on the screen with a clear and loud voice which eases the developers work as no significant modification needs to be made to the design of an application to make it work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' For example, Apple has built a “voice over” service inside IOS platform so it can work on almost all applications [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Software product line engineering One of the development approaches that are geared towards bringing down development cost is software product line engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The aim of this approach is to determine and group applications that are similar in functionality and reuse existing code / logic across them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This approach can be considered the most successful approach yet for efficient development in the era of multiplatform applications [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The book “Software product line engineering: a family-based software engineering process” highlights two main phases for software product line engineering which are domain engineering phase and application engineering phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Assuming that a software company has defined its product line, the domain phase then defines both the common and application specific requirements for the whole product line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Then comes the application phase where actual coding takes place, the common functionality 10 logic is shared and used across the entire product line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The product line might not consist of entirely different applications it may be the same application but on different platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This approach really encourages developers to focus more on the common elements in the product line such as requirements and design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' It helps developers understand the requirements regardless which specific platform the software is going to be written to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This approach also shifts the development process by putting requirement gathering and understand upfront rather than starting with design approaches which -in theory- should yield a faster development process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Self-adaptive requirements By nature, functional requirements tend to take all the attention and focus of developers and clients from non-functional requirements which are as critical as their counterparts [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Applications need to constantly “self-adapt” to provide users with the functionality they need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' In some situations, applications need to self- adapt to provide users with reduced functionality to provide a better support for dynamism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' A good approach is to use already existing self- adaptive system requirement specification languages such as “RELAX” [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The language “Relax” was designed as a middleware language that aims to explicitly express the behavioral and environmental uncertainties that are come attached to self-adaptive systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Relax takes a simple approach to dividing up requirements into those that must be satisfied completely (invariant) and those whose partial satisfaction / completion is not a necessity (variant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' All the requirements are then expressed using a combination of natural language and fuzzy logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The RELAX documentation then lists out the variant requirements and how the environment can affect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Integrating a self-adaptive specification language like RELAX into an existing agile software engineering approach would provide a better structure for requirement gathering and specification and would overall improve the quality of non-functional system requirements in the context of environment changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Adapting this approach will allow developers to better consider and adapt their application’s behavior in a non-optimal environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Shneiderman’s Eight Golden Rules to design better interfaces About Shneiderman’s Ben Shneiderman is an American computer science professor how specializes in human computer interaction in the University of Maryland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' He made an infamous book about human computer interaction called “Designing the User Interface: Strategies for effective 11 human-computer Interaction” in which he outlines eight golden principals for designing an interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Four of these principles can be directly translated into application design while the other four -while still relevant- would need some modifications to translate to mobile app development [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Strive for consistency Consistency in design relies on using same design elements through the entire application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Design elements are the icons, menu hierarchy, colors and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Users need to easily understand the flow of the application and how they can do a sequence of actions to achieve their goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The experience gained from interaction with a page needs to be transferrable and applied to the other pages within the same application without needing to learn a new skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Consistency in design helps with the feeling of familiarity with the software product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' A better way to implement a consistent design is not only to have consistency within all the pages of the application but also to have consistency with the design philosophy of the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' One example that has been adapted by many android developers is the charms menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' It is very often now on android applications is to find on the top left corner 3 parallel horizontal dashes that when pressed bring up a sliding charms menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This menu contains all the necessary options from settings to account management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The main reason android developers use the charms menu is because it is recommended by google as part of the “material design” interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Utilizing material design elements means that users can translate their experience from one application to another making the interaction easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Enable frequent users to use shortcuts An important demographic that must not be ignored when designing an application is the power users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Power users do not look or behave similar to casual users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' For power users the entire phone is just a utility to achieve multiple goals and sometimes multiple goals at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' So, for power users, there has to exist shortcuts that would enable them to achieve their goals in a shorter time and with less steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' A great implementation on shortcuts is Apple’s built in shortcut app on IOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The application enables users to automate almost any kind of tasks on IOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' These tasks can be combined in any order and be are contextually aware so they can be triggered by time or location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' These tasks can range from morning chores to congratulating someone on a special occasion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' A great example is to setup a morning routine that triggers every morning on an exact time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' It would start with an alarm to wake up the user then inform him / her with set day’s temperature and the probability of rain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' It could go on to then read the important news of the day, turn on the coffee machine and order an uber for work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' After the user leaves for 12 work, the phone then proceeds to lock the house doors using the smart lock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' This kind of automation was in recent years considered a future dream that has become a reality due to the advancements in mobile development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Offer informative feedback A proper engaging feedback should inform the user of what actions they made and also it should come in a proper timely manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' A good example for a good feedback is a processing status bar that would indicate to the user how long a certain process will take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' A very bad example, is to have an application that produces error codes whenever it encounters errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Error codes are not understood by users which makes the experience frustrating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Design dialogue to yield closure Communication is key, it is how humans understand each other and exchange information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The same principals should be applied when developing an application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' For example, a form of reassurance has to be present after completing a transaction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' It would be better if the application could display a message confirming that the transaction has been completed successfully or if it has failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Even if it has failed the application should display a message indicating that it had failed and explaining where exactly an error has happened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Offer simple error handling Developers should never expect that their applications are used by power users and experienced testers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' An application should be fool-proof and even in a rare case of an application breaking error, the user should be guided by an easy step by step solution that he / she could follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Permit easy reversal actions Continuing on with the idea of not expecting application users to be experts, there should always exist a way to reverse an action or a sequence of actions in the cases of accidental presses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The undo button is frequently used in photo editing applications where users might accidently apply the wrong filter on a picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Support internal locus of control A good way of interaction between users and an application is to give users sense of control and that they always initiate the actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Reduce short term memory load Application design interfaces should be simply organized with a proper layout and hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' The design should encourage the user to rely on recognition not recall as it is much easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Using the application should not feel like a puzzle or a brain exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' An application should always give users visual or auditory clues and provide relevant information at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' conclusion 13 Mobile platforms are moving towards more and more fragmentation whether its due to the variety of software platforms or even to the different flavors within each platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' An application can have the same name, design, look and feel on two different platforms but still be treated as two different applications due to the nature of development for each platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' So, to ensure consistency, functionalities of each app have to be manually checked against similar versions of it on the other platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Creating a general-purpose reusable graphical user interface depends on balancing compromises whether to be consistent or to follow the design language of each platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Testing is still considered as a huge challenge among the mobile app development industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' Unit testing is still not common among the mobile development community and the current testing structures are not mature enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' But not all the blame has to be put on developers as the current mobile testing tools look very powerless as they tend to lack the fundamental features of mobile phones like sensors, gesture navigation support and location support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQftvnR/content/2301.00602v1.pdf'} +page_content=' References [ 1 ] I.' metadata={'source': 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The originality of this certificate-based algorithm +is to achieve a tradeoff between simplicity and efficiency. +Simplicity is crucial in obtaining the proof of correctness +of the algorithm. This proof splits into the correctness of +an abstract algorithm stated over proof-oriented data types +and the correspondence with a low-level implementation +over computation-oriented data types. A special effort has +been made to reduce the algorithm to a small sequence of +elementary operations (e.g., matrix multiplications, basic +routines on sets and graphs), in order to make the derivation +of the correctness of the low-level implementation more +transparent. +Efficiency allows us to scale up to polytopes with a chal- +lenging combinatorics. For instance, we formally check the +two counterexamples of Matschke, Santos and Weibel to +the Hirsch conjecture, respectively 20- and 23-dimensional +polytopes with 36 425 and 73 224 vertices involving rational +coefficients with up to 40 digits in their numerator and de- +nominator. We also illustrate the performance of the method +by computing the list of vertices or the diameter of well- +known classes of polytopes, such as (polars of) cyclic poly- +topes involved in McMullen’s Upper Bound Theorem. +Keywords: polyhedra, polytopes, Hirsch Conjecture, proof +assistants, certified computation +1 +Introduction +1.1 +Motivations +The study of diameters of polyhedra is at the heart of the +following major problem in optimization: does the simplex +method terminate in polynomial time? This question is open +since Georg B. Dantzig introduced the simplex method in +the late 40s, and it has inspired to the Fields medalist Steve +Smale the ninth of his problems for the 21st century on the +existence of a strongly polynomial algorithm for linear pro- +gramming [41]. The simplex method [17] is certainly the +most standard technique to solve linear programs, i.e., prob- +lems of the form +Minimize +⟨𝑐,𝑥⟩ +subject to +⟨𝑎1,𝑥⟩ ≥ 𝑏1 , . . . , ⟨𝑎𝑚,𝑥⟩ ≥ 𝑏𝑚 , +𝑥 ∈ R𝑛 +for some vectors 𝑎1, . . . ,𝑎𝑚,𝑐 ∈ R𝑛 and reals 𝑏1, . . . ,𝑏𝑚, +where ⟨𝑦,𝑧⟩ � �𝑛 +𝑖=1 𝑦𝑖𝑧𝑖 denotes the Euclidean scalar prod- +uct of 𝑦,𝑧 ∈ R𝑛. It consists in minimizing the objective func- +tion 𝑥 ↦→ ⟨𝑐,𝑥⟩ over the convex polyhedron formed by the +points 𝑥 ∈ R𝑛 satisfying the constraints ⟨𝑎𝑖,𝑥⟩ ≥ 𝑏𝑖 for all +𝑖 = 1, . . . ,𝑚. From a geometric perspective, the principle of +the simplex method is to iteratively decrease the objective +function by visiting a subset of vertices of the polyhedron. +More precisely, at every iteration, the method selects a vertex +with smaller value among the vertices which are adjacent +(i.e., connected by an edge) to the current vertex. We refer +to Section 2 for the mathematical definitions of vertices and +edges of polyhedra, and to Figure 1 for an illustration. The +choice of the next vertex at every iteration is specified by the +so-called pivot rule. In consequence, every pivot rule makes +the simplex method draw a path in the vertex-edge graph of +the polyhedron, i.e., the graph defined by the vertices and +edges of the polyhedron (see Fig. 1). While a large number of +pivot rules have been described in the literature, all the rules +that have been mathematically analyzed have been shown to +exhibit (sub)exponential behavior in the worst case (see [3], +and [20] for a more recent account). +Recall that, in a graph, the distance between two vertices +is the length (i.e., the number of edges) of any shortest path +between them. The diameter of the graph is then defined as +the largest distance between any two vertices. Consequently, +the (combinatorial) diameter of a polyhedron, i.e., the diame- +ter of its graph, constitutes a lower bound on the number of +iterations performed by the simplex method with any pivot +rule. With this motivation, Hirsch formulated the following +conjecture in a letter to Dantzig in 1957: +Conjecture 1 (Hirsch conjecture over polytopes). The di- +ameter of any 𝑑-dimensional polytope with 𝑝 facets is bounded +by 𝑝 − 𝑑. +In this statement, a polytope refers to the convex hull +of finitely many points, or equivalently, a bounded polyhe- +dron. The facets of a 𝑑-dimensional polyhedron are the faces +of dimension 𝑑 − 1. (Originally, the conjecture was stated +arXiv:2301.04060v1 [cs.LO] 10 Jan 2023 + +Xavier Allamigeon, Quentin Canu, and Pierre-Yves Strub +−𝑐 +•𝐴 +𝐵 +𝐶 +𝐷 +𝐸 +𝐹 +𝐹 +𝐵 +𝐴 +𝐷 +𝐸 +𝐶 +Figure 1. Several possible execution traces of the simplex +method on the 3-dimensional cross polytope (see Section 6 +for a description). The execution of the simplex method +draws a path in the graph of the polytope from a starting +vertex to an optimal one (rightmost part of the picture). +over polyhedra rather than polytopes, but it was quickly +realized that the bound does not hold over unbounded poly- +hedra [35].) The study of the diameter of polytopes and poly- +hedra has received a tremendous attention over the years; +see e.g. [46] for a survey on the topic. Still, the conjecture +remained unsolved for more than fifty years, until Santos +exhibited a counterexample: +Theorem 1 ([39]). There exists a 43-dimensional polytope +with 86 facets and diameter larger than 43. +In a further work joint with Matschke and Weibel [36], San- +tos provided two other counterexamples to the conjecture, +respectively in dimension 20 and 23. All these constructions +critically rely on a computational argument. More precisely, +in all of them, the non-Hirsch polytope is obtained from a +smaller dimensional polytope with a special combinatorial +structure, that of a spindle, and in which the distance between +two distinguished vertices has to be proved larger than the +dimension. For the original counterexample, this spindle +has dimension 5, 48 facets and 322 vertices. Santos indicates +that the property has been checked thanks to the informal +software Polymake [24], and he provides two independent +proofs that are “computer-free (but not computation-free).” +Fortunately, exploiting the symmetry group of this spindle +makes the computations manageable by hand. +Since Santos’ breakthrough, the interest for diameters of +polytopes and polyhedra has remained intact. First, this is +still not understood to which extent a counterexample to the +Hirsch conjecture can be found in smaller dimensions. More- +over, from the perspective of the complexity of the simplex +method, the most important question is whether or not the +diameter of polyhedra can be bounded by a polynomial in +the dimension and the number of facets; this is known as the +polynomial Hirsch conjecture. It is likely that computations +will play a key role in any progress on these two questions, +just like they did in Santos’ construction. +In consequence, there is a strong motivation to develop +a framework in which computations over polyhedra are +performed in a proof assistant. This would considerably en- +large the scope of research for pathological polytopes (e.g., +giving the capacity to scale up to larger numbers of ver- +tices) while retaining (if not increasing) the level of trust +compared to pen-and-paper computations. Combinatorial +properties of polytopes are not the only topic such a contri- +bution would benefit to. For instance, polyhedra are a central +tool in critical applications such as software compilation or +verification [16], or invariant computation in the analysis of +dynamical systems [31]. In these applications, the computa- +tion of the vertices of a polyhedron is a core primitive, and +developing a formally proved algorithm that performs this +operation is a noteworthy goal. +1.2 +Contributions +We introduce an algorithm which computes the set of ver- +tices as well as the graph of polytopes, and we present its +implementation and proof of correctness in the proof as- +sistant Coq. The originality of our contribution lies in the +combination of simplicity of design and practical efficiency. +Simplicity is the key feature which makes the algorithm re- +alistically implementable and provable in a proof assistant. +Simplicity also has major advantages in terms of maintain- +ability, and portability to other proof assistants. +Our algorithm is based on certificates, i.e., it takes as input +a graph computed by some informal software, and formally +certifies that this graph is indeed correct. In more details, +the certificate is the graph of a perturbation of the polytope +that has remarkable properties such as being connected and +regular. Thanks to these properties, the certification consists +of a sequence of elementary steps, involving basic operations +such as matrix multiplications, set operations, etc. The graph +of the original polytope can be then deduced as the image of +the former graph by some mapping. +The algorithm is first implemented and proved correct by +using some proof-oriented types, e.g., dependent types rep- +resenting matrices, convex polyhedra, etc. However, these +types would have a prohibitive cost for computations: they +have been defined to ease the formal development of mathe- +matical theories, prioritizing the use of naive data-structures +and algorithms over computationally well-behaving ones. +As a consequence, we implement a second algorithm over +computation-oriented types, e.g., persistent arrays, native +integers, arbitrary precision rationals. An additional bene- +fit of the elementary structure of the algorithm is to make +the equivalence proof between the low-level and high-level +implementations easier. +In order to demonstrate the practical efficiency of our +approach, we make experiments on several classes of poly- +topes of interest. We manage to compute the graph of the 20- +and 23-dimensional counterexamples of [36] to the Hirsch +conjecture, respectively with 36 425 and 73 224 vertices, and + +A Formal Disproof of the Hirsch Conjecture +364 250 and 842 076 edges, and we deduce a formal disproof +of the conjecture. We also study classic polytopes such as +hypercubes, cross-polytopes, and polars of cyclic polytopes. +The latter are known to maximize the number of vertices +for fixed dimension and number of facets, as stated by Mc- +Mullen’s Upper Bound Theorem [37]. +1.3 +Related Work +Computing the vertices of polytopes and polyhedra is a noto- +riously difficult problem. As the number 𝑣 of vertices can be +exponential in the number 𝑚 of defining inequalities and the +dimension 𝑛, this complexity has to be measured as a func- +tion of 𝑚, 𝑛 and 𝑣. The worst-case complexity of the problem +is not fully understood: it is an NP-hard enumeration prob- +lem for unbounded polyhedra [34], but its status is still open +for polytopes. The two main vertex enumeration algorithms +are the double description method [23, 38] and the reverse +search method [5, 6]. While the complexity of the double +description method cannot be easily bounded in terms of +𝑚, 𝑛 and 𝑣, the complexity of the reverse search method is +in 𝑂(poly(𝑚,𝑛) 𝑣) for “nondegenerate” representations of +polytopes (we refer to Section 3 for the definition). This falls +in the same complexity class as our algorithm, since the two +approaches rely on the enumeration of simplex bases. The +double description method and the reverse search method +have standard implementations [4, 22] that are widely used +in software dedicated to computational mathematics [24, 43] +as well as software verification [8]. +Computational proofs, or proofs by reflection, is nowa- +days a common technique in the Coq proof assistant. For +instance, it is used in the implementation of tactics that rely +on symbolic computations [29], and has been widely used +for the formal proof of the four color theorem [25]. Proofs by +reflection is also at the root of the small scale reflection (SSRe- +flect [26]) proof methodology, where one makes a pervasive +use of computation for solving goals that involve decidable +predicates. We refer to [11, 13, 32, 33] for other examples +of computational based proofs in the Coq proof assistant. +Proof by reflection has also been used in many other sys- +tems: Agda [44], Isabelle/HOL [12], Lean [7], PVS [45]. In +relation with polyhedral computations, Farkas’ certification +techniques motivated by application to static analysis have +appeared in [10, 21]. In contrast with our work, this only +cover polyhedral computations using inequalities, and not +the computation of vertices or vertex-edge graphs. We fi- +nally mention that an implementation of a simplex-based +satisfiability procedure has been done by [42] in the proof +assistant Isabelle. +Last, our paper makes use of program and data structure +refinement techniques, a proof methodology that consists +in transforming a high-level program or data structure to a +lower-level, more efficient one, while preserving the main +properties (program specification and/or data structure in- +variants). Refinement techniques have already been used +in the context of the Coq proof assistant, notably in Co- +qEAL [14, 19], a Coq framework for easing the definition +of data structure refinements. Our formalization of data re- +finements closely follows the approach of CoqEAL. The +difference is that, for a first experiment, we have chosen not +to exploit the automated deduction of equivalence proofs +provided by CoqEAL (based on Coq typeclasses). In this +way, we have a finer-grained control of the low-level imple- +mentation of our algorithm. We point out that the burden of +proving the equivalence proof by hand was limited thanks +to the simplicity of our algorithm. +1.4 +Organization of the Paper +In Section 2, we recall basic notions on polyhedra and intro- +duce notation. In Section 3, we define the certificate-based +algorithm computing the graph of a given polytope. Section 4 +is dedicated to the implementation of the latter algorithm +using computation-oriented data types, and deals with the +proof of correctness of this implementation. In Section 5, we +bring the formal disproof of the Hirsch Conjecture. Finally, +we report on the experiments of our approach on several +classes of polytopes in Section 6. +The source files of the present submission can be found in +the git repository of the Coq-Polyhedra library1. We often +refer to these source files in the paper. On top of Coq, we +rely on the MathComp library [27] as well as the finmap and +bignums libraries.2,3 +We note that this work is a (slightly) extended version +of the conference paper [1] published in the proceedings of +CPP’23. +2 +Preliminaries and Notation +As discussed in the introduction, a (convex) polyhedron is +defined as the set of points 𝑥 ∈ R𝑛 satisfying finitely many +affine inequalities ⟨𝑎𝑖,𝑥⟩ ≥ 𝑏𝑖, where 𝑎1, . . . ,𝑎𝑚 ∈ R𝑛 and +𝑏1, . . . ,𝑏𝑚 ∈ R. We shall also write the system of constraints +as 𝐴𝑥 ≥ 𝑏, where 𝐴 ∈ R𝑚×𝑛 is the matrix with rows 𝑎𝑇 +𝑖 , 𝑏 = +(𝑏𝑖)𝑖 ∈ R𝑚, and ≥ denotes the entrywise ordering over R𝑚. +A notable subclass of polyhedra are polytopes. A polytope is +the convex hull +� �𝑝 +𝑖=1 𝜆𝑖𝑣𝑖 : 𝜆1, . . . , 𝜆𝑝 ≥ 0, �𝑝 +𝑖=1 𝜆𝑖 = 1 +� of +finitely many points 𝑣1, . . . , 𝑣𝑝 ∈ R𝑛. Minkowski Theorem +states that polytopes are precisely the bounded polyhedra. +The dimension of a polyhedron is defined as the dimen- +sion of its affine hull, i.e., the smallest (inclusionwise) affine +subspace containing the polyhedron. For instance, a point +has dimension 0, a line segment has dimension 1, etc. Let P +be a polyhedron. A (nonempty) face of P is the set of points +minimizing some linear function 𝑥 ↦→ ⟨𝑐,𝑥⟩ over P, where +𝑐 ∈ R𝑛. Equivalently, a face is the set of optimal solutions +of some linear program over P. The vertices and the edges +1https://github.com/Coq-Polyhedra/Coq-Polyhedra/tree/CPP-23 +2https://github.com/math-comp/finmap +3https://github.com/coq-community/bignums + +Xavier Allamigeon, Quentin Canu, and Pierre-Yves Strub +•𝑥 +𝑦 +𝑧 +𝑐1 +𝑐2 +Figure 2. The 3-dimensional cross polytope. The objective +vector 𝑐1 is minimized by only one vertex 𝑥, while objective +vector 𝑐2 is minimized by an edge [𝑦,𝑧] with 𝑦 and 𝑧. +of P are the faces of dimension 0 and 1 respectively; see +Figure 2. Any bounded edge writes as the line segment be- +tween two vertices 𝑣, 𝑣 ′ of P. In this case, the vertices 𝑣 and +𝑣 ′ are said to be adjacent. As described in the introduction, +the graph of P, denoted by 𝐺vert(P) (or simply 𝐺vert when +clear from context), is the combinatorial graph induced by +the adjacency relation over the vertices. +In the rest of the paper, we define [𝑝] � {1, . . . , 𝑝} for +all integer 𝑝 ≥ 1. The cardinality of a finite set 𝑆 is denoted +by #𝑆. Given a matrix 𝑀 ∈ R𝑝×𝑞, we denote by 𝑀𝑖 its 𝑖th +row (𝑖 ∈ [𝑝]), and, for all subset 𝐼 ⊂ [𝑝], by 𝑀𝐼 ∈ R#𝐼×𝑞 +the submatrix with rows 𝑀𝑖 for 𝑖 ∈ 𝐼. The identity matrix +of size 𝑝 × 𝑝 is denoted by Id𝑝. Given a (nonoriented) graph +𝐺 = (𝑉, 𝐸) (where 𝐸 ⊂ 𝑉 ×𝑉 ) and a vertex 𝑣 ∈ 𝑉 , we denote +by 𝑁𝐺 (𝑣) � {𝑤 ∈ 𝑉 : (𝑣,𝑤) ∈ 𝐸} the neighborhood of 𝑣 in +𝐺, i.e., the set of vertices 𝑤 adjacent to 𝑣. Given a function +𝑓 : 𝑉 → 𝑉 ′, the image of 𝐺 by 𝑓 , denoted by 𝑓 (𝐺), is defined +as the graph with vertices 𝑓 (𝑉 ) � {𝑓 (𝑣) : 𝑣 ∈ 𝑉 } and edges +� +(𝑓 (𝑣), 𝑓 (𝑤)) : (𝑣,𝑤) ∈ 𝐸 , 𝑓 (𝑣) ≠ 𝑓 (𝑤) +� +. +3 +Graph Certification Algorithm +3.1 +A First Algorithm for the Nondegenerate Setting +Our approach heavily relies on the notion of bases and ba- +sic points, which are the central ingredients of the simplex +method. Let P = {𝑥 ∈ R𝑛 : 𝐴𝑥 ≥ 𝑏} a polyhedron, where +𝐴 ∈ R𝑚×𝑛 and 𝑏 ∈ R𝑚. We recall that a basis is a subset +𝐼 ⊂ [𝑚] of cardinality 𝑛 such that the submatrix 𝐴𝐼 is non- +singular. In this case, the equality system 𝐴𝐼𝑥 = 𝑏𝐼 has a +unique solution 𝑥𝐼, which we call the basic point associated +with 𝐼. When 𝑥𝐼 belongs to the polyhedron P, the basis 𝐼 is +said to be feasible. By extension, the point 𝑥𝐼 is said to be a +feasible basic point. +It is a standard property that the vertices of P are precisely +the feasible basic points; see [40, Chapter 11]. However, in +general, the correspondence between them is not bijective: +every feasible basic point is a vertex, but a vertex may be the +basic point associated with more than one feasible basis; see +Figure 3 for an illustration. These bases are said to be degen- +erate. We say that we are in the nondegenerate setting when +2 +4 +1 +3 +•𝑧 + + +1 +𝑥1 + 𝑥2 + 𝑥3 +≥ −1 +2 +− 𝑥1 + 𝑥2 + 𝑥3 +≥ −1 +3 +𝑥1 − 𝑥2 + 𝑥3 +≥ −1 +4 +− 𝑥1 − 𝑥2 + 𝑥3 +≥ −1 +5 +𝑥1 + 𝑥2 − 𝑥3 +≥ −1 +6 +− 𝑥1 + 𝑥2 − 𝑥3 +≥ −1 +7 +𝑥1 − 𝑥2 − 𝑥3 +≥ −1 +8 +− 𝑥1 − 𝑥2 − 𝑥3 +≥ −1 +Figure 3. The 3-dimensional cross polytope has degenerate +bases. For instance, the bottom point 𝑧 (𝑥3 = −1) is a vertex +associated with four feasible bases: (2, 3, 4), (1, 3, 4), (1, 2, 4) +and (1, 2, 3). +there are no such bases, i.e., the correspondence between +feasible bases and vertices is one-to-one. +Given a linear program of the form +Minimize +⟨𝑐,𝑥⟩ +subject to +𝐴𝑥 ≥ 𝑏 , 𝑥 ∈ R𝑛 , +(1) +where 𝑐 ∈ R𝑛, the simplex method iterates over feasible +bases up to reaching a (feasible) basic point that minimizes +the function 𝑥 ↦→ ⟨𝑐,𝑥⟩. In this scheme, any two consecutive +bases 𝐼, 𝐼 ′ satisfy #(𝐼 ∩ 𝐼 ′) = 𝑛 − 1, i.e., they only differ +by one element. Such bases are said to be adjacent. This +adjacency relation gives rise to the graph of (feasible) bases, +that we denote by 𝐺bases. The relation between basic points +and vertices extends to the graph of the polyhedron and the +graph of bases as follows: +Proposition 2. The vertex-edge graph 𝐺vert is the image of +the graph 𝐺bases by the function 𝐼 ↦→ 𝑥𝐼 which maps any +feasible basis to its basic point. +Moreover, in the nondegenerate setting, the latter function +is an isomorphism between the two graphs. +This result elaborates on the geometric description of the +simplex that we made in the introduction, i.e., the simplex +method induces a path in the vertex-edge graph of the poly- +hedron. +In this section, we discuss the computation of the vertex- +edge graphs of polytopes in the nondegenerate case only. +The exposition of this special case is done to facilitate the +understanding of the general algorithm presented in Sec- +tion 3.2. We remark that, unless explicitly stated, we did not +have to formalize the results presented below. +In light of the second part of Proposition 2, we exploit +the isomorphism between 𝐺vert and 𝐺bases, and we sketch an +algorithm certifying that a given graph (computed a priori +by some informal procedure) coincides with the graph of +feasible bases. To this purpose, we exploit the following two +fundamental properties: +Proposition 3. The graph 𝐺bases is connected. + +A Formal Disproof of the Hirsch Conjecture +Algorithm 1 Graph certification algorithm (nondegenerate +setting and P = {𝑥 ∈ R𝑛 : 𝐴𝑥 ≥ 𝑏} bounded) +Require: 𝐴 ∈ R𝑚×𝑛, 𝑏 ∈ R𝑚 and 𝐺 = (𝑉, 𝐸) +1: assert 𝐺 is nonempty +2: for all 𝐼 ∈ 𝑉 do assert 𝐼 is a feasible basis +3: for all 𝐼 ∈ 𝑉 do +4: +for all 𝐽 ∈ 𝑁𝐺 (𝐼) do assert #(𝐼 ∩ 𝐽) = 𝑛 − 1 +5: +assert #𝑁𝐺 (𝐼) = 𝑛 +6: done +7: return true +Proposition 4. In the nondegenerate setting, and when P +is a polytope, the graph 𝐺bases is 𝑛-regular, i.e., every feasible +basis is adjacent to precisely 𝑛 feasible bases. +Proposition 3 holds in a general setting (even with de- +generate bases). It is a consequence of the fact that, for any +feasible basis 𝐼★, we can find a vector 𝑐 ∈ R𝑛 such that 𝑥𝐼★ is +the unique optimal solution of the linear program (1). Then, +the simplex method initialized with any feasible basis 𝐼 draws +a path to 𝐼★ in the graph 𝐺bases. +The algorithm covering the nondegenerate setting and +the assumption that P is a polytope is Algorithm 1. We use +the command assert as some syntactic sugar for the +block if not then return false. The vertices of the +input graph 𝐺 = (𝑉, 𝐸) are supposed to be sets 𝐼 of integers. +The algorithm consists in four steps: +(i) check that the graph is nonempty (Line 1); +(ii) check that every vertex 𝐼 is a feasible basis (Line 2); +(iii) for each vertex 𝐼 ∈ 𝑉 , check that its neighborhood +consists of adjacent bases (Line 4); +(iv) for each vertex 𝐼 ∈ 𝑉 , check that its neighborhood has +cardinality 𝑛 (Line 5). +The second and third steps ensures that 𝐺 is a subgraph of +𝐺bases. Moreover, as 𝐺bases is 𝑛-regular (Proposition 4), the +fourth step actually ensures that 𝑁𝐺 (𝐼) = 𝑁𝐺bases(𝐼) for all +𝐼 ∈ 𝑉 . In consequence, 𝐺 is a subgraph of 𝐺bases such that +the neighborhood of every vertex in 𝐺 agrees with that in +𝐺bases. As shown in the following lemma, the nonemptiness +of 𝐺 and the connectedness of 𝐺bases then ensure that 𝐺 and +𝐺bases are identical: +Lemma 5. Let𝐺 and 𝐻 two graphs such that𝐺 is a nonempty +subgraph of 𝐻. Suppose that 𝐻 is connected, and 𝑁𝐺 (𝑣) = +𝑁𝐻 (𝑣) for all vertices 𝑣 of 𝐺. Then 𝐺 = 𝐻. +This lemma is given by Lemma sub_gisof (see high_graph. +v) in the source of the project. (This result actually shows a +slightly more general though equivalent statement, where +𝐺 is replaced by an isomorphic graph.) The next result then +follows from Lemma 5, and shows that Algorithm 1 is correct: +Theorem 6. Suppose that we are in the nondegenerate setting, +and that P = {𝑥 ∈ R𝑛 : 𝐴𝑥 ≥ 𝑏} is a polytope. If Algorithm 1 +returns true, then 𝐺 = 𝐺bases. +• +• +• +• +Figure 4. The lexicographic perturbation approach. The top +vertex on the left-hand side is split into three distinct vertices +on the right-hand side. +The reader can also verify that if one of the assertions in +Algorithm 1 fails, then the graph 𝐺 cannot be equal to 𝐺bases. +3.2 +Dealing with the General Case +3.2.1 +A Perturbation Approach. The nondegenerate set- +ting does not hold in general, but we can reduce to it by +slightly perturbing the polytope, while still keeping a way to +recover the main combinatorial structure such as the set of +vertices or the vertex-edge graph. Our approach originates +from the so-called lexicographic pivot rule introduced by +Dantzig et al. [18], and later exploited by Avis [5] for his ver- +tex enumeration algorithm [4]. It consists in perturbing the +vector 𝑏 by replacing each entry 𝑏𝑖 (𝑖 ∈ [𝑚]) by the quantity +𝑏𝑖 − 𝜀𝑖, where 𝜀 > 0 is a sufficiently small real. Geometri- +cally, while the normal vectors to the hyperplanes delimiting +the polyhedron are still the same, the perturbation of the +vector 𝑏 breaks every vertex associated with several (degen- +erate) bases into distinct (and nondegenerate) basic points; +see Figure 4 for an illustration. +Instead of instantiating 𝜀 by some numerical values (which +would raise the problem of determining how small it should +be), the perturbation is achieved in a symbolic way, by think- +ing of each 𝑏𝑖 as a polynomial in 𝜀 of degree at most 𝑚. +This leads to considering a “polyhedron” where the entries +of the points are polynomials in 𝜀 of degree at most 𝑚 as +well. Such polynomials �𝑚 +𝑘=0 𝛼𝑘𝜀𝑘 can be encoded as row +vectors (𝛼0, . . . , 𝛼𝑚) of size 1 + 𝑚. In this case, the standard +order over the reals is replaced by the lexicographic order +≤lex over (1 + 𝑚)-tuples. Indeed, we have (𝛼0, . . . , 𝛼𝑚) ≤lex +(𝛽0, . . . , 𝛽𝑚) if and only if �𝑚 +𝑘=0 𝛼𝑘𝜀𝑘 ≤ �𝑚 +𝑖=0 𝛽𝑘𝜀𝑘 for all +0 < 𝜀 ≪ 1. This gives rise to the symbolically perturbed +polyhedron +�P � +� +𝑋 ∈ R𝑛×(1+𝑚) : 𝐴𝑋 ≥lex �𝑏 +� +(2) +where the matrix �𝑏 � +�𝑏 +−Id𝑚 +� +∈ R𝑚×(1+𝑚) corresponds +to the perturbation of the vector 𝑏 described above: the 𝑖th +row of �𝑏 encodes the polynomial 𝑏𝑖 − 𝜀𝑖. In (2), the relation +≥lex stands for the entrywise extension of the lexicographic +order: two matrices 𝑋,𝑌 ∈ R𝑝×(1+𝑚) satisfies 𝑋 ≥lex 𝑌 if +𝑋𝑖 ≥lex 𝑌𝑖 for all 𝑖 ∈ [𝑝]. The matrices 𝑋 ∈ R𝑛×(1+𝑚) in �P +correspond to vectors with perturbed entries (encoded as +polynomials in 𝜀 of degree at most 𝑚). + +Xavier Allamigeon, Quentin Canu, and Pierre-Yves Strub +The notion of feasible bases still makes sense in case of +such symbolically polyhedra. To avoid any confusion with +the bases of the original polyhedron, the feasible bases of �P +are referred to as lex-feasible bases. Formally, a set 𝐼 ⊂ [𝑚] is +a lex-feasible basis if 𝐼 has cardinality 𝑛, the matrix 𝐴𝐼 is non- +singular, and the basic point 𝑋 𝐼 � 𝐴−1 +𝐼 �𝑏𝐼 satisfies 𝐴𝑋 𝐼 ≥lex �𝑏. +We recall that lex-feasible bases are not degenerate: +Lemma 7 (see [2] for a formalization). Let 𝐼, 𝐼 ′ be two distinct +lex-feasible bases. Then 𝑋 𝐼 ≠ 𝑋 𝐼′. +3.2.2 +Formalizing the Properties of the Lex-Graph. In +what follows, we assume that P is a polytope. +The adjacency relation defined over feasible bases carries +over to lex-feasible bases in a straightforward way. This +induces the graph of lex-feasible bases, or lex-graph for short, +that we denote by 𝐺lex. +Lex-feasible bases form the cornerstone of the formaliza- +tion of the simplex method done by Allamigeon and Katz [2], +and provided in the library Coq-Polyhedra. In more details, +the latter work formalized the lex-simplex method, which +iterates over lex-feasible bases in order to avoid cycling +on degenerate bases. Coq-Polyhedra introduces the type +lex_feasible_basis A b of lex-feasible bases, where A : ' +M_(m,n) and b : 'cV_m correspond to the matrix 𝐴 and the +(unperturbed) vector 𝑏. (We recall that 'M_(m,n) and 'cV_m +are respectively the types of m × n-matrices and m-vectors +provided by the library MathComp [27].) We build on this +and start by defining the graph of lex-feasible bases, denoted +lex_graph: +Definition set_adjacence := (* adjacency relation *) +fun I I' : {set 'I_m} => +#| I :&: I' | == n.-1. +Definition lex_graph := mk_graph +[fset x | x : Simplex.lex_feasible_basis A b] +set_adjacence. +We then prove that, as expected, the properties of Proposi- +tions 3 and 4 hold in the case of lex-feasible bases: +Lemma lex_graph_connected : connected lex_graph. +Lemma lex_graph_regular : regular lex_graph n. +Their proofs are straightforward consequences of the formal- +ization of the lex-simplex method provided in Coq-Polyhedra. +One fundamental property of lex-feasible bases is that +they constitute a subset of the feasible bases of P. In more +details, we denote by 𝜋 the function which maps a matrix +𝑋 ∈ R𝑛×(1+𝑚) (i.e., a perturbed point) to its first column. The +latter corresponds to the unperturbed part of 𝑋, i.e., the value +of the perturbed point when 𝜀 = 0. The following result is +folklore (we refer to [2] for the formalization): +Proposition 8. Let 𝐼 be a lex-feasible basis. Then 𝐼 is a feasi- +ble basis of P, and 𝜋(𝑋 𝐼) is the associated basic point. +We extend the latter result to the following correspon- +dence between the lex-graph and the vertex-edge graph. +Theorem 9. The vertex-edge graph 𝐺vert of P is the image +of 𝐺lex by the function 𝜙 : 𝐼 ↦→ 𝜋(𝑋 𝐼). +In Coq, this statement is written as follows (see Module +enum_proof.v): +Theorem im_lex_graph_vert_graph : +poly_graph P = +(Simplex.point_of_basis b) @/ lex_graph. +where the term poly_graph P is the vertex-edge graph 𝐺vert, +the function Simplex.point_of_basis b corresponds to the +function 𝜙, and a term of the form f @/ G stands for the +image of a graph G by the function f. +We briefly comment on the formal proof of Theorem 9 +since, most often, its informal proof is not detailed in the +literature. The hardest part of the proof is to show that for +every edge [𝑣,𝑤] of P, there exist two adjacent lex-feasible +bases 𝐼, 𝐽 such that 𝑣 = 𝜙(𝐼) and 𝑤 = 𝜙(𝐽). We construct +these two bases by exploiting the lex-simplex method of +Coq-Polyhedra. More precisely, since [𝑣,𝑤] is an edge of P, +there exists a vector 𝑐 such that [𝑣,𝑤] is precisely the set +of points of P minimizing the function 𝑥 ↦→ ⟨𝑐,𝑥⟩. Calling +the lex-simplex method with the objective vector 𝑐 provides +a lex-feasible basis 𝐼0 such that the point 𝜙(𝐼0) reaches this +minimal value. Since 𝜙(𝐼0) is a vertex of P, this should be +either 𝑣 or 𝑤. Without loss of generality, we assume that +𝜙(𝐼0) = 𝑣. We now consider an objective vector 𝑐′ such that +𝑤 is the only point minimizing 𝑥 ↦→ ⟨𝑐′,𝑥⟩ over P (this +is possible by definition of a vertex), and introduce a third +objective vector 𝑑 � 𝑐 + 𝛿𝑐′, where 𝛿 > 0 is a sufficiently +small quantity. Intuitively, perturbing 𝑐 into 𝑑 in this way +should ensure that 𝑤 is the unique minimizer of 𝑥 ↦→ ⟨𝑑,𝑥⟩ +over P, and that 𝑣 is the second “best” vertex after 𝑤, i.e., +⟨𝑑,𝑤⟩ < ⟨𝑑, 𝑣⟩ < ⟨𝑑,𝑧⟩ for every vertex 𝑧 ∉ {𝑣,𝑤}.4 We +finally apply the lex-simplex method with objective function +𝑥 ↦→ ⟨𝑑,𝑥⟩, starting from the lex-feasible basis 𝐼0. Since the +objective function cannot increase along the way, the lex- +simplex method generates a sequence 𝐼0, . . . , 𝐼𝑝−1, 𝐼𝑝, . . . of +adjacent lex-feasible bases such that 𝜙(𝐼𝑘) = 𝑣 for all 𝑘 < 𝑝 +and 𝜙(𝐼𝑝) = 𝑤. Then, it suffices to take 𝐼 = 𝐼𝑝−1 and 𝐽 = 𝐼𝑝. +3.2.3 +Certification of the Lex-Graph and the Vertex- +Edge Graph. In light of the properties formalized in Lemma +lex_graph_connected and Lemma lex_graph_regular, we can +derive from Algorithm 1 a method certifying that an infor- +mally computed graph coincides with the lex-graph. Then, +Theorem 9 will allow us to recover the vertex-edge graph of +P from the latter by computing its image by 𝜙. +We describe the formalization of the certification proce- +dure for 𝐺lex. It takes as input a graph G whose vertices are +pairs of the form (I, X), where I : {set 'I_m} is a subset of +integers (less than 𝑚), and X : 'M_(n,1+m) is a (𝑛 × (1 +𝑚))- +matrix. These two components are intended to represent +a lex-feasible basis 𝐼 and the corresponding basic point 𝑋 𝐼 +4These relations actually imply how small 𝛿 needs to be chosen. + +A Formal Disproof of the Hirsch Conjecture +respectively. The algorithm is formalized as a program re- +turning a Boolean value (reminding that &&, [&& ...] and +[forall ..., ...] stand for Boolean conjunctions): +Definition high_enum_algo G : bool := +(G != graph0) (* G is nonempty *) && +[forall u : vertices G, (* u is a pair (I,X) *) +[&& card_verification u, +bas_verification u, +feas_verification u, +reg_verification u & +subset_verification u +] +]. +where we define +Definition card_verification u := #|u.1| == n. +Definition bas_verification u := +(row_submx A u.1) *m u.2 == row_submx b_pert u.1. +Definition feas_verification u := +u.2 \in Simplex.lex_polyhedron A b_pert. +Definition reg_verification u := +#|` successors G u| == n. +Definition subset_verification u := +[forall u' : successors G u, +set_adjacence u.1 u'.1]. +We recall that u.1 : T and u.2 : T' respectively correspond +to the first and second components of a pair u : T * T'. The +implementation of high_enum_algo follows the structure of +Algorithm 1. In particular, it starts by checking that the graph +G is nonempty, and then performs five consecutive tests on +every vertex u. The last two tests apply to the neighborhood +successors G u of u in G, and respectively check that it has +cardinality 𝑛 (reg_verification) and consists of adjacent +bases (subset_verification). The main difference with Al- +gorithm 1 is the way the first component 𝐼 of every vertex u +in G is verified to be a lex-feasible basis. This is the purpose +of the first three tests card_verification, bas_verification +and feas_verification. The first one checks that 𝐼 has cardi- +nality 𝑛, while the last two ones respectively make sure that +𝐴𝐼𝑋 = �𝑏𝐼 and 𝐴𝑋 ≥lex �𝑏. We note that, since �𝑏 = +�𝑏 +−Id𝑚 +� +, +the equality 𝐴𝐼𝑋 = �𝑏𝐼 ensures that 𝐴𝐼 is a nonsingular ma- +trix. Indeed, if 𝑌 is the submatrix of 𝑋 formed by its (1 +𝑖)th +columns for 𝑖 ∈ 𝐼, it can be verified that 𝐴𝐼𝑌 = −Id𝑛. In other +words, the matrix 𝑋 already carries a certificate that 𝐴𝐼 is +nonsingular. As a consequence, 𝐼 is a basis. Since 𝐴𝐼𝑋 = �𝑏𝐼 +then 𝑋 = 𝑋 𝐼, and the condition 𝐴𝑋 ≥lex �𝑏 finally ensures +that 𝐼 is a lex-feasible basis. +This is how we arrive at the proof of the correctness of +high_enum_algo: +Theorem repr_lex_graph G : +high_enum_algo G -> +gisof G lex_graph (fun u => u.1). +The gisof predicate in the rightmost part of the implica- +tion means that the function (𝐼,𝑋) ↦→ 𝐼 is an isomorphism +between G and the lex-graph. +The following statement is obtained by combining Theorem +im_lex_graph_vertex_graph and Theorem repr_lex_graph: +Theorem repr_poly_graph G : +high_enum_algo G -> poly_graph P = (phi @/ G). +where the function phi is defined as phi u = col 0 u.2, i.e., +if u = (I, X), then phi u is the first column of the matrix X. +4 +Efficient Implementation +The Coq function high_enum_algo introduced in Section 3 +works with dependent types (e.g., MathComp types) that +are adapted for proof but not for computation. For example, +natural numbers are expressed in unary form, and rationals +carry proof terms of the fact that their numerator and denom- +inator are coprime. Similarly, the implementations of finite +sets, graphs or matrices are based on MathComp sequences +(i.e., basic lists built by induction) that are not made for fast +computations, and are provided with multiple proof terms +for their well-formedness. As a consequence, the function +high_enum_algo cannot return within a reasonable amount +of time even on the simplest instances. To overcome this +problem, we exploit data types that are closer to machine +representations and thus practically more efficient. Based on +these, we implement a “low-level” version of the function +high_enum_algo; see Section 4.1. In Section 4.2, we describe +how we relate high-level data structures with low-level ones +by combining refinements. Finally, in Section 4.3, we deal +with the proof of equivalence of the low-level implementa- +tion of the function high_enum_algo. +4.1 +Low-Level Implementation +The main data types used in the low-level implementation of +the certification algorithm are the following: (i) the type int +of 63-bits integers (module Int63 in Coq) built on OCaml +integers; (ii) the type array of persistent arrays (module +PArray in Coq) built on OCaml arrays and based on the ideas +of [15, Section 2]; (iii) the type bigQ that represents arbitrarily +large rationals (module BigQ of Coq library bignums) built +on sequences of words based on 63-bits integers [30]. +The type bigQ is useful to manipulate polyhedra in which +the numerical entries of the inequality system or of the ver- +tices can be very large rationals, as in the counterexamples +to the Hirsch conjectures that we deal with in Section 5. As +we describe next, persistent arrays are involved in various +data structures in the low-level implementation. We point +out that we choose persistent arrays over Coq AVL trees, +because our early experiments have shown that the latter +suffer performance issues. Indeed, every tree comes with a +proof term for balancing, and this term can grow excessively +on large instances. + +Xavier Allamigeon, Quentin Canu, and Pierre-Yves Strub +Vectors and matrices with rational entries are implemented +using the types array bigQ and array (array bigQ) respec- +tively. A system of inequalities defining a polyhedron is then +represented by a term of type polyType := array (array +bigQ) * array bigQ. Bases, which are sets of row indices, +are encoded with the type array int. More precisely, the el- +ements of a basis are collected in a sorted array. This allows +us to compute the intersection of two bases in linear time. +Graphs whose vertices are labeled with some type t, i.e., +the low-level counterpart of graphs of type graph t, are +implemented using pairs of the form (g, lbl), where g : +array (array int) and lbl : array t. In more details, the +vertices of a low-level graph are indexed by integers (of type +int), and the term lbl.[i] corresponds to the label of the +vertex of index i. The term g represents the adjacency array +of the graph, i.e., g.[i] is the array containing the indices +of the neighbors of the vertex of index i. We use the nota- +tion graph_struct := array (array int). The indexing of +vertices is made in such a way that the labels in the array +lbl are sorted in nondecreasing order (to this extent, we +introduce a total order relation over labels). This ensures +that every label occurs only once in the graph. +Using these data structures, we build a low-level imple- +mentation, called enum_algo, of the function high_enum_algo; +see Module enum_algo.v. The function enum_algo is a trans- +parent adaptation of high_enum_algo on low-level data struc- +tures. In particular, every test in high_enum_algo has a coun- +terpart on the low-level side. Unlike high-level data struc- +tures based on dependent types, our low-level data structures +do not come with well-formedness invariants for free. In- +stead, these invariants have to be checked by extra functions. +For instance, we need to verify that arrays representing vec- +tors and matrices have a specific size (e.g., the dimension of +the ambient space, or the number of inequalities), that all +the vertex indices appearing in a graph belong to the right +range, and that arrays representing sets (such as bases or +graph labels) are sorted. The advantage of our certification +algorithm is that we do not need to prove that such invari- +ants are preserved throughout the function. Indeed, owing +to the simplicity of the algorithm, data structures are only +accessed for reading, and no new structure is produced. In +consequence, checking the consistency of data structures +occurs only once, before the call to the function enum_algo. +Once enum_algo has verified that a low-level graph is the +(low-level representation of) the lex-graph of the polytope, it +remains to deal with the low-level computation of the image +of the lex-graph by the function phi defined in Theorem +repr_poly_graph (cf. end of Section 3.2.3). This is the final +step to get the vertex-edge graph of the polytope. While +the image of a graph is a basic construction for high-level +graphs, the problem is slightly more involved on the low- +level side. Once again, we rely on certificates. In more details, +we define a function img_lex_graph that takes as input two +graphs g_lex g_vert : graph_struct and their respective +labelings lbl_lex and lbl_vert, and checks that (g_vert, +lbl_vert) is the image of (g_lex, lbl_lex) by the low-level +counterpart low_phi := fun u => (u.2).[0] of the function +phi. We give a short description of the implementation of +this function. +Additional certificates morph, morph_inv and edge_inv are +provided to the function img_lex_graph. The term morph : +array int corresponds to a mapping between the indices of +the vertices of the two graphs. The function img_lex_graph +first checks that this mapping is consistent with the function +low_phi over labels, in the sense that, for every index i of +g_lex, low_phi lbl_lex.[i] is equal to lbl_vert.[morph.[i +]]. The remaining part of img_lex_graph consists in verifying +that morph induces a graph morphism between the two inci- +dence structures described by g_lex and g_vert. To this pur- +pose, the algorithm checks that morph.[i] < length g_vert +for all indices i, which ensures morph to be well-formed. Then, +it makes sure that the mapping morph is surjective. This is +done by exploiting the certificate morph_inv : array int in- +tended to be a right-inverse of morph, and by checking that for +all indices i of g_vert, we have morph.[morph_inv.[i]] == i. +The algorithm then proceed with edges. It checks that if i j : +int are adjacent in g_lex, then morph.[i] and morph.[j] are +adjacent in g_vert as well, unless morph.[i] == morph.[j] +(by definition of the image). Conversely, it exploits the third +certificate edge_inv to verify for any two adjacent vertices i', +j' of g_vert, there exist two adjacent vertices i, j of g_lex +such that morph.[i'] == i and morph.[j'] == j.5 +4.2 +Data Refinements +In order to prove that the low-level implementations are +correct w.r.t. the high-level ones, we follow the approach +introduced in the project CoqEAL [14, 19] and use data re- +finements. Given a high-level type T and the corresponding +low-level type t, a refinement r : t -> T -> Prop is a re- +lation between terms that respectively correspond to the +low-level and the high-level representation of a same object. +In this setting, two functions are equivalent if they return +related outputs given related inputs. This is formalized as +follows: +Definition rel_func +(r1 : t -> T -> Prop) (r2 : u -> U -> Prop) +(f : t -> u) (g : T -> U) := +(forall x y, r1 x y -> r2 (f x) (g y)). +We use the notation (r1 =~> r2) f g for rel_func r1 r2 f +g. +The tree in Figure 5 describes the combination of refine- +ment relations performed in order to relate low-level and +high-level representations of lex-graphs, i.e., graphs over +5We warn that the preimages of i' and j' provided by morph_inv may +not be adjacent in g_lex, which is why we need the additional certificate +edge_inv. + +A Formal Disproof of the Hirsch Conjecture +bigQ ∼ rat +array bigQ ∼ array rat +array (array bigQ) ∼ array (array rat) +array (array T) ∼ 'M[T]_(p,q) +array (array bigQ) ∼ 'M[rat]_(n,1+m) +rel_array +rel_array +rel_comp +int ∼ 'I_m +array int ∼ array 'I_m +array T ∼{set T} +array int ∼{set 'I_m} +rel_array +rel_comp +low_lbl := array int * array (array bigQ) ∼ high_lbl := {set 'I_m} * 'M[rat]_(n,1+m) +rel_couple +graph_struct * array low_lbl ∼ graph high_lbl +rel_graph_r +Figure 5. Construction of the refinement relation rel_lex_graph between the low-level and high-level representations of +lex-graphs. We make use of the notation t ∼ T to state that there is a refinement relation between the low-level type t and the +high-level type T. +pairs of bases and matrices. (The formalization of the rela- +tions is carried out in the module refinement.v.) We com- +ment on this tree. +The leaves of the tree correspond to refinement relations +between atomic types. For every such relation, we need to +prove that the implementations of basic operations over the +low-level and the high-level types are equivalent. For in- +stance, the refinement rel_int_ord : int -> 'I_m -> Prop +relates computationally efficient integers with MathComp +ordinals of type 'I_m (i.e., unary integers less than a fixed +integer m), and we prove the equivalence of between the +low-level and high-level ordering. This states as: +Lemma rel_int_ord_lt : +(rel_int_ord =~> rel_int_ord =~> eq) +(fun i j : int => (i (i < j)%nat) +where eq is the identity relation. Other basic refinements +include the relation rel_array_set : array T -> {set T} +-> Prop between the implementation of finite sets with ar- +rays and the corresponding MathComp type {set T}, or the +relation rel_mx_col : array (array T) -> 'M[T]_(p,q) -> +Prop between array-based matrices and MathComp matri- +ces of size p × q. Finally, we use the refinement relation +rat_bigQ : bigQ -> rat -> Prop implemented in CoqEAL +between the type bigQ of computationally efficient rationals +and the MathComp type rat of rationals. +The edges of the tree correspond to functors. The latter +allow us to combine refinements in order to construct more +complex relations. For example, the functor rel_array lifts +a refinement r : t -> T -> Prop to another between the +types array t and array T: +Definition rel_array r a A := +length a = length A +/\ forall i, i < length a -> r a.[i] A.[i]. +It comes with the proof of equivalence for several basic prim- +itives, e.g., computing the image arr_map f a of some array +a by some function f: +Lemma rel_array_map r r' f F : +(r =~> r') f F -> +(rel_array r =~> rel_array r') +(arr_map f) (arr_map F). +Similarly, the functor rel_couple combine two refinement re- +lations r : t -> T -> Prop and r' : t' -> T' -> Prop into +a refinement relation between the product types t * t' and +T * T'. A third functor rel_comp allows us to compose a +refinement relation between t and T from two refinement re- +lations r : t -> tT -> Prop and r' : tT -> T -> Prop. For +example, the relation between the type array (array bigQ) +of low-level rational matrices and the type 'M[rat]_n of high- +level square matrices is provided by composing the relation +between low-level matrices of bigQ and rat with the relation +rel_mx_col previously discussed instanced with T := rat. +As shown in Figure 5, the refinement relation between low- +level and high-level lex-graphs is obtained by applying the +functor rel_graph_r to the refinement relation between the +types low_lbl and high_lbl (that respectively correspond to +the low-level and high-level representations of lex-graph la- +bels). The functor rel_graph_r is described in Figure 6. Given +a relation between a low-level type t and a high-level type T, +it builds a refinement relation between the type graph_struct +* array t of low-level graphs labeled by t and high-level +graphs labeled by T. It is built by applying the functors pre- +viously described to two atomic refinement relations. The +first one, called rel_structure, relates the low-level type +graph_struct := array (array int) of incidence arrays to +the type graph int high-level graphs over (low-level) inte- +gers. The second one relates the type graph int * array T, +which corresponds to a mixed representation of graphs in + +Xavier Allamigeon, Quentin Canu, and Pierre-Yves Strub +t ∼ T +array t ∼ array T +graph_struct ∼ graph int +graph_struct * array t ∼ graph int * array T +graph int * array T ∼ graph T +graph_struct * array t ∼ graph T +rel_array +rel_couple +rel_comp +Figure 6. Construction of the functor rel_graph_r between low-level and high-level graphs, from a refinement relation t ∼ T. +which the incidence structure is stored separately from the +mapping to labels, to the type graph T of high-level graphs. +4.3 +Proof of Equivalence +We are now ready to prove the equivalence between the low- +level and high-level implementations of the lex-graph certifi- +cation algorithm, i.e., enum_algo and high_enum_algo respec- +tively; see Module enum_equiv.v. It involves the refinement +relation rel_poly between low-level and high-level repre- +sentations of the inequality system defining the polyhedron +(between polyType and the type 'M[rat]_(m,n) * 'cV[rat] +_m of pairs (A,b)), and the refinement relation rel_lex_graph +between low-level and high-level lex-graphs described in +Section 4.2. The equivalence is stated as follows: +Lemma lex_certif_equiv : +(rel_poly =~> rel_lex_graph =~> eq) +enum_algo high_enum_algo. +If we put aside straightforward program transformations, the +proof essentially consists in unfolding the definition of every +test in the functions enum_algo and high_enum_algo, and to +use the equivalence between the basic operations involved +on the low-level and high-level sides, such as lexicographic +comparison of row vectors, matrix-vector multiplication, or +set intersection. +Analogously, the correctness of the low-level function +img_lex_graph certifying the image of the lex-graph by the +function phi is given by the following statement: +Lemma img_lex_graph_equiv (...) : +rel_lex_graph (g_lex, lbl_lex) G_lex -> +rel_vert_graph (g_vert, lbl_vert) G_vert -> +img_lex_graph morph morph_inv edge_inv +g_lex lbl_lex g_vert lbl_vert -> +G_vert = phi @/ G_lex. +where rel_vert_graph is an instantiation of the rel_graph_r +functor for graphs labelled with vectors. (We point out that +this statement is not an equivalence because of the addi- +tional certificates morph, morph_inv and edge_inv provided to +img_lex_graph.) +In practice, the certificates provided to the algorithms +are written using low-level types. Indeed, as explained in +Section 6, these certificates can be very large, and it would +impossible to compile them using high-level types. In order +to exploit the correctness statements that we previously +discussed, every refinement relation rel_t_T : t -> T -> +Prop comes with an extra function spec_t_T : t -> T which +builds a high-level term from a well-formed low-level one, +so that we have r x (spec_t_T x). In this way, we arrive at +the following statement, which +Theorem Validation Po (...) : +(...) +enum_algo Po g_lex lbl_lex -> +img_lex_graph morph morph_inv edge_inv +g_lex lbl_lex g_vert lbl_vert -> +poly_graph (poly_of_syst (spec_poly Po)) = +spec_vert_graph (g_vert, lbl_vert). +It takes as input the low-level certificates as well as the +hypotheses that they are certified by the low-level imple- +mentation of the algorithms. It proves that, for the high-level +polyhedron poly_of_syst (spec_poly Po) described by the +low-level inequality system Po, the (high-level) vertex-edge +graph corresponds to the low-level graph (g_vert, lbl_vert +). The proof is a combination of the correctness of the high- +level algorithm (Theorem repr_poly_graph) with that of the +low-level implementations (i.e., Lemma lex_certif_equiv and +Lemma img_lex_graph_equiv). +5 +Formal Disproof of Hirsch Conjecture +The purpose of this section is to describe how we arrive +at the formal disproof of the Hirsch Conjecture using the +certification algorithms and their low-level implementations +described in Section 4. Recall that the Hirsch conjecture +makes use of three notions: the diameter of the vertex-edge +graph, the number of facets, and the dimension of the poly- +tope. We disprove the conjecture by computing lower bounds +on the diameter and the dimension and an upper bound on +the number of facets, to conclude by transitivity. We first +explain how we deal with the computation of each quantity. +Considering a polyhedron P = {𝑥 ∈ R𝑛 : 𝐴𝑥 ≥ 𝑏} where +𝐴 ∈ R𝑚×𝑛 and 𝑏 ∈ R𝑚, it is well known that the number +of facets is bounded by the number 𝑚 of inequalities. This +statement has been added to the properties of polyhedra +established in Coq-Polyhedra (see Module poly_base.v), and +is expressed as follows: +Lemma facets_le {base : base_t[R,n]}: + +A Formal Disproof of the Hirsch Conjecture +(#|` facets 'P(base) | <= #|` base|)%nat. +The term base correspond to a set of inequalities defining +the polyhedron, and 'P(base) is the term representing the +polyhedron. +To deal with the dimension, we establish that the (infor- +mal) counterexamples to the Hirsch conjecture have dimen- +sion equal to𝑛 (i.e., the dimension of the ambient space). This +is achieved by exhibiting a set of 𝑛+1 vertices 𝑣0, . . . , 𝑣𝑛 ∈ R𝑛 +of the polytope that are affinely independent, i.e., the matrix +𝑀 = +�𝑣1 − 𝑣0 +. . . +𝑣𝑛 − 𝑣0� is nonsingular. These points +are provided by means of (low-level) certificates map_lbl : +array int and origin : int which represent the set of their +indices in the low-level vertex-edge graph, along with an +informally computed low-level matrix inv_lbl that we verify +to be the inverse of the matrix 𝑀. On the low-level side, the +verification of these certificates is performed by the function +dim_full_test (in enum_algo.v), and the correctness state- +ment is expressed as follows (see enum_high_algo.v): +Lemma high_dim_h : +dim_full_test map_lbl origin inv_lbl -> +\pdim P = n.+1. +where \pdim P stands for the dimension of the polytope P +shifted by one (Coq-Polyhedra uses the convention that the +emptyset have dimension 0, points have dimension 1, etc). +Finally, the lower bound on the diameter is computed by +providing a vertex 𝑣 of the graph whose eccentricity, i.e., the +maximal distance between 𝑣 and any other vertex 𝑤, reaches +the value of the diameter. In order to compute the eccen- +tricity, we exploit the Breadth-First Search (BFS) algorithm, +and implement it on low-level and high-level data structures +in the module diameter.v. The low-level implementation is +provided by the function Low.BFS : graph_struct -> int +-> NArith.BinNat which takes as input a low-level incidence +array and the index of a vertex, and returns the maximal +length of shortest paths from it. Similarly, the high-level +implementation is the function High.BFS working with the +type graph T. The correspondence with the low-level imple- +mentation is expressed as follows: +Lemma rel_struct_BFS (g : graph_struct) (G : graph +int): +rel_structure g G -> +forall x, mem_vertex g x-> +Low.BFS g x = High.BFS G x :> nat. +We are now ready to formally disprove the conjecture, +by running our certification algorithms on the two coun- +terexamples to the Hirsch conjecture provided in [36]. This +is done in the directories test/data/poly20dim21 and test +/data/poly23dim24 respectively (we refer to the README file +to extract them from the zipped archives in test/archives +). We exploit the explicit inequality representations of the +counterexamples provided by Weibel.6 They take the form +6https://sites.google.com/site/christopheweibel/research/hirsch- +conjecture +of input files for the library lrslib [4]. We use this library +together with additional Python scripts to generate the in- +formal certificates for our algorithms (they are stored in +test/data/polyXXdimYY/coq/). As explained in Section 4.3, +the latter are provided using low-level types. We give a short +description of every certificate: +• poly.v contains the description of the polytope by +inequalities; +• g_lex.v and lbl_lex.v (resp. g_vert.v and lbl_vert. +v) are intended to represent the lex-graph (resp. the +vertex-edge graph); +• morph.v, morph_inv.v and edge_inv.v are the certifi- +cates provided to the function img_lex_graph (see Sec- +tion 4.1); +• map_lbl.v, origin.v and inv_lbl.v are the certificates +for the dimension required by dim_full_test; +• cert.v provides certificates to check that the input +polyhedron is a polytope. These certificates are used +to show that nonnegative combinations of the inequal- +ities defining the polyhedron yield inequalities of the +form −𝐾 ≤ 𝑥𝑖 ≤ 𝐾 for all 𝑖 ∈ [𝑛], where 𝐾 is a suffi- +ciently large constant. The verification is performed by +the function bounded_Po_test (module enum_algo.v). +• start.v is the index of vertex that is used to get a lower +bound on the diameter. +Each file is compiled using coqc, and then imported in order +to get the formal disproof of the conjecture in test/data/ +polyXXdimYY/coq_Hirsch/Hirsch.v: +Theorem Hirsch_was_wrong : +exists (d : nat) (P : 'poly[rat]_d), +(High.diameter (poly_graph P) +> #|`facets P| - (\pdim P).-1)%nat. +Proof. +pose P := poly_of_syst (A, b)). +exists n'.+1, P. +apply/disprove_Hirsch. +- exact: well_formedness_ok. +- exact: enum_algo_ok. +- exact: img_graph_ok. +- exact: bounded_Po_test_ok. +- exact: dim_full_test_ok. +- exact: diameter_check_ok. +Qed. +(Note that the statement use (pdim P).-1 for the dimension +because of the shift-by-one convention in Coq-Polyhedra.) +The proof starts by exhibiting the witness of the existential +statements, i.e., the polyhedron provided in poly.v. It then +applies Theorem disprove_Hirsch which establishes that the +conjecture does not hold provided that all certification tests +return true (see file theories/enum_equiv.v). These latter +hypotheses are then verified in the last six lines of the proof. +They respectively correspond to the verification of (i) the +well-formedness of all input certificates, (ii) the lex-graph, + +Xavier Allamigeon, Quentin Canu, and Pierre-Yves Strub +(iii) the vertex-edge graph, (iv) the boundedness of the poly- +hedron, (v) the dimension of the polyhedron, (vi) the lower +bound on the diameter. Every test is achieved by using the +tactic vm_compute of Coq, which computes the (Boolean) re- +sult of each test. +As explained in the introduction, the approach of [39], also +used in [36], builds non-Hirsch polytopes by lifting special +low-dimensional spindles to higher dimension. This only +provides a lower bound on the diameter of the non-Hirsch +polytopes; see [39, Th. 1.5]. In addition to verifying the lower +bound, we formally certify the exact value of the diameter +of the 20- and 23-dimensional counterexamples of [36]: +Theorem poly20dim21_diameter : +diameter (poly_graph poly20dim21) = 21%nat. +Theorem poly23dim24_diameter : +diameter (poly_graph poly23dim24) = 24%nat. +Since the two polytopes respectively have 40 and 46 facets, +this entails that their diameter matches the lower bound. As +far as we know, this is the first proof of this fact. +6 +Practical Experiments +For the sake of reproducibility, experiments have been con- +ducted on two different architectures: (i) a machine with an +Apple M1 processor and 32 GB RAM running Mac OS 11.0.1 +(Architecture A), (ii) a machine with a 2.3 GHz Intel Core pro- +cessor and 64 GB RAM running Linux 5.4 (Architecture B). +Both use Coq 8.16.1. +As described in the introduction, the two counterexam- +ples of [36] are very involved polytopes, with several tens of +thousands vertices and hundreds of thousands edges. More- +over, the vertices have long rational coefficients, with up +to 40 digits long numerator and denominator. As a conse- +quence, explicitly writing the certificates in plain files lead +to very large .v files whose compilation is a challenging +(if not unrealistic) task for Coq. For instance, for the 23- +dimensional counterexample, we need about 600 MB to store +the term lbl_lex containing the labels of the lex-graph in +such a plain .v file. In this case, the memory used by coqc +during the compilation of the file turns out to be the main +limiting factor. The compilation runs out of memory and +fails with Architecture B. Thanks to a better memory man- +agement (memory compression and use of swap memory), +it succeeds on Architecture A, but it takes more than 8 000 s +to complete. To overcome these compilation issues, we have +implemented a Coq plugin called BinReader7 that provides a +command LoadData As building +a Coq term from a description read in the binary +file . The plugin handles all possible combi- +nations of the data types described in Section 4.1. In practice, +the time needed to build the term becomes negligible, and +the main consuming step is the typechecking of the term by +the kernel of Coq. As an illustration, we manage to build the +7https://github.com/Coq-Polyhedra/coq-binreader +term lbl_lex previously discussed in about 8 min on Archi- +tecture A and 13 min on Architecture B, with a total memory +usage limited to 50 GB. +Table 1 provides the total CPU time spent on each in- +stance using Architecture A (Architecture B is around twice +slower). The instances cube-n and cross-n respectively cor- +respond to the 𝑛-dimensional cubes and cross polytopes. The +instances cyclic-p-n are the polars of cyclic polytopes in +dimension 𝑛 defined by 𝑝 facets; see below for the formal +description. The instances spindle-n correspond to the orig- +inal small-dimensional spindles used in [36, 39] to build the +larger-dimensional counterexamples to the Hirsch conjec- +ture. Finally, poly20dim21 and poly23dim24 are respectively +the 20- and 23-dimensional counterexamples to the Hirsch +conjecture given in [36]. The second column of Table 1 pro- +vides the time taken by coqc to load the certificates using the +plugin BinReader (which includes the time needed to build +the terms from the binary files, and the time used by the ker- +nel to typecheck them). The third column provides the time +to check the certificates using the algorithms of Section 4. +The last column provides the time to compute the exact diam- +eter of the polytope in Coq. For the instances polyXXdimYY, +we also provide in italics the time taken to compute a lower +bound on the diameter, as described in Section 5. We point +out that, in practice, compilation time can be reduced using +parallelization (e.g., as provided by the build system dune), as +certificate checking and diameter computation can be done +independently (the computation of the diameter is performed +on the informal graph). +As we already mentioned, the computation of the results of +the certificate tests are carried out using the tactic vm_compute +in Coq [28].8 The later relies on an optimized bytecode based +virtual machine that performs better than the abstract re- +duction machinery of the Coq kernel when it comes to the +full evaluation of computationally intensive function appli- +cations. We did not manage to use the more efficient tactic +native_compute [9] (which consists in the extraction to some +OCaml file, its compilation and native execution). Indeed, +this tactic failed because of the large size of the terms, and +raised some exception. Using vm_compute, the formal disproof +of the Hirsch conjecture takes about 1 h 22 min with the 20- +dimensional counterexample, and about 1 h 55 min with the +23-dimensional counterexample. Since these are relatively +short times, we have reproduced the same experiments by +replacing vm_compute by the standard call-by-value conver- +sion tactic cbv of Coq. We manage to formally verify the two +counterexamples, with approximately a factor 20 slowdown. +We finally provide the mathematical descriptions of other +instances. The 𝑛-dimensional cube consists of the points +8More precisely, we use the tactic vm_cast_no_check which relies on +vm_compute, but does the computation only when the kernel typechecks +the proof term, i.e., during the execution of the command Qed. + +A Formal Disproof of the Hirsch Conjecture +Table 1. Total CPU time (in seconds) of the main steps of +the certification process on Architecture A. +instance +certificate +loading +certificate +checking +diameter +computation +cube_2 +7.85 +6.40 +1.07 +cube_3 +7.90 +6.30 +1.08 +cube_4 +7.93 +6.31 +1.09 +cube_5 +9.18 +6.31 +1.16 +cube_6 +8.32 +6.36 +1.15 +cube_7 +8.33 +6.45 +1.18 +cross_2 +8.54 +6.27 +1.21 +cross_3 +8.11 +6.73 +1.10 +cross_4 +8.17 +6.36 +1.14 +cross_5 +8.65 +6.99 +1.11 +cross_6 +14.97 +17.33 +1.19 +cross_7 +113.32 +265.15 +1.14 +cyclic_12_6 +8.37 +6.52 +1.22 +cyclic_20_10 +25.71 +45.87 +36.77 +spindle_25 +8.93 +7.06 +1.16 +spindle_28 +10.74 +6.90 +1.21 +spindle_48 +10.88 +8.70 +1.22 +poly20dim21 +404.01 +4490.89 +5295.91 +(17.57) +poly23dim24 +1147.00 +5736.19 +24 551.46 +(14.75) +𝑥 ∈ R𝑛 satisfying −1 ≤ 𝑥𝑖 ≤ 1 for all 𝑖 ∈ [𝑛]. It has 2𝑛 ver- +tices. The 𝑛-dimensional cross-polytope is the set of points +𝑥 ∈ R𝑛 satisfying �𝑛 +𝑖=1 |𝑥𝑖| ≤ 1. It can be described by 2𝑛 +inequalities of the form �𝑛 +𝑖=1 ±𝑥𝑖 ≥ −1. The challenge is +that it has many degenerate bases: every vertex is associated +with 2𝑛−1 feasible bases. This important degeneracy is re- +flected in the time required to compute the lex-graph. Given +𝑝 ≥ 1, the polar of the cyclic polytope, denoted 𝐶(𝑛, 𝑝), is +built by picking 𝑝 real values 𝑡1 < · · · < 𝑡𝑝, and by consid- +ering the inequalities ⟨𝑐𝑖 − ¯𝑐,𝑥⟩ ≤ 1 for all 𝑖 ∈ [𝑝], where +𝑐𝑖 � (𝑡𝑖,𝑡2 +𝑖 , . . . ,𝑡𝑛 +𝑖 ) and ¯𝑐 � +1 +𝑝 +�𝑝 +𝑖=1 𝑐𝑖. McMullen’s upper +bound theorem [37] states that the polytope 𝐶(𝑛, 𝑝) maxi- +mizes the number of vertices among the 𝑛-dimensional poly- +tope with 𝑝 facets. This number is given explicitly by: +�𝑝 − +�𝑛 +2 +� +� 𝑛 +2 +� +� ++ +�𝑝 − 1 − +�𝑛−1 +2 +� +� 𝑛−1 +2 +� +� +We have checked that the certified graph has precisely the +expected number of vertices. +7 +Conclusion +We have developed practically efficient algorithms that al- +low us to compute the vertex-edge graphs of polytopes by +checking the correctness of informally computed certificates. +Simplicity of design is a key feature to reach this goal. We +have brought a formal disproof of the well-known Hirsch +conjecture, by formally verifying the two counterexamples +of [36]. Despite their very involved combinatorial structure, +our approach manages to achieve the disproof in a few hours +within the proof assistant Coq. As far as we know, this is the +first work showing that computationally demanding proofs +on polyhedra can be practically carried out in a proof assis- +tant. +As a future work, we aim at significantly improving the +performance of our approach and get much closer to that of +informal software. To this purpose, we plan to exploit rank- +one updates of matrices and integer pivoting (see [5] and the +references therein) in order to reduce the size of certificates +and improve the computational complexity of matrix-vector +multiplications in our algorithms. We also hope that com- +putationally intensive approaches like the one presented in +this work will draw more attention to the computational +performance of proof assistants. To this extent, we aim at +exploiting the more efficient tactic native_compute [9] once +issues on handling large terms are solved. For the time being, +our approach deals only with polytopes. 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Extra Volume "Optimization Stories". + diff --git a/R9E2T4oBgHgl3EQfsgho/content/tmp_files/load_file.txt b/R9E2T4oBgHgl3EQfsgho/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1659acd65e53b20d5ab87215e29767fb66854ee9 --- /dev/null +++ b/R9E2T4oBgHgl3EQfsgho/content/tmp_files/load_file.txt @@ -0,0 +1,1133 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf,len=1132 +page_content='A Formal Disproof of the Hirsch Conjecture Xavier Allamigeon Inria CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris France xavier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='allamigeon@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='fr Quentin Canu Inria CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris France quentin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='canu@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='fr Pierre-Yves Strub Meta France strubpy@meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='com Abstract The purpose of this paper is the formal verification of a coun- terexample of Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' to the so-called Hirsch Conjecture on the diameter of polytopes (bounded convex polyhedra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In contrast with the pen-and-paper proof, our approach is entirely computational: we implement in Coq and prove cor- rect an algorithm that explicitly computes, within the proof assistant, vertex-edge graphs of polytopes as well as their diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The originality of this certificate-based algorithm is to achieve a tradeoff between simplicity and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Simplicity is crucial in obtaining the proof of correctness of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This proof splits into the correctness of an abstract algorithm stated over proof-oriented data types and the correspondence with a low-level implementation over computation-oriented data types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' A special effort has been made to reduce the algorithm to a small sequence of elementary operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', matrix multiplications, basic routines on sets and graphs), in order to make the derivation of the correctness of the low-level implementation more transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Efficiency allows us to scale up to polytopes with a chal- lenging combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' For instance, we formally check the two counterexamples of Matschke, Santos and Weibel to the Hirsch conjecture, respectively 20- and 23-dimensional polytopes with 36 425 and 73 224 vertices involving rational coefficients with up to 40 digits in their numerator and de- nominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We also illustrate the performance of the method by computing the list of vertices or the diameter of well- known classes of polytopes, such as (polars of) cyclic poly- topes involved in McMullen’s Upper Bound Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Keywords: polyhedra, polytopes, Hirsch Conjecture, proof assistants, certified computation 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='1 Motivations The study of diameters of polyhedra is at the heart of the following major problem in optimization: does the simplex method terminate in polynomial time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This question is open since Georg B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Dantzig introduced the simplex method in the late 40s, and it has inspired to the Fields medalist Steve Smale the ninth of his problems for the 21st century on the existence of a strongly polynomial algorithm for linear pro- gramming [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The simplex method [17] is certainly the most standard technique to solve linear programs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', prob- lems of the form Minimize ⟨𝑐,𝑥⟩ subject to ⟨𝑎1,𝑥⟩ ≥ 𝑏1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' , ⟨𝑎𝑚,𝑥⟩ ≥ 𝑏𝑚 , 𝑥 ∈ R𝑛 for some vectors 𝑎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' ,𝑎𝑚,𝑐 ∈ R𝑛 and reals 𝑏1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' ,𝑏𝑚, where ⟨𝑦,𝑧⟩ � �𝑛 𝑖=1 𝑦𝑖𝑧𝑖 denotes the Euclidean scalar prod- uct of 𝑦,𝑧 ∈ R𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' It consists in minimizing the objective func- tion 𝑥 ↦→ ⟨𝑐,𝑥⟩ over the convex polyhedron formed by the points 𝑥 ∈ R𝑛 satisfying the constraints ⟨𝑎𝑖,𝑥⟩ ≥ 𝑏𝑖 for all 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' ,𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' From a geometric perspective, the principle of the simplex method is to iteratively decrease the objective function by visiting a subset of vertices of the polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' More precisely, at every iteration, the method selects a vertex with smaller value among the vertices which are adjacent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', connected by an edge) to the current vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We refer to Section 2 for the mathematical definitions of vertices and edges of polyhedra, and to Figure 1 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The choice of the next vertex at every iteration is specified by the so-called pivot rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In consequence, every pivot rule makes the simplex method draw a path in the vertex-edge graph of the polyhedron, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', the graph defined by the vertices and edges of the polyhedron (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' While a large number of pivot rules have been described in the literature, all the rules that have been mathematically analyzed have been shown to exhibit (sub)exponential behavior in the worst case (see [3], and [20] for a more recent account).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Recall that, in a graph, the distance between two vertices is the length (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', the number of edges) of any shortest path between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The diameter of the graph is then defined as the largest distance between any two vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Consequently, the (combinatorial) diameter of a polyhedron, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', the diame- ter of its graph, constitutes a lower bound on the number of iterations performed by the simplex method with any pivot rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' With this motivation, Hirsch formulated the following conjecture in a letter to Dantzig in 1957: Conjecture 1 (Hirsch conjecture over polytopes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The di- ameter of any 𝑑-dimensional polytope with 𝑝 facets is bounded by 𝑝 − 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In this statement, a polytope refers to the convex hull of finitely many points, or equivalently, a bounded polyhe- dron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The facets of a 𝑑-dimensional polyhedron are the faces of dimension 𝑑 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' (Originally, the conjecture was stated arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='04060v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='LO] 10 Jan 2023 Xavier Allamigeon, Quentin Canu, and Pierre-Yves Strub −𝑐 𝐴 𝐵 𝐶 𝐷 𝐸 𝐹 𝐹 𝐵 𝐴 𝐷 𝐸 𝐶 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Several possible execution traces of the simplex method on the 3-dimensional cross polytope (see Section 6 for a description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The execution of the simplex method draws a path in the graph of the polytope from a starting vertex to an optimal one (rightmost part of the picture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' over polyhedra rather than polytopes, but it was quickly realized that the bound does not hold over unbounded poly- hedra [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=') The study of the diameter of polytopes and poly- hedra has received a tremendous attention over the years;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' [46] for a survey on the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Still, the conjecture remained unsolved for more than fifty years, until Santos exhibited a counterexample: Theorem 1 ([39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' There exists a 43-dimensional polytope with 86 facets and diameter larger than 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In a further work joint with Matschke and Weibel [36], San- tos provided two other counterexamples to the conjecture, respectively in dimension 20 and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' All these constructions critically rely on a computational argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' More precisely, in all of them, the non-Hirsch polytope is obtained from a smaller dimensional polytope with a special combinatorial structure, that of a spindle, and in which the distance between two distinguished vertices has to be proved larger than the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' For the original counterexample, this spindle has dimension 5, 48 facets and 322 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Santos indicates that the property has been checked thanks to the informal software Polymake [24], and he provides two independent proofs that are “computer-free (but not computation-free).” Fortunately, exploiting the symmetry group of this spindle makes the computations manageable by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Since Santos’ breakthrough, the interest for diameters of polytopes and polyhedra has remained intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' First, this is still not understood to which extent a counterexample to the Hirsch conjecture can be found in smaller dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' More- over, from the perspective of the complexity of the simplex method, the most important question is whether or not the diameter of polyhedra can be bounded by a polynomial in the dimension and the number of facets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' this is known as the polynomial Hirsch conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' It is likely that computations will play a key role in any progress on these two questions, just like they did in Santos’ construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In consequence, there is a strong motivation to develop a framework in which computations over polyhedra are performed in a proof assistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This would considerably en- large the scope of research for pathological polytopes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', giving the capacity to scale up to larger numbers of ver- tices) while retaining (if not increasing) the level of trust compared to pen-and-paper computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Combinatorial properties of polytopes are not the only topic such a contri- bution would benefit to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' For instance, polyhedra are a central tool in critical applications such as software compilation or verification [16], or invariant computation in the analysis of dynamical systems [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In these applications, the computa- tion of the vertices of a polyhedron is a core primitive, and developing a formally proved algorithm that performs this operation is a noteworthy goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='2 Contributions We introduce an algorithm which computes the set of ver- tices as well as the graph of polytopes, and we present its implementation and proof of correctness in the proof as- sistant Coq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The originality of our contribution lies in the combination of simplicity of design and practical efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Simplicity is the key feature which makes the algorithm re- alistically implementable and provable in a proof assistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Simplicity also has major advantages in terms of maintain- ability, and portability to other proof assistants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Our algorithm is based on certificates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', it takes as input a graph computed by some informal software, and formally certifies that this graph is indeed correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In more details, the certificate is the graph of a perturbation of the polytope that has remarkable properties such as being connected and regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Thanks to these properties, the certification consists of a sequence of elementary steps, involving basic operations such as matrix multiplications, set operations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The graph of the original polytope can be then deduced as the image of the former graph by some mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The algorithm is first implemented and proved correct by using some proof-oriented types, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', dependent types rep- resenting matrices, convex polyhedra, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' However, these types would have a prohibitive cost for computations: they have been defined to ease the formal development of mathe- matical theories, prioritizing the use of naive data-structures and algorithms over computationally well-behaving ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' As a consequence, we implement a second algorithm over computation-oriented types, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', persistent arrays, native integers, arbitrary precision rationals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' An additional bene- fit of the elementary structure of the algorithm is to make the equivalence proof between the low-level and high-level implementations easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In order to demonstrate the practical efficiency of our approach, we make experiments on several classes of poly- topes of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We manage to compute the graph of the 20- and 23-dimensional counterexamples of [36] to the Hirsch conjecture, respectively with 36 425 and 73 224 vertices, and A Formal Disproof of the Hirsch Conjecture 364 250 and 842 076 edges, and we deduce a formal disproof of the conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We also study classic polytopes such as hypercubes, cross-polytopes, and polars of cyclic polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The latter are known to maximize the number of vertices for fixed dimension and number of facets, as stated by Mc- Mullen’s Upper Bound Theorem [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='3 Related Work Computing the vertices of polytopes and polyhedra is a noto- riously difficult problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' As the number 𝑣 of vertices can be exponential in the number 𝑚 of defining inequalities and the dimension 𝑛, this complexity has to be measured as a func- tion of 𝑚, 𝑛 and 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The worst-case complexity of the problem is not fully understood: it is an NP-hard enumeration prob- lem for unbounded polyhedra [34], but its status is still open for polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The two main vertex enumeration algorithms are the double description method [23, 38] and the reverse search method [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' While the complexity of the double description method cannot be easily bounded in terms of 𝑚, 𝑛 and 𝑣, the complexity of the reverse search method is in 𝑂(poly(𝑚,𝑛) 𝑣) for “nondegenerate” representations of polytopes (we refer to Section 3 for the definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This falls in the same complexity class as our algorithm, since the two approaches rely on the enumeration of simplex bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The double description method and the reverse search method have standard implementations [4, 22] that are widely used in software dedicated to computational mathematics [24, 43] as well as software verification [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Computational proofs, or proofs by reflection, is nowa- days a common technique in the Coq proof assistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' For instance, it is used in the implementation of tactics that rely on symbolic computations [29], and has been widely used for the formal proof of the four color theorem [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Proofs by reflection is also at the root of the small scale reflection (SSRe- flect [26]) proof methodology, where one makes a pervasive use of computation for solving goals that involve decidable predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We refer to [11, 13, 32, 33] for other examples of computational based proofs in the Coq proof assistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Proof by reflection has also been used in many other sys- tems: Agda [44], Isabelle/HOL [12], Lean [7], PVS [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In relation with polyhedral computations, Farkas’ certification techniques motivated by application to static analysis have appeared in [10, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In contrast with our work, this only cover polyhedral computations using inequalities, and not the computation of vertices or vertex-edge graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We fi- nally mention that an implementation of a simplex-based satisfiability procedure has been done by [42] in the proof assistant Isabelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Last, our paper makes use of program and data structure refinement techniques, a proof methodology that consists in transforming a high-level program or data structure to a lower-level, more efficient one, while preserving the main properties (program specification and/or data structure in- variants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Refinement techniques have already been used in the context of the Coq proof assistant, notably in Co- qEAL [14, 19], a Coq framework for easing the definition of data structure refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Our formalization of data re- finements closely follows the approach of CoqEAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The difference is that, for a first experiment, we have chosen not to exploit the automated deduction of equivalence proofs provided by CoqEAL (based on Coq typeclasses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In this way, we have a finer-grained control of the low-level imple- mentation of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We point out that the burden of proving the equivalence proof by hand was limited thanks to the simplicity of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='4 Organization of the Paper In Section 2, we recall basic notions on polyhedra and intro- duce notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In Section 3, we define the certificate-based algorithm computing the graph of a given polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Section 4 is dedicated to the implementation of the latter algorithm using computation-oriented data types, and deals with the proof of correctness of this implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In Section 5, we bring the formal disproof of the Hirsch Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Finally, we report on the experiments of our approach on several classes of polytopes in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The source files of the present submission can be found in the git repository of the Coq-Polyhedra library1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We often refer to these source files in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' On top of Coq, we rely on the MathComp library [27] as well as the finmap and bignums libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='2,3 We note that this work is a (slightly) extended version of the conference paper [1] published in the proceedings of CPP’23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' 2 Preliminaries and Notation As discussed in the introduction, a (convex) polyhedron is defined as the set of points 𝑥 ∈ R𝑛 satisfying finitely many affine inequalities ⟨𝑎𝑖,𝑥⟩ ≥ 𝑏𝑖, where 𝑎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' ,𝑎𝑚 ∈ R𝑛 and 𝑏1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' ,𝑏𝑚 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We shall also write the system of constraints as 𝐴𝑥 ≥ 𝑏, where 𝐴 ∈ R𝑚×𝑛 is the matrix with rows 𝑎𝑇 𝑖 , 𝑏 = (𝑏𝑖)𝑖 ∈ R𝑚, and ≥ denotes the entrywise ordering over R𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' A notable subclass of polyhedra are polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' A polytope is the convex hull � �𝑝 𝑖=1 𝜆𝑖𝑣𝑖 : 𝜆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' , 𝜆𝑝 ≥ 0, �𝑝 𝑖=1 𝜆𝑖 = 1 � of finitely many points 𝑣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' , 𝑣𝑝 ∈ R𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Minkowski Theorem states that polytopes are precisely the bounded polyhedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The dimension of a polyhedron is defined as the dimen- sion of its affine hull, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', the smallest (inclusionwise) affine subspace containing the polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' For instance, a point has dimension 0, a line segment has dimension 1, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Let P be a polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' A (nonempty) face of P is the set of points minimizing some linear function 𝑥 ↦→ ⟨𝑐,𝑥⟩ over P, where 𝑐 ∈ R𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Equivalently, a face is the set of optimal solutions of some linear program over P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The vertices and the edges 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='com/Coq-Polyhedra/Coq-Polyhedra/tree/CPP-23 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='com/math-comp/finmap 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='com/coq-community/bignums Xavier Allamigeon, Quentin Canu, and Pierre-Yves Strub 𝑥 𝑦 𝑧 𝑐1 𝑐2 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The 3-dimensional cross polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The objective vector 𝑐1 is minimized by only one vertex 𝑥, while objective vector 𝑐2 is minimized by an edge [𝑦,𝑧] with 𝑦 and 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' of P are the faces of dimension 0 and 1 respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Any bounded edge writes as the line segment be- tween two vertices 𝑣, 𝑣 ′ of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In this case, the vertices 𝑣 and 𝑣 ′ are said to be adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' As described in the introduction, the graph of P, denoted by 𝐺vert(P) (or simply 𝐺vert when clear from context), is the combinatorial graph induced by the adjacency relation over the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In the rest of the paper, we define [𝑝] � {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' , 𝑝} for all integer 𝑝 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The cardinality of a finite set 𝑆 is denoted by #𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Given a matrix 𝑀 ∈ R𝑝×𝑞, we denote by 𝑀𝑖 its 𝑖th row (𝑖 ∈ [𝑝]), and, for all subset 𝐼 ⊂ [𝑝], by 𝑀𝐼 ∈ R#𝐼×𝑞 the submatrix with rows 𝑀𝑖 for 𝑖 ∈ 𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The identity matrix of size 𝑝 × 𝑝 is denoted by Id𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Given a (nonoriented) graph 𝐺 = (𝑉, 𝐸) (where 𝐸 ⊂ 𝑉 ×𝑉 ) and a vertex 𝑣 ∈ 𝑉 , we denote by 𝑁𝐺 (𝑣) � {𝑤 ∈ 𝑉 : (𝑣,𝑤) ∈ 𝐸} the neighborhood of 𝑣 in 𝐺, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', the set of vertices 𝑤 adjacent to 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Given a function 𝑓 : 𝑉 → 𝑉 ′, the image of 𝐺 by 𝑓 , denoted by 𝑓 (𝐺), is defined as the graph with vertices 𝑓 (𝑉 ) � {𝑓 (𝑣) : 𝑣 ∈ 𝑉 } and edges � (𝑓 (𝑣), 𝑓 (𝑤)) : (𝑣,𝑤) ∈ 𝐸 , 𝑓 (𝑣) ≠ 𝑓 (𝑤) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' 3 Graph Certification Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='1 A First Algorithm for the Nondegenerate Setting Our approach heavily relies on the notion of bases and ba- sic points, which are the central ingredients of the simplex method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Let P = {𝑥 ∈ R𝑛 : 𝐴𝑥 ≥ 𝑏} a polyhedron, where 𝐴 ∈ R𝑚×𝑛 and 𝑏 ∈ R𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We recall that a basis is a subset 𝐼 ⊂ [𝑚] of cardinality 𝑛 such that the submatrix 𝐴𝐼 is non- singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In this case, the equality system 𝐴𝐼𝑥 = 𝑏𝐼 has a unique solution 𝑥𝐼, which we call the basic point associated with 𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' When 𝑥𝐼 belongs to the polyhedron P, the basis 𝐼 is said to be feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' By extension, the point 𝑥𝐼 is said to be a feasible basic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' It is a standard property that the vertices of P are precisely the feasible basic points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' see [40, Chapter 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' However, in general, the correspondence between them is not bijective: every feasible basic point is a vertex, but a vertex may be the basic point associated with more than one feasible basis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' see Figure 3 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' These bases are said to be degen- erate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We say that we are in the nondegenerate setting when 2 4 1 3 𝑧 \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 1 𝑥1 + 𝑥2 + 𝑥3 ≥ −1 2 − 𝑥1 + 𝑥2 + 𝑥3 ≥ −1 3 𝑥1 − 𝑥2 + 𝑥3 ≥ −1 4 − 𝑥1 − 𝑥2 + 𝑥3 ≥ −1 5 𝑥1 + 𝑥2 − 𝑥3 ≥ −1 6 − 𝑥1 + 𝑥2 − 𝑥3 ≥ −1 7 𝑥1 − 𝑥2 − 𝑥3 ≥ −1 8 − 𝑥1 − 𝑥2 − 𝑥3 ≥ −1 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The 3-dimensional cross polytope has degenerate bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' For instance, the bottom point 𝑧 (𝑥3 = −1) is a vertex associated with four feasible bases: (2, 3, 4), (1, 3, 4), (1, 2, 4) and (1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' there are no such bases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', the correspondence between feasible bases and vertices is one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Given a linear program of the form Minimize ⟨𝑐,𝑥⟩ subject to 𝐴𝑥 ≥ 𝑏 , 𝑥 ∈ R𝑛 , (1) where 𝑐 ∈ R𝑛, the simplex method iterates over feasible bases up to reaching a (feasible) basic point that minimizes the function 𝑥 ↦→ ⟨𝑐,𝑥⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In this scheme, any two consecutive bases 𝐼, 𝐼 ′ satisfy #(𝐼 ∩ 𝐼 ′) = 𝑛 − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', they only differ by one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Such bases are said to be adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This adjacency relation gives rise to the graph of (feasible) bases, that we denote by 𝐺bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The relation between basic points and vertices extends to the graph of the polyhedron and the graph of bases as follows: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The vertex-edge graph 𝐺vert is the image of the graph 𝐺bases by the function 𝐼 ↦→ 𝑥𝐼 which maps any feasible basis to its basic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Moreover, in the nondegenerate setting, the latter function is an isomorphism between the two graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This result elaborates on the geometric description of the simplex that we made in the introduction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', the simplex method induces a path in the vertex-edge graph of the poly- hedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In this section, we discuss the computation of the vertex- edge graphs of polytopes in the nondegenerate case only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The exposition of this special case is done to facilitate the understanding of the general algorithm presented in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We remark that, unless explicitly stated, we did not have to formalize the results presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In light of the second part of Proposition 2, we exploit the isomorphism between 𝐺vert and 𝐺bases, and we sketch an algorithm certifying that a given graph (computed a priori by some informal procedure) coincides with the graph of feasible bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' To this purpose, we exploit the following two fundamental properties: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The graph 𝐺bases is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' A Formal Disproof of the Hirsch Conjecture Algorithm 1 Graph certification algorithm (nondegenerate setting and P = {𝑥 ∈ R𝑛 : 𝐴𝑥 ≥ 𝑏} bounded) Require: 𝐴 ∈ R𝑚×𝑛, 𝑏 ∈ R𝑚 and 𝐺 = (𝑉, 𝐸) 1: assert 𝐺 is nonempty 2: for all 𝐼 ∈ 𝑉 do assert 𝐼 is a feasible basis 3: for all 𝐼 ∈ 𝑉 do 4: for all 𝐽 ∈ 𝑁𝐺 (𝐼) do assert #(𝐼 ∩ 𝐽) = 𝑛 − 1 5: assert #𝑁𝐺 (𝐼) = 𝑛 6: done 7: return true Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In the nondegenerate setting, and when P is a polytope, the graph 𝐺bases is 𝑛-regular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', every feasible basis is adjacent to precisely 𝑛 feasible bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Proposition 3 holds in a general setting (even with de- generate bases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' It is a consequence of the fact that, for any feasible basis 𝐼★, we can find a vector 𝑐 ∈ R𝑛 such that 𝑥𝐼★ is the unique optimal solution of the linear program (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Then, the simplex method initialized with any feasible basis 𝐼 draws a path to 𝐼★ in the graph 𝐺bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The algorithm covering the nondegenerate setting and the assumption that P is a polytope is Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We use the command assert as some syntactic sugar for the block if not then return false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The vertices of the input graph 𝐺 = (𝑉, 𝐸) are supposed to be sets 𝐼 of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The algorithm consists in four steps: (i) check that the graph is nonempty (Line 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' (ii) check that every vertex 𝐼 is a feasible basis (Line 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' (iii) for each vertex 𝐼 ∈ 𝑉 , check that its neighborhood consists of adjacent bases (Line 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' (iv) for each vertex 𝐼 ∈ 𝑉 , check that its neighborhood has cardinality 𝑛 (Line 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The second and third steps ensures that 𝐺 is a subgraph of 𝐺bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Moreover, as 𝐺bases is 𝑛-regular (Proposition 4), the fourth step actually ensures that 𝑁𝐺 (𝐼) = 𝑁𝐺bases(𝐼) for all 𝐼 ∈ 𝑉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In consequence, 𝐺 is a subgraph of 𝐺bases such that the neighborhood of every vertex in 𝐺 agrees with that in 𝐺bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' As shown in the following lemma, the nonemptiness of 𝐺 and the connectedness of 𝐺bases then ensure that 𝐺 and 𝐺bases are identical: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Let𝐺 and 𝐻 two graphs such that𝐺 is a nonempty subgraph of 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Suppose that 𝐻 is connected, and 𝑁𝐺 (𝑣) = 𝑁𝐻 (𝑣) for all vertices 𝑣 of 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Then 𝐺 = 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This lemma is given by Lemma sub_gisof (see high_graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' v) in the source of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' (This result actually shows a slightly more general though equivalent statement, where 𝐺 is replaced by an isomorphic graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=') The next result then follows from Lemma 5, and shows that Algorithm 1 is correct: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Suppose that we are in the nondegenerate setting, and that P = {𝑥 ∈ R𝑛 : 𝐴𝑥 ≥ 𝑏} is a polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' If Algorithm 1 returns true, then 𝐺 = 𝐺bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The lexicographic perturbation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The top vertex on the left-hand side is split into three distinct vertices on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The reader can also verify that if one of the assertions in Algorithm 1 fails, then the graph 𝐺 cannot be equal to 𝐺bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='2 Dealing with the General Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='1 A Perturbation Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The nondegenerate set- ting does not hold in general, but we can reduce to it by slightly perturbing the polytope, while still keeping a way to recover the main combinatorial structure such as the set of vertices or the vertex-edge graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Our approach originates from the so-called lexicographic pivot rule introduced by Dantzig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' [18], and later exploited by Avis [5] for his ver- tex enumeration algorithm [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' It consists in perturbing the vector 𝑏 by replacing each entry 𝑏𝑖 (𝑖 ∈ [𝑚]) by the quantity 𝑏𝑖 − 𝜀𝑖, where 𝜀 > 0 is a sufficiently small real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Geometri- cally, while the normal vectors to the hyperplanes delimiting the polyhedron are still the same, the perturbation of the vector 𝑏 breaks every vertex associated with several (degen- erate) bases into distinct (and nondegenerate) basic points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' see Figure 4 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Instead of instantiating 𝜀 by some numerical values (which would raise the problem of determining how small it should be), the perturbation is achieved in a symbolic way, by think- ing of each 𝑏𝑖 as a polynomial in 𝜀 of degree at most 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This leads to considering a “polyhedron” where the entries of the points are polynomials in 𝜀 of degree at most 𝑚 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Such polynomials �𝑚 𝑘=0 𝛼𝑘𝜀𝑘 can be encoded as row vectors (𝛼0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' , 𝛼𝑚) of size 1 + 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In this case, the standard order over the reals is replaced by the lexicographic order ≤lex over (1 + 𝑚)-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Indeed, we have (𝛼0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' , 𝛼𝑚) ≤lex (𝛽0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' , 𝛽𝑚) if and only if �𝑚 𝑘=0 𝛼𝑘𝜀𝑘 ≤ �𝑚 𝑖=0 𝛽𝑘𝜀𝑘 for all 0 < 𝜀 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This gives rise to the symbolically perturbed polyhedron �P � � 𝑋 ∈ R𝑛×(1+𝑚) : 𝐴𝑋 ≥lex �𝑏 � (2) where the matrix �𝑏 � �𝑏 −Id𝑚 � ∈ R𝑚×(1+𝑚) corresponds to the perturbation of the vector 𝑏 described above: the 𝑖th row of �𝑏 encodes the polynomial 𝑏𝑖 − 𝜀𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In (2), the relation ≥lex stands for the entrywise extension of the lexicographic order: two matrices 𝑋,𝑌 ∈ R𝑝×(1+𝑚) satisfies 𝑋 ≥lex 𝑌 if 𝑋𝑖 ≥lex 𝑌𝑖 for all 𝑖 ∈ [𝑝].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The matrices 𝑋 ∈ R𝑛×(1+𝑚) in �P correspond to vectors with perturbed entries (encoded as polynomials in 𝜀 of degree at most 𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Xavier Allamigeon, Quentin Canu, and Pierre-Yves Strub The notion of feasible bases still makes sense in case of such symbolically polyhedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' To avoid any confusion with the bases of the original polyhedron, the feasible bases of �P are referred to as lex-feasible bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Formally, a set 𝐼 ⊂ [𝑚] is a lex-feasible basis if 𝐼 has cardinality 𝑛, the matrix 𝐴𝐼 is non- singular, and the basic point 𝑋 𝐼 � 𝐴−1 𝐼 �𝑏𝐼 satisfies 𝐴𝑋 𝐼 ≥lex �𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We recall that lex-feasible bases are not degenerate: Lemma 7 (see [2] for a formalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Let 𝐼, 𝐼 ′ be two distinct lex-feasible bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Then 𝑋 𝐼 ≠ 𝑋 𝐼′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='2 Formalizing the Properties of the Lex-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In what follows, we assume that P is a polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The adjacency relation defined over feasible bases carries over to lex-feasible bases in a straightforward way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This induces the graph of lex-feasible bases, or lex-graph for short, that we denote by 𝐺lex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Lex-feasible bases form the cornerstone of the formaliza- tion of the simplex method done by Allamigeon and Katz [2], and provided in the library Coq-Polyhedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In more details, the latter work formalized the lex-simplex method, which iterates over lex-feasible bases in order to avoid cycling on degenerate bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=" Coq-Polyhedra introduces the type lex_feasible_basis A b of lex-feasible bases, where A : ' M_(m,n) and b : 'cV_m correspond to the matrix 𝐴 and the (unperturbed) vector 𝑏." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=" (We recall that 'M_(m,n) and 'cV_m are respectively the types of m × n-matrices and m-vectors provided by the library MathComp [27]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=") We build on this and start by defining the graph of lex-feasible bases, denoted lex_graph: Definition set_adjacence := (* adjacency relation *) fun I I' : {set 'I_m} => #| I :&: I' | == n." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Definition lex_graph := mk_graph [fset x | x : Simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='lex_feasible_basis A b] set_adjacence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We then prove that, as expected, the properties of Proposi- tions 3 and 4 hold in the case of lex-feasible bases: Lemma lex_graph_connected : connected lex_graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Lemma lex_graph_regular : regular lex_graph n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Their proofs are straightforward consequences of the formal- ization of the lex-simplex method provided in Coq-Polyhedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' One fundamental property of lex-feasible bases is that they constitute a subset of the feasible bases of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In more details, we denote by 𝜋 the function which maps a matrix 𝑋 ∈ R𝑛×(1+𝑚) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', a perturbed point) to its first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The latter corresponds to the unperturbed part of 𝑋, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', the value of the perturbed point when 𝜀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The following result is folklore (we refer to [2] for the formalization): Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Let 𝐼 be a lex-feasible basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Then 𝐼 is a feasi- ble basis of P, and 𝜋(𝑋 𝐼) is the associated basic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We extend the latter result to the following correspon- dence between the lex-graph and the vertex-edge graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The vertex-edge graph 𝐺vert of P is the image of 𝐺lex by the function 𝜙 : 𝐼 ↦→ 𝜋(𝑋 𝐼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In Coq, this statement is written as follows (see Module enum_proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v): Theorem im_lex_graph_vert_graph : poly_graph P = (Simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='point_of_basis b) @/ lex_graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' where the term poly_graph P is the vertex-edge graph 𝐺vert, the function Simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='point_of_basis b corresponds to the function 𝜙, and a term of the form f @/ G stands for the image of a graph G by the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We briefly comment on the formal proof of Theorem 9 since, most often, its informal proof is not detailed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The hardest part of the proof is to show that for every edge [𝑣,𝑤] of P, there exist two adjacent lex-feasible bases 𝐼, 𝐽 such that 𝑣 = 𝜙(𝐼) and 𝑤 = 𝜙(𝐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We construct these two bases by exploiting the lex-simplex method of Coq-Polyhedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' More precisely, since [𝑣,𝑤] is an edge of P, there exists a vector 𝑐 such that [𝑣,𝑤] is precisely the set of points of P minimizing the function 𝑥 ↦→ ⟨𝑐,𝑥⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Calling the lex-simplex method with the objective vector 𝑐 provides a lex-feasible basis 𝐼0 such that the point 𝜙(𝐼0) reaches this minimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Since 𝜙(𝐼0) is a vertex of P, this should be either 𝑣 or 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Without loss of generality, we assume that 𝜙(𝐼0) = 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We now consider an objective vector 𝑐′ such that 𝑤 is the only point minimizing 𝑥 ↦→ ⟨𝑐′,𝑥⟩ over P (this is possible by definition of a vertex), and introduce a third objective vector 𝑑 � 𝑐 + 𝛿𝑐′, where 𝛿 > 0 is a sufficiently small quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Intuitively, perturbing 𝑐 into 𝑑 in this way should ensure that 𝑤 is the unique minimizer of 𝑥 ↦→ ⟨𝑑,𝑥⟩ over P, and that 𝑣 is the second “best” vertex after 𝑤, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', ⟨𝑑,𝑤⟩ < ⟨𝑑, 𝑣⟩ < ⟨𝑑,𝑧⟩ for every vertex 𝑧 ∉ {𝑣,𝑤}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='4 We finally apply the lex-simplex method with objective function 𝑥 ↦→ ⟨𝑑,𝑥⟩, starting from the lex-feasible basis 𝐼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Since the objective function cannot increase along the way, the lex- simplex method generates a sequence 𝐼0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' , 𝐼𝑝−1, 𝐼𝑝, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' of adjacent lex-feasible bases such that 𝜙(𝐼𝑘) = 𝑣 for all 𝑘 < 𝑝 and 𝜙(𝐼𝑝) = 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Then, it suffices to take 𝐼 = 𝐼𝑝−1 and 𝐽 = 𝐼𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='3 Certification of the Lex-Graph and the Vertex- Edge Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In light of the properties formalized in Lemma lex_graph_connected and Lemma lex_graph_regular, we can derive from Algorithm 1 a method certifying that an infor- mally computed graph coincides with the lex-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Then, Theorem 9 will allow us to recover the vertex-edge graph of P from the latter by computing its image by 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We describe the formalization of the certification proce- dure for 𝐺lex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=" It takes as input a graph G whose vertices are pairs of the form (I, X), where I : {set 'I_m} is a subset of integers (less than 𝑚), and X : 'M_(n,1+m) is a (𝑛 × (1 +𝑚))- matrix." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' These two components are intended to represent a lex-feasible basis 𝐼 and the corresponding basic point 𝑋 𝐼 4These relations actually imply how small 𝛿 needs to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' A Formal Disproof of the Hirsch Conjecture respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The algorithm is formalized as a program re- turning a Boolean value (reminding that &&, [&& .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='] and [forall .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='] stand for Boolean conjunctions): Definition high_enum_algo G : bool := (G !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='= graph0) (* G is nonempty *) && [forall u : vertices G, (* u is a pair (I,X) *) [&& card_verification u, bas_verification u, feas_verification u, reg_verification u & subset_verification u ] ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' where we define Definition card_verification u := #|u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='1| == n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Definition bas_verification u := (row_submx A u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='1) *m u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='2 == row_submx b_pert u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Definition feas_verification u := u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='2 \\in Simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='lex_polyhedron A b_pert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Definition reg_verification u := #|` successors G u| == n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=" Definition subset_verification u := [forall u' : successors G u, set_adjacence u." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content="1 u'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We recall that u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='1 : T and u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content="2 : T' respectively correspond to the first and second components of a pair u : T * T'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The implementation of high_enum_algo follows the structure of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In particular, it starts by checking that the graph G is nonempty, and then performs five consecutive tests on every vertex u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The last two tests apply to the neighborhood successors G u of u in G, and respectively check that it has cardinality 𝑛 (reg_verification) and consists of adjacent bases (subset_verification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The main difference with Al- gorithm 1 is the way the first component 𝐼 of every vertex u in G is verified to be a lex-feasible basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This is the purpose of the first three tests card_verification, bas_verification and feas_verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The first one checks that 𝐼 has cardi- nality 𝑛, while the last two ones respectively make sure that 𝐴𝐼𝑋 = �𝑏𝐼 and 𝐴𝑋 ≥lex �𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We note that, since �𝑏 = �𝑏 −Id𝑚 � , the equality 𝐴𝐼𝑋 = �𝑏𝐼 ensures that 𝐴𝐼 is a nonsingular ma- trix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Indeed, if 𝑌 is the submatrix of 𝑋 formed by its (1 +𝑖)th columns for 𝑖 ∈ 𝐼, it can be verified that 𝐴𝐼𝑌 = −Id𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In other words, the matrix 𝑋 already carries a certificate that 𝐴𝐼 is nonsingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' As a consequence, 𝐼 is a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Since 𝐴𝐼𝑋 = �𝑏𝐼 then 𝑋 = 𝑋 𝐼, and the condition 𝐴𝑋 ≥lex �𝑏 finally ensures that 𝐼 is a lex-feasible basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This is how we arrive at the proof of the correctness of high_enum_algo: Theorem repr_lex_graph G : high_enum_algo G -> gisof G lex_graph (fun u => u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The gisof predicate in the rightmost part of the implica- tion means that the function (𝐼,𝑋) ↦→ 𝐼 is an isomorphism between G and the lex-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The following statement is obtained by combining Theorem im_lex_graph_vertex_graph and Theorem repr_lex_graph: Theorem repr_poly_graph G : high_enum_algo G -> poly_graph P = (phi @/ G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' where the function phi is defined as phi u = col 0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', if u = (I, X), then phi u is the first column of the matrix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' 4 Efficient Implementation The Coq function high_enum_algo introduced in Section 3 works with dependent types (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', MathComp types) that are adapted for proof but not for computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' For example, natural numbers are expressed in unary form, and rationals carry proof terms of the fact that their numerator and denom- inator are coprime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Similarly, the implementations of finite sets, graphs or matrices are based on MathComp sequences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', basic lists built by induction) that are not made for fast computations, and are provided with multiple proof terms for their well-formedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' As a consequence, the function high_enum_algo cannot return within a reasonable amount of time even on the simplest instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' To overcome this problem, we exploit data types that are closer to machine representations and thus practically more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Based on these, we implement a “low-level” version of the function high_enum_algo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='2, we describe how we relate high-level data structures with low-level ones by combining refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Finally, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='3, we deal with the proof of equivalence of the low-level implementa- tion of the function high_enum_algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='1 Low-Level Implementation The main data types used in the low-level implementation of the certification algorithm are the following: (i) the type int of 63-bits integers (module Int63 in Coq) built on OCaml integers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' (ii) the type array of persistent arrays (module PArray in Coq) built on OCaml arrays and based on the ideas of [15, Section 2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' (iii) the type bigQ that represents arbitrarily large rationals (module BigQ of Coq library bignums) built on sequences of words based on 63-bits integers [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The type bigQ is useful to manipulate polyhedra in which the numerical entries of the inequality system or of the ver- tices can be very large rationals, as in the counterexamples to the Hirsch conjectures that we deal with in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' As we describe next, persistent arrays are involved in various data structures in the low-level implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We point out that we choose persistent arrays over Coq AVL trees, because our early experiments have shown that the latter suffer performance issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Indeed, every tree comes with a proof term for balancing, and this term can grow excessively on large instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Xavier Allamigeon, Quentin Canu, and Pierre-Yves Strub Vectors and matrices with rational entries are implemented using the types array bigQ and array (array bigQ) respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' A system of inequalities defining a polyhedron is then represented by a term of type polyType := array (array bigQ) * array bigQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Bases, which are sets of row indices, are encoded with the type array int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' More precisely, the el- ements of a basis are collected in a sorted array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This allows us to compute the intersection of two bases in linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Graphs whose vertices are labeled with some type t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', the low-level counterpart of graphs of type graph t, are implemented using pairs of the form (g, lbl), where g : array (array int) and lbl : array t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In more details, the vertices of a low-level graph are indexed by integers (of type int), and the term lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' [i] corresponds to the label of the vertex of index i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The term g represents the adjacency array of the graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='[i] is the array containing the indices of the neighbors of the vertex of index i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We use the nota- tion graph_struct := array (array int).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The indexing of vertices is made in such a way that the labels in the array lbl are sorted in nondecreasing order (to this extent, we introduce a total order relation over labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This ensures that every label occurs only once in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Using these data structures, we build a low-level imple- mentation, called enum_algo, of the function high_enum_algo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' see Module enum_algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The function enum_algo is a trans- parent adaptation of high_enum_algo on low-level data struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In particular, every test in high_enum_algo has a coun- terpart on the low-level side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Unlike high-level data struc- tures based on dependent types, our low-level data structures do not come with well-formedness invariants for free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In- stead, these invariants have to be checked by extra functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' For instance, we need to verify that arrays representing vec- tors and matrices have a specific size (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', the dimension of the ambient space, or the number of inequalities), that all the vertex indices appearing in a graph belong to the right range, and that arrays representing sets (such as bases or graph labels) are sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The advantage of our certification algorithm is that we do not need to prove that such invari- ants are preserved throughout the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Indeed, owing to the simplicity of the algorithm, data structures are only accessed for reading, and no new structure is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In consequence, checking the consistency of data structures occurs only once, before the call to the function enum_algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Once enum_algo has verified that a low-level graph is the (low-level representation of) the lex-graph of the polytope, it remains to deal with the low-level computation of the image of the lex-graph by the function phi defined in Theorem repr_poly_graph (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This is the final step to get the vertex-edge graph of the polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' While the image of a graph is a basic construction for high-level graphs, the problem is slightly more involved on the low- level side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Once again, we rely on certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In more details, we define a function img_lex_graph that takes as input two graphs g_lex g_vert : graph_struct and their respective labelings lbl_lex and lbl_vert, and checks that (g_vert, lbl_vert) is the image of (g_lex, lbl_lex) by the low-level counterpart low_phi := fun u => (u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' [0] of the function phi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We give a short description of the implementation of this function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Additional certificates morph, morph_inv and edge_inv are provided to the function img_lex_graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The term morph : array int corresponds to a mapping between the indices of the vertices of the two graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The function img_lex_graph first checks that this mapping is consistent with the function low_phi over labels, in the sense that, for every index i of g_lex, low_phi lbl_lex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' [i] is equal to lbl_vert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='[morph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' [i ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The remaining part of img_lex_graph consists in verifying that morph induces a graph morphism between the two inci- dence structures described by g_lex and g_vert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' To this pur- pose, the algorithm checks that morph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' [i] < length g_vert for all indices i, which ensures morph to be well-formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Then, it makes sure that the mapping morph is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This is done by exploiting the certificate morph_inv : array int in- tended to be a right-inverse of morph, and by checking that for all indices i of g_vert, we have morph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='[morph_inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' [i]] == i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The algorithm then proceed with edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' It checks that if i j : int are adjacent in g_lex, then morph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' [i] and morph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' [j] are adjacent in g_vert as well, unless morph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' [i] == morph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' [j] (by definition of the image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=" Conversely, it exploits the third certificate edge_inv to verify for any two adjacent vertices i', j' of g_vert, there exist two adjacent vertices i, j of g_lex such that morph." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=" [i'] == i and morph." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=" [j'] == j." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='2 Data Refinements In order to prove that the low-level implementations are correct w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' the high-level ones, we follow the approach introduced in the project CoqEAL [14, 19] and use data re- finements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Given a high-level type T and the corresponding low-level type t, a refinement r : t -> T -> Prop is a re- lation between terms that respectively correspond to the low-level and the high-level representation of a same object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In this setting, two functions are equivalent if they return related outputs given related inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This is formalized as follows: Definition rel_func (r1 : t -> T -> Prop) (r2 : u -> U -> Prop) (f : t -> u) (g : T -> U) := (forall x y, r1 x y -> r2 (f x) (g y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We use the notation (r1 =~> r2) f g for rel_func r1 r2 f g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The tree in Figure 5 describes the combination of refine- ment relations performed in order to relate low-level and high-level representations of lex-graphs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=", graphs over 5We warn that the preimages of i' and j' provided by morph_inv may not be adjacent in g_lex, which is why we need the additional certificate edge_inv." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=" A Formal Disproof of the Hirsch Conjecture bigQ ∼ rat array bigQ ∼ array rat array (array bigQ) ∼ array (array rat) array (array T) ∼ 'M[T]_(p,q) array (array bigQ) ∼ 'M[rat]_(n,1+m) rel_array rel_array rel_comp int ∼ 'I_m array int ∼ array 'I_m array T ∼{set T} array int ∼{set 'I_m} rel_array rel_comp low_lbl := array int * array (array bigQ) ∼ high_lbl := {set 'I_m} * 'M[rat]_(n,1+m) rel_couple graph_struct * array low_lbl ∼ graph high_lbl rel_graph_r Figure 5." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Construction of the refinement relation rel_lex_graph between the low-level and high-level representations of lex-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We make use of the notation t ∼ T to state that there is a refinement relation between the low-level type t and the high-level type T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' pairs of bases and matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' (The formalization of the rela- tions is carried out in the module refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=') We com- ment on this tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The leaves of the tree correspond to refinement relations between atomic types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' For every such relation, we need to prove that the implementations of basic operations over the low-level and the high-level types are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=" For in- stance, the refinement rel_int_ord : int -> 'I_m -> Prop relates computationally efficient integers with MathComp ordinals of type 'I_m (i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', unary integers less than a fixed integer m), and we prove the equivalence of between the low-level and high-level ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This states as: Lemma rel_int_ord_lt : (rel_int_ord =~> rel_int_ord =~> eq) (fun i j : int => (i (i < j)%nat) where eq is the identity relation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=" Other basic refinements include the relation rel_array_set : array T -> {set T} > Prop between the implementation of finite sets with ar- rays and the corresponding MathComp type {set T}, or the relation rel_mx_col : array (array T) -> 'M[T]_(p,q) -> Prop between array-based matrices and MathComp matri- ces of size p × q." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Finally, we use the refinement relation rat_bigQ : bigQ -> rat -> Prop implemented in CoqEAL between the type bigQ of computationally efficient rationals and the MathComp type rat of rationals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The edges of the tree correspond to functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The latter allow us to combine refinements in order to construct more complex relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' For example, the functor rel_array lifts a refinement r : t -> T -> Prop to another between the types array t and array T: Definition rel_array r a A := length a = length A /\\ forall i, i < length a -> r a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' [i] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' [i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' It comes with the proof of equivalence for several basic prim- itives, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=", computing the image arr_map f a of some array a by some function f: Lemma rel_array_map r r' f F : (r =~> r') f F -> (rel_array r =~> rel_array r') (arr_map f) (arr_map F)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=" Similarly, the functor rel_couple combine two refinement re- lations r : t -> T -> Prop and r' : t' -> T' -> Prop into a refinement relation between the product types t * t' and T * T'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=" A third functor rel_comp allows us to compose a refinement relation between t and T from two refinement re- lations r : t -> tT -> Prop and r' : tT -> T -> Prop." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=" For example, the relation between the type array (array bigQ) of low-level rational matrices and the type 'M[rat]_n of high- level square matrices is provided by composing the relation between low-level matrices of bigQ and rat with the relation rel_mx_col previously discussed instanced with T := rat." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' As shown in Figure 5, the refinement relation between low- level and high-level lex-graphs is obtained by applying the functor rel_graph_r to the refinement relation between the types low_lbl and high_lbl (that respectively correspond to the low-level and high-level representations of lex-graph la- bels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The functor rel_graph_r is described in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Given a relation between a low-level type t and a high-level type T, it builds a refinement relation between the type graph_struct array t of low-level graphs labeled by t and high-level graphs labeled by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' It is built by applying the functors pre- viously described to two atomic refinement relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The first one, called rel_structure, relates the low-level type graph_struct := array (array int) of incidence arrays to the type graph int high-level graphs over (low-level) inte- gers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The second one relates the type graph int * array T, which corresponds to a mixed representation of graphs in Xavier Allamigeon, Quentin Canu, and Pierre-Yves Strub t ∼ T array t ∼ array T graph_struct ∼ graph int graph_struct * array t ∼ graph int * array T graph int * array T ∼ graph T graph_struct * array t ∼ graph T rel_array rel_couple rel_comp Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Construction of the functor rel_graph_r between low-level and high-level graphs, from a refinement relation t ∼ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' which the incidence structure is stored separately from the mapping to labels, to the type graph T of high-level graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='3 Proof of Equivalence We are now ready to prove the equivalence between the low- level and high-level implementations of the lex-graph certifi- cation algorithm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', enum_algo and high_enum_algo respec- tively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' see Module enum_equiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=" It involves the refinement relation rel_poly between low-level and high-level repre- sentations of the inequality system defining the polyhedron (between polyType and the type 'M[rat]_(m,n) * 'cV[rat] _m of pairs (A,b)), and the refinement relation rel_lex_graph between low-level and high-level lex-graphs described in Section 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The equivalence is stated as follows: Lemma lex_certif_equiv : (rel_poly =~> rel_lex_graph =~> eq) enum_algo high_enum_algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' If we put aside straightforward program transformations, the proof essentially consists in unfolding the definition of every test in the functions enum_algo and high_enum_algo, and to use the equivalence between the basic operations involved on the low-level and high-level sides, such as lexicographic comparison of row vectors, matrix-vector multiplication, or set intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Analogously, the correctness of the low-level function img_lex_graph certifying the image of the lex-graph by the function phi is given by the following statement: Lemma img_lex_graph_equiv (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=') : rel_lex_graph (g_lex, lbl_lex) G_lex -> rel_vert_graph (g_vert, lbl_vert) G_vert -> img_lex_graph morph morph_inv edge_inv g_lex lbl_lex g_vert lbl_vert -> G_vert = phi @/ G_lex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' where rel_vert_graph is an instantiation of the rel_graph_r functor for graphs labelled with vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' (We point out that this statement is not an equivalence because of the addi- tional certificates morph, morph_inv and edge_inv provided to img_lex_graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=') In practice, the certificates provided to the algorithms are written using low-level types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Indeed, as explained in Section 6, these certificates can be very large, and it would impossible to compile them using high-level types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In order to exploit the correctness statements that we previously discussed, every refinement relation rel_t_T : t -> T -> Prop comes with an extra function spec_t_T : t -> T which builds a high-level term from a well-formed low-level one, so that we have r x (spec_t_T x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In this way, we arrive at the following statement, which Theorem Validation Po (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=') : (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=') enum_algo Po g_lex lbl_lex -> img_lex_graph morph morph_inv edge_inv g_lex lbl_lex g_vert lbl_vert -> poly_graph (poly_of_syst (spec_poly Po)) = spec_vert_graph (g_vert, lbl_vert).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' It takes as input the low-level certificates as well as the hypotheses that they are certified by the low-level imple- mentation of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' It proves that, for the high-level polyhedron poly_of_syst (spec_poly Po) described by the low-level inequality system Po, the (high-level) vertex-edge graph corresponds to the low-level graph (g_vert, lbl_vert ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The proof is a combination of the correctness of the high- level algorithm (Theorem repr_poly_graph) with that of the low-level implementations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', Lemma lex_certif_equiv and Lemma img_lex_graph_equiv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' 5 Formal Disproof of Hirsch Conjecture The purpose of this section is to describe how we arrive at the formal disproof of the Hirsch Conjecture using the certification algorithms and their low-level implementations described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Recall that the Hirsch conjecture makes use of three notions: the diameter of the vertex-edge graph, the number of facets, and the dimension of the poly- tope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We disprove the conjecture by computing lower bounds on the diameter and the dimension and an upper bound on the number of facets, to conclude by transitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We first explain how we deal with the computation of each quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Considering a polyhedron P = {𝑥 ∈ R𝑛 : 𝐴𝑥 ≥ 𝑏} where 𝐴 ∈ R𝑚×𝑛 and 𝑏 ∈ R𝑚, it is well known that the number of facets is bounded by the number 𝑚 of inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This statement has been added to the properties of polyhedra established in Coq-Polyhedra (see Module poly_base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content="v), and is expressed as follows: Lemma facets_le {base : base_t[R,n]}: A Formal Disproof of the Hirsch Conjecture (#|` facets 'P(base) | <= #|` base|)%nat." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=" The term base correspond to a set of inequalities defining the polyhedron, and 'P(base) is the term representing the polyhedron." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' To deal with the dimension, we establish that the (infor- mal) counterexamples to the Hirsch conjecture have dimen- sion equal to𝑛 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', the dimension of the ambient space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This is achieved by exhibiting a set of 𝑛+1 vertices 𝑣0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' , 𝑣𝑛 ∈ R𝑛 of the polytope that are affinely independent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', the matrix 𝑀 = �𝑣1 − 𝑣0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' 𝑣𝑛 − 𝑣0� is nonsingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' These points are provided by means of (low-level) certificates map_lbl : array int and origin : int which represent the set of their indices in the low-level vertex-edge graph, along with an informally computed low-level matrix inv_lbl that we verify to be the inverse of the matrix 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' On the low-level side, the verification of these certificates is performed by the function dim_full_test (in enum_algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v), and the correctness state- ment is expressed as follows (see enum_high_algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v): Lemma high_dim_h : dim_full_test map_lbl origin inv_lbl -> \\pdim P = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' where \\pdim P stands for the dimension of the polytope P shifted by one (Coq-Polyhedra uses the convention that the emptyset have dimension 0, points have dimension 1, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Finally, the lower bound on the diameter is computed by providing a vertex 𝑣 of the graph whose eccentricity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', the maximal distance between 𝑣 and any other vertex 𝑤, reaches the value of the diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In order to compute the eccen- tricity, we exploit the Breadth-First Search (BFS) algorithm, and implement it on low-level and high-level data structures in the module diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The low-level implementation is provided by the function Low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='BFS : graph_struct -> int > NArith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='BinNat which takes as input a low-level incidence array and the index of a vertex, and returns the maximal length of shortest paths from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Similarly, the high-level implementation is the function High.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='BFS working with the type graph T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The correspondence with the low-level imple- mentation is expressed as follows: Lemma rel_struct_BFS (g : graph_struct) (G : graph int): rel_structure g G -> forall x, mem_vertex g x-> Low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='BFS g x = High.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='BFS G x :> nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We are now ready to formally disprove the conjecture, by running our certification algorithms on the two coun- terexamples to the Hirsch conjecture provided in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This is done in the directories test/data/poly20dim21 and test /data/poly23dim24 respectively (we refer to the README file to extract them from the zipped archives in test/archives ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We exploit the explicit inequality representations of the counterexamples provided by Weibel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='6 They take the form 6https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='com/site/christopheweibel/research/hirsch- conjecture of input files for the library lrslib [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We use this library together with additional Python scripts to generate the in- formal certificates for our algorithms (they are stored in test/data/polyXXdimYY/coq/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' As explained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='3, the latter are provided using low-level types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' We give a short description of every certificate: poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v contains the description of the polytope by inequalities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' g_lex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v and lbl_lex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' g_vert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v and lbl_vert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' v) are intended to represent the lex-graph (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' the vertex-edge graph);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' morph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v, morph_inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v and edge_inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v are the certifi- cates provided to the function img_lex_graph (see Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' map_lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v, origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v and inv_lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v are the certificates for the dimension required by dim_full_test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v provides certificates to check that the input polyhedron is a polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' These certificates are used to show that nonnegative combinations of the inequal- ities defining the polyhedron yield inequalities of the form −𝐾 ≤ 𝑥𝑖 ≤ 𝐾 for all 𝑖 ∈ [𝑛], where 𝐾 is a suffi- ciently large constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The verification is performed by the function bounded_Po_test (module enum_algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v is the index of vertex that is used to get a lower bound on the diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Each file is compiled using coqc, and then imported in order to get the formal disproof of the conjecture in test/data/ polyXXdimYY/coq_Hirsch/Hirsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content="v: Theorem Hirsch_was_wrong : exists (d : nat) (P : 'poly[rat]_d), (High." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='diameter (poly_graph P) > #|`facets P| - (\\pdim P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='-1)%nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' pose P := poly_of_syst (A, b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=" exists n'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='+1, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' apply/disprove_Hirsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' exact: well_formedness_ok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' exact: enum_algo_ok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' exact: img_graph_ok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' exact: bounded_Po_test_ok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' exact: dim_full_test_ok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' exact: diameter_check_ok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Qed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' (Note that the statement use (pdim P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='-1 for the dimension because of the shift-by-one convention in Coq-Polyhedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=') The proof starts by exhibiting the witness of the existential statements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=', the polyhedron provided in poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' It then applies Theorem disprove_Hirsch which establishes that the conjecture does not hold provided that all certification tests return true (see file theories/enum_equiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' These latter hypotheses are then verified in the last six lines of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' They respectively correspond to the verification of (i) the well-formedness of all input certificates, (ii) the lex-graph, Xavier Allamigeon, Quentin Canu, and Pierre-Yves Strub (iii) the vertex-edge graph, (iv) the boundedness of the poly- hedron, (v) the dimension of the polyhedron, (vi) the lower bound on the diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Every test is achieved by using the tactic vm_compute of Coq, which computes the (Boolean) re- sult of each test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' As explained in the introduction, the approach of [39], also used in [36], builds non-Hirsch polytopes by lifting special low-dimensional spindles to higher dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' This only provides a lower bound on the diameter of the non-Hirsch polytopes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' see [39, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In addition to verifying the lower bound, we formally certify the exact value of the diameter of the 20- and 23-dimensional counterexamples of [36]: Theorem poly20dim21_diameter : diameter (poly_graph poly20dim21) = 21%nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Theorem poly23dim24_diameter : diameter (poly_graph poly23dim24) = 24%nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Since the two polytopes respectively have 40 and 46 facets, this entails that their diameter matches the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' As far as we know, this is the first proof of this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' 6 Practical Experiments For the sake of reproducibility, experiments have been con- ducted on two different architectures: (i) a machine with an Apple M1 processor and 32 GB RAM running Mac OS 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='1 (Architecture A), (ii) a machine with a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='3 GHz Intel Core pro- cessor and 64 GB RAM running Linux 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='4 (Architecture B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Both use Coq 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' As described in the introduction, the two counterexam- ples of [36] are very involved polytopes, with several tens of thousands vertices and hundreds of thousands edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' More- over, the vertices have long rational coefficients, with up to 40 digits long numerator and denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' As a conse- quence, explicitly writing the certificates in plain files lead to very large .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v files whose compilation is a challenging (if not unrealistic) task for Coq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' For instance, for the 23- dimensional counterexample, we need about 600 MB to store the term lbl_lex containing the labels of the lex-graph in such a plain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content='v file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' In this case, the memory used by coqc during the compilation of the file turns out to be the main limiting factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' The compilation runs out of memory and fails with Architecture B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' Thanks to a better memory man- agement (memory compression and use of swap memory), it succeeds on Architecture A, but it takes more than 8 000 s to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfsgho/content/2301.04060v1.pdf'} +page_content=' To overcome these compilation issues, we have implemented a Coq plugin called BinReader7 that provides a command LoadData 1034 erg/s) are kept for about 104 years, but then quickly drop +below 1032 erg/s (and therefore objects become undetectable) after +a few tens of kyr. Simulations using "old" non-magnetized models +(Gudmundsson et al. 1983) do not allow to capture this behaviour. +4 CONCLUSIONS +Understanding how young and middle-aged magnetars cool is of great +importance for a correct interpretation of the observational data. In +this work, we have revisited the cooling curves of NSs, focusing +on the effect of the assumed envelope models (typically used as a +boundary condition), and considering two extreme field topologies, +crustal and core field. We noticed that the 𝑇𝑏 − 𝑇𝑠 relation is very +sensitive to the magnetic field strength. Although for relatively low +magnetic fields, different magnetized iron envelopes predict similar +effective surface temperatures. For relatively strong magnetic fields +in the magnetar range there are substantial differences. For a given +temperature at the base of the envelope (𝑇𝑏), the most recent models +(that incorporate better microphysics) predict surface temperatures +(𝑇𝑠) about a factor 2-3 higher than their predecessors. Since the +flux scales as 𝑇4𝑠 , this correction significantly enlarges the photon +luminosity, which in turn leads to a very fast transition from a high +luminosity epoch (during the neutrino cooling era) to a very low +luminosity (soon after we enter the photon cooling era). This trend is +very pronounced in models where the magnetic field threads the star’s +core and most electric currents circulate there. Conversely, in crustal- +confined models, the additional energy released by Joule heating +close to the star surface is very effective and governs the energy +balance equation, which counterbalances the effect. Thus, depending +on where the bulk of the electrical currents circulates, one can expect +middle-aged magnetars which are relatively bright or sources with +very low luminosities (< 1032 erg/s) which persistent emission is +essentially undetectable as X-ray sources. We stress again that a 104 +yr, high field NS with a core field (light and heavy element envelopes) +can actually be much cooler than a similar NS with a pulsar-like +field, only because of the effect of magnetic field in the envelope (see +Fig. 4 upper-right panel). This has potentially strong implications for +population synthesis studies of the pulsar and magnetar populations +because observational biases introduced by the lack of detectability +of some class of sources affect the predictions of birth rates and field +distributions. We plan to incorporate these effects in future works. +To briefly compare our results with observational data, one should +only concentrate on objects with "Real ages" and that are at the +extremes of our cooling curves: A) 1E 2259+586 (middle-aged mag- +netar) can only be explained with a crustal-field and magnetized light +elements. B) All XDINS cannot be explained with core-fields. They +necessarily need that the crustal-field has a strong component but the +envelope can be light or heavy, magnetic or non-magnetic. C) CCOs +are in an age and luminosity range that do not allow distinguishing +between envelope models or magnetic topology. D) Middle-age faint +pulsar such as PSR B2334+61 might be explained only with the fast +decay of the light element envelope curves, since for low magnetic +field NSs, light envelopes might produce cooler NSs than heavy ele- +ments for older ages (> 104 yr), regardless of the field configuration +(see Fig. 4 bottom panels). Ultimately, the existence of strongly mag- +netized neutron stars with detectable thermal emission at later times +would be a strong argument in favor of a crustal magnetic field. +Our study highlights the importance of treating carefully all in- +gredients in the complex theory of NS cooling. Boundary conditions +neglecting the role of the envelope, or using non-magnetized en- +velopes, can lead to discrepancies as large as one order of magnitude +relative to observational data. On the other hand, an accurate esti- +mation of surface luminosity is important to constrain any source +property (e.g. surface B-field or age). +ACKNOWLEDGEMENTS +JAP acknowledges support from the Generalitat Valenciana grants +PROMETEO/2019/071 and ASFAE/2022/026 (with funding from +NextGenerationEU PRTR-C17.I1) and the AEI grant PID2021- +127495NB-I00. CD and NR are supported by the ERC Consolida- +tor Grant “MAGNESIA” No. 817661 (PI: Rea) and this work has +been carried out within the framework of the doctoral program in +Physics of the Universitat Autònoma de Barcelona and it is partially +MNRAS 000, 1–5 (2022) + +How bright can old magnetars be? +5 +102 +103 +104 +105 +106 +Time (yr) +1032 +1033 +1034 +1035 +Thermal luminosity (erg s 1) +Crustal Field +Heavy-env, no B +Light-env, no B +Light-env, B +Heavy-env, B +102 +103 +104 +105 +106 +Time (yr) +1032 +1033 +1034 +1035 +Thermal luminosity (erg s 1) +Core Field +102 +103 +104 +105 +106 +Time (yr) +1032 +1033 +1034 +1035 +Thermal luminosity (erg s 1) +Crustal Field +102 +103 +104 +105 +106 +Time (yr) +1032 +1033 +1034 +1035 +Thermal luminosity (erg s 1) +Core Field +1013 +1014 +1015 +Surface dipolar B-field at pole (Gauss) +Figure 4. Luminosity curves of four studied envelope models: Heavy-env, no B (Gudmundsson et al. 1983), Light-env, no B (Potekhin et al. 1997), Light-env, B +(Potekhin et al. 2003) and Heavy-env, B (Potekhin et al. 2015). On the left-hand side, we show the results of models with crustal-confined magnetic field, and on +the right, those with core-dominant field topology. The results are represented at two initial magnetic field intensities at the polar surface), e.g., at 𝐵 = 5 × 1014 +G (upper panels) and at 𝐵 = 1013 G (bottom panels). +supported by the program Unidad de Excelencia María de Maeztu +CEX2020-001058-M. DV is supported by the European Research +Council (ERC) under the European Union’s Horizon 2020 research +and innovation programme (ERC Starting Grant "IMAGINE" No. +948582, PI: DV). +DATA AVAILABILITY +Data available on request. +REFERENCES +Aguilera D. N., Pons J. A., Miralles J. A., 2008, The Astrophysical Journal, +673, L167 +Akgün T., Cerdá-Durán P., Miralles J. A., Pons J. 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Y., Gnedin O., Chabrier G., 2004, +Astronomy & Astrophysics, 417, 169 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–5 (2022) + diff --git a/UtE0T4oBgHgl3EQfVQBb/content/tmp_files/load_file.txt b/UtE0T4oBgHgl3EQfVQBb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f8347079335ec330f8709d1ed12142498b3fc631 --- /dev/null +++ b/UtE0T4oBgHgl3EQfVQBb/content/tmp_files/load_file.txt @@ -0,0 +1,450 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf,len=449 +page_content='MNRAS 000, 1–5 (2022) Preprint 9 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='0 How bright can old magnetars be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Assessing the impact of magnetized envelopes and field topology on neutron star cooling Clara Dehman,1,2★ José A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Pons,3 Daniele Viganò,1,2,4 Nanda Rea1,2 1Institute of Space Sciences (ICE-CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193, Barcelona, Spain 2Institut d’Estudis Espacials de Catalunya (IEEC), Carrer Gran Capità 2–4, 08034 Barcelona, Spain 3Departament de Física Aplicada, Universitat d’Alacant, 03690 Alicante, Spain 4Institute of Applied Computing & Community Code (IAC3), University of the Balearic Islands, Palma, 07122, Spain Accepted 2023 January 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Received 2022 December 23;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' in original form 2022 November 4 ABSTRACT Neutron stars cool down during their lifetime through the combination of neutrino emission from the interior and photon cooling from the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Strongly magnetised neutron stars, called magnetars, are no exception, but the effect of their strong fields adds further complexities to the cooling theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Besides other factors, modelling the outermost hundred meters (the envelope) plays a crucial role in predicting their surface temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In this letter, we revisit the influence of envelopes on the cooling properties of neutron stars, with special focus on the critical effects of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' We explore how our understanding of the relation between the internal and surface temperatures has evolved over the past two decades, and how different assumptions about the neutron star envelope and field topology lead to radically different conclusions on the surface temperature and its cooling with age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In particular, we find that relatively old magnetars with core-threading magnetic fields are actually much cooler than a rotation-powered pulsar of the same age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' This is at variance with what is typically observed in crustal-confined models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Our results have important implications for the estimates of the X-ray luminosities of aged magnetars, and the subsequent population study of the different neutron star classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Key words: stars: neutron – stars: magnetars – stars: interiors – stars: magnetic field – stars: evolution 1 INTRODUCTION It has long been hoped that observations of direct thermal emis- sion from the surface of neutron stars (NSs), confronted to theoreti- cal cooling curves (the temperature-age or luminosity-age relation), could yield valuable information about star interior, such as the nu- clear equation of state and chemical composition (Yakovlev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Pons & Viganò 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' However, NSs are also known to be endowed with strong magnetic fields, and therefore an appropriate treatment of the coupled ther- mal and magnetic field evolution in detail is of great importance to understand the observed emissions from the surface of NSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In this respect, recent works have devoted a significant effort to extend re- alistic simulations to 3D (Wood & Hollerbach 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Gourgouliatos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' De Grandis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Igoshev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2021a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Dehman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Within this context, the relevance of envelope models is usually overlooked, since it enters in multidimensional simulations "only" as a boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' However, as we will show, it actually plays a key role to connect the internal properties with the observable quantities (effective temperature, luminosity), especially for highly magnetised objects, such as magnetars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The very different thermal relaxation timescales of the envelope and the crust of NSs make computationally unfeasible any attempt ★ E-mail: c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='dehman@csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='es to perform cooling simulations in a numerical grid that includes all layers up to the star surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Thus, the usual approach is to compute envelope models separately, and then use a phenomenological fit predicting the value of the local surface temperature (𝑇𝑠) as a function of the temperature at the base of the envelope (𝑇𝑏), to be used as a boundary condition (the 𝑇𝑏 − 𝑇𝑠 relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Among the many early studies of the thermal structure of NSs, we must mention the seminal works of Tsuruta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (1972), Gudmunds- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (1983), Hernquist (1985) or Schaaf (1990), who pointed out that regions with tangential magnetic field are much colder than the regions where the field is nearly radial (see Yakovlev & Kaminker (1994) and references therein for a review of the early works).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Later, Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (1997) constructed a more general fit valid for different compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' These envelope models used improved calculations on the equation of state and opacities in the outer NS layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In particular, the so-called accreted envelopes contain layers of different chemical elements (H, He, C, O shells) created from accreted matter from the supernova fallback material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Page (1994) and Page & Sarmiento (1996) were the first to de- scribe realistic surface temperature distributions with dipolar and dipolar+quadrupolar magnetic fields, the latter presenting "𝑇𝑏 − 𝑇𝑠" relationships with such configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The thermal structure of NSs with magnetized envelopes was also studied by Potekhin & Yakovlev (2001), and later improved in Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' It includes the effect of magnetic fields on the 𝑇𝑏 −𝑇𝑠 relation, providing analytical fits valid for a magnetic field strength up to 1016 G and arbitrary © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='02261v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='HE] 5 Jan 2023 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Dehman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 𝑇𝑏 −𝑇𝑠 relations of different envelope models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The non-accreted models are illustrated on the left and the fully accreted ones in the panel on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The studied envelopes are: (i) non-magnetised (in black) (Gudmundsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 1997), and (ii) magnetised (in color) (Potekhin & Yakovlev 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Pons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' For the latter, we consider two different values of a purely radial magnetic field strength (then, suitable for a polar 𝑇𝑠 if the topology is a simple dipole): 𝐵 = 1013 (in blue) and 1015 G (in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' inclination angles of the field lines with respect to the normal to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Similar studies exploring other field topologies were done by Geppert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (2004, 2006) and Pérez-Azorín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Subsequent calculations in Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (2007) included the effect of the neutrino emissivity in the outer crust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Pons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (2009) re- visited the magnetized envelope problem with two motivations: (i) upgrading the microphysical inputs (thermal conductivity) because the contribution of ions or phonons to the thermal conductivity of the envelope can reduce the anisotropy of heat conduction (Chugunov & Haensel 2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (ii) estimating the accuracy of the plane-parallel ap- proximation since its spherical symmetry assumption does not allow meridional heat fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The state-of-the-art models can be found in the thorough review by Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' They present new fits for non-accreted magnetised envelopes, including both the effects of neutrino emission and the effects of non-radial heat transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In this paper, we aim at comparing a set of the different envelope models studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Our main goal is to assess how the evolution of theoretical cooling models for different magnetic fields intensities and geometries are affected by the choice of the envelope and its treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In light of this analysis, to determine under which circumstances we can use observational X-ray data to constrain the cooling models, and consequently the NS parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The letter is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In §2, we recap the 𝑇𝑏 − 𝑇𝑠 re- lation of the different envelope models existing in the literature with different magnetic field intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In §3, we perform cooling simula- tions using the last version of our 2D magneto-thermal code (Viganò et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' We examine crustal-confined and core-dominant field topologies considering both iron and fully-accreted light envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' We discuss our results and draw our main conclusions in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2 ENVELOPE MODELS Since NSs are observed both as isolated sources or as part of binary systems, it is common to consider two different compositions for the envelope: either iron, arguably expected in the case of catalyzed matter in isolated systems, or light elements, mainly thought as prod- ucts of accretion from a companion star or in a newly born systems that witness fall-back accretion after a supernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In this study, we explore four models composed of iron (non-accreted matter) and two models of fully accreted (light) envelopes (Gudmundsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Potekhin & Yakovlev 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Aguilera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Pons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In essence, an envelope model is simply a stationary solution for the heat transfer equation and it is then fitted to give an empirical relation between the surface temperature 𝑇𝑠, which determines the radiation flux, and the interior temperature 𝑇𝑏 at the crust/envelope boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The location of 𝑇𝑏 is generally chosen to correspond to some density between the neutron drip point 𝜌 = 3 × 1011 g cm−3 and 𝜌 = 1010 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' At such a low density, the neutrino emission is usually negligible, as long as 𝑇𝑏 < 109 K (which happens very soon, only a few decades after the NS birth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Thus, we have omitted corrections due to neutrino emissivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The 𝑇𝑏 −𝑇𝑠 relations of the studied envelope models are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In the panel on the left we show the iron envelopes, whereas the light ones are displayed on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The studied envelopes are: (i) two models with no magnetic field dependence (in black), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=', Gudmundsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (1983) in the left panel (solid lines) and Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (1997) in the panel on the right (dots);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (ii) other magnetised envelopes, for which we show the 𝑇𝑠 for two values of a purely radial surface magnetic field strength (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=', suitable for a pole in a dipolar topology): 𝐵 = 1013 G (in blue) and 𝐵 = 1015 G (in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Potekhin & Yakovlev (2001) is illustrated with dots (left panel), Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (2003) with dashed lines (right panel), Pons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (2009) with dashed lines (left panel), and finally Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (2015) with dashdotdotted lines (left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The lowest effective temperature 𝑇𝑠 among all models is displayed by Gudmundsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Every new effect incorporated in later works (composition, magnetic field) results in a higher predicted surface temperatures 𝑇𝑠 for a given 𝑇𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Let us briefly review the main conclusions from a quick comparison of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In general, it is well known that assuming light-elements envelopes appreciably affect the NS luminosity (Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Com- pared to iron models, we have a higher𝑇𝑠 for the same𝑇𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Concerning magnetic fields, as long as the average intensity is 𝐵 ≲ 1013 G, we expect a surface temperature similar to the non-magnetised case (for the same given composition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' On the other hand, for magnetar condi- tions, the general trend is that higher fields lead to have higher surface temperature, everything else being equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Interestingly, the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 1 shows that the more recent calculations (incorporating more accurate physics) have revised the predicted 𝑇𝑠 to higher values than any of the previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' MNRAS 000, 1–5 (2022) B=0 B = 1013G B = 1015GB=0 B = 1013G B = 1015GHow bright can old magnetars be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 3 100 101 102 103 104 105 106 Time (yr) 1032 1033 1034 1035 Thermal luminosity (erg s 1) Crustal Field Gudmundsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 1983 Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2001 Pons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2009 Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2015 100 101 102 103 104 105 106 Time (yr) 1032 1033 1034 1035 Thermal luminosity (erg s 1) Core Field 1013 1014 1015 Surface dipolar B-field at pole (Gauss) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Luminosity curves of the four studied iron envelopes (Gudmundsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Potekhin & Yakovlev 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Pons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2015) with an initial magnetic field intensity at the polar surface of 𝐵 = 5 × 1014 G (hereafter, the colorbar indicates its evolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The left panel corresponds to crust-confined topology, whereas the right one to core-dominant field topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Thus, it is expected that the state-of-the-art envelope models pre- dict different cooling curves from the models used two decades ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' This motivates us to revisit the results for cooling curves and consider different magnetic field topologies and strengths, as an important step in understanding the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 3 NEUTRON STAR COOLING MODELS The cooling history of a magnetar is a delicate balance between neutrino and photon emissivity on one side and Joule heating in the star’s crust on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' If the currents are dissipated in the outer crust, the heat deposited is more effectively transported to the surface and has an impact on the star luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' On the contrary, heat dissipated in the inner crust or the core is very inefficient in modifying the surface temperature, because it is essentially lost via neutrino emission, as first discussed in Kaminker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (2006) to explain the high thermal luminosities of magnetars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' To compare the different envelopes existing in the literature, we have used the 2D magneto-thermal code (the latest version is de- scribed in Viganò et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2021) to run a set of cooling models using different initial configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The NS background model is a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='4𝑀⊙ NS built with the Sly41 equation of state (Douchin & Haensel 2001), and we assume the superfluid models of Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (2015), which is the reason for the abrupt change in the slope of the cooling curves at ages ∼ 300 yrs in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=', the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The rapid cooling during the photon cooling era is also caused by the low core heat capacity, which in turn depends on the assumed pairing details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' A compre- hensive revision of the microphysics embedded in magneto-thermal models can be found in Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' We considered two families of magnetic field topologies to study in detail the two extreme configurations: (i) crust-confined field con- sisting of a poloidal dipole and a toroidal quadrupole with steep radial gradients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (ii) core-dominated twisted-torus magnetic fields as in Akgün et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (2017), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=', a dipolar topology, with the currents cir- culating almost only in the core, and Gyr-long decay timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' We stress that, for our purposes, we choose these two extreme topologies mean to cover a wide range of values for the crustal Ohmic dissi- pation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' For each topology, we consider two different field strengths 1 https://compose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='obspm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='fr/ (1013 G and 5 × 1014 G) for the initial value of the dipolar field at the polar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The maximum initial toroidal field is fixed to 1013 G in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The magnetic field at the surface is always matched continuously with a current-free magnetic field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' the electric cur- rents do not leak into the magnetosphere ∇ × 𝑩 = 0, with vanishing field at infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' We have used the different envelope models presented in §2 cou- pled with the NS cooling models, to study the dependence of the NS cooling curves on the assumed envelope, in two given magnetic field topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Magnetar cooling curves obtained using different iron envelopes and a field strength of 𝐵 = 5 × 1014 G are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In the left panel, we consider a crustal-confined topology, and in the right panel we have a core-dominant field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' For a high field in- tensity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=', magnetar-like scenario, there are significant qualitative differences between crustal-confined and core-dominant field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Let us summarize the main findings: At early times, during the neutrino cooling era (say 𝑡 < 104 yr), both models are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The interior temperature evolves in- dependently of the envelope model (photon radiation is negligible), and the different 𝑇𝑏 − 𝑇𝑠 relation translates directly in the surface temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Interestingly, the most recent models show the highest luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' This is a direct consequence of the results of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 1 (left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Later, once we enter the photon cooling era, the situation is inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The envelope models that provide a higher surface tem- perature actually radiate photons (which now govern the evolution) more efficiently, and the star cools down faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In this epoch, the difference between crustal-confined and core- threading magnetic field becomes more evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In the first case, heat dissipation occurs relatively close to the surface, which keeps the stellar crust warmer and delays the drop of the luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In the second case, Joule heating is completely inefficient (currents are mostly in the core), and the effect mentioned above, with a very fast drop of luminosity for high field models becomes evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' To illustrate more clearly these differences, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 3 we compare cooling curves adopting the Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' (2015) envelope but now varying both, the field topology and strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' We show the results with the two different magnetic field intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' For a relatively low magnetic field, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=', 1013 G, the crustal-confined and core-dominated simulations have a very similar behavior and the magnetic field does MNRAS 000, 1–5 (2022) 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Dehman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 102 103 104 105 106 Time (yr) 1032 1033 1034 1035 Thermal luminosity (erg s 1) Crustal Field Core Field 1013 1014 1015 Surface dipolar B-field at pole (Gauss) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Luminosity curves of the latest magnetised iron envelope (Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2015) with different initial magnetic field intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Solid lines corre- spond to crustal field models and dashed-curves to core field ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' not dissipate much (the curves keep the blue color throughout the evolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' For the strong field case, 5 × 1014 G, the crustal-confined models show a significant dissipation of the magnetic field, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=', the magnetic field has dissipated from 5 × 1014 G (red) to a few 1013 G (turquoise) after 1 million year of evolution (colorbar of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' As a consequence, the impact of Joule heating is essential in the crust- confined, while it is almost negligible for the core-dominated model here considered, since the crustal currents are orders of magnitude less intense and the core currents have much longer Ohmic timescales (moreover, the little they dissipate converts into neutrinos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The most relevant difference is that core-threaded field simulations with mag- netized envelopes show faster cooling after 104 yr than low field models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Therefore, the observational appearance of a magnetar at late times essentially depends on where currents are located and how much magnetic flux penetrates the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' We also note that, for a strong enough magnetic field in the core of a NS, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=', 𝐵 = 1015 G, an additional cooling channel via neutrino synchrotron (Kaminker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 1997) is activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' It provides further cooling of the NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' However, we found that this effect is subdominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' We now extend our analysis to accreted (light element) envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The results of the comparison between light and heavy elements are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In the left panels, we display the results for models with crustal-confined magnetic fields and in the right panels those for core-dominant fields, for the two dipolar intensities 𝐵 = 5 × 1014 G (upper panels) and at 𝐵 = 1013 G (bottom panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The same qualitative features discussed for iron envelopes are valid, but with luminosities shifted to slightly higher values (up to an order of magnitude) for accreted envelopes during the neutrino cool- ing epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Instead, it drops faster as soon as we enter in the photon cooling era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' That is due to the even higher 𝑇𝑠 resulting from light el- ements in the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' A strong magnetic field enhances this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In the top-right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 4, we clearly see how high luminosities (> 1034 erg/s) are kept for about 104 years, but then quickly drop below 1032 erg/s (and therefore objects become undetectable) after a few tens of kyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Simulations using "old" non-magnetized models (Gudmundsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 1983) do not allow to capture this behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 4 CONCLUSIONS Understanding how young and middle-aged magnetars cool is of great importance for a correct interpretation of the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' In this work, we have revisited the cooling curves of NSs, focusing on the effect of the assumed envelope models (typically used as a boundary condition), and considering two extreme field topologies, crustal and core field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' We noticed that the 𝑇𝑏 − 𝑇𝑠 relation is very sensitive to the magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Although for relatively low magnetic fields, different magnetized iron envelopes predict similar effective surface temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' For relatively strong magnetic fields in the magnetar range there are substantial differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' For a given temperature at the base of the envelope (𝑇𝑏), the most recent models (that incorporate better microphysics) predict surface temperatures (𝑇𝑠) about a factor 2-3 higher than their predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Since the flux scales as 𝑇4𝑠 , this correction significantly enlarges the photon luminosity, which in turn leads to a very fast transition from a high luminosity epoch (during the neutrino cooling era) to a very low luminosity (soon after we enter the photon cooling era).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' This trend is very pronounced in models where the magnetic field threads the star’s core and most electric currents circulate there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Conversely, in crustal- confined models, the additional energy released by Joule heating close to the star surface is very effective and governs the energy balance equation, which counterbalances the effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Thus, depending on where the bulk of the electrical currents circulates, one can expect middle-aged magnetars which are relatively bright or sources with very low luminosities (< 1032 erg/s) which persistent emission is essentially undetectable as X-ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' We stress again that a 104 yr, high field NS with a core field (light and heavy element envelopes) can actually be much cooler than a similar NS with a pulsar-like field, only because of the effect of magnetic field in the envelope (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 4 upper-right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' This has potentially strong implications for population synthesis studies of the pulsar and magnetar populations because observational biases introduced by the lack of detectability of some class of sources affect the predictions of birth rates and field distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' We plan to incorporate these effects in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' To briefly compare our results with observational data, one should only concentrate on objects with "Real ages" and that are at the extremes of our cooling curves: A) 1E 2259+586 (middle-aged mag- netar) can only be explained with a crustal-field and magnetized light elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' B) All XDINS cannot be explained with core-fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' They necessarily need that the crustal-field has a strong component but the envelope can be light or heavy, magnetic or non-magnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' C) CCOs are in an age and luminosity range that do not allow distinguishing between envelope models or magnetic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' D) Middle-age faint pulsar such as PSR B2334+61 might be explained only with the fast decay of the light element envelope curves, since for low magnetic field NSs, light envelopes might produce cooler NSs than heavy ele- ments for older ages (> 104 yr), regardless of the field configuration (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 4 bottom panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Ultimately, the existence of strongly mag- netized neutron stars with detectable thermal emission at later times would be a strong argument in favor of a crustal magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Our study highlights the importance of treating carefully all in- gredients in the complex theory of NS cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Boundary conditions neglecting the role of the envelope, or using non-magnetized en- velopes, can lead to discrepancies as large as one order of magnitude relative to observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' On the other hand, an accurate esti- mation of surface luminosity is important to constrain any source property (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' surface B-field or age).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' ACKNOWLEDGEMENTS JAP acknowledges support from the Generalitat Valenciana grants PROMETEO/2019/071 and ASFAE/2022/026 (with funding from NextGenerationEU PRTR-C17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='I1) and the AEI grant PID2021- 127495NB-I00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' CD and NR are supported by the ERC Consolida- tor Grant “MAGNESIA” No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 817661 (PI: Rea) and this work has been carried out within the framework of the doctoral program in Physics of the Universitat Autònoma de Barcelona and it is partially MNRAS 000, 1–5 (2022) How bright can old magnetars be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 5 102 103 104 105 106 Time (yr) 1032 1033 1034 1035 Thermal luminosity (erg s 1) Crustal Field Heavy-env,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' no B Light-env,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' no B Light-env,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' B Heavy-env,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' B 102 103 104 105 106 Time (yr) 1032 1033 1034 1035 Thermal luminosity (erg s 1) Core Field 102 103 104 105 106 Time (yr) 1032 1033 1034 1035 Thermal luminosity (erg s 1) Crustal Field 102 103 104 105 106 Time (yr) 1032 1033 1034 1035 Thermal luminosity (erg s 1) Core Field 1013 1014 1015 Surface dipolar B-field at pole (Gauss) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' Luminosity curves of four studied envelope models: Heavy-env, no B (Gudmundsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 1983), Light-env, no B (Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 1997), Light-env, B (Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2003) and Heavy-env, B (Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' On the left-hand side, we show the results of models with crustal-confined magnetic field, and on the right, those with core-dominant field topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' The results are represented at two initial magnetic field intensities at the polar surface), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=', at 𝐵 = 5 × 1014 G (upper panels) and at 𝐵 = 1013 G (bottom panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' supported by the program Unidad de Excelencia María de Maeztu CEX2020-001058-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' DV is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC Starting Grant "IMAGINE" No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' 948582, PI: DV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQfVQBb/content/2301.02261v1.pdf'} +page_content=' DATA AVAILABILITY Data 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Bongiorno +,13 Raffaella Bonino +,25, 34 Alessandro Brez +,7 +Niccol`o Bucciantini +,42, 43, 44 Fiamma Capitanio +,12 Simone Castellano +,7 Elisabetta Cavazzuti +,45 +Chien-Ting Chen +,20 Stefano Ciprini +,46, 18 Alessandra De Rosa +,12 Ettore Del Monte +,12 +Laura Di Gesu +,45 Immacolata Donnarumma +,45 Victor Doroshenko +,47 Michal Dov˘ciak +,48 +Teruaki Enoto +,49 Yuri Evangelista +,12 Sergio Fabiani +,12 Riccardo Ferrazzoli +,12 Javier A. Garcia +,50 +Kiyoshi Hayashida,51 Jeremy Heyl +,52 Wataru Iwakiri +,53 Svetlana G. Jorstad +,32, 54 Philip Kaaret +,13, 55 +Vladimir Karas +,48 Fabian Kislat +,56 Takao Kitaguchi,49 Jeffery J. Kolodziejczak +,13 +Henric Krawczynski +,57 Simone Maldera +,25 Fr´ed´eric Marin +,58 Andrea Marinucci +,45 Ikuyuki Mitsuishi,59 +Tsunefumi Mizuno +,60 C.-Y. Ng +,37 Chiara Oppedisano +,25 Alessandro Papitto +,19 George G. Pavlov +,61 +Abel L. Peirson +,4 Melissa Pesce-Rollins +,7 Pierre-Olivier Petrucci +,62 Maura Pilia +,41 +Andrea Possenti +,41 Brian D. Ramsey +,13 John Rankin +,12 Gloria Spandre +,7 Douglas A. Swartz +,20 +Toru Tamagawa +,49 Roberto Taverna +,63 Yuzuru Tawara,59 Allyn F. Tennant +,13 Nicholas E. Thomas +,13 +Francesco Tombesi +,64, 46, 65 Alessio Trois +,41 Sergey S. Tsygankov +,27 Roberto Turolla +,63, 39 +Jacco Vink +,66 Kinwah Wu +,39 and Fei Xie +67, 12 +1University of Maryland, Baltimore County, Baltimore, MD 21250, USA +2NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA +3Center for Research and Exploration in Space Science and Technology, NASA/GSFC, Greenbelt, MD 20771, USA +4Department of Physics and Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, California 94305, +USA +5Department of Space Science, University of Alabama in Huntsville, 320 Sparkman Drive, Huntsville, AL 35899 +6Center for Space Plasma and Aeronomic Research, University of Alabama in Huntsville, 320 Sparkman Drive, Huntsville, AL 35899, +USA +7Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy +8Universit`a di Pisa, Dipartimento di Fisica Enrico Fermi, Largo B. Pontecorvo 3, 56127 Pisa, Italy +9Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Strada Comunale Cinthia, 80126 Napoli, Italy +10Department of Physics & Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA +11Dipartimento di Fisica, Universit`a di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy +12INAF Istituto di Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133 Roma, Italy +13NASA Marshall Space Flight Center, Huntsville, AL 35812, USA +14Department of Astronomy and Astrophysics, The Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA +15Finnish Centre for Astronomy with ESO, 20014 University of Turku, Finland +16MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, +MA 02139, USA +17INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica, Via A. Corti 12, I-20133 Milano, Italy +18Space Science Data Center, Agenzia Spaziale Italiana, Via del Politecnico snc, 00133 Roma, Italy +Corresponding author: Michela Negro +mnegro1@umbc.edu +arXiv:2301.01798v1 [astro-ph.HE] 4 Jan 2023 + +ID2 +19INAF Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monte Porzio Catone (RM), Italy +20Science and Technology Institute, Universities Space Research Association, Huntsville, AL 35805, USA +21Department of Physics, Yamagata University, 1-4-12 Kojirakawa-machi, Yamagata-shi 990-8560, Japan. +22Scuola Universitaria Superiore IUSS, Piazza della Vittoria 15, I-27100 Pavia, Italy +23Universit`a di Pisa, Dipartimento di Fisica Enrico Fermi, Largo B. Pontecorvo 3, I-56127 Pisa, Italy +24Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo B. Pontecorvo 3, I-56127 Pisa, Italy +25Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy +26Dipartimento di Matematica e Fisica, Universit`a degli Studi Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy +27Department of Physics and Astronomy, 20014 University of Turku, Finland +28Center for Astrophysics — Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA +29Nordic Optical Telescope, ES-38711 Bre˜na Baja, Spain +30Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark +31Tartu Observatory, University of Tartu, 61602 T˜oravere, Estonia +32Institute for Astrophysical Research, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA +33INAF Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807 Merate (LC), Italy +34Dipartimento di Fisica, Universit`a degli Studi di Torino, Via Pietro Giuria 1, 10125 Torino, Italy +35Yamagata University,1-4-12 Kojirakawa-machi, Yamagata-shi 990-8560, Japan +36International Space Science Institute (ISSI), Hallerstrasse 6, 3012 Bern, Switzerland +37Department of Physics, The University of Hong Kong, Pokfulam, Hong Kong +38Laboratory for Space Research, The University of Hong Kong, Hong Kong +39Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK +40Instituto de Astrof´ısica de Andaluc´ıa—CSIC, Glorieta de la Astronom´ıa s/n, 18008 Granada, Spain +41INAF Osservatorio Astronomico di Cagliari, Via della Scienza 5, 09047 Selargius (CA), Italy +42INAF Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125 Firenze, Italy +43Dipartimento di Fisica e Astronomia, Universit`a degli Studi di Firenze, Via Sansone 1, 50019 Sesto Fiorentino (FI), Italy +44Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, Via Sansone 1, 50019 Sesto Fiorentino (FI), Italy +45ASI - Agenzia Spaziale Italiana, Via del Politecnico snc, 00133 Roma, Italy +46Istituto Nazionale di Fisica Nucleare, Sezione di Roma ”Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy +47Institut f¨ur Astronomie und Astrophysik, Universit¨at T¨ubingen, Sand 1, 72076 T¨ubingen, Germany +48Astronomical Institute of the Czech Academy of Sciences, Bo˘cn´ı II 1401/1, 14100 Praha 4, Czech Republic +49RIKEN Cluster for Pioneering Research, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan +50California Institute of Technology, Pasadena, CA 91125, USA +51Osaka University, 1-1 Yamadaoka, Suita, Osaka 565-0871, Japan +52University of British Columbia, Vancouver, BC V6T 1Z4, Canada +53International Center for Hadron Astrophysics, Chiba University, Chiba 263-8522, Japan +54Department of Astrophysics, St. Petersburg State University, Universitetsky pr. 28, Petrodvoretz, 198504 St. Petersburg, Russia +55Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242, USA +56Department of Physics and Astronomy and Space Science Center, University of New Hampshire, Durham, NH 03824, USA +57Physics Department and McDonnell Center for the Space Sciences, Washington University in St. Louis, St. Louis, MO 63130, USA +58Universit´e de Strasbourg, CNRS, Observatoire Astronomique de Strasbourg, UMR 7550, 67000 Strasbourg, France +59Graduate School of Science, Division of Particle and Astrophysical Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi +464-8602, Japan +60Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan +61Department of Astronomy and Astrophysics, Pennsylvania State University, University Park, PA 16802, USA +62Universit´e Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France +63Dipartimento di Fisica e Astronomia, Universit`a degli Studi di Padova, Via Marzolo 8, 35131 Padova, Italy +64Dipartimento di Fisica, Universit`a degli Studi di Roma ”Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy +65Department of Astronomy, University of Maryland, College Park, Maryland 20742, USA +66Anton Pannekoek Institute for Astronomy & GRAPPA, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The +Netherlands +67Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical Science and Technology, Guangxi University, Nanning +530004, China +ABSTRACT +We present the IXPE observation of GRB 221009A which includes upper limits on the linear polar- +ization degree of both prompt and afterglow emission in the soft X-ray energy band. GRB 221009A + +3 +is an exceptionally bright gamma-ray burst (GRB) that reached Earth on 2022 October 9 after trav- +elling through the dust of the Milky Way. The Imaging X-ray Polarimetry Explorer (IXPE) pointed +at GRB 221009A on October 11 to observe, for the first time, the 2–8 keV X-ray polarization of a +GRB afterglow. We set an upper limit to the polarization degree of the afterglow emission of 13.8% +at a 99% confidence level. This result provides constraints on the jet opening angle and the viewing +angle of the GRB, or alternatively, other properties of the emission region. Additionally, IXPE cap- +tured halo-rings of dust-scattered photons which are echoes of the GRB prompt emission. The 99% +confidence level upper limit of the prompt polarization degree is about 55%, consistent with a scenario +involving synchrotron emission in an ordered magnetic field. This single IXPE pointing provides both +the first assessment of X-ray polarization of a GRB afterglow and the first GRB study with polarization +observations of both the prompt and afterglow phases. +Keywords: GRB — X-ray — polarization +1. INTRODUCTION +Gamma-Ray Bursts (GRBs) are among the most energetic events in the Universe. These events are characterized by +a “prompt” gamma-ray emission, the most luminous phase of the burst, followed by a temporally decaying “afterglow” +that can last for days or even years and is observed across the whole electromagnetic spectrum whenever the needed +sensitivity is available. GRBs are conventionally classified by duration of the prompt phase into the short (< 2 sec) +and long (> 2 sec) class, with distinct physical origins (Galama et al. 1998; Abbott et al. 2017; Fong et al. 2015; Burns +et al. 2021). Long GRBs originates from collapsars (Woosley & Bloom 2006), a rare sub-type of type Ic supernovae. In +the standard model, a fast spinning core collapses into a rapidly spinning black hole which devours some of the massive +star progenitor. This results in a hyper-accreting process that powers bipolar, ultrarelativistic, collimated jets which +ultimately release the prompt GRB signature (Kumar & Zhang 2015). Despite detecting more than 10,000 GRBs in +their prompt phase, our understanding of these events and the underlying physical processes is still limited (Zhang +2018). We also notice that in the X-Ray band covered by IXPE (2−8 keV), measurements of the prompt emission are +meager. In fact, our knowledge relies on less than 100 detections by BeppoSAX and HETE-2 (see, e.g., Frontera 2019; +Costa & Frontera 2011; Tamagawa et al. 2003). +Advances in understanding require new diagnostics, e.g. observations of polarization or multiple messengers. How- +ever, so far, only upper limits on neutrino emission from either prompt or early-afterglow emission have been set, +suggesting a leptonic composition of the jet bulk (Abbasi et al. 2022). Only a few GRBs (before GRB 221009A) have +been detected in the very-high-energy regime and only one (short GRB) coincident with gravitational waves (Abbott +et al. 2017). Polarization measurements of the prompt emission of GRBs can represent a unique observable to constrain +the outflow composition and dynamics, to determine the structure of the magnetic fields at the jet formation, and +provide insights on the radiation mechanisms behind the observed GRB spectra as well as on our viewing angle within +the jet opening angle (see, e.g., Gill et al. (2021) for a recent overview). Thus far, GRB polarization observations in +the prompt phase have only occurred in the hard X-ray / soft gamma-ray band, reporting generally high polarization +degrees, but never an unambiguous detection (see, e.g., McConnell 2017, for a critical review). The largest catalog +of prompt GRB gamma-ray polarization measurements comes from the POLAR mission, with 14 observations but no +clear detection. The picture is further complicated because the time-integrated polarization seems to be affected by +polarization angle swing in time (Kole et al. 2020). The forthcoming POLAR-2 (Hulsman 2020) and COSI (Tom- +sick et al. 2021) missions are designed for significantly larger detection catalogs, as is the proposed LEAP mission +(McConnell et al. 2021). +After the prompt emission the jet propagates and interacts with the ambient medium, developing a shock, where +electrons are accelerated and produce synchrotron emission, referred to as afterglow, throughout the whole electro- +magnetic spectrum, from radio to very-high energy gamma-rays. Observations of polarization in the afterglow phase +can also provide insight into jet physics and structure (Rossi et al. 2004). Models in the literature (Kuwata et al. 2022) +predict a progressive loss of coherence of the propagating jet magnetic fields, which results in an expected low polar- +ization degree (below 5–3%) of the late-phases of the GRB afterglow emission. These predictions are largely consistent +with results from time-resolved GRB afterglow measurements of optical polarization (Mundell et al. 2013; Stringer & +Lazzati 2020). Observations have also been made in the radio wavelengths, with typically a lower polarization degree + +4 +than in optical (Urata et al. 2019; Urata et al. 2022). No observations of afterglow polarization have been reported so +far at X-ray energies. +On 2022 October 9 an exceptionally bright transient event outshone the rest of the high-energy sky. The first trigger +was recorded in the gamma-ray band by the Fermi Gamma-ray Burst Monitor (GBM) at 13:16:59.988 UTC (Veres +et al. 2022; Lesage et al. 2022), and the same event was also strongly detected by the Fermi Large Area Telescope +(LAT) up to a hundred GeV (Pillera et al. 2022). The Large High Altitude Air Shower Observatory (LHAASO) also +reported the detection of gamma-rays up to 18 TeV (Huang et al. 2022). After about an hour from the initial trigger, +as soon as the source was observed, the Burst Alert Telescope on board of the Neil Gehrels Swift Observatory triggered +on the same event and the Swift X-Ray Telescope (XRT)(Burrows et al. 2005) was on target 143 seconds later and the +Swift Ultraviolet/Optical Telescope (UVOT) located it at (RA(J2000), DEC(J2000)) = (288.26452◦, 19.77350◦) with +a 90%-confidence error radius of about 0.61 arcsec (Dichiara et al. 2022). The event, soon classified as a gamma-ray +burst (Veres et al. 2022), happened at a redshift of 0.151 as reported by X-shooter/VLT (Ugarte Postigo et al. 2022), +and is the brightest (at Earth) ever recorded by any gamma-ray burst monitor by a large margin. Furthermore, the +detection of dust-scattered soft X-ray rings was reported (Tiengo et al. 2022) through Swift/XRT observations in the +two days after the prompt emission. Such rings are produced by X-rays from the extremely bright prompt emission +efficiently scattered at small angles by interstellar dust grains in our Galaxy (e.g., Miralda-Escud´e 1999). The scattered +X-rays are delayed with respect to direct ones, due to their longer path length from the source to the observer, with a +delay that depends on the distance of the dust cloud traversed by the X-ray radiation. Thus, the rings are echoes of +the prompt emission. +The Imaging X-ray Polarimetry Explorer (IXPE) is a space observatory with three identical telescopes designed +to measure the polarization of astrophysical X-rays (Weisskopf et al. 2022; O’Dell et al. 2019; Soffitta et al. 2020). +Launched on 2021 December 9, IXPE is an international collaboration between NASA and the Italian Space Agency +(ASI), and it has been operating since January of 2022. IXPE measures polarization using the photo-electric effect +of X-rays absorbed in the gas gap of a Gas Pixel Detector (GPD) (Bellazzini et al. 2006). On 2022 October 11 at +23:35:35.184 UTC IXPE started the observation of GRB 221009A in response to a Target of Opportunity request +(Negro et al. 2022a). The location position was provided by a Swift/UVOT observation (Dichiara et al. 2022). The +observation ended on 2022 October 14 at 00:46:44.184 UTC with an effective exposure of 94,122 s, and a preliminary +quick-look data analysis, image-, time-, and energy-integrated, was available already on October 14, showing a 99% +confidence level (C.L.) upper limit of 11.1% in polarization degree (Negro et al. 2022b). +In this work we present the results of the IXPE observation carried out with the fully processed data and through +a careful data analysis. +This represents the first observation of X-ray polarization of a GRB afterglow, the first +measurement of soft X-ray polarization of GRB prompt emission, and the first time we observe polarization properties +in both prompt and afterglow phases of the same GRB. +After a brief general introduction on IXPE data analysis provided in Section 2, we devote Section 3 to the data anal- +ysis and interpretation of the GRB afterglow emission, while Section 4 illustrates the data analysis and interpretation +of the rings in association to the GRB prompt emission. Summary and conclusions are offered in Section 5. +2. IXPE POLARIZATION ANALYSIS +We analyze IXPE Level 2 processed data1, combining the data collected by the three identical detector units (DUs). +The time-integrated radial profile reveals inconsistency with the expectation from a point-like source, showing a profile +that deviates from the instrument point spread function (PSF). In particular two excesses around the peak emission +are visible (see Figure 7 and Figure 10 in the Appendix). Such excess appears as rings around a bright core emission +and are associated with dust-scattering halos (e.g., Hayakawa 1970; Miralda-Escud´e 1999). To utilize the full potential +of this observation an image- and time-resolved analysis has been carried out, as described in the next section. +Prior to the data analysis, we perform a first background rejection removing a fraction of background events, mostly +composed of cosmic rays interacting in the sensitive area of the instrument. An irreducible background component +remains and needs to be estimated and subtracted from the data as well. The correct modeling of such a component +is particularly relevant to study the fainter extended emission of the dust-scattering rings. As a suitable background +region cannot be extracted directly from this observation, due to the presence of the rings, we assess the expected X- +ray background rate from previous IXPE observations. In particular, we consider three IXPE observations of low-rate +1 IXPE data are publicly available on the HEASARC archive. + +5 +Figure 1. Left: Background-rejected radial profile around the core emission for DU1; as shown in the lower panel, the source +profile starts deviating more than 20% from the instrumental PSF at around 0.43 arcmin. The equivalent plots for DU2 and +DU3 are not reported here as they carry the same information. Right: Q/I versus U/I plot; in orange we show the distribution +resulting from the spectropolarimetric analysis and the 50%, 90% and 99% C.L. contours in black. The blue cross and circle +show the PCUBE analysis result and the related 1 sigma error. +point-like sources: the observation of 1ES 1959+650 carried out between 2022 June 9 and 2022 June 12 (BKG1); the +observation of BL Lacertae (BL Lac) which happened between 2022 July 7 and 2022 July 09 (BKG2); the observation +of 3C 279 performed between 2022 June 12 and 2022 June 18 (BKG3). From each of these observations we extract the +background spectrum and we simulate a long exposure (1 Ms) IXPE observation with the ixpeobssim simulation tool +(Baldini et al. 2022). The three selected observations provide a good bracketing of the background emission. More +details on the particle background rejection, the residual background simulation, scaling, and subtraction are reported +in Appendix A. +Typically, for IXPE observations, the polarization information is extracted via two types of analyses: a polarimetric +analysis and a spectropolarimetric analysis. For the former, we use the xpbin routine of ixpeobssim with the flag +--algorithm PCUBE. This routine computes the I-normalized Stokes parameters Q and U from the pseudo-Stokes +parameters of the sample of selected events. The algorithm supports the calculation of the background-subtracted +Stokes parameters, if a background template is provided. The polarization degree (PD) and polarization angle (PA) +with associated errors are calculated from the Q/I and U/I parameters following the recipe of Kislat et al. (2015). +The spectropolarimetric analysis, as opposed to the simpler polarimetric analysis, accounts for the shape of the +intensity spectrum. This analysis consists of the joint fit of the I, Q and U spectra and, for this work, we make use +of the Multi-Mission Maximum Likelihood (3ML) framework 2 (Vianello et al. 2015), which is publicly available and +allows for both frequentist and Bayesian analysis approaches. Here we report the results of the frequentist analysis, +but we verified that the Bayesian approach leads to the same results. +Hereafter, we refer to the central region as the core, while the inner and the outer rings are denoted r1 and r2, +respectively. In the next sections we will illustrate the data analyses and results for these different regions. +3. THE CORE / AFTERGLOW EMISSION +3.1. Data analysis +We start with the analysis of the core, which arises from the burst afterglow. We select the region as a disc centered +on the brightest pixel of the IXPE image and radius of 26 arcsec (0.43 arcmin). Beyond this radius, the radial profile +of the emission deviates from the PSF of the instrument by more than 20%, as shown in Figure 1 (left panel). Such +a deviation informs us on possible contamination from the emission of dust-scattered X-rays (from the GRB prompt +2 https://threeml.readthedocs.io/en/stable/index.html + +104 +Counts/bin +103 +102 +PSF）／PSF +4 +PSF +1.2 × PSF +2 +Data +(Data +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Angular sep. [arcmin]0.20 +Spec.pol. ++ +0.15 +PCUBE(1o) +X +0.10 +10 +9 +0.05 +0.00 +50 % +-0.05 +-0.10 +-0.15 +-0.20 +-0.20 -0.15 -0.10 -0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +Q/I6 +Table 1. Spectropolarimetric core analysis +Parameter +Value +PD +(6.1 ± 3.0)% +PD U.L. 99% C.L. +< 13.8% +PD U.L. 95% C.L. +< 12.1% +Γcore +1.98 ± 0.03 +Note—Summary table of the spectropolarimetric +analysis of the core. +The spectropolarimetric fit +is performed in the 2–8 keV energy band and it +assumes Gaussian statistics. +The PA is uncon- +strained. The best-fit Q/I and U/I constants are +(5.6 ± 2.9) × 10−2 and (2.8 ± 2.9) × 10−2, respec- +tively. +and/or afterglow emission) that we cannot fully resolve. We verified that the bright core emission from the central +point-like source dominates the final result as we find consistent numbers within the one sigma uncertainty when +varying slightly the selecting radius. According to the IXPE PSF, cutting at a radius of 26 arcsec eliminates less than +∼15% of the total source emission. +Given the high photon statistics of the core, the results of the analysis are not affected by the choice of the +background spectrum. Here, we report the results for the background template BKG2. Through the PCUBE analysis, +we find an unconstrained polarization in the 2–8 keV energy range and derive a 99% C.L. upper limit of 16.1%. No +evolution with time or energy is observed. For the spectropolarimetric analysis we model the observed spectrum with +an absorbed power law decreasing in energy, with intrinsic parameters fixed to the values of the Swift/XRT automated +online analysis (Evans et al. 2007), which are consistent with other reported values (Kennea et al. 2022). In particular, +the intrinsic absorption parameters are fixed to 1.36 × 1022cm−2 (Evans et al. 2007) at a redshift of 0.151 (Ugarte +Postigo et al. 2022), while the Galactic absorption value is fixed to 5.38×1021cm−2 (Willingale et al. 2013). To account +for mismatches of inter-calibration among the different IXPE telescopes, a constant normalization is left free to vary +for DU2 and DU3 with respect to DU1. +The best-fit values of the spectropolarimetric analysis are provided in Table 1. We find a best-fit power-law index of +Γcore = 1.98 ± 0.03, in agreement with expectations from a late afterglow emission (see Section 3.2). The polarization +results are slightly more constraining than, but consistent with, the PCUBE analysis with a polarization degree of +6.1 ± 3.0%. The right panel of Figure 1 shows the Q/I versus U/I distribution of the core emission. +The Stokes parameters Q/I and U/I are expected to be normally distributed with respective means Q/I and U/I, +and equal standard deviations. An error contour in (Q/I, U/I) space is a circle of radius ϵ centered on (Q/I, U/I), +where (ϵ/σ)2 is distributed as χ2(d.o.f=2). The probability that the observed polarization exceeds the measured value, +under the null hypothesis of unpolarized emission, is 9.7%. This is inconsistent with a zero degree of polarization at +90% C.L. We therefore set a upper limit to the polarization degree (1D distribution) of 13.8% at 99% C.L. (12.1% at +95% C.L.). For completeness, the I, Q, U spectra are reported in Figure 12 (first row) in Appendix B.3. +3.2. Interpretation +According to current models, once beyond the early flaring stages, GRB afterglows arise via synchrotron processes +from electrons accelerated through interactions of the GRB jet with the circumstellar material. This is consistent with +observations from radio to high energies of GRB afterglows (Kumar & Zhang 2015). The physics is well understood +and follows a set of closure relations (e.g., Sari & M´esz´aros 2000) which, when observations fit a self-consistent picture, +can be used to infer properties of the underlying emitting region through observables such as the spectral indices and +rate of temporal decay. The synchrotron spectrum is described by a set of power laws with different spectral indices, +each with its own closure relation depending on the particle density distribution of the circumstellar environment. +We model the core X-ray emission as observed by IXPE with this interpretation, in order to utilize the polarimetric +observation to constrain intrinsic properties of the jet and our viewing angle. + +7 +We start by investigating the density of the interstellar matter around the GRB progenitor. The density profile +in units of cm−3, n(R), where R is the distance from the central engine, is parameterized by the index k, such that +n(R) ∝ R−k. For example, k = 0 corresponds to a constant density medium and k = 2 describes a wind medium, and +in-between values are also possible. A wind medium may be expected around long GRBs since they arise in the deaths +of massive stars. The density profile affects the time evolution of the synchrotron break frequencies. We assume that +the IXPE energy range lies between the typical (νm) and the cooling (νc) synchrotron frequencies, and show that this +assumption yields a consistent picture. +The time and energy evolution of GRB afterglow emission is described by (see, e.g., Granot & Sari 2002) +Fν ∝ t−αν−β . +(1) +The core spectrum is well fit by an absorbed power law with a photon index of 1.98±0.03, which yields β = Γcore−1 = +0.98 ± 0.03. The temporal evolution of GRB afterglow usually shows a break, which causes the index α to increase. +This steepening is proportional to (Γjθj)2, where Γj is the jet Lorentz factor and θj the jet half opening angle. Taking +into account the time evolution of the Lorentz factor, the increase in the temporal index will be ∆α = (k − 3)/(4 − k). +From the closure relations (e.g., Sari & M´esz´aros 2000) between the temporal and spectral indices, we can express the +index of the density profile k as +k = 2(4α − 6β + 3) +2α − 3β + 3 . +(2) +For α = 1.634 ± 0.015, measured by Swift-XRT3 and β = 0.98 ± 0.03, we get k = 2.20 ± 0.05. We note that using the +Swift-XRT spectral index of 0.88 ± 0.15, we get k = 2.00 ± 0.06. In our model we assume that the IXPE observation +was preceded by an achromatic jet break at ∼1 day (D’Avanzo et al. 2022). +We will thus assume that the forward shock propagates in a wind medium with density n(R) = AR−2, where +A = 3.02 × 1035A⋆ cm−1. To estimate A⋆ we introduce fiducial or base values for the energy density fraction in +electrons and in magnetic fields: +ϵe = 10−1ϵe,−1 , +ϵB = 10−3ϵB,−3 . +(3) +Furthermore, we set the kinetic energy of the outflow to Ek,iso ≈ 1055 erg and we use the Q = 10xQx scaling +convention for quantity Q in cgs units. With the above choice of parameters, and neglecting the effect of inverse +Compton scattering on the cooling, we have (e.g., Granot & Sari 2002): +νm = 3.6 × 1012 E1/2 +k,55 ϵ2 +e,−1 ϵ1/2 +B,−3 (t/3.5 d)−3/2 Hz , +νc = 2 × 1018 E1/2 +k,55 A−2 +⋆,−1.5 ϵ−3/2 +B,−3 (t/3.5 d)1/2 Hz , +(4) +confirming that indeed νm < νIXPE ≲ νc at the time of the IXPE observation, and this ordering persists at later times +because νm ∝ t−3/2 and νc ∝ t1/2. In this spectral regime, the energy spectral index is given by β = (p−1)/2, where p +is the power law index of the electron energy distribution (dNe/dγe ∝ γ−p +e , where γe is the electron’s Lorentz factor). +Using the β derived from IXPE observation, we find p = 2.96 ± 0.06. +We can now estimate A⋆ by comparing the observed flux density at 10 keV, Fν,obs ≈ 10−6 Jy, to the synchrotron +model prediction (e.g., Granot & Sari 2002), valid after the jet break: +A⋆ ≈ 3 × 10−1 E−(3+p)/2 +k,55 +ϵ2−2p +e,−1 ϵ−(p+1)/2 +B,−3 +cm−1. +(5) +We note that A⋆ depends strongly on the ϵe parameter (A⋆ ∝ ϵ−4 +e +for p ≈ 3). +A separate constraint for our afterglow model comes from the measured jet break time, tjet, which scales as tjet ∝ +Ekθ4 +jA−1 +⋆ +if the ratio between the jet opening angle θj and viewing angle θv is known. This parameter, and its position +in time with respect to the time of the observation, is relevant for polarization, as it can be broadly associated with +the time when the polarization degree lightcurve has a zero point and the polarization angle rotates by 90 degrees. In +fact, for uniform (top-hat) jet structure with no sideways expansion, significant polarization arises from the break in +3 Swift-XRT data were analyzed in the IXPE observation time window through the online tool: https://www.swift.ac.uk/xrt live cat/ +01126853/. IXPE’s light curve shows a consistent time evolution, but we find a less precise estimation of the power index. Therefore, we +adopt the Swift value in our model. + +8 +10 +1 +100 +101 +t [day] +0 +5 +10 +15 +20 +25 +PD [%] +base values +v/ +j=0.5 +v/ +j=0.8 +j = 1.7 deg +j = 1.3 deg +2 = 3/8 +2 = 3/4 +IXPE - UL (99% C.L.) +Optical - UL (99% C.L.) +Figure 2. +Polarization lightcurves using a set of base parameter values (θj = 1.5 deg, θv = +2 +3θj, p=2.96, Ek = 1055 erg, +A⋆ = 3 × 10−1 cm−1, ξ = 0). We show the effect of changing the θv/θj ratio, the jet opening angle, θj and the magnetic field +ratio, ξ. The IXPE upper limits are shown in teal, while the black upper limit marks the upper limit of the contemporaneous +optical observation. The shaded band shows a Gaussian modulation centered on the PD (darker shade) and width equal to +the one sigma uncertainty on the PD (from the spectropolarimetric fit). We stress that we do not claim a measurement, which +would require at least a 99% C.L. significance. +symmetry of the visible surface. This surface is typically an annulus when projected to the plane of the sky. As the +annulus grows, it encompasses a progressively larger fraction of the jet surface. Eventually, for an off-axis observer, the +annulus will grow beyond the size of the jet on one side, while still collecting emission from the opposite side, resulting +in net polarization. The polarization lightcurve exhibits the typical two-bump structure (Ghisellini & Lazzati 1999), +where the jet break time approximately corresponds to the minimum between the bumps. Our model is constructed +so that the PD zero point between the two bumps is at ≈ 1 day, to match the estimated jet break time (D’Avanzo +et al. 2022). We derive the expected polarization degree by integrating the intensity and polarization of the comoving +volume elements of the jet over the equal arrival time surfaces (Sari 1999; Granot & K¨onigl 2003; Shimoda & Toma +2021; Pedreira et al. 2022). For each comoving volume element, the maximum PD of a synchrotron-emitting, shock +accelerated electron population with power-law distribution with index p will be: PD = (p + 1)/(p + 7/3) ≲ 75% +(Rybicki & Lightman 1979). The observed polarization will be reduced from this value by integrating over all the +parts of the jet that contribute to the flux at a given observer time (see e.g., Lyutikov et al. 2003). +The evolution of the polarization as a function of time depends strongly on a variety of parameters. We take a set +of parameters (base values) that give a polarization consistent with the IXPE spectropolarimetric measurement: jet +opening angle θj = 1.5 deg, viewing angle θv = 2 +3θj (Ghisellini & Lazzati 1999), electron energy distribution index +p=2.96, kinetic energy Ek = 1055 erg, density parameter A⋆ = 3 × 10−1 cm−1 and ξ = 0. The parameter ξ is the +ratio of the magnetic field strength in two directions defined as: ξ2 = 2⟨B2 +||⟩/⟨B2 +⊥⟩. Here, B|| and B⊥ are the magnetic +field parallel and perpendicular to the shock normal, respectively. The case ξ = 0 yields the maximum attainable +polarization for any given set of afterglow parameters. Our model with base values and several additional realizations +is presented in Figure 2. In general terms, all realizations have zero points anchored at ≈ 1 day and yield increasing +PD at the time of the IXPE observations. For a given θv/θj ratio, we can choose a set of parameters (Ek, A⋆ and θj) +so that tjet = 1 day is satisfied. A higher θv/θj ratio results in a higher peak polarization and earlier jet break time, +due to the higher level of asymmetry as we move away from the jet axis. +All presented models in Figure 2, except the low jet opening angle, are consistent with the upper limit. Taking the +PD=6.1 ± 3.0% at face value, models with jet opening angle θj < 1.5 deg (while keeping all other base values fixed) +are disfavored. Similarly, models with θv/θj > 2/3 tend to overpredict the IXPE measurement. Assuming a magnetic +field ratio, ξ, closer to 1 simply scales down the PD. In principle, any model that overpredicts the observations can be + +9 +made consistent by appropriate choice of ξ. The IXPE measurement, considering the base values, favors cases where +ξ2 ≲ 1/2. +Optical polarization observations occurred during the IXPE observation window at the Nordic Optical Telescope +(Lindfors et al. 2022). The sky conditions allowed an estimation of an upper limit to the optical polarization degree +of 8.3% at 99% C.L. (5.1% at 95% C.L., Lindfors et al. 2022). In Appendix B.3 we provide more details about the +optical data reduction. The optical band falls in the same spectral regime as the X-rays (νm < νoptical < νc) for most +choices of parameters around the base values. Thus the optical upper limit can be used to constrain the models. The +optical limit is slightly stronger, but gives qualitatively the same constraints as the X-ray limit. +4. THE RINGS / PROMPT EMISSION +4.1. Data Analysis +As mentioned in the introduction, the observed rings are the result of a known effect involving Galactic dust along +the line-of-sight of a bright transient event. A fraction of the photons emitted in the prompt phase of the GRB are +scattered by dust clouds in the Milky Way. Those scattered inwards towards the line of sight arrive at Earth after +traveling a longer path length with respect to the unscattered ones. This results in a later arrival time of the scattered +photons, with a delay that depends on the distance of the dust cloud to Earth and the scattering angle θs. The angular +size of the halos θh is related to the scattering angle, as θs(1 − x), where x is the ratio between the distance of the +cloud and the distance of the source. Since we are dealing with a transient event at cosmological distance (x ≪ 1), +the approximation θh ∼ θs applies (Miralda-Escud´e 1999; Draine 2003a). +Being produced by a short transient event, the rings expand radially in time. This arises as photons with different +scattering angles travel different path lengths. In order to study the radial evolution of the rings and correctly select +prompt, scattered photons as the rings expand, we devise a method inspired by the procedure described in Tiengo & +Mereghetti (2006). For each photon i detected at a time Ti and at a sky coordinate (ai, δi), we define the following +variables +ti = Ti − T0 +and +Ki = [(ai − aB)2 + (δi − δB)2]/2cti = θ2 +i /2cti , +(6) +where (aB, δB) are the coordinates of the unscattered emission (the center of the core in the IXPE image) and θi is the +angular distance (in arcsec) of the photon i from the point (aB, δB). The trigger time of the prompt emission is taken +Figure 3. Study of the ring evolution in the IXPE observation. Left: distribution of the variable Ki = 1/Di in time bins (in +terms of seconds after the trigger time). Middle: Equivalent to the figure on the left panel, but for the BKG2 template. Right: +Background subtracted distribution of 1/Di for the specific case of the BKG2 template (the equivalent distributions for BKG1 +and BKG3 are provided in Appendix B.3). + +0.7 +GRB clean data +0.6 +0? /(2c t) [1 / kpc] +0.5 +0.4 +0.3 +0.2 +0.1 +2.5 +3.0 +3.5 +t;[s] +1e50.7 +BKG2 +0.6 +0? / (2c t) [1 / kpc] +0.5 - +0.4 +0.3 +0.2 - +0.1 +2.5 +3.0 +3.5 +t;[s] +1e5400 +Best-fit curve +350 +Background subtracted (BKG2) +300 +250 +Counts/bin +200 +150 +100 +50. ++++ ++++++++ +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +O? / (2c t)[1 /kpc]10 +Figure 4. Left: Q/I versus U/I distribution of the polarization of the rings resulting from the spectropolarimetric analysis. The +50%, 90% and 99% C.L. contours are shown in black. The red circle reports, for comparison, the 95% upper limit to the PD of +the core/afterglow emission. Right: Results of the spectropolarimetric fit for the PD assuming different background templates. +The colors violet, teal and green correspond to the different assumed backgrounds BKG1, BKG2 and BKG3, respectively. The +orange band is centered on the average of the best-fit values weighted by their uncertainties, and has a width representative of +the mean relative statistical error. +from the Fermi/GBM4 (Veres et al. 2022). The advantage of this approach is that in the plane Ki vs ti, shown in the +left panel of Figure 3, the expanding rings appear as horizontal bands, facilitating the event selection. We remove the +dominant emission from the core by removing events within 0.85 arcmin from the center to avoid contamination from +the bright core (afterglow) emission. We estimate that the contamination from the core emission at radial distances +larger than 0.85 arcmin is less than 4%. The distribution n(Ki), after subtracting the simulated background events, +is shown in the right panel of Figure 3: the contribution of the two rings is prominent. We fit the distribution around +the peaks with the sum of two Lorentzian functions, which approximate well the observed distribution. +We define the event selection cut on the Ki distribution as illustrated in Figure 3 (right panel). The area under each +best-fit Lorentzian between Ri +min and Ri +max (orange areas in the plot), where i = 1, 2 denotes r1 and r2 respectively, +is at least a factor of twenty larger than the area under the other Lorentzian in the same range (gray areas in the +plot). This ensures a negligible contamination from the emission of one ring onto the other. The edges of the selection +for the wider ring are symmetric with respect to the peak, while the innermost edge of the smaller ring is naturally +defined by our region cut off at 0.85 arcmin to exclude the core emission. +Similar to the core analysis, we proceed with the PCUBE polarization analysis in the 2–8 keV energy band. The +observed spectra of the two rings are expected to be different because they are generated from the same prompt +emission scattered at different angles. +As discussed later on, given a scattering angle, the scattering efficiency of +X-rays by dust grains is energy dependent (Draine 2003b). This leads to the realization that combining the two ring +selections into one single PCUBE analysis would be inaccurate. +Furthermore, the low statistics of the individual +ring selections prevents a proper background template subtraction for the PCUBE analysis. This implies that the +estimated uncertainties are not accurate because they are computed on a boosted statistic that includes background +events. The results of the PCUBE analysis for the individual rings are reported in Table 4 of Appendix B.3. We find a +PDr1 = 19.6±8.7% and PDr2 = 17.2±8.8%, in agreement with each other. These values indicate a higher polarization +degree than what observed in the core, though never exceeding the 99% C.L. required to claim a detection.5 +The spectropolarimetric fit, allowing a proper combination of the rings selections, can give a more accurate estimation +of the underlying polarization. The phenomenological model we define to describe the rings emission allows for the +4 The time difference between the GBM trigger and the beginning on the IXPE observation is 209848 s. Fermi-GBM triggered on a precursor +event, about 210 s before the main brighter peak. We reasonably assume that the rings emission is an echo of the brightest part of the +prompt phase. Hence we use the GBM trigger time plus 210 seconds. In any case, a difference of 210 seconds on the total time-distance +between IXPE observation and the GBM trigger does not affect our results. +5 The minimum detectable polarization at 99% C.L. (for non background subtracted data) is MDP99% = 26.5% and MDP99% = 26.8% for +r1 and r2, respectively + +1.00 +r1+r2 (BKG2) +Spec.pol. ++ +0.75 +Core (95% C.L.i u.I.) +0.50 +0.25 +9 +0.00 +0 +-0.25 +50% +-0.50 +-0.75 +-1.00 +1.00 -0.75 -0.50 -0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Q/I100 +BKG2 +ri + r2 (sta.) +BKG1 +BKG3 +80- +99% C.L +95% C.L +Pol. Deg. [%] +99% C.L. +60 +95% C.L +99% C.L +95% C.L +40 +20 +011 +Table 2. Spectropolarimetric rings analysis +Parameter +Value +PD +27.2±14.7 (sta.) +±4.0 (sys.) +% +PD U.L. 99% C.L. +< [54.6% − 81.5%] +PD U.L. 95% C.L. +< [47.1% − 71.4%] +Γr1 +2.89±0.20 (sta.) +±0.07 (sys.) +Γr2 +3.98±0.25 (sta.) +±0.30 (sys.) +Note—Summary table of the spectropolarimetric anal- +ysis for the rings emission. +The spectropolarimet- +ric fit is performed assuming Poissonian statistics +of the background-rejected data with background- +subtraction applied. +We report the statistical-error- +weighted average of the three measurements (each as- +suming a different background) along with the associ- +ated statistical and systematic errors. The upper lim- +its are strongly dependent on the assumed subtracted +background: we report the range of values defined by +the minimum and maximum value obtained. The PA +is unconstrained. +spectral parameters of the rings to be different while sharing common polarization parameters. The spectra of both +rings are modeled as absorbed simple power laws, while we assume constant polarization parameters. The intrinsic +and Galactic absorption parameters are kept fixed to the same values adopted for the core analysis. As opposed to the +PCUBE analysis, we perform the subtraction of the background spectrum. We test different background assumptions, +subtracting the spectra derived from BKG1, BKG2, and BKG3 templates, to which we applied the analogous event +selection as for r1 and r2. +The results are summarized in Table 2. +We find that r1 has a best-fit photon index, averaged over the values +obtained assuming different backgrounds, Γr1 = 2.89, with a relative statistical error of about 7% and negligible +systematic uncertainty due to the choice of the background spectrum. r2 has a steeper spectrum, with photon index +Γr2 = 3.98 with a relative statistical error of about 6% as well as a 7% relative systematic error associated with +different background assumptions. This is also illustrated in Figure 14 in Appendix B.3. As we will discuss in the +next section, such a difference in spectral index between the two rings is expected owing to the energy dependence +of the X-ray scattering cross section. We note that, for the case of r2, the estimated photon index found assuming +BKG1 shows a larger statistical error: the softer spectrum of the emission from this ring with respect to r1 makes the +measurement more sensitive to the spectral characteristics of the subtracted background (Figure 8 in Appendix B.3 +shows that BKG1 has the hardest spectrum). +As for the polarization, we find a PD of (27.2 ± 14.7 (sta.) ± 4.0 (sys.))%, where the systematic uncertainty is +given by the assumption regarding the background spectrum. Figure 4 shows the results for the different background +subtractions. The significance of this result, tested against the null hypothesis of unpolarized emission, is about 81% +C.L.. The 1D 99% C.L. upper limit on the PD varies between 54.6% and 81.5%, depending on the assumed background. +Such a difference is due to the low-statistic regime we have for the rings data selection, which causes the statistical +uncertainty to be strongly affected by small changes of the subtracted background. +The comparison of the best-fit PDs found assuming different backgrounds is illustrated in the right plot of Figure 4. +For completeness, the Q/I versus U/I distributions obtained for the different assumed background are provided in +Figure 15 in Appendix B.3. Note that, given the different approaches and handling of the background, the PCUBE +and spectropolarimetric analyses are not directly comparable in this case. +Figure 15 in Appendix B.3 reports the Q/I versus U/I plots for the different background assumptions. Additionally, +we show in the same figure the equivalent plots for the spectropolarimetric fit of the individual rings. Furthermore, +Figure 11 and Table 4 report the results of the PCUBE analysis of the individual rings resolved in two logarithmic +energy bins between 2 and 8 keV. We refer the reader to the Appendix for further discussion on this matter. + +12 +Figure 5. Left: Best-fit average distance of the dust clouds. The orange bands are centered on the average of the best-fit values +weighted by their uncertainties, and have a width representative of the mean relative statistical error for the two dust clouds +responsible for r1 and r2 emissions. Right: Derived intrinsic GRB prompt spectrum. The light-grey regions cover the energy +ranges excluded in the fitting procedure. In both plots, the colors violet, teal and green correspond to the different assumed +backgrounds BKG1, BKG2 and BKG3, respectively. +4.2. Interpretation +As discussed in Draine (2003b), the effect of the dust scattering at such a small angles is not expected to alter +the intrinsic polarization of the incoming radiation (see their Figure 5). However, an explicit demonstration of this +statement in the X-ray band is not directly discussed in the literature. +Therefore we investigated the effect on +polarization from reflection, scattering and transmission considering the common dust compounds. All of the above +processes lead to a negligible effect on the polarization of the X-ray radiation, as discussed in Appendix B.2. Therefore, +we can reasonably assume that any polarization observed from the X-ray scattering halos is attributable to the original +emission. +A high polarization degree (PD≳ 20%) in the prompt phase, when viewing the jet at angles smaller than the opening +angle, θv < θj, can be achieved by synchrotron emission in an ordered, toroidal magnetic field configuration (Toma +et al. 2009). Alternatively, high polarization can be achieved by random magnetic fields or Compton drag models, in +a geometry where we are viewing the jet close to its edge, θj ≲ θv < θj + 1/Γj (Granot 2003). This scenario will result +in a very early jet break and potentially high PD in the afterglow, which is disfavored by the observations. In what +follows, therefore, we will focus on the ordered synchrotron scenario. +We estimate the polarization degree integrated over the duration of the prompt emission. The PD mainly depends +on the photon index, the viewing angle, and the product of the jet opening angle and the Lorentz factor, yj = (θjΓj)2. +For the IXPE observation, only the low-energy photon index is relevant. We consider the two extreme values 0.62 +and 1.25, which correspond to the minimum and maximum best-fit values of the prompt GRB intrinsic spectral index +given by the assumed background bracketing (see Section 4.3). For Γj = 700 (Liu et al. 2022), θj = 1.5 deg, and +θv = 2 +3θj we obtain 16% and 36%, respectively. This range is consistent with the measured upper limits. +4.3. Dust clouds and intrinsic GRB prompt emission +In this section we derive some constraints on the dust clouds’ distance and optical depth. We attempt to derive the +intrinsic spectrum of the GRB prompt emission from the observed rings spectra. However, such considerations are +limited by the imaging capabilities of our instrument with respect to other missions that were observing the burst at +the same time. We therefore anticipate that the higher angular resolution and wider field of view of XMM-Newton +and Swift/XRT data, possibly resolving the presence of more than 2 rings, will better determine the characteristics of +the dust clouds visible at the time of the IXPE observation. Constraining the spectral parameters of the rings from +independent observations might help constrain the IXPE polarization parameters. This will be explored in a follow-up +paper. +The dust cloud distance Ddust is given by +Ddust = 2c ∆t +θ2 +h +, +(7) + +16 +14 +12 +r1 +10 +r2 +BKG1 +BKG2 +8 +BKG3 +6 +4 +210-9 +vF,[erg/cm?/s] +BKG1 +BKG2 +BKG3 +10-10 +Konus-WIND (GCN 32668) +1 +2 +3 +4 +6 +10 +Energy[keV]13 +where c is the speed of light, ∆t is the difference between the time of the burst T0 and the time of the observation of +the rings. Hence, from the fit of the distribution Ki defined in Eq. 6, we can easily derive the distance of the clouds. In +fact, the best-fit center of each Lorentzian is the inverse of the distance in kpc of the related dust clouds. We find the +dust cloud associated with r1 to be at a distance of 14.41 kpc with a relative statistical error of 6%. The second cloud, +responsible for r2 emission, is estimated to be at a distance of 3.75 kpc with a relative statistical error of 1%.6 The +variance of this measurement, given by the different assumed backgrounds, is negligible with respect to the statistical +error, as shown in Figure 5 (left panel). The half-width at half-maximum of the two Lorentzian curves correspond to +∼ (9.5±1.1) kpc and ∼ (0.7±0.1) kpc for r1 and r2, respectively. Such values are larger than the values expected from +the effect of the IXPE PSF7 by a factor of ∼3 (for r1) and ∼2 (for r2). This could be symptomatic of a non-negligible +thickness of the dust clouds, or, more likely, the presence of several rings within r1 and r2, which we do not resolve. +As mentioned in the previous section, the X-ray emission of the ring is the echo of the prompt GRB emission +scattered by Galactic dust clouds. Therefore, assuming the characteristics of the dust to be known, it is possible +to infer the intrinsic spectrum of the prompt soft X-ray emission of the GRB from the dust-scattered spectra. The +relation between the scattered spectra and the intrinsic one reads (Tiengo & Mereghetti 2006): +Φr(E, θ1, θ2) = Φ(E)τd(E)(g(θ1, E) − g(θ2, E)) , +(8) +where the intrinsic spectrum is modeled with an absorbed power law decreasing in energy, θ1 and θ2 are the rings +extents at the beginning and at the end of the observations, and the function +g(θ, E) = +(θ/θs(E))2 +1 + (θ/θs(E))2 , +θs(E) = 360′′ +� E +keV +�−1 +(9) +accounts for both the fraction of halo we do not observe (because it lies outside the IXPE observing window) and the +dependence of the scattering angle on the energy8. In fact, larger (smaller) scattering angles correspond to a higher +probability of scattering lower (higher) energy X-ray photons, which results in a steeper (harder) spectrum (Draine +2003b). +According to Draine & Bond (2004), the total scattering optical depth of the dust for photons between 0.8 and 10 +keV can be estimated as +τd = τd1 + τd2 + τforeground = 0.15Av(E/keV)−1.8 +(10) +with τd1 and τd2 being the optical depths associated with the two dust clouds that produce the echo rings r1 and r2, +respectively, and τforeground is the total optical depth between us and the first dust cloud we detect. Av is the V-band +total Galactic extinction in magnitudes in the direction of the GRB. We assume Av = 4.2 mag as reported in the +circulars by the VLT and JWST groups (Izzo et al. 2022; Levan et al. 2022). +According to the measurements of Neckel & Klare (1980), the total Av up to 3 kpc in the direction of GRB 221009A +is about 3.3 mag. As the first cloud is at 3.75 kpc, this sum of three terms should be reasonably valid. Note that +variations of the assumed value of Av affect the normalization of the intrinsic GRB prompt spectrum, but do not affect +the slope of the power law. +We perform a maximum likelihood analysis by simultaneously fitting both rings spectra with the model in Eq. 8. As +for the spectropolarimetric analysis, we have repeated the analysis three times for the different background models. +Figure 5 (right panel) illustrates the results of this fit. +The fit is performed between 2 and 5 keV to avoid the low count statistics part of the ring spectra at high energy. +The best-fit estimate of the fraction of the total optical depth associated with the farther and closer dust clouds is +n ∼ 0.36 and (1 − n) ∼ 0.64 respectively, and is not affected by the choice of the background. This translates into +extinction values of Av,d1 ∼ 0.33 and Av,d2 ∼ 0.57. +As for the intrinsic parameters of the GRB, we find that the prompt GRB power law has a photon index between +0.62 and 1.25 with a relative statistical uncertainty of the order of 10%. This spectral index could be directly compared +to the index inferred by the STIX observation (Xiao et al. 2022) and to the extrapolation of the lower-end of the energy +spectrum measured by Fermi/GBM. Konus-Wind, which detects photons down to 20 keV in energy, has released a +6 Note that the farthest dust cloud is different from the (closer) ones producing the rings observed in the earlier Swift-XRT observations +reported in Tiengo et al. (2022). The closer cloud in IXPE observation is consistent with the farthest one in Swift observation. +7 DU-averaged half-power diameter of ∼26”, (Weisskopf et al. 2022)) would correspond to a full-width at half-maximum computed at the +median time of the observation of ∼5.7 kpc and ∼0.8 kpc for r1 and r2, respectively. +8 θs(E) is the median scattering angle for photons of energy E, and the equation is a good approximation for photons of energy > 0.5 keV +(Draine 2003a). + +14 +preliminary analysis of the prompt emission of this burst (Frederiks et al. 2022). These authors find a time-averaged +spectrum at the onset of the brightest phase of the event with a low-energy photon index of 1.09±0.01. Such value lies +within range defined by the best-fit values we find assuming different background models (see Figure 5, right panel). +Once confirmed, the direct observations of the prompt emission spectrum can provide a potential way to determine the +best IXPE background model to use. In fact, the most representative background model could be selected based on +agreement of these IXPE-inferred values with an externally measured index value. At the time of this work, however, +such information is not publicly available, so we leave these considerations to a future work. Considering the bracketing +given by the intrinsic spectra derived assuming different backgrounds, we infer a total fluence in the 1–10 keV band of +F = [1.6 − 6.1] × 10−4 erg/cm2. The fluence is obtained from the integrated flux between 1 and 10 keV and multiplied +by the IXPE total time of the observation, to account for the missing fluence due to Earth occultation time. The +range of values we report for the fluence is based on the best-fit parameters of the intrinsic power-law model, ignoring +statistical uncertainties which are of the order of 10–15%. +5. CONCLUSION +IXPE observed GRB 221009A from October 11 at 23:35:35 UTC to October 14 at 00:46:44 UTC for an effective +exposure to the target of 94,122 s. The imaging capability of the instrument revealed the presence of a bright core +emission, associated with the GRB afterglow, and the extended emission of two expanding dust-scattering halo rings. +Such emission is an echo of the GRB prompt emission and therefore carries information about the latter. +We studied the linear polarization properties of the core/afterglow emission, and derived an upper limit on the +polarization degree of 13.8% at the 99% C.L. The temporal and spectral parameters of the afterglow at the time of +the IXPE observation are consistent with a forward shock propagating in a wind-like medium, with X-ray emission +arising from synchrotron processes. The observed upper limit on the polarization degree favors a jet opening angle +to be wider than 1.5 degrees, and a viewing angle wider than 2/3 of the jet opening angle (with some underlying +assumptions). Also, scenarios with an equal magnetic field strength in the two directions parallel and perpendicular +to the shock normal seem to be disfavored. +The polarization analysis of the combined dust-scattering rings revealed a polarization degree of (27.2 ± 14.7(sta.) ± +4.0(sys.))% with 99% C.L. upper limit ranging between 54.6% and 81.5% depending on the assumed background. We +also derive a photon index for the intrinsic GRB prompt spectrum between 0.62 and 1.25, depending on the background +model considered. We note that this range includes the Konus-WIND low-energy spectral index derived at energies +above 20 keV. Considering the best-fit spectra, a scenario involving toroidal, ordered magnetic fields when the viewing +angle is smaller than the jet opening angle, predicts high polarization degree up to 36%, compatible with the observed +upper limits. The upper limits on polarization from the IXPE observation exclude the case where we are observing +close to the edge of a sharp transition in the jet. +Aside from the polarization properties of GRB 221009A, the main focus of this work, we could derive some constraints +on the Galactic dust clouds distance. Through the time evolution of the emission from the two dust-scattering halos +that we resolve, we estimated an average distance of the clouds to be about 14.41 and 3.75 kpc for the inner and outer +ring, respectively. The width of the halos compared to the width expected from the effect of PSF suggests the presence +of several unresolved halos within the two halos observed by IXPE. Contemporaneous observations by instruments +with better angular resolution can inform us whether or not this is true. +Future joint analyses exploiting contemporaneous observations from different instruments could be beneficial to +constrain the spectral parameters and, therefore, better single out the polarization signature of the rings/prompt +emission. Furthermore, independent polarization measurements from other instruments assessing a different energy +regime, for either afterglow or prompt emission, will help to understand the full phenomenology behind this exceptional +event. Works along these lines are already ongoing and will be the subject of upcoming publications. +On a final note, we remark that the IXPE observation of GRB 221009A is, on its own account, exceptional and +unique. +We assessed, for the first time, the observation of soft X-ray linear polarization from the late afterglow +emission of a GRB. Also for the first time, thanks to the peculiar location of GRB 221009A in the sky – so close to +the Galactic plane – we were able to assess the polarization properties of the prompt emission in the same observation +through the radiation scattered off the Galactic dust. Aside from providing valuable information about this peculiar +event, this IXPE observation is a proof of observational feasibility for future nearby bright transient events. This, +several years from now, could inspire new directions for the IXPE mission and widen IXPE’s science portfolio to + +15 +include fast-transient events, opening a new door for time-domain high-energy astrophysics. +ACKNOWLEDGEMENTS +We thank Hintz Amenitsch for fruitful discussions on X-ray scattering at small angles. We also acknowledge the +developers of the Slack team-work platform, which played a crucial role in enabling fast and efficient communication +among several different teams. We thank I. Negueruela for the careful optical polarization observations at the Nordic +Optical Telescope. Based on observations made with the Nordic Optical Telescope, owned in collaboration by the +University of Turku and Aarhus University, and operated jointly by Aarhus University, the University of Turku and +the University of Oslo, representing Denmark, Finland and Norway, the University of Iceland and Stockholm University +at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrof´ısica de Canarias. The +data presented here were obtained with ALFOSC, which is provided by the Instituto de Astrof´ısica de Andaluc´ıa +(IAA) under a joint agreement with the University of Copenhagen and NOT. MN acknowledges the support by NASA +under award number 80GSFC21M0002. PV acknowledges support from NASA grant NNM11AA01A. IXPE-related +research at Boston University is supported in part by U.S. National Science Foundation grant AST-2108622. SM and +AT acknowledge financial support from the Italian MUR through grant PRIN 2017LJ39LM. +The Imaging X ray Polarimetry Explorer (IXPE) is a joint US and Italian mission. The US contribution is supported +by the National Aeronautics and Space Administration (NASA) and led and managed by its Marshall Space Flight +Center (MSFC), with industry partner Ball Aerospace (contract NNM15AA18C). The Italian contribution is supported +by the Italian Space Agency (Agenzia Spaziale Italiana, ASI) through contract ASI-OHBI-2017-12-I.0, agreements +ASI-INAF-2017-12-H0 and ASI-INFN-2017.13-H0, and its Space Science Data Center (SSDC) with agreements ASI- +INAF-2022-14-HH.0 and ASI-INFN 2021-43-HH.0, and by the Istituto Nazionale di Astrofisica (INAF) and the Istituto +Nazionale di Fisica Nucleare (INFN) in Italy. 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The remaining fraction is indistinguishable from genuine X-ray-triggered +events and constitutes an irreducible background that must be treated statistically. +We adopted a two-step strategy to remove the background events: first we apply a background rejection and then +a background subtraction, as detailed here below. +Background rejection —Typical X-ray events, compared to charge cosmic-ray events, display a higher fraction of energy +deposit associated to the main track9 over the total energy of the event. Based on such a difference, a rejection cut can +be devised to remove the portion of events that are of clear cosmic-ray nature. The left panel of Figure 6 illustrates +the energy fraction deposited in the main track of the event as a function of the reconstructed energy: the blue line +marks the rejection event cut we apply for events between 2 and 8 keV. +We verify that the rejected events do not manifest any trace of the observed target (see the comparison between the +middle and right panels of Figure 6) and that they do not carry any significant polarization. In the region of the point +source the fraction of the rejected background events reach at most 0.6% in the case of DU2 (see also Fig. 7). +Figure 6. Left: background rejection cut based on the energy fraction contained in the main cluster as a function of the +reconstructed energy. The events that are below the blue line are removed. The gray shaded areas mark the energy ranges +outside the fiducial range for IXPE data analyses. Center and right: diagnostic maps produced to check the efficiency of the +cut: total events map (middle) and rejected events map (right). The maps are in detector coordinates, so the central source +appears blurred following a pattern caused by deliberate dithering of the satellite (Weisskopf et al. 2022). No apparent residuals +of the central source are visible in the background map on the right. +Background subtraction —The residual irreducible background needs to be estimated, simulated, and subtracted. The +standard approach consists of selecting a region of the image in the field of view far from the point source, avoiding +the edges where the sensitivity of the instrument degrades. However, in the case of extended sources (e.g. the dust- +scattering rings we detect in this observation), this method cannot be applied. To address the issue, we estimate the +residual background from a previous IXPE observation of a relatively faint source. For this work we considered: 1) +the observation of 1ES 1959+650 carried out between 2022 June 9 and 2022 June 12; 2) the observation of 3C 279 +performed between 2022 June 12 and 2022 June 18 3) the observation of BL Lacertae (BL Lac) which happened +between 2022 July 7 and 2022 July 09. Due to changes in IXPE operations, the observations prior to June 09 would +9 The first step of IXPE reconstruction algorithm is a clustering stage meant to identify a group of adjacent pixels that recorded a charge +value above a noise-rejection threshold. For typical X-ray-induced events, the charge deposit associated with the photo-electron produces a +single main cluster, while additional, spurious clusters are caused by noise fluctuations. The reconstruction algorithm assumes the cluster +with the higher charge deposit to correspond to the main track. On the other side, charged cosmic-ray-induced events may produce several, +disconnected clusters of charge inside the detector with similar energy deposit. It follows that cosmic rays display a lower fraction of energy +deposit associated to the main track with respect to the total energy of the event. + +35 +8 +102 +7 +30 +1.2 +Y absorption point (clean) +Y absorption point (bkg) +6 +25 +1.0 +2 +20 +0.8 +0 +0 +4 +101 +0.6 +15 +3 +-2 +-2 +0.4 +10 +2 +0.2 +5 +1 +6 +0.0 +100 +0 +0 +2 +4 +6 +8 +10 +-6 +-2 +0 +6 +2 +6 +Energy[kev] +X absorption point (clean) +X absorption point (bkg)19 +Figure 7. Comparison of the radial profiles of the GRB observation (water green) with the sum of the rejected background +(dotted red) and the simulated residual background from BL Lac (hatched yellow) for the three IXPE detector units, zoomed +on the vertical scale to exclude the large central peak in correspondence of the core and better show the region of the two rings. +not provide background estimations suitable for this data analysis, and therefore have not been considered. These +particular sources are point-like and have a small count rate (< 0.2 Hz), which gives us a wide region of high noise to +signal ratio to characterize the background. The first two observations are close in time and show a similar background +spectrum, while the background obtained from the BL Lac observation shows a lower background rate: this provides +a good bracketing for our analysis. +The same background rejection procedure is applied to the data of all the observations considered. The residual +background spectrum is derived by selecting the events in an annulus centered on the source with inner and outer radius +of 1.2’ and 5.5’ respectively. For each background spectrum we simulate an IXPE observation using the ixpeobssim +simulation tool. Events are generated uniformly on the surface of the detectors and then projected in the sky using a +realistic model for the pointing history that accounts for satellite dithering (Weisskopf et al. 2022). In order to reduce +the statistical uncertainty, background templates are simulated with a longer exposure (1 Ms) compared to the GRB +observation, then re-weighted appropriately to the respective livetime ratio before the subtraction. +Figure 7 shows the radial profiles for the three detector units in celestial coordinates: the data of the observation of +GRB 221009A are compared to the rejected background and the simulated background (here we show the case for the +background extracted from the BL Lac observation, as an example). +Background scaling —Due to statistical fluctuations in the low-count regime of the GRB rings data, the simulated +backgrounds need to be scaled in order to never overshoot the data at high energies and at the edges of the field of +view, namely where the background is expected to dominate. To define the scaling factor for each background, we +estimate 1) the integral of the background spectra for both r1 and r2 selections between 5 and 8 keV and 2) the integral +of the radial profile above 6’, then we derive their ratio with the corresponding values of the GRB data. The ratios +are reported in the label of Figure 8 and the horizontal lines show visually how the value of the integrals compare +to each other. The right panel of Figure 8 shows the radial profile of all the three simulated background templates, +appropriately scaled to the livetime of the GRB observation. The scaling factor for each background is defined by the +most extreme values among the ratios of r1 spectra, r2 spectra and radial profile. The background derived from the +1ES 1959+65 observation is hence scaled down by a factor of 1.07, the one derived from the BL Lac observation by a +factor of 1.05, and the one from 3C 279 by a factor of 1.10. Table 3 reports the number of counts for the GRB and +for simulated background templates (re-weighted to account for the different live times) in the 2–8 keV band for the +three region selections of our analysis. +B. ADDITIONAL CONSIDERATIONS +B.1. Optical polarization data analysis +During the IXPE pointing we also performed optical polarization observations in the R-band at the Nordic Optical +Telescope (Lindfors et al. 2022). The observations were obtained using the Alhambra Faint Object Spectrograph and +Camera (ALFOSC) in the standard linear polarimetric mode that includes a λ/2 retarder followed by calcite. At +the time of the observations (2022 October 12 at 20:15UT) the sky conditions were clear with 1.2 arcsecond seeing. +However, GRB221009A is located in a crowded Galactic field. This resulted in the extraordinary beam of a nearby + +100 +GRB +Simulated residual background +Rejectedbackgroundcomponent +80 +Entries/bin +60 +40 +20 +0 +0 +1 +2 +3 +4 +5 +6 +Angularseparation[arcmin]100 +GRB +Simulated residual background +... +Rejectedbackgroundcomponent +80 +Entries/bin +60 +40 +20 +0 +2 +3 +4 +5 +6 +Angular separation [arcmin]100 +GRB +Simulated residual background +... +Rejectedbackgroundcomponent +80 +Entries/bin +60 +40 +20 +0 +0 +1 +2 +3 +4 +5 +6 +Angular separation [arcmin]20 +Figure 8. +Left and Middle: simulated background spectra of r1 and r2 event selections compared to the GRB observed ring +spectra. The horizontal lines show the values of the integral of the spectra between 5 and 8 keV. Right: the radial profile of +the simulated background templates compared to the GRB data profile for DU1. The horizontal lines show the values of the +integral of the profiles above 6 arcmin. In all plots the gray shaded areas mark the regions of the parameters space excluded +from the analysis. +Table 3. Count Statisitcs +Tot. counts for +r1 +core +r2 +GRB data +16121 +5450 +5502 +Bkg 1ES 1959+65 +135 +4099 +4313 +Bkg BL Lac +105 +3263 +3497 +Bkg 3C 279 +124 +3776 +4017 +Note—Total and background counts in the 2–8 +keV band for the core, r1, and r2 selections. +These numbers refer to the background rejected +data. The background counts are computed by +multiplying the background rate by the GRB ob- +servation live time. +bright star to overlap with the ordinary beam of the source. As such, the standard polarimetric analysis was not +possible (Hovatta et al. 2016; Nilsson et al. 2018, see e.g.). Instead, we performed careful modelling of the point spread +function. We used the second brightest star within the ALFOSC field of view to create a model of the PSF, which was +then subtracted from each image separately. This process, however, can result in background artifacts. To mitigate +the effect of any artifact we used a small aperture of 1.5 arcsec radius to perform the measurements using standard +formulas. +B.2. Effect of dust scattering on X-rays polarization +We investigated the effect on polarization from reflection, scattering and transmission considering the dominant +dust compounds, Carbon and silicates (see e.g. Costantini & Corrales 2022, for a recent discussion of the topic). At +the small angles that we observe, even assuming a coherent reflection angle, any polarization induced by reflection of +X-rays would result in a negligible modulation of less than 10−5, or a PD∼0.001%. These values were obtained using +the Center for X-Ray Optics database and online tools10. Polarization from transmission is expected to be negligible +as well for X-rays of energies at the peak of IXPE sensitivity, given that the common dust compounds do not show K +or L shell edges there. A fraction of the scattered light might have a polarization status affected by big spheroidal dust +grains via Mie scattering. We checked this by using the python package Miepython11, which calculates light scattering +according to the Mie theory and Rayleigh–Gans approximation, and adopting the X-ray refraction index for silicates +provided by Draine & Lee (1984) and Laor & Draine (1993). We find that at the scattering angles we are considering, +the PD due to refraction is less than 5.5 × 10−5 at 2 keV for a binary population of grains (e.g.: perfectly aligned, +10 https://henke.lbl.gov/optical constants/ +11 https://miepython.readthedocs.io/ + +ri spectrum +10-3 +Rate [Hz] +10- +Bkg 1ES 1959+65 +Bkg BL Lac +Bkg 3C 279 (x 0.96) +GRB +2 × 100 +3 × 100 +4 × 100 +6 × 100 +Energy[keV]r2 spectrum +10-3 +Rate [Hz] +10-4 +Bkg 1ES 1959+65 +Bkg BL Lac (x 0.95) +Bkg 3C 279 (x 0.90) +GRB +2 × 100 +3 × 100 +4 × 100 +6 × 100 +Energy[keV]60 +Bkg 1ES 1959+65 (x 0.93) +Bkg BL Lac +50 +Bkg 3C 279 +data +40 +Counts/bin +30 +20 +10 +2 +3 +6 +1 +4 +5 +8 +9 +Anaular sep. [arcminl21 +elongated and not aligned, spherical grains) with a power-law size distribution with an index of -3.5 (Costantini & +Corrales 2022). Therefore, we can reasonably assume that any polarization observed from the X-ray scattering halos +is attributable to the original emission. +B.3. Additional plots +Figure 9. IXPE counts map combining the 3 DU observations obtained with the xpbin routine of ixpeobssim with the flag +--algorithm CMAP. The core/afterglow emission dominates the image, however the fainter halos are already visible to the +attentive eye. + +19°55' +103 +50' +Declination(2000) +102 +Counts/pixel +45' +101 +40' +100 +1gh13m30s +15s +005 +12m45s +30s +RightAscension(2000)22 +Figure 10. Time evolution of the dust-scattering halos. The counts maps are generated in three time bins combining the data +of the 3 DUs. The core region has been removed to better show the rings. The images have been smoothed with a Gaussian +beam for visualization purposes. To guide the eye, we added a thin circle in the first and last images of the sequence to mark +the minimum of the counts gap between the two evolving rings. +Table 4. Summary table of the PCUBE rings analysis +r1 +r2 +∆E +PD +PD u.l.(99%) +PD +PD u.l.(99%) +keV +[%] +[%] +[%] +[%] +2–8 +19.6 ± 8.7 +<42.0 +17.2 ± 8.8 +<39.9 +2–4 +27.2 ± 8.3 +<48.6 +5.5 ± 8.3 +<26.9 +4–8 +15.5 ± 11.5 +<45.1 +25.1 ± 11.6 +<55.0 +Note—Results of the PCUBE analysis between 2 and 8 keV and resolved in 2 loga- +rithmic energy bins. This analysis is performed on the background-rejected (not +background subtracted) data. This implies that 1) the estimated uncertainties +are not accurate because they are computed on a boosted statistic that includes +background events (a big fraction of the total, see Table 3); and 2) the results +of the PCUBE analysis are not directly comparable to those resulting form the +spectropolarimetric analysis. The latter represents a more accurate analysis. For +the 2−8 keV PCUBE analysis the minimum detectable polarization at 99% C.L. +is MDP99% = 26.5% and MDP99% = 26.8% for r1 and r2, respectively. +Note that in the 2–4 keV bin for r1 the PD might seem to exceed the 99% C.L.. +However, in this case, the test-statistic follows a χ2 distribution with 4 d.o.f, ac- +counting for the two energy bins considered and 2D Q-U space. This gives a 3% +probability of finding a χ2 value equal to or exceeding the observed one in case +unpolarized emission, which means that we are compatible with the null hypoth- +esis within the 97% C.L.. Such significance is even lower if we account for the +trials due to both rings selections: in this case we should derive the significance +from a χ2 distribution with 8 d.o.f.. + +T- To = 209-268 ks +T- To = 268-327 ks +T- To = 327-386 ks23 +Figure 11. +Polarization PCUBE analysis results for r1 (top row) and r2 (middle row), for one energy bin 2–8 keV (left +column) and two logarithmic energy bins 2–4 keV and 4–8 keV (right column). Tab.4 provides the values and 1σ errors on +the PD and PA for the energy-resolved PCUBE analysis. Note that the PCUBE analysis of the rings emission is performed +without a proper background subtraction (we refer to the caption of Tab.4 for further discussion on the caveats of this point). +The PAs of the two rings seem to be significantly different, however: as mentioned in the main text, the uncertainties are +underestimated. Polarization measurements with significance below the 99% C.L. are not considered as a detectios and the +PAs are to be considered unconstrained. +Furthermore, under the assumption that the two rings originate from the same +prompt emission, there is no (known) reason to expect the polarization of the two rings to be intrinsically rotated. This fact is +symptomatic of fluctuations due to the low photon statistics of the signal, and further justifies our approach of combining the +two rings for a more accurate spectropolarimetric analysis. + +ri +0.3 +45° +60° +30° +0.30 +0.2 +75° +15° +0.20 +0.1 - +0.10 +2.00-8.00 kev +.06 +-0.1 - +-75° +-15 +-0.2 +09- +-30° +-45° +-0.3 +-0.3 +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.3 +Q/I0.4 +ri +45° +09 +30 +0.40 +0.30 +0.2 +75° +15° +0.20. +0.10 +0.0 +.06 +0° +.00-4.00 kev +N00:8.00 keN +-0.2 +-75° +-15° +-60° +-45° +30 +-0.4 +-0.4 +-0.2 +0.0 +0.2 +0.4 +Q/Ir2 +0.3 +45° +.09 +30° +0.30 +0.2 +75° +15° +0.20 +0.1 - +0.10 +0.0 +90. +0。 +2.00-8.0Q kev +-0.1 - +-75° +-15° +-0.2 +09- +-30° +45° +-0.3 +-0.3 +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.3 +Q/Ir2 +0.4 +45° +.09 +30° +0.40 +0.30 +0.2 +75° +15° +0.20 +0.10 +0.0 +200/4.00.kel +.06 +0° +4.00-8.00K +-0.2 +-75° +-15° +.09- +-30° +-45° +-0.4 +-0.4 +-0.2 +0.0 +0.2 +0.4 +Q/I24 +Figure 12. I, Q and U spectra (background rejected and subtracted) for the core region (top row), r1 region (middle row), +and r2 region (bottom row). All spectropolarimetric fits have a χ2 +red ∼ 1. +Figure 13. Equivalent plots to that in Figure 3 right panel, but subtracting a different background. + +10 +ixpe DUl I Model +ixpe_DU2_I Model +ixpe_DU3_I Model +ixpe_DUl_I +(counts s-1 keV-1) +ixpe_DU2_l +ixpe_DU3_l +Net rate +10-3 +10-4 +2 +Residuals +0 ++I++ ++Ti+'Ti +2 +-4 +6 +2 × 100 +3 × 100 +4 × 100 +6 × 100 +Energy +(keV)0.015 +ixpe DUl Q Model +ixpe DU2 Q Model +0.010 +ixpe_DU3_Q Model +ixpe DUl Q +ixpe DU2 Q +keV-1) +0.005 +ixpe_DU3_Q +Net rate +(counts s-1 +0.000 +-0.005 +-0.010 +-0.015 +Residuals +9 +2 × 100 +3 × 100 +4× 100 +6 × 100 +Energy +(keV)0.015 +0.010 +0.005 +0.000 +Net rate +-0.005 +-0.010 +ixpe_DUl_U Model +ixpe DU2 U Model +0.015 +ixpe DU3 U Model +ixpe_DUl_U +-0.020 +ixpe_DU2_U +ixpe_DU3_U +-0.025 +Residuals +9 +'tit! +2 × 100 +3 × 100 +4 × 100 +6 × 100 +Energy +(keV)10- +ixpe_DU1_I Model +ixpe DU2 I Model +ixpe_DU3_I Model +10-2 +ixpe_DUl_! +ixpe_DU2_I +(counts s-1 keV-1 +#+tttt. +ixpe_DU3_ +Net rate +10 +10-4 +10-5 +10-6 +2 +Residuals +0 +6 +-4 +-6 +2 × 100 +3 × 100 +4 × 100 +6× 100 +Energy +(keV)0.0100 +ixpe DUl Q Model +ixpe_DU2_Q Model +0.0075 +ixpe DU3 Q Model +ixpe_DU1_Q +0.0050 +ixpe_DU2_Q +keV-1) +ixpe_DU3_Q +0.0025 +(counts s-1 +0.0000 +0.0025 +-0.0050 +-0.0075 +-0.0100 +2 +Residuals +1 +0 +-1 +2 × 100 +3 × 100 +4 × 100 +6 × 100 +Energy +(keV)0.008 +ixpe DUl U Model +0.006 +ixpe_DU2_U Model +ixpe_DU3_U Model +0.004 +ixpe_DUl_U +ixpe_DU2_U +keV-1 +0.002 +ixpe_DU3_ U +0.000 +counts s- +0.002 +0.004 +-0.006 +-0.008 +-0.010 +1 +Residuals +0 +-2 +-3 +2 × 100 +3 × 100 +4 × 100 +6× 100 +Energy +(keV)10-1 +ixpe DUl I Model +ixpe_DU2_I Model +ixpe_DU3_I Model ++++t+ +10-2 +ixpe_DUl_! +++ +ixpe_DU2_ +ixpe_DU3 I +Net rate +10 +10-4 +10-5 +10-6 +2 +Residuals +2 × 100 +3 × 100 +4 × 100 +6× 100 +Energy +(keV)0.010 +ixpe DUl Q Model +ixpe_DU2_Q Model +ixpe_DU3_Q Model +0.005 +ixpe_DU1_Q +ixpe_DU2_Q +keV-1) ++++ +ixpe_DU3_Q +0.000 +counts s- +0.005 +-0.010 +1.5 +Residuals ++++++++ +b +0.0 +++++++ +-1.5 +-3.0 +2 × 100 +3 × 100 +4 × 100 +6× 100 +Energy +(keV)0.010 +ixpe DUl U Model +0.008 +ixpe_DU2_U Model +ixpe_DU3_U Model +0.006 +ixpe_DU1_U +ixpe_DU2_U +keV-1) +0.004 +ixpe_DU3_U +0.002 +counts s- +0.000 +0.002 +-0.004 +-0.006 +-0.008 +1.5 +Residuals +++++ +0.0 +++! +-1.5 +3.0 +2 × 100 +3 × 100 +4 × 100 +6× 100 +Energy +(keV)400 +Best-fit curve +350 +Background subtracted (1ES 1959+65) +300 +250 +Counts/bin +200 +150 +100 +50 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +O? / (2c t;)[1 / kpc]400 +Best-fit curve +350 +Background subtracted (3C 279) +300 +250 +Counts/bin +200 +150 +100 +50 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +O? / (2c t)[1 / kpc]25 +Figure 14. Left: Comparison of the best-fit spectral indices of the two rings emissions. Right: Effect of the corrections for the +optical depth and for the energy-dependent scattering efficiency. The light-gray regions cover the energy ranges excluded in the +fitting procedure. + +-2.5 +-3.0 +Photon index +-3.5 +-4.0 +-4.5 +ri (sta.+sys.) +BKG1 +BKG3 +r2 (sta.+sys.) +BKG2 +-5.0100 +10-1 +10-2 +10-3 +10-4 +Φr (observed) +Φr (observed) +10-5 +Φr/ Td +Φr2/ T d2 +10-6. +中n/(g(02, E) - g(01, E) +中r/(g(02, E) - g(01, E) +10-7 +100 +2 × 100 +3 × 100 +4× 100 +6× 100 +101 +Energy[keV]26 +Figure 15. Q/I versus U/I plots resulting from the spectropolarimetric analysis for the combined analysis of r1 + r2 (top row) +and for r1 and r2 individual analyses (middle and bottom row, respectively), shown for completeness. The three columns refer +to different background assumptions: BKG1 (extracted from IXPE observations of 1ES 1959+65) on the left column, BKG2 +(from BL Lac observation) on the central column, and BKG3 (from 3C 279 observation) on the right column. This shows that +the result of the combined analysis is mostly driven by r1, while r2 seems more unpolarized. However, within the 50% contours +r1 and r2 are compatible. + +1.00 +r1+r2 (BKG1) +Spec.pol. ++ +0.75 +0.50 +0.25 +% +0.00 +6 +6 +% +-0.25 +50 % +-0.50 +-0.75 +-1.00 +1.00 -0.75 -0.50 -0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Q/I1.00 +r1+r2 (BKG2) +Spec.pol. ++ +0.75 +0.50 +0.25 +30% +0.00 +0 +% +50% +-0.25 +-0.50 +-0.75 +-1.00 +1.00 -0.75 -0.50 -0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Q/I1.00 +r1+r² (BKG3) +Spec.pol. ++ +0.75 +0.50 +% +.99 +0.25 +30% +0.00 +% +0 +-0.25 +50% +-0.50 +-0.75 +-1.00 +1.00 -0.75 -0.50 -0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Q/I1.00 +r1(BKG1) +Spec.pol. ++ +0.75 +0.50 +0.25 +0% +0.00 +-0.25 +50 % +do +06 +-0.50 +-0.75 +99 % +-1.00 +1.00 -0.75 -0.50 -0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Q/I1.00 +r1 (BKG2) +Spec.pol. +0.75 +0.50 +0.25 +0.00 +-0.25 +50% +-0.50 +-0.75 +-1.00 +1.00 -0.75 -0.50 -0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Q/I1.00 +r1 (BKG3) +Spec.pol. ++ +0.75 +0.50 +0.25 +0% +% +50 +0.00 +-0.25 +o +-0.50 +66 +-0.75 +-1.00 +1.00 -0.75 -0.50 -0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Q/I1.00 +r2 (BKG1) +Spec.pol. ++ +0.75 +0.50 +5g% +0.25 +50% +0.00 +-0.25 +-0.50 +-0.75 +99 % +-1.00 +1.00 -0.75 -0.50 -0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Q/I1.00 +r2 (BKG2) +Spec.pol. ++ +0.75 +0.50 +0.25 +10% +0.00 +9 +6 ++ +-0.25 +% +50% +-0.50 +-0.75 +-1.00 +1.00 -0.75 -0.50 -0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Q/I1.00 +r² (BKG3) +Spec.pol. ++ +0.75 +0.50 +0.25 +0 +1007 +0.00 +9 +-0.25 +50% +-0.50 +-0.75 +-1.00 +1.00 -0.75 -0.50 -0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Q/I27 +Figure 16. Corner plots resulting from the Bayesian spectropolarimetric fit performed with 3ML of the core region (top) and +the combined fit of r1 and r2 (bottom). This shows how the Bayesian approach yields results consistent with the frequentist +approach adopted in this work. It also shows the nice convergence of the fit for all the parameters of interest. In both plots, +we show the case of BKG2 assumed as background model for the background subtraction (the cases of BKG1 and BKG3 yield +analogous results). + +ndex +$6'T- +k = 0.055±0.829 +0.023±0.033 +cons_ixpe_DU2 +0.96 +0.8> +ndex +consixpeDU2 +cons_ixpe_DU3_l0.00055 +index +-2.8+0.17 +ndex.3 +-2.> +3.0 +0.0015 +index.3 +-3.80.21 +3.2 +ndex +-0.1±0:13 +2? +00 +2.4 +3.6 +0.3 +index 3 +K 3 +index_3 \ No newline at end of file diff --git a/VtAzT4oBgHgl3EQf1P70/content/tmp_files/load_file.txt b/VtAzT4oBgHgl3EQf1P70/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6dd3cb9f9f8b03559871420099939577f91686bd --- /dev/null +++ b/VtAzT4oBgHgl3EQf1P70/content/tmp_files/load_file.txt @@ -0,0 +1,2029 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf,len=2028 +page_content='Draft version January 6, 2023 Typeset using LATEX default style in AASTeX631 The IXPE view of GRB 221009A Michela Negro ,1, 2, 3 Niccol`o Di Lalla ,4 Nicola Omodei ,4 P´eter Veres ,5, 6 Stefano Silvestri ,7, 8 Alberto Manfreda ,9, 7 Eric Burns ,10 Luca Baldini ,7, 11 Enrico Costa ,12 Steven R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Ehlert ,13 Jamie A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Kennea ,14 Ioannis Liodakis ,15 Herman L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Marshall ,16 Sandro Mereghetti ,17 Riccardo Middei ,18, 19 Fabio Muleri ,12 Stephen L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' O’Dell ,13 Oliver J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} 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GRAPPA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' University of Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Science Park 904,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 1098 XH Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The Netherlands 67Guangxi Key Laboratory for Relativistic Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' School of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Guangxi University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Nanning 530004,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' China ABSTRACT We present the IXPE observation of GRB 221009A which includes upper limits on the linear polar- ization degree of both prompt and afterglow emission in the soft X-ray energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' GRB 221009A 3 is an exceptionally bright gamma-ray burst (GRB) that reached Earth on 2022 October 9 after trav- elling through the dust of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The Imaging X-ray Polarimetry Explorer (IXPE) pointed at GRB 221009A on October 11 to observe, for the first time, the 2–8 keV X-ray polarization of a GRB afterglow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We set an upper limit to the polarization degree of the afterglow emission of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='8% at a 99% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This result provides constraints on the jet opening angle and the viewing angle of the GRB, or alternatively, other properties of the emission region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Additionally, IXPE cap- tured halo-rings of dust-scattered photons which are echoes of the GRB prompt emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The 99% confidence level upper limit of the prompt polarization degree is about 55%, consistent with a scenario involving synchrotron emission in an ordered magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This single IXPE pointing provides both the first assessment of X-ray polarization of a GRB afterglow and the first GRB study with polarization observations of both the prompt and afterglow phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Keywords: GRB — X-ray — polarization 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' INTRODUCTION Gamma-Ray Bursts (GRBs) are among the most energetic events in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' These events are characterized by a “prompt” gamma-ray emission, the most luminous phase of the burst, followed by a temporally decaying “afterglow” that can last for days or even years and is observed across the whole electromagnetic spectrum whenever the needed sensitivity is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' GRBs are conventionally classified by duration of the prompt phase into the short (< 2 sec) and long (> 2 sec) class, with distinct physical origins (Galama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Fong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Burns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Long GRBs originates from collapsars (Woosley & Bloom 2006), a rare sub-type of type Ic supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In the standard model, a fast spinning core collapses into a rapidly spinning black hole which devours some of the massive star progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This results in a hyper-accreting process that powers bipolar, ultrarelativistic, collimated jets which ultimately release the prompt GRB signature (Kumar & Zhang 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Despite detecting more than 10,000 GRBs in their prompt phase, our understanding of these events and the underlying physical processes is still limited (Zhang 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We also notice that in the X-Ray band covered by IXPE (2−8 keV), measurements of the prompt emission are meager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In fact, our knowledge relies on less than 100 detections by BeppoSAX and HETE-2 (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', Frontera 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Costa & Frontera 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Tamagawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Advances in understanding require new diagnostics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' observations of polarization or multiple messengers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' How- ever, so far, only upper limits on neutrino emission from either prompt or early-afterglow emission have been set, suggesting a leptonic composition of the jet bulk (Abbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Only a few GRBs (before GRB 221009A) have been detected in the very-high-energy regime and only one (short GRB) coincident with gravitational waves (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Polarization measurements of the prompt emission of GRBs can represent a unique observable to constrain the outflow composition and dynamics, to determine the structure of the magnetic fields at the jet formation, and provide insights on the radiation mechanisms behind the observed GRB spectra as well as on our viewing angle within the jet opening angle (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', Gill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' (2021) for a recent overview).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Thus far, GRB polarization observations in the prompt phase have only occurred in the hard X-ray / soft gamma-ray band, reporting generally high polarization degrees, but never an unambiguous detection (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', McConnell 2017, for a critical review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The largest catalog of prompt GRB gamma-ray polarization measurements comes from the POLAR mission, with 14 observations but no clear detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The picture is further complicated because the time-integrated polarization seems to be affected by polarization angle swing in time (Kole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The forthcoming POLAR-2 (Hulsman 2020) and COSI (Tom- sick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2021) missions are designed for significantly larger detection catalogs, as is the proposed LEAP mission (McConnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' After the prompt emission the jet propagates and interacts with the ambient medium, developing a shock, where electrons are accelerated and produce synchrotron emission, referred to as afterglow, throughout the whole electro- magnetic spectrum, from radio to very-high energy gamma-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Observations of polarization in the afterglow phase can also provide insight into jet physics and structure (Rossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Models in the literature (Kuwata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022) predict a progressive loss of coherence of the propagating jet magnetic fields, which results in an expected low polar- ization degree (below 5–3%) of the late-phases of the GRB afterglow emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' These predictions are largely consistent with results from time-resolved GRB afterglow measurements of optical polarization (Mundell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Stringer & Lazzati 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Observations have also been made in the radio wavelengths, with typically a lower polarization degree 4 than in optical (Urata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Urata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' No observations of afterglow polarization have been reported so far at X-ray energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' On 2022 October 9 an exceptionally bright transient event outshone the rest of the high-energy sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The first trigger was recorded in the gamma-ray band by the Fermi Gamma-ray Burst Monitor (GBM) at 13:16:59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='988 UTC (Veres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Lesage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022), and the same event was also strongly detected by the Fermi Large Area Telescope (LAT) up to a hundred GeV (Pillera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The Large High Altitude Air Shower Observatory (LHAASO) also reported the detection of gamma-rays up to 18 TeV (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' After about an hour from the initial trigger, as soon as the source was observed, the Burst Alert Telescope on board of the Neil Gehrels Swift Observatory triggered on the same event and the Swift X-Ray Telescope (XRT)(Burrows et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2005) was on target 143 seconds later and the Swift Ultraviolet/Optical Telescope (UVOT) located it at (RA(J2000), DEC(J2000)) = (288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='26452◦, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='77350◦) with a 90%-confidence error radius of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='61 arcsec (Dichiara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The event, soon classified as a gamma-ray burst (Veres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022), happened at a redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='151 as reported by X-shooter/VLT (Ugarte Postigo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022), and is the brightest (at Earth) ever recorded by any gamma-ray burst monitor by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Furthermore, the detection of dust-scattered soft X-ray rings was reported (Tiengo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022) through Swift/XRT observations in the two days after the prompt emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Such rings are produced by X-rays from the extremely bright prompt emission efficiently scattered at small angles by interstellar dust grains in our Galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', Miralda-Escud´e 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The scattered X-rays are delayed with respect to direct ones, due to their longer path length from the source to the observer, with a delay that depends on the distance of the dust cloud traversed by the X-ray radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Thus, the rings are echoes of the prompt emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The Imaging X-ray Polarimetry Explorer (IXPE) is a space observatory with three identical telescopes designed to measure the polarization of astrophysical X-rays (Weisskopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' O’Dell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Soffitta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Launched on 2021 December 9, IXPE is an international collaboration between NASA and the Italian Space Agency (ASI), and it has been operating since January of 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' IXPE measures polarization using the photo-electric effect of X-rays absorbed in the gas gap of a Gas Pixel Detector (GPD) (Bellazzini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' On 2022 October 11 at 23:35:35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='184 UTC IXPE started the observation of GRB 221009A in response to a Target of Opportunity request (Negro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The location position was provided by a Swift/UVOT observation (Dichiara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The observation ended on 2022 October 14 at 00:46:44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='184 UTC with an effective exposure of 94,122 s, and a preliminary quick-look data analysis, image-, time-, and energy-integrated, was available already on October 14, showing a 99% confidence level (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=') upper limit of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1% in polarization degree (Negro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In this work we present the results of the IXPE observation carried out with the fully processed data and through a careful data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This represents the first observation of X-ray polarization of a GRB afterglow, the first measurement of soft X-ray polarization of GRB prompt emission, and the first time we observe polarization properties in both prompt and afterglow phases of the same GRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' After a brief general introduction on IXPE data analysis provided in Section 2, we devote Section 3 to the data anal- ysis and interpretation of the GRB afterglow emission, while Section 4 illustrates the data analysis and interpretation of the rings in association to the GRB prompt emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Summary and conclusions are offered in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' IXPE POLARIZATION ANALYSIS We analyze IXPE Level 2 processed data1, combining the data collected by the three identical detector units (DUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The time-integrated radial profile reveals inconsistency with the expectation from a point-like source, showing a profile that deviates from the instrument point spread function (PSF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In particular two excesses around the peak emission are visible (see Figure 7 and Figure 10 in the Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Such excess appears as rings around a bright core emission and are associated with dust-scattering halos (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', Hayakawa 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Miralda-Escud´e 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' To utilize the full potential of this observation an image- and time-resolved analysis has been carried out, as described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Prior to the data analysis, we perform a first background rejection removing a fraction of background events, mostly composed of cosmic rays interacting in the sensitive area of the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' An irreducible background component remains and needs to be estimated and subtracted from the data as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The correct modeling of such a component is particularly relevant to study the fainter extended emission of the dust-scattering rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' As a suitable background region cannot be extracted directly from this observation, due to the presence of the rings, we assess the expected X- ray background rate from previous IXPE observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In particular, we consider three IXPE observations of low-rate 1 IXPE data are publicly available on the HEASARC archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Left: Background-rejected radial profile around the core emission for DU1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' as shown in the lower panel, the source profile starts deviating more than 20% from the instrumental PSF at around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='43 arcmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The equivalent plots for DU2 and DU3 are not reported here as they carry the same information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Right: Q/I versus U/I plot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' in orange we show the distribution resulting from the spectropolarimetric analysis and the 50%, 90% and 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' contours in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The blue cross and circle show the PCUBE analysis result and the related 1 sigma error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' point-like sources: the observation of 1ES 1959+650 carried out between 2022 June 9 and 2022 June 12 (BKG1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' the observation of BL Lacertae (BL Lac) which happened between 2022 July 7 and 2022 July 09 (BKG2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' the observation of 3C 279 performed between 2022 June 12 and 2022 June 18 (BKG3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' From each of these observations we extract the background spectrum and we simulate a long exposure (1 Ms) IXPE observation with the ixpeobssim simulation tool (Baldini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The three selected observations provide a good bracketing of the background emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' More details on the particle background rejection, the residual background simulation, scaling, and subtraction are reported in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Typically, for IXPE observations, the polarization information is extracted via two types of analyses: a polarimetric analysis and a spectropolarimetric analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' For the former, we use the xpbin routine of ixpeobssim with the flag --algorithm PCUBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This routine computes the I-normalized Stokes parameters Q and U from the pseudo-Stokes parameters of the sample of selected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The algorithm supports the calculation of the background-subtracted Stokes parameters, if a background template is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The polarization degree (PD) and polarization angle (PA) with associated errors are calculated from the Q/I and U/I parameters following the recipe of Kislat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The spectropolarimetric analysis, as opposed to the simpler polarimetric analysis, accounts for the shape of the intensity spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This analysis consists of the joint fit of the I, Q and U spectra and, for this work, we make use of the Multi-Mission Maximum Likelihood (3ML) framework 2 (Vianello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2015), which is publicly available and allows for both frequentist and Bayesian analysis approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Here we report the results of the frequentist analysis, but we verified that the Bayesian approach leads to the same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Hereafter, we refer to the central region as the core, while the inner and the outer rings are denoted r1 and r2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In the next sections we will illustrate the data analyses and results for these different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' THE CORE / AFTERGLOW EMISSION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Data analysis We start with the analysis of the core, which arises from the burst afterglow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We select the region as a disc centered on the brightest pixel of the IXPE image and radius of 26 arcsec (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='43 arcmin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Beyond this radius, the radial profile of the emission deviates from the PSF of the instrument by more than 20%, as shown in Figure 1 (left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Such a deviation informs us on possible contamination from the emission of dust-scattered X-rays (from the GRB prompt 2 https://threeml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='io/en/stable/index.' metadata={'source': 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+page_content='10 10 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='00 50 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='20 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='20 Q/I6 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Spectropolarimetric core analysis Parameter Value PD (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0)% PD U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' < 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='8% PD U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' < 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1% Γcore 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='03 Note—Summary table of the spectropolarimetric analysis of the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The spectropolarimetric fit is performed in the 2–8 keV energy band and it assumes Gaussian statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The PA is uncon- strained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The best-fit Q/I and U/I constants are (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='9) × 10−2 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='9) × 10−2, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' and/or afterglow emission) that we cannot fully resolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We verified that the bright core emission from the central point-like source dominates the final result as we find consistent numbers within the one sigma uncertainty when varying slightly the selecting radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' According to the IXPE PSF, cutting at a radius of 26 arcsec eliminates less than ∼15% of the total source emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Given the high photon statistics of the core, the results of the analysis are not affected by the choice of the background spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Here, we report the results for the background template BKG2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Through the PCUBE analysis, we find an unconstrained polarization in the 2–8 keV energy range and derive a 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' upper limit of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' No evolution with time or energy is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' For the spectropolarimetric analysis we model the observed spectrum with an absorbed power law decreasing in energy, with intrinsic parameters fixed to the values of the Swift/XRT automated online analysis (Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2007), which are consistent with other reported values (Kennea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In particular, the intrinsic absorption parameters are fixed to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='36 × 1022cm−2 (Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2007) at a redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='151 (Ugarte Postigo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022), while the Galactic absorption value is fixed to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='38×1021cm−2 (Willingale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' To account for mismatches of inter-calibration among the different IXPE telescopes, a constant normalization is left free to vary for DU2 and DU3 with respect to DU1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The best-fit values of the spectropolarimetric analysis are provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We find a best-fit power-law index of Γcore = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='03, in agreement with expectations from a late afterglow emission (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The polarization results are slightly more constraining than, but consistent with, the PCUBE analysis with a polarization degree of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The right panel of Figure 1 shows the Q/I versus U/I distribution of the core emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The Stokes parameters Q/I and U/I are expected to be normally distributed with respective means Q/I and U/I, and equal standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' An error contour in (Q/I, U/I) space is a circle of radius ϵ centered on (Q/I, U/I), where (ϵ/σ)2 is distributed as χ2(d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='f=2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The probability that the observed polarization exceeds the measured value, under the null hypothesis of unpolarized emission, is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This is inconsistent with a zero degree of polarization at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We therefore set a upper limit to the polarization degree (1D distribution) of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='8% at 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1% at 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' For completeness, the I, Q, U spectra are reported in Figure 12 (first row) in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Interpretation According to current models, once beyond the early flaring stages, GRB afterglows arise via synchrotron processes from electrons accelerated through interactions of the GRB jet with the circumstellar material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This is consistent with observations from radio to high energies of GRB afterglows (Kumar & Zhang 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The physics is well understood and follows a set of closure relations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', Sari & M´esz´aros 2000) which, when observations fit a self-consistent picture, can be used to infer properties of the underlying emitting region through observables such as the spectral indices and rate of temporal decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The synchrotron spectrum is described by a set of power laws with different spectral indices, each with its own closure relation depending on the particle density distribution of the circumstellar environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We model the core X-ray emission as observed by IXPE with this interpretation, in order to utilize the polarimetric observation to constrain intrinsic properties of the jet and our viewing angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 7 We start by investigating the density of the interstellar matter around the GRB progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The density profile in units of cm−3, n(R), where R is the distance from the central engine, is parameterized by the index k, such that n(R) ∝ R−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' For example, k = 0 corresponds to a constant density medium and k = 2 describes a wind medium, and in-between values are also possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' A wind medium may be expected around long GRBs since they arise in the deaths of massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The density profile affects the time evolution of the synchrotron break frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We assume that the IXPE energy range lies between the typical (νm) and the cooling (νc) synchrotron frequencies, and show that this assumption yields a consistent picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The time and energy evolution of GRB afterglow emission is described by (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', Granot & Sari 2002) Fν ∝ t−αν−β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' (1) The core spectrum is well fit by an absorbed power law with a photon index of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='98±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='03, which yields β = Γcore−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The temporal evolution of GRB afterglow usually shows a break, which causes the index α to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This steepening is proportional to (Γjθj)2, where Γj is the jet Lorentz factor and θj the jet half opening angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Taking into account the time evolution of the Lorentz factor, the increase in the temporal index will be ∆α = (k − 3)/(4 − k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' From the closure relations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', Sari & M´esz´aros 2000) between the temporal and spectral indices, we can express the index of the density profile k as k = 2(4α − 6β + 3) 2α − 3β + 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' (2) For α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='634 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='015, measured by Swift-XRT3 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='03, we get k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We note that using the Swift-XRT spectral index of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='15, we get k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In our model we assume that the IXPE observation was preceded by an achromatic jet break at ∼1 day (D’Avanzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We will thus assume that the forward shock propagates in a wind medium with density n(R) = AR−2, where A = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='02 × 1035A⋆ cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' To estimate A⋆ we introduce fiducial or base values for the energy density fraction in electrons and in magnetic fields: ϵe = 10−1ϵe,−1 , ϵB = 10−3ϵB,−3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' (3) Furthermore, we set the kinetic energy of the outflow to Ek,iso ≈ 1055 erg and we use the Q = 10xQx scaling convention for quantity Q in cgs units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' With the above choice of parameters, and neglecting the effect of inverse Compton scattering on the cooling, we have (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', Granot & Sari 2002): νm = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='6 × 1012 E1/2 k,55 ϵ2 e,−1 ϵ1/2 B,−3 (t/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 d)−3/2 Hz , νc = 2 × 1018 E1/2 k,55 A−2 ⋆,−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 ϵ−3/2 B,−3 (t/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 d)1/2 Hz , (4) confirming that indeed νm < νIXPE ≲ νc at the time of the IXPE observation, and this ordering persists at later times because νm ∝ t−3/2 and νc ∝ t1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In this spectral regime, the energy spectral index is given by β = (p−1)/2, where p is the power law index of the electron energy distribution (dNe/dγe ∝ γ−p e , where γe is the electron’s Lorentz factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Using the β derived from IXPE observation, we find p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We can now estimate A⋆ by comparing the observed flux density at 10 keV, Fν,obs ≈ 10−6 Jy, to the synchrotron model prediction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', Granot & Sari 2002), valid after the jet break: A⋆ ≈ 3 × 10−1 E−(3+p)/2 k,55 ϵ2−2p e,−1 ϵ−(p+1)/2 B,−3 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' (5) We note that A⋆ depends strongly on the ϵe parameter (A⋆ ∝ ϵ−4 e for p ≈ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' A separate constraint for our afterglow model comes from the measured jet break time, tjet, which scales as tjet ∝ Ekθ4 jA−1 ⋆ if the ratio between the jet opening angle θj and viewing angle θv is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This parameter, and its position in time with respect to the time of the observation, is relevant for polarization, as it can be broadly associated with the time when the polarization degree lightcurve has a zero point and the polarization angle rotates by 90 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In fact, for uniform (top-hat) jet structure with no sideways expansion, significant polarization arises from the break in 3 Swift-XRT data were analyzed in the IXPE observation time window through the online tool: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='swift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='uk/xrt live cat/ 01126853/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' IXPE’s light curve shows a consistent time evolution, but we find a less precise estimation of the power index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Therefore, we adopt the Swift value in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 8 10 1 100 101 t [day] 0 5 10 15 20 25 PD [%] base values v/ j=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 v/ j=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='8 j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='7 deg j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3 deg 2 = 3/8 2 = 3/4 IXPE - UL (99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=') Optical - UL (99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=') Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Polarization lightcurves using a set of base parameter values (θj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 deg, θv = 2 3θj, p=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='96, Ek = 1055 erg, A⋆ = 3 × 10−1 cm−1, ξ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We show the effect of changing the θv/θj ratio, the jet opening angle, θj and the magnetic field ratio, ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The IXPE upper limits are shown in teal, while the black upper limit marks the upper limit of the contemporaneous optical observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The shaded band shows a Gaussian modulation centered on the PD (darker shade) and width equal to the one sigma uncertainty on the PD (from the spectropolarimetric fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We stress that we do not claim a measurement, which would require at least a 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' symmetry of the visible surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This surface is typically an annulus when projected to the plane of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' As the annulus grows, it encompasses a progressively larger fraction of the jet surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Eventually, for an off-axis observer, the annulus will grow beyond the size of the jet on one side, while still collecting emission from the opposite side, resulting in net polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The polarization lightcurve exhibits the typical two-bump structure (Ghisellini & Lazzati 1999), where the jet break time approximately corresponds to the minimum between the bumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Our model is constructed so that the PD zero point between the two bumps is at ≈ 1 day, to match the estimated jet break time (D’Avanzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We derive the expected polarization degree by integrating the intensity and polarization of the comoving volume elements of the jet over the equal arrival time surfaces (Sari 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Granot & K¨onigl 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Shimoda & Toma 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Pedreira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' For each comoving volume element, the maximum PD of a synchrotron-emitting, shock accelerated electron population with power-law distribution with index p will be: PD = (p + 1)/(p + 7/3) ≲ 75% (Rybicki & Lightman 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The observed polarization will be reduced from this value by integrating over all the parts of the jet that contribute to the flux at a given observer time (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', Lyutikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The evolution of the polarization as a function of time depends strongly on a variety of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We take a set of parameters (base values) that give a polarization consistent with the IXPE spectropolarimetric measurement: jet opening angle θj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 deg, viewing angle θv = 2 3θj (Ghisellini & Lazzati 1999), electron energy distribution index p=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='96, kinetic energy Ek = 1055 erg, density parameter A⋆ = 3 × 10−1 cm−1 and ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The parameter ξ is the ratio of the magnetic field strength in two directions defined as: ξ2 = 2⟨B2 ||⟩/⟨B2 ⊥⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Here, B|| and B⊥ are the magnetic field parallel and perpendicular to the shock normal, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The case ξ = 0 yields the maximum attainable polarization for any given set of afterglow parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Our model with base values and several additional realizations is presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In general terms, all realizations have zero points anchored at ≈ 1 day and yield increasing PD at the time of the IXPE observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' For a given θv/θj ratio, we can choose a set of parameters (Ek, A⋆ and θj) so that tjet = 1 day is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' A higher θv/θj ratio results in a higher peak polarization and earlier jet break time, due to the higher level of asymmetry as we move away from the jet axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' All presented models in Figure 2, except the low jet opening angle, are consistent with the upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Taking the PD=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0% at face value, models with jet opening angle θj < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 deg (while keeping all other base values fixed) are disfavored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Similarly, models with θv/θj > 2/3 tend to overpredict the IXPE measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Assuming a magnetic field ratio, ξ, closer to 1 simply scales down the PD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In principle, any model that overpredicts the observations can be 9 made consistent by appropriate choice of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The IXPE measurement, considering the base values, favors cases where ξ2 ≲ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Optical polarization observations occurred during the IXPE observation window at the Nordic Optical Telescope (Lindfors et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The sky conditions allowed an estimation of an upper limit to the optical polarization degree of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3% at 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1% at 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', Lindfors et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3 we provide more details about the optical data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The optical band falls in the same spectral regime as the X-rays (νm < νoptical < νc) for most choices of parameters around the base values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Thus the optical upper limit can be used to constrain the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The optical limit is slightly stronger, but gives qualitatively the same constraints as the X-ray limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' THE RINGS / PROMPT EMISSION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Data Analysis As mentioned in the introduction, the observed rings are the result of a known effect involving Galactic dust along the line-of-sight of a bright transient event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' A fraction of the photons emitted in the prompt phase of the GRB are scattered by dust clouds in the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Those scattered inwards towards the line of sight arrive at Earth after traveling a longer path length with respect to the unscattered ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This results in a later arrival time of the scattered photons, with a delay that depends on the distance of the dust cloud to Earth and the scattering angle θs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The angular size of the halos θh is related to the scattering angle, as θs(1 − x), where x is the ratio between the distance of the cloud and the distance of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Since we are dealing with a transient event at cosmological distance (x ≪ 1), the approximation θh ∼ θs applies (Miralda-Escud´e 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Draine 2003a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Being produced by a short transient event, the rings expand radially in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This arises as photons with different scattering angles travel different path lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In order to study the radial evolution of the rings and correctly select prompt, scattered photons as the rings expand, we devise a method inspired by the procedure described in Tiengo & Mereghetti (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' For each photon i detected at a time Ti and at a sky coordinate (ai, δi), we define the following variables ti = Ti − T0 and Ki = [(ai − aB)2 + (δi − δB)2]/2cti = θ2 i /2cti , (6) where (aB, δB) are the coordinates of the unscattered emission (the center of the core in the IXPE image) and θi is the angular distance (in arcsec) of the photon i from the point (aB, δB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The trigger time of the prompt emission is taken Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Study of the ring evolution in the IXPE observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Left: distribution of the variable Ki = 1/Di in time bins (in terms of seconds after the trigger time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Middle: Equivalent to the figure on the left panel, but for the BKG2 template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Right: Background subtracted distribution of 1/Di for the specific case of the BKG2 template (the equivalent distributions for BKG1 and BKG3 are provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='7 GRB clean data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='6 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' /(2c t) [1 / kpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='[s] 1e50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='7 BKG2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='6 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' / (2c t) [1 / kpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='[s] 1e5400 Best-fit curve 350 Background subtracted (BKG2) 300 250 Counts/bin 200 150 100 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' +++ ++++++++ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='7 O?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' / (2c t)[1 /kpc]10 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Left: Q/I versus U/I distribution of the polarization of the rings resulting from the spectropolarimetric analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The 50%, 90% and 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' contours are shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The red circle reports, for comparison, the 95% upper limit to the PD of the core/afterglow emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Right: Results of the spectropolarimetric fit for the PD assuming different background templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The colors violet, teal and green correspond to the different assumed backgrounds BKG1, BKG2 and BKG3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The orange band is centered on the average of the best-fit values weighted by their uncertainties, and has a width representative of the mean relative statistical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' from the Fermi/GBM4 (Veres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The advantage of this approach is that in the plane Ki vs ti, shown in the left panel of Figure 3, the expanding rings appear as horizontal bands, facilitating the event selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We remove the dominant emission from the core by removing events within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='85 arcmin from the center to avoid contamination from the bright core (afterglow) emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We estimate that the contamination from the core emission at radial distances larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='85 arcmin is less than 4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The distribution n(Ki), after subtracting the simulated background events, is shown in the right panel of Figure 3: the contribution of the two rings is prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We fit the distribution around the peaks with the sum of two Lorentzian functions, which approximate well the observed distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We define the event selection cut on the Ki distribution as illustrated in Figure 3 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The area under each best-fit Lorentzian between Ri min and Ri max (orange areas in the plot), where i = 1, 2 denotes r1 and r2 respectively, is at least a factor of twenty larger than the area under the other Lorentzian in the same range (gray areas in the plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This ensures a negligible contamination from the emission of one ring onto the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The edges of the selection for the wider ring are symmetric with respect to the peak, while the innermost edge of the smaller ring is naturally defined by our region cut off at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='85 arcmin to exclude the core emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Similar to the core analysis, we proceed with the PCUBE polarization analysis in the 2–8 keV energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The observed spectra of the two rings are expected to be different because they are generated from the same prompt emission scattered at different angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' As discussed later on, given a scattering angle, the scattering efficiency of X-rays by dust grains is energy dependent (Draine 2003b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This leads to the realization that combining the two ring selections into one single PCUBE analysis would be inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Furthermore, the low statistics of the individual ring selections prevents a proper background template subtraction for the PCUBE analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This implies that the estimated uncertainties are not accurate because they are computed on a boosted statistic that includes background events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The results of the PCUBE analysis for the individual rings are reported in Table 4 of Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We find a PDr1 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='6±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='7% and PDr2 = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='8%, in agreement with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' These values indicate a higher polarization degree than what observed in the core, though never exceeding the 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' required to claim a detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 The spectropolarimetric fit, allowing a proper combination of the rings selections, can give a more accurate estimation of the underlying polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The phenomenological model we define to describe the rings emission allows for the 4 The time difference between the GBM trigger and the beginning on the IXPE observation is 209848 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Fermi-GBM triggered on a precursor event, about 210 s before the main brighter peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We reasonably assume that the rings emission is an echo of the brightest part of the prompt phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Hence we use the GBM trigger time plus 210 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In any case, a difference of 210 seconds on the total time-distance between IXPE observation and the GBM trigger does not affect our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 5 The minimum detectable polarization at 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' (for non background subtracted data) is MDP99% = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5% and MDP99% = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='8% for r1 and r2, respectively 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='00 r1+r2 (BKG2) Spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='75 Core (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='i u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='25 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='00 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='25 50% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='00 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='75 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='50 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='00 Q/I100 BKG2 ri + r2 (sta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=') BKG1 BKG3 80- 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L Pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' [%] 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 60 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L 40 20 011 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Spectropolarimetric rings analysis Parameter Value PD 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='7 (sta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=') ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0 (sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=') % PD U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' < [54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='6% − 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5%] PD U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' < [47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1% − 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='4%] Γr1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='89±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='20 (sta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=') ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='07 (sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=') Γr2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='98±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='25 (sta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=') ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='30 (sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=') Note—Summary table of the spectropolarimetric anal- ysis for the rings emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The spectropolarimet- ric fit is performed assuming Poissonian statistics of the background-rejected data with background- subtraction applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We report the statistical-error- weighted average of the three measurements (each as- suming a different background) along with the associ- ated statistical and systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The upper lim- its are strongly dependent on the assumed subtracted background: we report the range of values defined by the minimum and maximum value obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The PA is unconstrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' spectral parameters of the rings to be different while sharing common polarization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The spectra of both rings are modeled as absorbed simple power laws, while we assume constant polarization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The intrinsic and Galactic absorption parameters are kept fixed to the same values adopted for the core analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' As opposed to the PCUBE analysis, we perform the subtraction of the background spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We test different background assumptions, subtracting the spectra derived from BKG1, BKG2, and BKG3 templates, to which we applied the analogous event selection as for r1 and r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The results are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We find that r1 has a best-fit photon index, averaged over the values obtained assuming different backgrounds, Γr1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='89, with a relative statistical error of about 7% and negligible systematic uncertainty due to the choice of the background spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' r2 has a steeper spectrum, with photon index Γr2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='98 with a relative statistical error of about 6% as well as a 7% relative systematic error associated with different background assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This is also illustrated in Figure 14 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' As we will discuss in the next section, such a difference in spectral index between the two rings is expected owing to the energy dependence of the X-ray scattering cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We note that, for the case of r2, the estimated photon index found assuming BKG1 shows a larger statistical error: the softer spectrum of the emission from this ring with respect to r1 makes the measurement more sensitive to the spectral characteristics of the subtracted background (Figure 8 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3 shows that BKG1 has the hardest spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' As for the polarization, we find a PD of (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='7 (sta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=') ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0 (sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' ))%, where the systematic uncertainty is given by the assumption regarding the background spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Figure 4 shows the results for the different background subtractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The significance of this result, tested against the null hypothesis of unpolarized emission, is about 81% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='. The 1D 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' upper limit on the PD varies between 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='6% and 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5%, depending on the assumed background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Such a difference is due to the low-statistic regime we have for the rings data selection, which causes the statistical uncertainty to be strongly affected by small changes of the subtracted background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The comparison of the best-fit PDs found assuming different backgrounds is illustrated in the right plot of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' For completeness, the Q/I versus U/I distributions obtained for the different assumed background are provided in Figure 15 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Note that, given the different approaches and handling of the background, the PCUBE and spectropolarimetric analyses are not directly comparable in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Figure 15 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3 reports the Q/I versus U/I plots for the different background assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Additionally, we show in the same figure the equivalent plots for the spectropolarimetric fit of the individual rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Furthermore, Figure 11 and Table 4 report the results of the PCUBE analysis of the individual rings resolved in two logarithmic energy bins between 2 and 8 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We refer the reader to the Appendix for further discussion on this matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 12 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Left: Best-fit average distance of the dust clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The orange bands are centered on the average of the best-fit values weighted by their uncertainties, and have a width representative of the mean relative statistical error for the two dust clouds responsible for r1 and r2 emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Right: Derived intrinsic GRB prompt spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The light-grey regions cover the energy ranges excluded in the fitting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In both plots, the colors violet, teal and green correspond to the different assumed backgrounds BKG1, BKG2 and BKG3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Interpretation As discussed in Draine (2003b), the effect of the dust scattering at such a small angles is not expected to alter the intrinsic polarization of the incoming radiation (see their Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' However, an explicit demonstration of this statement in the X-ray band is not directly discussed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Therefore we investigated the effect on polarization from reflection, scattering and transmission considering the common dust compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' All of the above processes lead to a negligible effect on the polarization of the X-ray radiation, as discussed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Therefore, we can reasonably assume that any polarization observed from the X-ray scattering halos is attributable to the original emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' A high polarization degree (PD≳ 20%) in the prompt phase, when viewing the jet at angles smaller than the opening angle, θv < θj, can be achieved by synchrotron emission in an ordered, toroidal magnetic field configuration (Toma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Alternatively, high polarization can be achieved by random magnetic fields or Compton drag models, in a geometry where we are viewing the jet close to its edge, θj ≲ θv < θj + 1/Γj (Granot 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This scenario will result in a very early jet break and potentially high PD in the afterglow, which is disfavored by the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In what follows, therefore, we will focus on the ordered synchrotron scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We estimate the polarization degree integrated over the duration of the prompt emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The PD mainly depends on the photon index, the viewing angle, and the product of the jet opening angle and the Lorentz factor, yj = (θjΓj)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' For the IXPE observation, only the low-energy photon index is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We consider the two extreme values 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='62 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='25, which correspond to the minimum and maximum best-fit values of the prompt GRB intrinsic spectral index given by the assumed background bracketing (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' For Γj = 700 (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022), θj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 deg, and θv = 2 3θj we obtain 16% and 36%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This range is consistent with the measured upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Dust clouds and intrinsic GRB prompt emission In this section we derive some constraints on the dust clouds’ distance and optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We attempt to derive the intrinsic spectrum of the GRB prompt emission from the observed rings spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' However, such considerations are limited by the imaging capabilities of our instrument with respect to other missions that were observing the burst at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We therefore anticipate that the higher angular resolution and wider field of view of XMM-Newton and Swift/XRT data, possibly resolving the presence of more than 2 rings, will better determine the characteristics of the dust clouds visible at the time of the IXPE observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Constraining the spectral parameters of the rings from independent observations might help constrain the IXPE polarization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This will be explored in a follow-up paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The dust cloud distance Ddust is given by Ddust = 2c ∆t θ2 h , (7) 16 14 12 r1 10 r2 BKG1 BKG2 8 BKG3 6 4 210-9 vF,[erg/cm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='/s] BKG1 BKG2 BKG3 10-10 Konus-WIND (GCN 32668) 1 2 3 4 6 10 Energy[keV]13 where c is the speed of light, ∆t is the difference between the time of the burst T0 and the time of the observation of the rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Hence, from the fit of the distribution Ki defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 6, we can easily derive the distance of the clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In fact, the best-fit center of each Lorentzian is the inverse of the distance in kpc of the related dust clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We find the dust cloud associated with r1 to be at a distance of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='41 kpc with a relative statistical error of 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The second cloud, responsible for r2 emission, is estimated to be at a distance of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='75 kpc with a relative statistical error of 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='6 The variance of this measurement, given by the different assumed backgrounds, is negligible with respect to the statistical error, as shown in Figure 5 (left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The half-width at half-maximum of the two Lorentzian curves correspond to ∼ (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1) kpc and ∼ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1) kpc for r1 and r2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Such values are larger than the values expected from the effect of the IXPE PSF7 by a factor of ∼3 (for r1) and ∼2 (for r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This could be symptomatic of a non-negligible thickness of the dust clouds, or, more likely, the presence of several rings within r1 and r2, which we do not resolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' As mentioned in the previous section, the X-ray emission of the ring is the echo of the prompt GRB emission scattered by Galactic dust clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Therefore, assuming the characteristics of the dust to be known, it is possible to infer the intrinsic spectrum of the prompt soft X-ray emission of the GRB from the dust-scattered spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The relation between the scattered spectra and the intrinsic one reads (Tiengo & Mereghetti 2006): Φr(E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' θ2) = Φ(E)τd(E)(g(θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' E) − g(θ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' E)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' (8) where the intrinsic spectrum is modeled with an absorbed power law decreasing in energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' θ1 and θ2 are the rings extents at the beginning and at the end of the observations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' and the function g(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' E) = (θ/θs(E))2 1 + (θ/θs(E))2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' θs(E) = 360′′ � E keV �−1 (9) accounts for both the fraction of halo we do not observe (because it lies outside the IXPE observing window) and the dependence of the scattering angle on the energy8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In fact, larger (smaller) scattering angles correspond to a higher probability of scattering lower (higher) energy X-ray photons, which results in a steeper (harder) spectrum (Draine 2003b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' According to Draine & Bond (2004), the total scattering optical depth of the dust for photons between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='8 and 10 keV can be estimated as τd = τd1 + τd2 + τforeground = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='15Av(E/keV)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='8 (10) with τd1 and τd2 being the optical depths associated with the two dust clouds that produce the echo rings r1 and r2, respectively, and τforeground is the total optical depth between us and the first dust cloud we detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Av is the V-band total Galactic extinction in magnitudes in the direction of the GRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We assume Av = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2 mag as reported in the circulars by the VLT and JWST groups (Izzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Levan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' According to the measurements of Neckel & Klare (1980), the total Av up to 3 kpc in the direction of GRB 221009A is about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' As the first cloud is at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='75 kpc, this sum of three terms should be reasonably valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Note that variations of the assumed value of Av affect the normalization of the intrinsic GRB prompt spectrum, but do not affect the slope of the power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We perform a maximum likelihood analysis by simultaneously fitting both rings spectra with the model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' As for the spectropolarimetric analysis, we have repeated the analysis three times for the different background models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Figure 5 (right panel) illustrates the results of this fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The fit is performed between 2 and 5 keV to avoid the low count statistics part of the ring spectra at high energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The best-fit estimate of the fraction of the total optical depth associated with the farther and closer dust clouds is n ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='36 and (1 − n) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='64 respectively, and is not affected by the choice of the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This translates into extinction values of Av,d1 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='33 and Av,d2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' As for the intrinsic parameters of the GRB, we find that the prompt GRB power law has a photon index between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='62 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='25 with a relative statistical uncertainty of the order of 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This spectral index could be directly compared to the index inferred by the STIX observation (Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022) and to the extrapolation of the lower-end of the energy spectrum measured by Fermi/GBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Konus-Wind, which detects photons down to 20 keV in energy, has released a 6 Note that the farthest dust cloud is different from the (closer) ones producing the rings observed in the earlier Swift-XRT observations reported in Tiengo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The closer cloud in IXPE observation is consistent with the farthest one in Swift observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 7 DU-averaged half-power diameter of ∼26”, (Weisskopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022)) would correspond to a full-width at half-maximum computed at the median time of the observation of ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='7 kpc and ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='8 kpc for r1 and r2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 8 θs(E) is the median scattering angle for photons of energy E, and the equation is a good approximation for photons of energy > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 keV (Draine 2003a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 14 preliminary analysis of the prompt emission of this burst (Frederiks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' These authors find a time-averaged spectrum at the onset of the brightest phase of the event with a low-energy photon index of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Such value lies within range defined by the best-fit values we find assuming different background models (see Figure 5, right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Once confirmed, the direct observations of the prompt emission spectrum can provide a potential way to determine the best IXPE background model to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In fact, the most representative background model could be selected based on agreement of these IXPE-inferred values with an externally measured index value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' At the time of this work, however, such information is not publicly available, so we leave these considerations to a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Considering the bracketing given by the intrinsic spectra derived assuming different backgrounds, we infer a total fluence in the 1–10 keV band of F = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='6 − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1] × 10−4 erg/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The fluence is obtained from the integrated flux between 1 and 10 keV and multiplied by the IXPE total time of the observation, to account for the missing fluence due to Earth occultation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The range of values we report for the fluence is based on the best-fit parameters of the intrinsic power-law model, ignoring statistical uncertainties which are of the order of 10–15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' CONCLUSION IXPE observed GRB 221009A from October 11 at 23:35:35 UTC to October 14 at 00:46:44 UTC for an effective exposure to the target of 94,122 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The imaging capability of the instrument revealed the presence of a bright core emission, associated with the GRB afterglow, and the extended emission of two expanding dust-scattering halo rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Such emission is an echo of the GRB prompt emission and therefore carries information about the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We studied the linear polarization properties of the core/afterglow emission, and derived an upper limit on the polarization degree of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='8% at the 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The temporal and spectral parameters of the afterglow at the time of the IXPE observation are consistent with a forward shock propagating in a wind-like medium, with X-ray emission arising from synchrotron processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The observed upper limit on the polarization degree favors a jet opening angle to be wider than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 degrees, and a viewing angle wider than 2/3 of the jet opening angle (with some underlying assumptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Also, scenarios with an equal magnetic field strength in the two directions parallel and perpendicular to the shock normal seem to be disfavored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The polarization analysis of the combined dust-scattering rings revealed a polarization degree of (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='7(sta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=') ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0(sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' ))% with 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' upper limit ranging between 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='6% and 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5% depending on the assumed background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We also derive a photon index for the intrinsic GRB prompt spectrum between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='62 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='25, depending on the background model considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We note that this range includes the Konus-WIND low-energy spectral index derived at energies above 20 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Considering the best-fit spectra, a scenario involving toroidal, ordered magnetic fields when the viewing angle is smaller than the jet opening angle, predicts high polarization degree up to 36%, compatible with the observed upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The upper limits on polarization from the IXPE observation exclude the case where we are observing close to the edge of a sharp transition in the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Aside from the polarization properties of GRB 221009A, the main focus of this work, we could derive some constraints on the Galactic dust clouds distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Through the time evolution of the emission from the two dust-scattering halos that we resolve, we estimated an average distance of the clouds to be about 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='41 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='75 kpc for the inner and outer ring, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The width of the halos compared to the width expected from the effect of PSF suggests the presence of several unresolved halos within the two halos observed by IXPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Contemporaneous observations by instruments with better angular resolution can inform us whether or not this is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Future joint analyses exploiting contemporaneous observations from different instruments could be beneficial to constrain the spectral parameters and, therefore, better single out the polarization signature of the rings/prompt emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Furthermore, independent polarization measurements from other instruments assessing a different energy regime, for either afterglow or prompt emission, will help to understand the full phenomenology behind this exceptional event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Works along these lines are already ongoing and will be the subject of upcoming publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' On a final note, we remark that the IXPE observation of GRB 221009A is, on its own account, exceptional and unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We assessed, for the first time, the observation of soft X-ray linear polarization from the late afterglow emission of a GRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Also for the first time, thanks to the peculiar location of GRB 221009A in the sky – so close to the Galactic plane – we were able to assess the polarization properties of the prompt emission in the same observation through the radiation scattered off the Galactic dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Aside from providing valuable information about this peculiar event, this IXPE observation is a proof of observational feasibility for future nearby bright transient events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This, several years from now, could inspire new directions for the IXPE mission and widen IXPE’s science portfolio to 15 include fast-transient events, opening a new door for time-domain high-energy astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank Hintz Amenitsch for fruitful discussions on X-ray scattering at small angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We also acknowledge the developers of the Slack team-work platform, which played a crucial role in enabling fast and efficient communication among several different teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We thank I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Negueruela for the careful optical polarization observations at the Nordic Optical Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Based on observations made with the Nordic Optical Telescope, owned in collaboration by the University of Turku and Aarhus University, and operated jointly by Aarhus University, the University of Turku and the University of Oslo, representing Denmark, Finland and Norway, the University of Iceland and Stockholm University at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrof´ısica de Canarias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The data presented here were obtained with ALFOSC, which is provided by the Instituto de Astrof´ısica de Andaluc´ıa (IAA) under a joint agreement with the University of Copenhagen and NOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' MN acknowledges the support by NASA under award number 80GSFC21M0002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' PV acknowledges support from NASA grant NNM11AA01A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' IXPE-related research at Boston University is supported in part by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' National Science Foundation grant AST-2108622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' SM and AT acknowledge financial support from the Italian MUR through grant PRIN 2017LJ39LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The Imaging X ray Polarimetry Explorer (IXPE) is a joint US and Italian mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The US contribution is supported by the National Aeronautics and Space Administration (NASA) and led and managed by its Marshall Space Flight Center (MSFC), with industry partner Ball Aerospace (contract NNM15AA18C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The Italian contribution is supported by the Italian Space Agency (Agenzia Spaziale Italiana, ASI) through contract ASI-OHBI-2017-12-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0, agreements ASI-INAF-2017-12-H0 and ASI-INFN-2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='13-H0, and its Space Science Data Center (SSDC) with agreements ASI- INAF-2022-14-HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0 and ASI-INFN 2021-43-HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0, and by the Istituto Nazionale di Astrofisica (INAF) and the Istituto Nazionale di Fisica Nucleare (INFN) in Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2006, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', 44, 507 Xiao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', Krucker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022, GCN, 32661, 1 Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2018, The physics of gamma-ray bursts (Cambridge University Press) 18 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' BACKGROUND HANDLING The vast majority of the background events for the IXPE telescope are instrumental in origin, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=', cosmic rays that trigger the detector and are reconstructed as photons by the reconstruction algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' On top of those events, a weak X-ray background is also expected (Lumb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' A fraction of the background events can be identified and rejected by looking at the track morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The remaining fraction is indistinguishable from genuine X-ray-triggered events and constitutes an irreducible background that must be treated statistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We adopted a two-step strategy to remove the background events: first we apply a background rejection and then a background subtraction, as detailed here below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Background rejection —Typical X-ray events, compared to charge cosmic-ray events, display a higher fraction of energy deposit associated to the main track9 over the total energy of the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Based on such a difference, a rejection cut can be devised to remove the portion of events that are of clear cosmic-ray nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The left panel of Figure 6 illustrates the energy fraction deposited in the main track of the event as a function of the reconstructed energy: the blue line marks the rejection event cut we apply for events between 2 and 8 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We verify that the rejected events do not manifest any trace of the observed target (see the comparison between the middle and right panels of Figure 6) and that they do not carry any significant polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In the region of the point source the fraction of the rejected background events reach at most 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='6% in the case of DU2 (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Left: background rejection cut based on the energy fraction contained in the main cluster as a function of the reconstructed energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The events that are below the blue line are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The gray shaded areas mark the energy ranges outside the fiducial range for IXPE data analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Center and right: diagnostic maps produced to check the efficiency of the cut: total events map (middle) and rejected events map (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The maps are in detector coordinates, so the central source appears blurred following a pattern caused by deliberate dithering of the satellite (Weisskopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' No apparent residuals of the central source are visible in the background map on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Background subtraction —The residual irreducible background needs to be estimated, simulated, and subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The standard approach consists of selecting a region of the image in the field of view far from the point source, avoiding the edges where the sensitivity of the instrument degrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' However, in the case of extended sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' the dust- scattering rings we detect in this observation), this method cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' To address the issue, we estimate the residual background from a previous IXPE observation of a relatively faint source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' For this work we considered: 1) the observation of 1ES 1959+650 carried out between 2022 June 9 and 2022 June 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2) the observation of 3C 279 performed between 2022 June 12 and 2022 June 18 3) the observation of BL Lacertae (BL Lac) which happened between 2022 July 7 and 2022 July 09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Due to changes in IXPE operations, the observations prior to June 09 would 9 The first step of IXPE reconstruction algorithm is a clustering stage meant to identify a group of adjacent pixels that recorded a charge value above a noise-rejection threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' For typical X-ray-induced events, the charge deposit associated with the photo-electron produces a single main cluster, while additional, spurious clusters are caused by noise fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The reconstruction algorithm assumes the cluster with the higher charge deposit to correspond to the main track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' On the other side, charged cosmic-ray-induced events may produce several, disconnected clusters of charge inside the detector with similar energy deposit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' It follows that cosmic rays display a lower fraction of energy deposit associated to the main track with respect to the total energy of the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 35 8 102 7 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2 Y absorption point (clean) Y absorption point (bkg) 6 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0 2 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='8 0 0 4 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='6 15 3 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='4 10 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2 5 1 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0 100 0 0 2 4 6 8 10 6 2 0 6 2 6 Energy[kev] X absorption point (clean) X absorption point (bkg)19 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Comparison of the radial profiles of the GRB observation (water green) with the sum of the rejected background (dotted red) and the simulated residual background from BL Lac (hatched yellow) for the three IXPE detector units, zoomed on the vertical scale to exclude the large central peak in correspondence of the core and better show the region of the two rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' not provide background estimations suitable for this data analysis, and therefore have not been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' These particular sources are point-like and have a small count rate (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2 Hz), which gives us a wide region of high noise to signal ratio to characterize the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The first two observations are close in time and show a similar background spectrum, while the background obtained from the BL Lac observation shows a lower background rate: this provides a good bracketing for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The same background rejection procedure is applied to the data of all the observations considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The residual background spectrum is derived by selecting the events in an annulus centered on the source with inner and outer radius of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2’ and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5’ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' For each background spectrum we simulate an IXPE observation using the ixpeobssim simulation tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Events are generated uniformly on the surface of the detectors and then projected in the sky using a realistic model for the pointing history that accounts for satellite dithering (Weisskopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In order to reduce the statistical uncertainty, background templates are simulated with a longer exposure (1 Ms) compared to the GRB observation, then re-weighted appropriately to the respective livetime ratio before the subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Figure 7 shows the radial profiles for the three detector units in celestial coordinates: the data of the observation of GRB 221009A are compared to the rejected background and the simulated background (here we show the case for the background extracted from the BL Lac observation, as an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Background scaling —Due to statistical fluctuations in the low-count regime of the GRB rings data, the simulated backgrounds need to be scaled in order to never overshoot the data at high energies and at the edges of the field of view, namely where the background is expected to dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' To define the scaling factor for each background, we estimate 1) the integral of the background spectra for both r1 and r2 selections between 5 and 8 keV and 2) the integral of the radial profile above 6’, then we derive their ratio with the corresponding values of the GRB data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The ratios are reported in the label of Figure 8 and the horizontal lines show visually how the value of the integrals compare to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The right panel of Figure 8 shows the radial profile of all the three simulated background templates, appropriately scaled to the livetime of the GRB observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The scaling factor for each background is defined by the most extreme values among the ratios of r1 spectra, r2 spectra and radial profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The background derived from the 1ES 1959+65 observation is hence scaled down by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='07, the one derived from the BL Lac observation by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='05, and the one from 3C 279 by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Table 3 reports the number of counts for the GRB and for simulated background templates (re-weighted to account for the different live times) in the 2–8 keV band for the three region selections of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' ADDITIONAL CONSIDERATIONS B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Optical polarization data analysis During the IXPE pointing we also performed optical polarization observations in the R-band at the Nordic Optical Telescope (Lindfors et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The observations were obtained using the Alhambra Faint Object Spectrograph and Camera (ALFOSC) in the standard linear polarimetric mode that includes a λ/2 retarder followed by calcite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' At the time of the observations (2022 October 12 at 20:15UT) the sky conditions were clear with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2 arcsecond seeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' However, GRB221009A is located in a crowded Galactic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This resulted in the extraordinary beam of a nearby 100 GRB Simulated residual background Rejectedbackgroundcomponent 80 Entries/bin 60 40 20 0 0 1 2 3 4 5 6 Angularseparation[arcmin]100 GRB Simulated residual background .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Rejectedbackgroundcomponent 80 Entries/bin 60 40 20 0 2 3 4 5 6 Angular separation [arcmin]100 GRB Simulated residual background .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Rejectedbackgroundcomponent 80 Entries/bin 60 40 20 0 0 1 2 3 4 5 6 Angular separation [arcmin]20 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Left and Middle: simulated background spectra of r1 and r2 event selections compared to the GRB observed ring spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The horizontal lines show the values of the integral of the spectra between 5 and 8 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Right: the radial profile of the simulated background templates compared to the GRB data profile for DU1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The horizontal lines show the values of the integral of the profiles above 6 arcmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In all plots the gray shaded areas mark the regions of the parameters space excluded from the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Count Statisitcs Tot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' counts for r1 core r2 GRB data 16121 5450 5502 Bkg 1ES 1959+65 135 4099 4313 Bkg BL Lac 105 3263 3497 Bkg 3C 279 124 3776 4017 Note—Total and background counts in the 2–8 keV band for the core, r1, and r2 selections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' These numbers refer to the background rejected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The background counts are computed by multiplying the background rate by the GRB ob- servation live time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' bright star to overlap with the ordinary beam of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' As such, the standard polarimetric analysis was not possible (Hovatta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Nilsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2018, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Instead, we performed careful modelling of the point spread function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We used the second brightest star within the ALFOSC field of view to create a model of the PSF, which was then subtracted from each image separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This process, however, can result in background artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' To mitigate the effect of any artifact we used a small aperture of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 arcsec radius to perform the measurements using standard formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Effect of dust scattering on X-rays polarization We investigated the effect on polarization from reflection, scattering and transmission considering the dominant dust compounds, Carbon and silicates (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Costantini & Corrales 2022, for a recent discussion of the topic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' At the small angles that we observe, even assuming a coherent reflection angle, any polarization induced by reflection of X-rays would result in a negligible modulation of less than 10−5, or a PD∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='001%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' These values were obtained using the Center for X-Ray Optics database and online tools10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Polarization from transmission is expected to be negligible as well for X-rays of energies at the peak of IXPE sensitivity, given that the common dust compounds do not show K or L shell edges there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' A fraction of the scattered light might have a polarization status affected by big spheroidal dust grains via Mie scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We checked this by using the python package Miepython11, which calculates light scattering according to the Mie theory and Rayleigh–Gans approximation, and adopting the X-ray refraction index for silicates provided by Draine & Lee (1984) and Laor & Draine (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' We find that at the scattering angles we are considering, the PD due to refraction is less than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 × 10−5 at 2 keV for a binary population of grains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' : perfectly aligned, 10 https://henke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='gov/optical constants/ 11 https://miepython.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='io/ ri spectrum 10-3 Rate [Hz] 10- Bkg 1ES 1959+65 Bkg BL Lac Bkg 3C 279 (x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='96) GRB 2 × 100 3 × 100 4 × 100 6 × 100 Energy[keV]r2 spectrum 10-3 Rate [Hz] 10-4 Bkg 1ES 1959+65 Bkg BL Lac (x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='95) Bkg 3C 279 (x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='90) GRB 2 × 100 3 × 100 4 × 100 6 × 100 Energy[keV]60 Bkg 1ES 1959+65 (x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='93) Bkg BL Lac 50 Bkg 3C 279 data 40 Counts/bin 30 20 10 2 3 6 1 4 5 8 9 Anaular sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' [arcminl21 elongated and not aligned, spherical grains) with a power-law size distribution with an index of -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 (Costantini & Corrales 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Therefore, we can reasonably assume that any polarization observed from the X-ray scattering halos is attributable to the original emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Additional plots Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' IXPE counts map combining the 3 DU observations obtained with the xpbin routine of ixpeobssim with the flag --algorithm CMAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The core/afterglow emission dominates the image, however the fainter halos are already visible to the attentive eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=" 19°55' 103 50' Declination(2000) 102 Counts/pixel 45' 101 40' 100 1gh13m30s 15s 005 12m45s 30s RightAscension(2000)22 Figure 10." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Time evolution of the dust-scattering halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The counts maps are generated in three time bins combining the data of the 3 DUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The core region has been removed to better show the rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The images have been smoothed with a Gaussian beam for visualization purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' To guide the eye, we added a thin circle in the first and last images of the sequence to mark the minimum of the counts gap between the two evolving rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Summary table of the PCUBE rings analysis r1 r2 ∆E PD PD u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' (99%) PD PD u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' (99%) keV [%] [%] [%] [%] 2–8 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='6 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='7 <42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='8 <39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='9 2–4 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='2 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3 <48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3 <26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='9 4–8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 <45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='1 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='6 <55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0 Note—Results of the PCUBE analysis between 2 and 8 keV and resolved in 2 loga- rithmic energy bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This analysis is performed on the background-rejected (not background subtracted) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This implies that 1) the estimated uncertainties are not accurate because they are computed on a boosted statistic that includes background events (a big fraction of the total, see Table 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' and 2) the results of the PCUBE analysis are not directly comparable to those resulting form the spectropolarimetric analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The latter represents a more accurate analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' For the 2−8 keV PCUBE analysis the minimum detectable polarization at 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' is MDP99% = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5% and MDP99% = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='8% for r1 and r2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Note that in the 2–4 keV bin for r1 the PD might seem to exceed the 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='. However, in this case, the test-statistic follows a χ2 distribution with 4 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='f, ac- counting for the two energy bins considered and 2D Q-U space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This gives a 3% probability of finding a χ2 value equal to or exceeding the observed one in case unpolarized emission, which means that we are compatible with the null hypoth- esis within the 97% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='. Such significance is even lower if we account for the trials due to both rings selections: in this case we should derive the significance from a χ2 distribution with 8 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='. T- To = 209-268 ks T- To = 268-327 ks T- To = 327-386 ks23 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Polarization PCUBE analysis results for r1 (top row) and r2 (middle row), for one energy bin 2–8 keV (left column) and two logarithmic energy bins 2–4 keV and 4–8 keV (right column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='4 provides the values and 1σ errors on the PD and PA for the energy-resolved PCUBE analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Note that the PCUBE analysis of the rings emission is performed without a proper background subtraction (we refer to the caption of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='4 for further discussion on the caveats of this point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The PAs of the two rings seem to be significantly different, however: as mentioned in the main text, the uncertainties are underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Polarization measurements with significance below the 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' are not considered as a detectios and the PAs are to be considered unconstrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Furthermore, under the assumption that the two rings originate from the same prompt emission, there is no (known) reason to expect the polarization of the two rings to be intrinsically rotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This fact is symptomatic of fluctuations due to the low photon statistics of the signal, and further justifies our approach of combining the two rings for a more accurate spectropolarimetric analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' ri 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='3 45° 60° 30° 0.' 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Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Equivalent plots to that in Figure 3 right panel, but subtracting a different background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=" 10 ixpe DUl I Model ixpe_DU2_I Model ixpe_DU3_I Model ixpe_DUl_I (counts s-1 keV-1) ixpe_DU2_l ixpe_DU3_l Net rate 10-3 10-4 2 Residuals 0 +I++ ++Ti+'Ti 2 4 6 2 × 100 3 × 100 4 × 100 6 × 100 Energy (keV)0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='015 ixpe DUl Q Model ixpe DU2 Q Model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='010 ixpe_DU3_Q Model ixpe DUl Q ixpe DU2 Q keV-1) 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='000 Net rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='010 ixpe_DUl_U Model ixpe DU2 U Model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='015 ixpe DU3 U Model ixpe_DUl_U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='020 ixpe_DU2_U ixpe_DU3_U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content="025 Residuals 9 'tit!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2 × 100 3 × 100 4 × 100 6 × 100 Energy (keV)10- ixpe_DU1_I Model ixpe DU2 I Model ixpe_DU3_I Model 10-2 ixpe_DUl_!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' ixpe_DU2_I (counts s-1 keV-1 #+tttt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' ixpe_DU3_ Net rate 10 10-4 10-5 10-6 2 Residuals 0 6 4 6 2 × 100 3 × 100 4 × 100 6× 100 Energy (keV)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0100 ixpe DUl Q Model ixpe_DU2_Q Model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0075 ixpe DU3 Q Model ixpe_DU1_Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0050 ixpe_DU2_Q keV-1) ixpe_DU3_Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0025 (counts s-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0100 2 Residuals 1 0 1 2 × 100 3 × 100 4 × 100 6 × 100 Energy (keV)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='008 ixpe DUl U Model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='006 ixpe_DU2_U Model ixpe_DU3_U Model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='004 ixpe_DUl_U ixpe_DU2_U keV-1 0.' metadata={'source': 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100 3 × 100 4 × 100 6× 100 Energy (keV)10-1 ixpe DUl I Model ixpe_DU2_I Model ixpe_DU3_I Model +++t+ 10-2 ixpe_DUl_!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' ++ ixpe_DU2_ ixpe_DU3 I Net rate 10 10-4 10-5 10-6 2 Residuals 2 × 100 3 × 100 4 × 100 6× 100 Energy (keV)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='010 ixpe DUl Q Model ixpe_DU2_Q Model ixpe_DU3_Q Model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='005 ixpe_DU1_Q ixpe_DU2_Q keV-1) +++ ixpe_DU3_Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='000 counts s- 0.' metadata={'source': 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spectral indices of the two rings emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' Right: Effect of the corrections for the optical depth and for the energy-dependent scattering efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' The light-gray regions cover the energy ranges excluded in the fitting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0 Photon index 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='5 ri (sta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='+sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=') BKG1 BKG3 r2 (sta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='+sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=') BKG2 5.' metadata={'source': 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(extracted from IXPE observations of 1ES 1959+65) on the left column, BKG2 (from BL Lac observation) on the central column, and BKG3 (from 3C 279 observation) on the right column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This shows that the result of the combined analysis is mostly driven by r1, while r2 seems more unpolarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' However, within the 50% contours r1 and r2 are compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content='00 r1+r2 (BKG1) Spec.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' This shows how the Bayesian approach yields results consistent with the frequentist approach adopted in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' It also shows the nice convergence of the fit for all the parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=' In both plots, we show the case of BKG2 assumed as background model for the background subtraction (the cases of BKG1 and BKG3 yield analogous results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} +page_content=" ndex $6'T- k = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQf1P70/content/2301.01798v1.pdf'} 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Morris,1 Artem Bohdan,1, 2 Martin S. Weidl,2 Michelle Tsirou,1 Karol Fulat,3 and Martin Pohl1, 3 +1Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany +2Max-Planck-Institut für Plasmaphysik, Boltzmannstr. 2, DE-85748 Garching, Germany +3Institute of Physics and Astronomy, University of Potsdam, D-14476 Potsdam, Germany +ABSTRACT +Thermal electrons have gyroradii many orders of magnitude smaller than the finite width of a shock, +thus need to be pre-accelerated before they can cross it and be accelerated by diffusive shock acceler- +ation. One region where pre-acceleration may occur is the inner foreshock, which upstream electrons +must pass through before any potential downstream crossing. In this paper, we perform a large scale +particle-in-cell simulation that generates a single shock with parameters motivated from supernova +remnants. Within the foreshock, reflected electrons excite the oblique whistler instability and produce +electromagnetic whistler waves, which co-move with the upstream flow and as non-linear structures +eventually reach radii of up to 5 ion-gyroradii. We show that the inner electromagnetic configuration +of the whistlers evolves into complex non-linear structures bound by a strong magnetic field around +4 times the upstream value. +Although these non-linear structures do not in general interact with +co-spatial upstream electrons, they resonate with electrons that have been reflected at the shock. +We show that they can scatter, or even trap, reflected electrons, confining around 0.8% of the total +upstream electron population to the region close to the shock where they can undergo substantial +pre-acceleration. This acceleration process is similar to, yet approximately 3 times more efficient than, +stochastic shock drift acceleration. +Keywords: acceleration of particles, instabilities, ISM – supernova remnants, methods – numerical, +plasmas, shock waves +1. INTRODUCTION +It was suggested by Fermi (1949) that hadrons could +be accelerated by magnetic mirrors and give rise to the +well-documented observed cosmic-ray power-law spec- +trum (e.g. Hillas 1984; Nagano 2009). +The original +Fermi acceleration assumes interactions between parti- +cles and magnetic mirrors, which move at speed U, oc- +cur with an isotropic distribution of incident angles be- +tween the two (as measured by a stationary observer). +The relative orientation of these interactions can either +cause particles to gain (when head-on) or lose (if head- +tail) energy, with a slight preference (of order U/c) for +the former. The expected angle-averaged fractional en- +ergy gain per collision is ∝ (U/c)2, where c is the speed +of light. Astrophysical shocks occurring in nature are +more accurately described by diffusive shock accelera- +tion (DSA) (Krymskii 1977; Axford et al. 1977; Bell +Corresponding author: Paul J. Morris +paul.morris@desy.de +1978; Blandford & Ostriker 1978), where the interac- +tions always occur head-on in the rest-frames both up- +stream (unshocked plasma) and downstream (shocked +plasma) of the shock front. DSA analytically predicts +a more efficient fractional energy-gain-per-crossing of +∝ (U/c) and power law spectrum. DSA has been gener- +ally successful in explaining a wide variety of astrophys- +ical sources such as active galactic nuclei (Marchenko +et al. 2017) and supernova remnants (SNRs) (Reynolds +2008), where we observe non-thermal emission that is +often characterized by a power-law. The radiative prop- +erties of these objects can be interpreted as originating +from an underlying population of high-energy particles +(protons, electrons etc.), providing evidence to support +DSA. +Despite its numerous successes, aspects of the under- +lying micro-physics necessary for DSA to work are yet to +be conclusively determined (Amano et al. 2022). This is +because in DSA it is assumed that the shock is a perfect +discontinuity, when in reality it has a finite width of the +order of the gyroradius of a proton traveling with the +arXiv:2301.00872v1 [physics.plasm-ph] 2 Jan 2023 + +2 +Morris et al. +shock speed, rgi. While this does not pose challenging +for thermal ions to cross into the downstream, thermal +electrons require a significant amount of pre-acceleration +before their gyroradii are sufficiently enlarged that they +can easily cross the shock transition from upstream to +downstream, or vice-versa. +To accelerate particles to +cosmic-ray energies, DSA requires them to cross the +shock multiple times, thus only electrons that have al- +ready undergone sufficient pre-acceleration can “be in- +jected" into DSA and undergo further acceleration by +this mechanism. +Particle-in-cell (PIC) simulations are an excellent tool +with an eminent track record when it comes to investi- +gating electron pre-acceleration. These simulations are +fully kinetic, containing individual electrons and ions, +thus allow for a self-consistent treatment when these +particles move in their self-generated electromagnetic +fields. From these, we can obtain time- and spatially +dependent information concerning both individual par- +ticles and the fields they experience which allow us to un- +veil the underlying physical processes (Pohl et al. 2020). +In this work, we use physical parameters appropri- +ate for supernova remnants, which are characterized by +non-relativistic outflows with sonic and Alfvénic Mach +numbers of MS, MA ≈ 20 − 2000 (Wang et al. 2009). +In contrast, the low Mach number regime is associated +with the Earth’s bow shock (Ms, MA < 10). Focusing +on SNR parameters is advantageous for many reasons. +First and foremost, it has been established for almost +50 years that cosmic rays (CRs) can be accelerated by +SNRs, with the majority of Galactic CRs believed to +originate from these objects (Axford et al. 1977; Krym- +skii 1977; Drury 1983; Bell 1978; Blandford & Ostriker +1978). Additionally the close proximity of SNRs permits +the study of non-thermal radiation in radio, X-, and γ- +rays. This radiation is often attributed to a population +of accelerated electrons, thus understanding their accel- +eration is essential to comprehend the radiative proper- +ties of SNRs. +A crucial parameter in governing the behavior of a +shock is the obliquity angle, θBn, which subtends the +upstream magnetic field with the shock normal. Typ- +ically perpendicular shocks, where θBn = 90◦, have +well defined shock transitions, with a small (of order +∼ rgi) shock foot region leading up to the ramp. These +shocks have been thoroughly studied over the last decade +using 2D PIC simulations (Amano & Hoshino 2009; +Kato & Takabe 2010; Matsumoto et al. 2012, 2013, +2015; Wieland et al. 2016; Bohdan et al. 2017, 2019a,b, +2020b,a, 2021). Conversely, decreasing the shock obliq- +uity angle more freely permits the escape of energetic +particles back upstream as their trajectories are tied to +the magnetic field lines, allowing them to outrun the +shock if sufficient energization has taken place. +The +extended regions containing the reflected particles are +known as the electron and ion foreshocks (depending +on the particle species) (e.g. Burgess 1995; Fitzenreiter +1995; Treumann 2009) where the energy transported up- +stream by these reflected particles can excite instabili- +ties and generate turbulence which can in turn influence, +and possibly pre-accelerate upstream electrons. All up- +stream electrons that eventually encounter the shock +must first pass through the foreshock, thus a physical +description of these regions is essential to fully compre- +hend the overall description of electron pre-acceleration. +Prior work has demonstrated that energetic electrons +that have been pre-energized by shock surfing accel- +eration are more likely to be reflected back upstream +(Amano & Hoshino 2007), where mirror reflection (also +called shock drift acceleration, SDA) (Wu 1984; Leroy +& Mangeney 1984) is the mechanism responsible for the +reflection (Honda & Honda 2005). This latter mecha- +nism operates on electrons gyrating close enough to the +shock ramp so that part of their gyrational orbit en- +close the region with enhanced magnetic field, causing a +temporary orbital tightening and causing them to drift +along the shock (Wu 1984; Leroy & Mangeney 1984). +Results obtained from using 1-dimensional PIC simu- +lations demonstrated that the energy content of these +reflected electrons was sufficient to power electrostatic +and electromagnetic waves in the shock foot, which ef- +fectively trap electrons allowing them to undergo more +cycles of shock drift acceleration (SDA), gaining more +energy and eventually cross into the downstream region +(Xu et al. 2020; Kumar & Reville 2021). +Bohdan et al. (2022) used 2D3V (2 spatial dimensions +and all three velocity and field components) PIC simu- +lations, with a combination of large scale and periodic +boundary condition simulations, accompanied by ana- +lytically solving the dispersion to elucidate the exact in- +stabilities excited by reflected electrons in the electron +foreshock. The first of these are electrostatic electron +acoustic waves (EAW). In paper I of this series, Morris +et al. (2022, hereafter Paper I) investigated the effect +of changing the orientation of the upstream magnetic +field on the foreshock structure by performing a series +of narrow box simulations. +It was found that EAWs +are quickly excited within a few ion gyro-radii, and the +EAWs are stronger for decreasing θBn. It was shown that +these waves can interact with, and in ∼ 1% of cases, di- +vert upstream electrons away from the shock. Bohdan +et al. (2022) further identified electromagnetic whistler +waves in the inner foreshock region, which require a com- +paratively larger energy density of reflected electrons rel- + +Electron-Whistler Interactions in Oblique Shocks +3 +ative to EAWs, subsequently excited at later times than +EAWs. These whistler waves occur on spatial scales ap- +proaching ion length scales, as opposed to the EAWs +where the characteristic size is similar to the electron +inertial length. +In this paper we focus on the micro- +physics of individual electrons which encounter these +whistler waves, and interpret their behavior in the con- +text of electron pre-acceleration in astrophysical shocks. +In Section 2 we outline our simulation setup before pro- +viding an overview of the shock structure in Section 3. +Section 4 contains the main discussion, where we outline +the properties and development of whistler waves before +detailing their evolution and interaction with electrons +present in the foreshock. +2. SIMULATION SETUP +In Paper I, we investigated the effect of changing +the obliquity angle, θBn and the plane-angle, φ, which +characterize the orientation of the initial large-scale +upstream magnetic field, on the electron foreshock at +short times. The run-time of these simulations can be +quantified in terms of the ion-gyrofrequency, defined as +Ωci = |e|B0/mi, for electron charge magnitude |e|, mag- +netic field amplitude B0 and ion mass mi, with the total +run-time tsim = 7.8Ω−1 +ci . They further employed a nar- +row box, spanning 4.8λsi transversely, where λsi is the +ion skin length. +This relatively small transverse size +reduced the computational expense of a single simula- +tion, and therefore permitted multiple simulations to be +performed. The chosen parameters also additionally al- +lowed for a comparison to the 3D simulations of Mat- +sumoto et al. (2017). +Conversely, in this paper, we investigate electron pre- +acceleration as a consequence of the electromagnetic +foreshock, which begins to emerge at around tsim ∼ +10Ω−1 +ci . This is characterized by the presence of whistler +waves, which at late times develop into non-linear struc- +tures that can reach up to approximately 1 − 10λsi in +diameter, which are better captured by our 2D3V simu- +lations in the out-of-plane (φ = 90◦) case. Accordingly, +to adequately resolve these structures we perform a sim- +ulation with a wider box, of transverse size 41.6λsi and +with a longer total run-time of tsim = 51.5Ω−1 +ci , allowing +us to follow their long-term evolution. This computa- +tionally expensive simulation featured in Bohdan et al. +(2022), and the setup will be briefly outlined below. +Our code is a modified version of TRISTAN (Buneman +1993), which cyclically solves Maxwell’s equations for +fields defined on a simulation grid and updates the posi- +tions of individual particles located in the grid cells ac- +cording to Lorentz forces via the Vay solver (Vay 2008). +This version allows us to track the progression and prop- +erties of individual particles and measure the local field +strengths they encounter to better elucidate the physical +processes they experience. +At the beginning of the simulation, we initialize a +plasma slab by injecting ions and electrons, where mi +and me are the ion and electron mass and mi/me = 50, +co-spatially into the simulation box with the number of +particles per cell per species given as n0 = 40. The two +particle species are initialized in thermal equilibrium, +such that kBTe = kBTi = 9.86 · 10−4 mec2, where kB is +the Boltzmann constant and c the speed of light. This +defines the sound speed as cs = +� +2ΓkBTi/mi = 0.0081c, +for adiabatic index Γ = 5/3. Across this plasma slab, +we apply a large-scale, uniform, magnetic field accord- +ing to ⃗B0 = B0(cos θBn, sin θBn cos ϕ, sin θBn sin ϕ) = +B0(0.5, 0, +√ +3/2), where θBn = 60◦ and φ = 90◦. +Such a setup defines the important temporal scales +in our simulation, such as the electron plasma and gy- +rofrequencies, ωpe and Ωce respectively. Their ratio is +quantified by Ωce = |e|B0/me = 0.06 ωpe, where ωpe is +the electron plasma frequency. From this, we can define +the electron skin length, λse = c/ωpe = 8∆, that is re- +solved by 8 grid cells (denoted by ∆). For ions, their +inertial length scales as λsi = +� +mi/me λse. We ensure +that these relevant frequencies are sufficiently resolved +in our simulation by advancing it in time-step units of +δt = 1/16 ω−1 +pe . +We move our plasma slab with a bulk velocity as mea- +sured in the simulation frame of ⃗vup/c = −0.20ˆx. Con- +sequentially, a motional electric field is produced and de- +fined by ⃗E0 = −⃗vup × ⃗B0. Such a setup leads to a large +value of ∇ × ⃗E, and a correspondingly large ∂ ⃗B/∂t, at +x = 0 which can induce a large initial transient. We mit- +igate this by tapering the initial upstream field values +to zero over the region x ≤ 50∆ (Wieland et al. 2016). +The nonzero value of ∇ × ⃗B is exactly compensated by +a drift current carried by the ions, which is removed at +x = 0. +Constituent particles within the plasma that reach +the boundary at x = 0 encounter a reflecting wall, +which performs the transformation vx → −vx (vy and +vz are unaffected) (Quest 1985; Burgess et al. 1989), +and so they propagate back upstream. +The mag- +netic field behind this upstream-moving plasma com- +pletely isotropizes after approximately a few ion gyro +times, with a compression ratio of nd/n0 ∼ 4.0 rela- +tive to the undisturbed upstream plasma. In the sim- +ulation frame, a quasi-stationary shock with velocity +⃗v∗ +sh/c = 0.067 ˆx propagates upstream, which is equiv- +alent to a shock velocity of ⃗vsh/c = 0.263 when mea- +sured in the upstream rest frame. The properties of the +shock can be further quantified by the Alfvén velocity, + +4 +Morris et al. +Figure 1. +Reflecting wall setup of the simulation. +φ is +the angle B0 makes relative to the simulation plane. θBn +is the angle subtended between the shock normal, ˆn, and +the magnetic field vector, B0. +Here we use θBn = 60◦ +and φ = 90◦, which define the upstream magnetic field as +B0 = B0(0.5, 0, +√ +3/2). The dotted line indicates the trans- +verse size of the simulation box, which is 41.6 ion skin +lengths. +vA = B0/ +� +µ0(neme + nimi), for vacuum permeabil- +ity µ0 and ni = ne = n0 are the ion and the electron +number densities. This gives rise to the Alfvénic Mach +number, MA = vsh/vA = 30, whereas the sonic Mach +number is MS = vsh/cs = 32.5. The plasma beta value, +which denotes the thermal-to-magnetic energy density +ratio in the upstream region is β = 1. Our simulation +setup is shown in Fig. 1. As the shock propagates up- +stream, the domain length in the x-direction increases, +with new plasma injected into the new regions with the +same properties outlined above. The extension of the +simulation box is essential to ensure that that we main- +tain all reflected electrons within the boundaries of our +simulation. +3. SHOCK STRUCTURE AND SUMMARY OF +FORESHOCK CHARACTERISTICS +The late-time (Ωcit = 49.9) shock structure is shown +in Fig. 2. For illustrative purposes, we show only the +inner electromagnetic foreshock, containing the whistler +waves, and the beginning of the outer electrostatic fore- +shock containing the EAWs. The full simulation box at +this timestep extends to ∼ 2000λsi. As explained in Bo- +hdan et al. (2022) and Paper I, the EAWs here are not +well captured because the simulation setup employs an +out-of-plane field angle (φ = 90◦), with EAWs propa- +gating in the direction of the upstream magnetic field. +We therefore cannot see them so easily because they do +not lie in the simulation plane. In contrast to Paper I, +slightly ahead of the shock transition at x/λsi ⪆ 410, +we see large electromagnetic irregularities, which have +developed from the oblique whistler instability. Their +onset begins here at Ωcit ∼ 10, beyond the simulation +time in Paper I, and they are associated with under- +dense electron cavities (panel (a), also present in ions ) +as well as magnetic- and electric-field turbulence (pan- +els (b) and (c)). The physical scales of these cavities +is of order λsi, justifying the use of a larger transverse +simulation box to allow a robust investigation of these +phenomena. We further note that these magnetic and +electric field inhomogeneities extend into the upstream +beyond the shock ramp (for x > 410λsi) in the field +profiles in panels (d) and (e), but decrease in strength +with increasing distance from the shock. +From these +panels, where the field is averaged across the transverse +direction of the simulation box, we see that (B/B0)2 +approaches unity more quickly than (E/E0)2, indicat- +ing the end of the inner electromagnetic foreshock and +the beginning of the outer electrostatic foreshock, the +latter of which is the subject of Paper I. +In Bohdan et al. (2022), it was demonstrated via +means of periodic boundary condition simulations that +the differences in the inner and outer foreshocks can be +explained by differences in the reflected electron pop- +ulations that excite them. The latter, which leads to +the excitation of electrostatic EAWs, has in comparison +to the inner foreshock a reflected electron beam density +a factor of 10 lower. Furthermore, the thermal spread +of this electron beam is approximately 25% lower than +that of the inner electromagnetic foreshock. These dis- +crepancies are enough such that in the inner foreshock +the electromagnetic oblique whistler instability is the +dominant excited instability, as opposed to the electron +acoustic instability which is prevalent in the outer re- +gions. +The analysis presented in Paper I focused on the be- +havior of upstream electrons within the outer electro- +static foreshock. In the remainder of this paper, we fo- +cus on the inner electromagnetic foreshock, focusing on +the properties of the waves, how they affect upstream +electrons, and whether they can lead to electron pre- +acceleration. +3.1. Shock Reflection Rate +We first quantify the energy content in the reflected +electrons which are essential to excite the electromag- +netic oblique whistler instability. We do so by first es- +timating the shock location, xsh, taken as the location +where ni/n0 = 2 for ion number density ni and the +subscript 0 denotes the far upstream value. We define +this based on ion density as that of the ions reflected at +the shock is around 10−4n0 and is more stable in the +foreshock relative to the electron number densities (due + +Reflecting wall +Bo +n +41.6入si +Ba +Simulation PlaneElectron-Whistler Interactions in Oblique Shocks +5 +Figure 2. +Late time shock structure for Ωcit = 49.9. Panel (a) shows the electron density map, (b) and (c) show fluctuations +in the Bz and Ey components of the magnetic- and electric-fields, respectively, and (d) and (e) the magnetic and electric field +profiles, respectively. We see the transition from the downstream to upstream regions at x/λsi ∼ 400, with the immediate +upstream characterized with electromagnetic structures. +Figure 3. The reflection rate, R = Ne,ref/(Ne,ref + Ne,ups) +where Ne,ref is the number of reflected electrons and Ne,ups +the number of upstream electrons that have not yet reached +the shock, (blue dashed line), mean value of γ − 1 for the +reflected electrons (red dotted line, simulation frame) and +normalized kinetic energy contained within the reflected +electron beam (black line). +The latter quantity is nor- +malized relative to the upstream kinetic energy such that +UK,ref/UK0 = Rγref − 1/(γups − 1) for reflection rate R and +γups = (1 − (vups/c)2)−1/2. +Quantities are calculated im- +mediately ahead of the shock ramp between xsh + 2λsi ≤ +x ≤ xsh + 10λsi. Although the reflection rate falls with the +onset of whistler waves at Ωcit ∼ 10, the overall energy den- +sity increases as the reflected electrons are more energetic on +average. +to higher electron reflection rates) and electromagnetic +field amplitudes, which are disturbed by the whistler +waves. +In practice, this definition places xsh on the +shock ramp, so we measure the reflection rate in the +region defined by xsh + 2λsi ≤ x ≤ xsh + 10λsi to ensure +we are measuring it for a region within the electron fore- +shock. Note that the parameters of the region of interest +do not affect the presented results, so long as it resides +within the electron foreshock. The chosen fixed region +nevertheless lies close to the shock, and at late times +is completely occupied by whistler waves, thus enables +us to measure if they have any tangible effect on the +reflection rate. +As measured in the simulation frame, the electron +spectrum in the defined upstream region has a double +peak structure in Γ − 1, where the low energy peak cor- +responds to the thermal population moving with the +upstream bulk flow and the high energy peak corre- +sponds to reflected electrons. +As in Paper I, Fig. +4, +we use the local minimum between these peaks to dis- +tinguish between reflected and upstream electrons. At +each timestep, reflected electrons are those within the +defined region where Γ − 1 exceeds the value for which +d(Ne(Γ−1))/d(Γ−1) = 0 and d2(Ne(Γ−1))/d(Γ−1)2 > +0 in the electron spectra. Those with Γ − 1 lower than +this threshold are considered to be upstream electrons +traveling with the bulk flow of the incoming plasma. +Note that this definition has no dependency on the di- +rection in which the electron is traveling, thus an ener- +getic reflected electron that is re-directed towards the +shock is still considered reflected. + +(a) +1 +25 +log10 +0 +0 +-0.8 +10 +25 +(b) +(Bz Boz) +0 +[Bo +0 +10 +10 +(Ey Eoy) +0 +[Eo +07 +-10 +(E/Eo)2 (B/Bo)2 +(d) +(e) +400 +500 +600 +700 +800 +x/Λsi14 +0.10 +6 +12 +0.08 +5 +Fraction of reflected electrons, +10 +. ref/Uko +8 +0.06 +Uk, +6 +3 +0.04 +4 +2 +0.02 +2 +1 +0.00 +0 +10 +20 +30 +40 +50 +time (Qcit)6 +Morris et al. +Fig. 3 indicates that the onset of whistlers may indeed +affect the reflection rate. The blue dashed line shows +that the reflection rate falls after the onset of whistler +waves at around Ωcit ∼ 10, but is relatively stable at +around 4% for Ωcit > 30. During the simulation, de- +spite variations and a slow decline in the reflection rate, +the energy density of the reflected beam increases at a +roughly linear rate as indicated by the black solid line +in Fig. 3. Possible causes of this include more efficient +acceleration of reflected particles or an acceleration re- +gion which has a size that increases with time. +The +red-dotted line shows that this can be explained by the +fact that the mean Lorentz factor of reflected electrons +also increases approximately linearly with time. +We have verified that the reflection rate calculation is +robust to our choice of region. The changes in reflection +rate for the region shown in Fig. 3 occur further up- +stream in the same manner, although with a time-lag as +it takes the reflected electrons longer to travel upstream +along ⃗B0 and reach those regions. Accordingly, the en- +ergy density of the reflected electron beam increases +throughout the simulation and reaches the threshold +value to excite whistlers further from xsh as the sim- +ulation progresses, hence explaining why the size of the +whistler region increases with simulation run-time. +4. WHISTLER WAVES +4.1. Wave Properties/Summary of Bohdan 2022 +A study of the instabilities driving the waves that +arise in the electron foreshock was undertaken in Bohdan +et al. (2022), with results based on the same large-scale +simulation that is presented here. In this earlier work, +the electromagnetic waves present in the inner foreshock +that we focus on in this paper were subject to a linear +dispersion analysis, where it was established that they +arise as a result of the oblique whistler instability. Ev- +idence for this came from the fact that the waves in +question have approximately the same parallel and per- +pendicular wavenumbers as well as growth rate as pre- +dicted by linear theory for the fastest-growing oblique +whistler mode. +Defining the wavevectors parallel and +perpendicular to the upstream magnetic field as k∥ and +k⊥, respectively, and noting that the oblique whistler +instability is excited by a beam of electrons moving par- +allel to the upstream flow with velocity vb, we summarize +the results of Bohdan et al. (2022) as: +1. The dependence of the perpendicular wave num- +ber, k⊥, of the fastest growing oblique whistler +mode on the parallel velocity of the reflected elec- +tron beam is weak. For the parameters used here, +the peak growth rate occurs at k⊥λse ≈ 0.2. +2. The excited waves are in resonance with electrons +reflected at the shock. Hence, k∥ is extremely sen- +sitive to v∥, with the lth order gyroresonance given +by, +ϖres = k∥vb − ℓ |Ωce| +γb +, +l ≥ 1 +(1) +for beam Lorentz factor γb and electron gyrofre- +quency Ωce. +3. The beam can only excite fluctuations with suffi- +ciently small phase speed (O(10−3c)), i.e. waves +with angular frequencies satisfying ϖW < ϖres. +4. Decreasing the beam number density of the re- +flected electrons results in a smaller growth rate +of the whistler mode. +From this latter point, and from Bohdan et al. (2022, +Fig. 9), we note that the dependence of ω on k for the +whistlers is approximately linear, and hence the group +velocity, vg, is of the same order of magnitude as the +phase velocity, meaning vg << vup. Accordingly, we see +the linear structures co-move with the upstream bulk +flow when viewed in the frame of reference of our simu- +lation. +4.2. Non-linear Structures +Although the behavior of the initial whistler wave +structure can be appropriately described by linear anal- +ysis, at later times in the simulations they develop into +non-linear wave packets, with a complex internal struc- +ture. In this section we outline the structural evolution +of the whistler waves into non-linear wave packets as +they propagate towards the shock from the upstream. +Fig. 4 shows the electromagnetic field structure of a +particularly prominent non-linear structure (developing +from a whistler wave) occurring in the inner electro- +magnetic foreshock of the simulation at Ωcit = 42.9. +Quantities have been measured in the upstream rest +frame, which removes the large scale motional electric +field, thus the electric field structure displayed is domi- +nated by that associated with the non-linear structures. +The top row illustrates the magnetic field structure, with +the middle row showing the electric field. From left to +right the figure shows the x-, y-, and z- components. We +note that while all three magnetic field components have +roughly similar peak magnitudes, the maximum abso- +lute values of the Ex and Ey components are around an +order of magnitude higher than for Ez. +We further note that the characteristic size of the +whistlers increases as they approach the shock, which +is consistent with our previous studies in Bohdan et al. + +Electron-Whistler Interactions in Oblique Shocks +7 +Figure 4. +The structure of a prominent whistler wave +packet present in the electromagnetic foreshock at Ωcit = +42.1 is shown as measured in the upstream rest frame. For vi- +sual clarity, quantities are normalized relative to background +simulation-frame field values (note there is no motional E- +field in this frame). Panels (a), (b) and (c) show Bx, By and +Bz, panels (d), (e) and (f) show Ex, Ey and Ez. Panels (g), +(h) and (i) show the Lorentz force for a test particle moving +at ⃗v = (−ve,th/ +√ +3) · (1, 1, 1), for electron thermal velocity +ve,th. Here, the Lorentz force components are normalized to +the modulus of F0 = eve,thB0. +(2022). It is from within these waves at late times that +highly non-linear structures develop. From panels (a) +- (c) of Fig. 4, we see that the B-fields are in general +strongest in magnitude at the edge of the whistlers, and +progressively weaker towards the central region, such +that in 2D space the value of |B/B0| reaches a maximum +in a ring shape encircling the nonlinear wave structure, +as depicted in Fig. 5. The range of spatial radii from +1 − 5λsi of these structures is similar to that of short +large amplitude magnetic structures (SLAMS) (Mann +& Classen 1995; Wang et al. 2020), which have been +detected in the bow shocks of the Earth (Mann et al. +1994), Venus (Wang et al. 2020), and Jupiter (Tsu- +rutani et al. 1993). +However, in contrast to SLAMs, +where the density is amplified by a factor of a few rel- +ative to the upstream plasma, we see from Fig. 2 that +the non-linear structures discussed here are associated +with under-dense cavities, with under-densities as low +as n/n0 ≈ 0.15. +We further note that observational +evidence for whistler-mode induced structures has been +provided by analysing satellite data from the Magne- +tospheric Multiscale mission (He et al. 2021; Shi et al. +2022). These observational data apply to the Earth’s +Figure 5. The structure of |B|/|B0| for the non-linear struc- +tures. The magnetic energy density is strongest in a ring-like +shape encircling the structure, with the field strength declin- +ing towards the center and radially outside of the area of +maximum strength. +bow shock, thus the parameters used here are not con- +sistent with or supernova remnant based simulation. +To comprehend any pre-acceleration that arises con- +sequentially from the interactions of electrons with +whistler waves, we must understand the forces they +experience when interacting with them. In general, an +electron with charge qe immersed in both electric (E) +and magnetic (B) fields experiences a Lorentz force, F , +which is defined as, +F = qe(E + v × B) +(2) +where bold quantities represent vectors with x-, y- and +z- components in Cartesian space. From Eqn. 2, the +velocity of an electron, both in terms of direction and +magnitude, can influence the resulting behavior when in- +teracting with a region of strong electromagnetic fields, +such as those within the non-linear structures. +We trace a sample of 10,000 upstream electrons to +probe any interactions with the whistler waves and non- +linear structures. These electrons are selected randomly +from the upstream population at Ωcit = 30 from a re- +gion between 360λsi and 410λsi ahead of the shock, and +traced for the remainder of the simulation (over 20 Ω−1 +ci ). +Such a sample is representative of the global upstream +population, and our sampled region permits an adequate +duration for them to pass through the foreshock and in- +teract with the shock itself. +4.3. Interaction of upstream (bulk flow) electrons with +whistlers +We have already established that the non-linear struc- +tures have phase and group velocities of O(10−3c). Be- + +10 +(b) +(c) +(a) +20 +1 +S. +B-Bo +0 +[Bol +10 +-1 +10 +40 +(d) +e +20 +2 +CC +S +E +0 +[Eo] +10 +-2 +40 +(g) +101 +20 +100 +S. +O F/\Fol +-100 +10 +-101 +370 +380 +370 +380 +370 +380 +x/Λsi +x/Λsi +x/^si4 +20 +2 +[B/Bol +5. +15 +1 +10 +0.5 +0.2 +370 +380 +x/Λsi8 +Morris et al. +cause of this, they are quasi-stationary when viewed +in the upstream rest frame, and any force components +arising from their motion relative to the upstream bulk +plasma is negligible. For these reasons, the upstream +rest frame is appropriate to analysis interactions be- +tween the structures and upstream electrons. +In this frame, the upstream electron population is +thermal, with a most probable speed of ve,th = 0.044c. +This corresponds to an electron gyroradius of rge = +ve,thΩ−1 +ce += 5.7∆ ≈ λsi/10, which is around two or- +ders of magnitude smaller than the size of the largest +non-linear structures such as that shown in Fig. 4. As +By = 0 and Bz > Bx, the By terms in the Lorentz force +(see Eqn. 2) can be neglected. The colormap of panels +(g), (h) and (i) in Fig. 4 show the Cartesian compo- +nent of the Lorentz force as measured in the upstream +rest frame that would be experienced by a test electron +moving with the upstream bulk flow such that its ve- +locity is given by ⃗vtest = −ve,th/ +√ +3(1, 1, 1). We choose +Cartesian velocities of vtest such that the thermal veloc- +ity magnitude is divided equally between them. +Thermal upstream electrons that are spatially coin- +cident with the growing non-linear structure will expe- +rience forces according to Eqn. 2. In the x-direction, +the direction of Ex oscillates and Fig. 4 shows that this +oscillation dominates the structure of Fx. This in com- +bination with gyroradii typically much smaller than the +radial extend of the structure ensures that the electrons +will in general remain co-moving with the non-linear +structure in the x-direction. Additionally, the force in +the z-direction is comparatively weak relative to other +components. +The interaction of upstream co-spatial electrons with +the non-linear structures is much more interesting in the +y-component. From Eqn. 2, Fy = qe(Ey+vzBx−vxBz). +Firstly, when considering cool thermal upstream elec- +trons, the qeEy term dominates, thus the overall Lorentz +force directs them outwards and away from the non- +linear structure, helping to carve out low density cav- +ities. To understand the behavior of hotter upstream +electrons we need to account for the signs of the three +terms that constitute Fy and the field geometry shown +in Fig. 4. Here, we note that the cross terms are in the +same direction as the qeEy term if they are both nega- +tive, as is the case for our test particle. In fact, this sce- +nario is both plausible and likely as from Fig. 4 Bx and +Bz are generally diametrically opposed (thus vx and vz +share the same sign, and are positive and negative each +for half of one gyration period). In the negative case, +as for cool electrons the Lorentz force is again directed +outwards. However, when vx and vz are both positive, +the magnitude of this outwards force reduces. Despite +Figure 6. +Probability distribution along the transverse +direction before and after the wave shown in Figs 4 and 5. +Due to the shape and strength of the Ey component of the +whistlers, electrons spatially coincident with the top half ex- +perience an upwards force, while those approaching the lower +half an downwards force. In these plots, we see the initial dis- +tribution (top, at Ωcit = 41.3) and distribution after (lower +panel, at Ωcit = 42.4), which is now bimodal. The vertical +dashed red lines indicate the radial extent of the non-linear +structures. +this, the overwhelming majority of upstream electrons +are too cool for the force to ever point inwards, with +the overall force away from the structure center when +averaged over the electron gyro-period. +In general, as shown in panel (h) of Fig. 4, there is a +net force away from the center of the non-linear struc- +tures on co-spatial upstream electrons. Using a sub-set +of our traced electrons that are located within 2λsi of the +radial extent of the structure, we see that the direction +of the Lorentz force leads to a bi-modal distribution of +these electrons, as shown in Fig. 6. Here, we see a slight +preference for the electrons to be present beyond the +radial extent of the structure, as opposed to centrally +within it. This effect is only noticeable for non-linear +waves with particularly large amplitudes, and in general +it preferentially expels relatively cooler electrons. +Crucially, we note that for upstream electrons the +force associated with the non-linear structures is small, +and not generally towards the wave center. This means +that upstream electrons are not likely to resonate with +these waves, making the scenario where upstream elec- +trons are trapped in the non-linear structures that have +developed within the whistler potential highly unlikely. +4.4. Interaction of Reflected Electrons with Non-linear +Structures + +0.10 +(a) +0.05 +ity +0.00 + probabil +(b) +0.10 +0.05 +0.00 +0.05 +0.00 +-0.05 +0 +5 +10 +15 +20 +y/ΛsiElectron-Whistler Interactions in Oblique Shocks +9 +Figure 7. Panels (a) - (d) show the trajectory of a reflected electron though the simulation, with the current location at +Ωcit = 49 marked by the circle center. Its path over the previous Ω−1 +ci +is shown by the black and white line. The background +in panels (a) - (b) show fluctuations in Ex and Ey, while (c) and (d) show the Lorentz force components Fx, and Fy, using +the particle velocity at Ωcit = 49 of v/c = (0.47, 0.04, 0.86) normalized relative to F1 = evupB0, respectively. We see that these +force components are now directed towards the wave center. Panel (e) shows the work done on the electron by all Cartesian +components of the electric field in its own rest frame. The interval shown here corresponds to the cyan-shaded area of panel (f) +which shows the electron Lorentz factor as a function of time. Here, the dashed green line shows the current time of Ωcit = 49. +The solid lines within the cyan and salmon colored panels show the change in Lorentz factor vs ∆y for the color-indicated +regions, and the analytically expected ∆γ vs ∆y from SDA is plotted with a black dashed line. For aesthetic purposes, the color +bar label has been moved inside the figure for panels (a)-(d). We see two prominent kinks in the particle trajectory. The first +of these occurs at Ωcit = 48.44, which is caused by the electron deflecting around the lower half of a non-linear structure (note +that the structure centered at (415, 12) here is not responsible, as it was further upstream at this timestep). The second kink +occurs at Ωcit = 48.75, at which point the electron is trapped by the non-linear structure we see it co-move with afterwards. +In general, as measured in the upstream reference +frame, reflected electrons have positive values for at +least two Cartesian velocity components. +A positive +vz is needed as they travel along the upstream mag- +netic field lines, of which the strongest component lies +along ˆz. An additional magnetic field component in ˆx, +in combination with the fact that a positive vx is re- +quired so the electron can outrun the shock (such that +vx > vsh cos θBn = vsh/2) ensures vx is also positive. If +this latter criterion is not met, it cannot be reflected. +When considering the vy component, we first note +that reflected electrons require some pre-acceleration to +be reflected from the shock. These mechanism tend to +provide acceleration due to work done by the motional +electric field, which here lies in the −ˆy direction, lead- +ing to electron acceleration in the ˆy direction by virtue +of their negative charges. +This velocity component is +purely gyrational, so oscillates around zero at the elec- +tron gyrofrequency. +From Eqn. 2, we see that this changes the forces a +reflected electron experiences during an encounter with +a whistler wave relative to an upstream electron. Fig. 7 +shows such an interaction in the simulation frame. Pan- +els (a) and (b) show fluctuations in Ex and Ey (to visu- +alize the other field components of the whistler see Fig. +4), while (c) and (d) show Fx and Fy, respectively. Here, +they are normalized relative to modulus of F1 = evupB0. +Panel (e) shows the Cartesian components of accelera- +tion felt by the electron in its own rest frame. (f) shows +the Lorentz factor of the electron. The red dashed line +in this panel indicates the timestep corresponding to the +images. For panels (a) - (d) the trajectory of the elec- +tron during the previous ion gyroperiod is shown by the +black-and-white lines. +Immediately we see that the trajectory of the reflected +electron is deflected away from the direction of ⃗B0 and +is influenced by the presence of the developed non-linear +structures. Furthermore, the electron appears to have +been trapped by the (prominent) upper structure lo- +cated at (x/λsi, y/λsi) = (410, 27). +Crucially we can +determine from Eqn. 2 that, in contrast to upstream +electrons, the Fx and Fy forces are now directed towards +the center of the non-linear structure. If we consider the +y-direction in the simulation frame, in comparison to an +interaction with an upstream electron (where vx < 0), +the vxBz term is now aligned with the Ey term, with + +10 +400 +400 +40 +a +0.002 +e) +Ex +32 +30 +Eo +30||Foul +0.001 +1 +16 +W +S. +0.000 +M20 +0 +0 +y +f-16-0.001 +-1 +10 +10 +-32 +-0.002 +Wx +Wy +Wz +sum +0 +-10 +0 +-400 +41.0 +41.5 +42.0 +42.5 +380 +400 +420 +380 +400 +420 +x/Λsi +x/Λsi +10 +400 +40 +40F +10 +f) +Ey - Eyo +2.5 +32 +S +M +30 +Eo +301 +|Fou +8 +16 +0.0 +6 +-5 +0 +5 +0 +0 +Ay/si +2 +-16 +4 +-1 +10 +10 +0 +. +-32 +25 +2 +0 +ww +Ay/Asi +-10 +0 +-400 +380 +400 +420 +380 +400 +420 +35 +40 +45 +50 +x/Λsi +x/si +timestep, Qcit10 +Morris et al. +each of these pointing inwards. +If all three velocity +components are positive, no matter how the electron +approaches the whistler, all three terms in Fy (Eqn. 2) +point inwards, enabling trapping. The three Cartesian +components of the Lorentz force generally point inwards +for reflected electrons. As measured in the upstream rest +frame and relative to the upstream magnetic field, typ- +ical Larmor radii for reflected electrons are around 1-5 +λsi, but can be up to ≈ 10λsi, which from Fig. 5 are +compressed by a factor of around 4 when encountering +the strong magnetic field associated with a non-linear +structure, thus reflected electrons can typically be con- +tained within them. +4.5. Interaction Probability +We can estimate the probability that a reflected elec- +tron will interact with a non-linear structure by consid- +ering the path taken by such a particle. The whistler +waves and resulting non-linear structures move with +group velocity vg << vup, hence we perform this calcu- +lation in the upstream rest frame using the approxima- +tion that the non-linear structures are stationary. We +use primed quantities here to represent the upstream +rest frame, and further assume the size of the whistler- +containing electromagnetic foreshock from which the +non-linear structures derive to be a constant size of x′ +w. +Primed quantities with the subscript ∥ and ⊥ refer to +components measured parallel or perpendicular to the +magnetic field vector in the upstream rest frame, re- +spectively. +Since the reflected electrons are gyrating, we can con- +sider the path length to be a sum of 2 components. +These consist of a linear component, R′ +lin as a result of +the path of reflected electrons following the large-scale +upstream magnetic field structure, and an oscillatory +component, R′ +osc, as a result of the gyration around the +magnetic field. The total path length can be considered +to be, +R′ +path = +� +R′2 +lin + R′2 +osc +(3) +where R′ +osc dominates if the electron Larmor radius is +significantly larger than x′ +w. Otheriwse, R′ +path ≈ x′ +w. +R′ +lin is simply the size of the whistler-containing re- +gion, such that R′ +lin = x′ +w. To compute the time an elec- +tron takes to traverse it, we consider that they travel in +the direction of the upstream rest frame magnetic field, +⃗B0 +′ which now makes an angle θ′ +Bn with the simulation +plane. We therefore only see their propagation projected +onto the simulation (xy) plane, with the time taken to +traverse R′ +lin = x′ +w defined as, +t′ +lin = +x′ +w +v′ +∥ cos θ′ +Bn +, +(4) +where the cos θ′ +Bn term provides the necessary path cor- +rection to compensate for the inclination of the upstream +magnetic field with respect to the simulation plane. +During time t′ +lin, assuming it does not interact with +any non-linear structures, a reflected electron completes +t′ +lin/τge oscillations, where τge = 2πrge/v⊥ is the elec- +tron gyro-period. As a result of our magnetic field orien- +tation with respect to simulation plane, the shape made +by the gyrational orbit of the electron on the simulation +plane is elliptical. On account of B′ +y = 0, the size of +the semi-major axis is a = rge, where rge is the electron +gyroradius. The semi-minor axis appears contracted in +the direction of motion, such that its value is given by +b = rge sin θ′ +Bn. For the sake of presenting a more easily +interpretable solution, we approximate the perimeter of +the elliptical path, p, as, +p ≈ 2π +� +a2 + b2 +2 += +√ +2πrge +� +1 + sin2 θ′ +Bn, +(5) +which is typically accurate to better than 5% assum- +ing the ellipse is not too elongated (Muir 1902). Other, +more accurate, approximations can be found in the lit- +erature (Ramanujan et al. 2015). The total gyrational +path length is therefore given by R′ +osc = pt′ +lin/τge. +Combining these, we can rewrite Eqn. 3 as, +R′ +path = x′ +w +� +1 + +v′ +⊥ +√ +2v′ +∥ +� +1 + sin2 θ′ +Bn +1 − sin2 θ′ +Bn +� +, +(6) +where this equation must also satisfy the reflection con- +straint that v′ +∥ cos θ′ +Bn ≥ vsh to remain valid. +The probability of interaction with a non-linear struc- +ture follows, +Pinteract ≈ nnlsσnlsR′ +path += nnlsπr′2 +nlsx′ +w +� +�1 + 1 +2 +� +v′ +⊥ +v′ +∥ +�2 �1 + sin2 θ′ +Bn +1 − sin2 θ′ +Bn +�� +� +1/2 +, +(7) +for non-linear structure number density nnls and cross- +sectional area σnls = πrnls′2 as measured in the upstream +rest frame. We can interpret our results within the con- +text of Eqn. 7. +Firstly, we recover the intuitively expected results that +both a larger number density of non-linear structures +and their cross-sectional area linearly increase the in- +teraction probability. In the limit v′ +∥ → c, v′ +⊥ → 0 to +prevent the electron becoming superluminal, and we re- +cover the expected solution that R′ +path = x′ +w (the same +is true if v′ +⊥ is small). Eqn. 7 also recovers the expected +solution that the path length approaches ∞ in the case + +Electron-Whistler Interactions in Oblique Shocks +11 +of a perpendicular shock where θ′ +Bn → 90◦ because the +reflected particles are unable to escape upstream. +More significantly, we note that the second term in +Eqn. 7 indicates that the probability of interaction is +proportional to v⊥, but inversely proportional to v∥. +One may conclude that this may favor reflected elec- +trons with v′ +⊥ >> v′ +∥, however this is not accurate. For +a more realistic picture, we must again consider that +for a reflected electron to outrun the shock it must sat- +isfy v′ +∥ cos θ′ +Bn ≥ vsh. This, in addition to the constraint +that v′2 = v′2 +⊥ + v′2 +∥ ≤ c, restricts the value of v′ +⊥ to be +v′ +⊥ ≤ 0.848c. +The electron shown in Fig. 7 interacts with a non- +linear structure, and becomes trapped. +During this +time, it is carried towards the shock as it is unable to +escape, until the wave ‘breaks’ when encountering the +shock ramp. +We note that the Larmor radius of re- +flected electrons that are trapped by whistler waves need +to be smaller in size relative to the whistler wave. Typi- +cally, when measured relative to the upstream magnetic +field, reflected electrons have gyroradii in the range of +1 − 5λsi, though can approach 10λsi. This comparison +means the ratio of reflected electron gyroradii to the ra- +dius of a non-linear structure lies in the range 0.1 − 1, +as the non-linear structures measure ≈ 5λsi radially. Al- +though the magnetic field amplification associated with +these structures will contract electron gyroradii that en- +counter them, this may not be sufficient to allow the +non-linear structures to trap the most energetic reflected +electrons. In reality, we might expect the probability of +trapping to fall off exponentially as the gyroradius ap- +proaches the radius of the non-linear structure, rnls, such +that Ptrap ∝ exp(−rge/rnls). +With a simplifying assumption that reflected electrons +would move with a constant v∥ in a calm upstream re- +gion, we can calculate the most probable value by as- +suming Ptrap ∝ Pinteract · exp(−rge/rnls). Numerically +solving this equation and accounting for the aforemen- +tioned constraints, under our simulation setup we ob- +tain a peak trapping probability for v′ +⊥ ≈ 0.40c and +v′ +∥ ≈ 0.54c, which corresponds to reflected electrons that +are around 18 times as energetic as thermal electrons in +the far upstream. +While this analysis has allowed us to estimate trap- +ping conditions for reflected electrons, it only provides +a snapshot over a small time-frame. More realistically, +Bohdan et al. (2022) show that the size of the whistler +region from which the non-linear structures arise from +grows with time. By integrating the energy density of +the reflected electrons up to the point that the whistler +waves become detectable and extrapolating this to late +times, Bohdan et al. (2022) estimate that the size of this +Figure 8. Plot showing perpendicular vs parallel momen- +tum for the electron in Fig. 7, as measured in the upstream +rest frame relative to the local magnetic field. Arcs of con- +stant momentum indicate pitch angle scattering, while in- +creases in p′ +⊥ are suggestive of SDA. +region would reach a steady state at Ωcit ≈ 125. At this +time, the size of the whistler containing region would ex- +tent to around 2000λsi ahead of the shock. Within the +context of our analysis, from Eqn. 7 we would expect +the trapping probability to increase up until Ωcit ≈ 125 +for all reflected electrons with gyroradii small enough +to be contained by the non-linear structures. Further +upstream beyond this region, the energy density of the +electron acoustic waves (which are not well captured in +this out-of-plane simulation) that are the subject of Pa- +per I would dominate, hence the non-linear structures +discussed here would cease to be important beyond this +limit. +4.6. Stochastic Shock Drift Acceleration +From Eqn. 7, electrons with some perpendicular accel- +eration are more likely to be trapped by the non-linear +structures, assuming that they have enough parallel ac- +celeration to escape the shock front and under the condi- +tion that their gyroradii are smaller than the character- +istic radius of the non-linear structures. Trapped elec- +trons will be returned to shock where they may undergo +further pre-acceleration. Identifying a mechanism that +provided perpendicular acceleration can explain much +about their behavior. +Known mechanisms in this region that increase v⊥ +include shock surfing acceleration (SSA) and stochas- +tic shock drift acceleration (SSDA). The former of these +processes is dependent on the presence of electrostatic +Buneman waves, which are excited by a velocity dif- +ference between incoming electrons and reflected ions, +which results in their production near the shock ramp +(Buneman 1958; Gary 1987). In perpendicular shocks, +ion gyration at the shock is sufficient to excite them in +the shock foot (Bohdan et al. 2019a). In oblique shocks, +the Buneman instability is strongly modified due to the + +10 +50 +pl/mec +45 +5 +40 +35 +0 +-5 +0 +5 +10 +p /mec12 +Morris et al. +presence of whistler waves and so the overall efficiency +of SSA might be different compared to perpendicular +shocks. Note that ions propagating back upstream can- +not drive Buneman waves in the foreshock region since +the ion reflection rate is too small, nion,ref/n0 = 10−4, +and the growth rate predicted for Buneman waves is +over two orders of magnitude smaller than that for the +whistler waves (Bohdan et al. 2022). +Another candidate acceleration mechanism is shock +drift acceleration (SDA). The original theory of SDA in- +dicated that it could efficiently accelerate charged par- +ticles (Wu 1984; Leroy & Mangeney 1984). +It occurs +if the electron gyrates close enough to the shock ramp +such that part of its orbit overlaps the region with en- +hanced magnetic field, tightening its gyro-radius during +these regions. The gradient in the magnetic field results +in a drift analogous to ∇B drift with work done by, and +in the direction of, the motional electric field, which is +perpendicular to ⃗B0 by definition (Ball & Melrose 2001). +Despite the efficient energization, +Vandas (2001) +demonstrated that SDA alone is not efficient enough to +account for the observed power-law spectrum and fluxes +of accelerated electrons in astrophysical sources. Phys- +ically, this occurs because in the original SDA theory, +candidate electrons are not confined to the shock transi- +tion region where acceleration occurs, thus limiting the +efficiency of the mechanism. One such way of overcom- +ing this impediment is to add pitch angle scattering, +with electrons scattering off whistlers being observed +in the Earth’s bow shock (Oka et al. 2017). Katou & +Amano (2019) proposed the stochastic shock drift ac- +celeration (SSDA) mechanism which incorporates pitch +angle scattering into the SDA model, increasing the +time in the acceleration region and accordingly the en- +ergy gain. Additional evidence of SSDA has been found +in both in 3D PIC simulations (Matsumoto et al. 2017) +and in observations which support electron scattering by +whistler waves (Oka et al. 2019). This latter work con- +cludes that the energization directly via the whistlers is +low, but the electrons, as here, are confined within the +acceleration region and become more energetic. +This +picture is consistent with our results. Fig. 8 plots p′ +⊥ +vs p′ +∥ (where the primes again represent the upstream +rest frame) for the electron shown in Fig. 7. We see +increases in p′ +⊥, supporting SDA as the mechanism that +provided the acceleration, and variations in pitch an- +gle for constant p′, which are indicative of scattering +(Matsumoto et al. 2017; Ha et al. 2021). The vertical +red line at around (p′ +∥, p′ +⊥) = (5, 7.5) corresponds to the +salmon pink region of Fig. +7 panel (f). +This occurs +when Ωcit ≈ 49, when the electron has been returned to +the shock by the non-linear structure that had trapped +it. +Another signature of SSDA is that the change in +energy is directly proportional to the motional electric +field. For our magnetic field configuration electrons will +drift in the +ˆy direction (Krauss-Varban & Wu 1989). +We see from panel (f) that for the salmon-pink region, +the predicted ∆γ (dashed black line) agrees closely with +measured values (solid red line), verifying that SSDA is +observed in our simulation. +However, the presence of the non-linear structures fur- +ther complicates this picture, and the acceleration of the +most energetic electrons cannot be fully described by +SSDA alone. Panel (f) in Fig. 7 indicates that this par- +ticular electron undergoes two periods of rapid and effi- +cient acceleration, with relative quiescence in between. +At around Ωcit = 41, the electron first encounters the +shock and is accelerated, with this region indicated by +the cyan panel and corresponding to the color-matched +sub-panel. The phase of constant energy corresponds to +the electron traveling upstream, and includes the time- +period when it is trapped by the non-linear structure, +indicating the primary role of such structures is to keep +electrons confined to the acceleration region. The cyan +panel shows the measured ∆γ (blue solid line) and ana- +lytical ∆γ (black dashed line) as a function of ∆y for the +first acceleration period. Although there are incidences +where the energization rate can be attributed to SSDA, +∆γ is shown to also increase for ∆y ≤ 0. We see that +this behavior is also evident in the subset of the most +energetic traced electrons. Fig. 9 indicates that they +generally have a linear relationship between ∆γ and ∆y, +as for the trace electron shown in Fig. 7 is shown on Fig. +9 by the red circle. However, on average, the accelera- +tion is between three and four times more efficient than +can be accounted from SSDA via the motional electric +field alone, with this prediction indicated by the purple +dashed line. This requires further investigation. +To explain this, we compute the total work done, W, +by electric field on the electron in its own rest frame. +This is a sum of the Cartesian components, such that +W = Wx + Wy + Wz, with these displayed in Fig. 7 +panel (e) for the interval Ωcet = 41 − 43 which cor- +responds to the cyan area in panel (f). +Noting that +(∆y/λsi, ∆γ) = (0, 0) on the latter plot corresponds +to Ωcit = 41 on panel (e), we see that initially when +Wy ≤ 0 the change in y-coordinate is ∆y ≤ 0. Closer to +Ωcit = 43, Wy becomes positive, in which regions we see +acceleration consistent with SSDA (the blue solid line +is quasi parallel to the black dashed line in panel (f), +cyan background). However, the important distinction +to SSDA is that the overall acceleration during this pe- +riod is typically greater than zero. We assess the relative +importance of each component by computing the mean + +Electron-Whistler Interactions in Oblique Shocks +13 +Figure 9. +The change in Lorentz factor as a function of +∆y/λsi at Ωcit = 51.5 for the traced electrons. +The neg- +ative average ∆y typically occurs for electrons that have +passed into the downstream. +The location of the electron +shown in Fig. +7 is indicated by the red circle. +Under +the assumption that the work done to cause the accelera- +tion comes from the motional electric field, SSDA predicts +∆γSSDA ≈ qeE0y∆y/(mec2), thus the energy gain for the +most energetic electrons is 3-4 times greater than can be ac- +counted for by SSDA from the motional electric field alone. +work done across this time interval. For this electron, +the Wz component is particularly strong, and has a rel- +ative contribution to the overall particle energy change +that is comparable to that from Wy, with Wz ≈ 2.1Wy. +Wx is weaker at Wx ≈ 0.17Wy. Typically for reflected +electrons, we see similar levels of work from Wz and +Wy with a smaller Wx, unlike in SSDA where we would +expect Wy to drive the acceleration exclusively. +The +reason we see more efficient acceleration overall rela- +tive to SSDA is therefore because all Cartesian compo- +nents can in principle contribute, thus we would expect +∆γ/∆γSSDA ≥ 2 on account of the comparable Wy and +Wz, which is consistent with what we show in Fig. 9. +The precise details of the underlying microphysics of +the acceleration are likely a consequence of the complex +non-linear structures arriving at the shock. Indeed, pre- +vious studies have shown that changes in local conditions +can lead to more efficient electron acceleration (Kobzar +et al. 2021). For the electron considered in Fig. 7, we +compute the dot-product of ⃗B0 and the time averaged lo- +cal magnetic field from Ωcit = 41.4 to Ωcit = 41.5, when +Wz dominates the work done by the electric field in the +particle rest frame (panel (f)). The cosine of the angle +between then is about 0.9, whereas for the pure SDA +consistent region (i.e. pink shaded region of Fig. 7(f) ) +this number is ≈ 1. This means that the local magnetic +field in these regions on average subtends an angle with +the upstream field of around 26◦ during this period. As +the work done during acceleration via SDA occurs via +purely perpendicular electric fields, this changing of the +local field orientation permits acceleration in additional +directions. Indeed, the non-linear structures arriving at +the shock also perturbate the local electric fields, with +Ey/E0y ranging from ±15 in the cyan region of Fig. 7f, +yet averaging at Ey/E0y ≈ 1. The overall acceleration +is therefore highly sensitive on local values and orienta- +tions of the electromagnetic fields. +4.7. Pitch Angle Distribution +On the one hand, an electron with a larger v′ +⊥ will +have a longer path length via Eqn. 6, yet if v′ +⊥ is too +large its gyroradius becomes too large for trapping to +occur. Similarly, v′ +∥ is constrained by the necessity for +reflected particles to be able to outrun the shock. The +pitch angle, α′ = arctan(v′ +⊥/v′ +∥), is a useful quantity +that can provide important information about the sub- +sample of reflected electrons. +We select all reflected electrons that reach γ ≥ 5 at +the end of the simulation at Ωcit = 51.5 and compute +their pitch angles as measured in the upstream reference +frame relative to B′ +0 over the final two Ω−1 +ci in the simula- +tion. We show the probability distribution of these pitch +angles in Fig. 10. By definition, electrons with pitch an- +gles α > π/2 are heading back the shock, however, in +this frame recall that v′ +∥ cos θ′ +Bn > vsh is required for elec- +trons to outrun the shock and travel upstream. This, in +addition to the speed limit of c, sets v′ +∥,min = 0.53c and +v′ +⊥,max ≈ 0.85c, from which we can calculate a maximum +permitted pitch angle of α′ +max = arctan(v′ +⊥,max/v′ +∥,min) +for reflected electrons that are traveling towards the up- +stream in the upstream rest frame. Accordingly, all elec- +trons with pitch angles α′ ≳ π/3 will be caught up to +by the shock. +Integrating the probability density shown in Fig. 10 +gives the probability of α′ > π/3 as P(α′ > π/3) = 0.20, +where this probability represents an estimate of the frac- +tion of reflected electrons that are re-directed back to +the shock as a consequence of interactions with the non- +linear structures. Considering the stable late-time elec- +tron reflection rate of ≈ 4%, we estimate that around +0.8% of all the incoming upstream electrons can be ener- +gized by the enhanced SSDA mechanism discussed here. +This argument assumes that the reflected electrons con- +sidered here are located within the upstream region con- +taining the non-linear structures, which is true for the +majority of reflected electrons within this sample, but +the numbers presented here should nevertheless be con- +sidered as an approximate upper limit. +4.8. The electron downstream distribution +Figure 11 shows the electron downstream distribution. +The low energy part of the spectra is represented by a + +△YsSDA +15.0 +102 +3△YsSDA +12.5 +4△YsSDA +10.0 +7.5 +5.0 +2.5 +0.0 +100 +-50 +-25 +0 +25 +50 +AylAsi (from injection)14 +Morris et al. +Figure 10. Pitch angle distribution for the most energetic +70 traced particles which have γ ≥ 5 over the final 2Ω−1 +ci +of +the simulation. In this frame, all electrons with pitch angle +α ≳ π/3 will be caught up by, or are heading towards, the +shock. +Maxwellian distribution. The electron to ion tempera- +ture ratio is about Te/Ti = 0.17 which is close to that in +a perpendicular shock simulation with similar shock pa- +rameters (Bohdan et al. 2020a). High energy electrons +follow a power-law with an approximate index up to -2.8. +The fraction of electrons not covered by the thermal dis- +tribution is about 0.8% and they hold about 7% of the +downstream electron energy. As shown in Fig. 11, at the +end of the simulation, the most energetic downstream +electrons present at the very tail of the cut-off already +reach gyroradii that are comparable to the shock width +or the ion upstream gyroradius, γinj,e ≈ mi +me +vsh +c +≈ 13. +The threshold for injection into DSA will be a factor +of a few higher than that. Unfortunately, the simula- +tion time is still too short to properly capture the DSA +phase of acceleration both for electrons and ions, as well +as formation of a nonthermal tail in the ion distribution. +5. CONCLUSIONS +In this paper, we have presented results on the elec- +tron foreshock of an oblique collision-less shock, based +on a large scale PIC simulation with a total run-time +characterized by Ωcit = 51.5. The physical properties +of this simulation are such that a shock front is gener- +ated in the high Mach number regime, and thus consis- +tent with the environments of supernova remnants. We +have discussed the onset and characteristic properties +of whistler waves, and outlined their development into +complex non-linear structures which carve out density +cavities in the plasma. These non-linear structures are +capable of trapping and scattering reflected electrons, +confining them to the region close to the shock where +Figure 11. The electron energy distribution in the shock +downstream. The dotted line is the Maxwellian fit to the +low-energy part of the electron distribution. The dashed line +represents the slope of the nonthermal electron tail. +they can be efficiently pre-accelerated, which increases +the likelihood of being injected into DSA and being fur- +ther energized. Our main conclusions can be summa- +rized as follows: +• Reflected electrons at the shock excite the oblique +whistler instability, +generating electromagnetic +whistlers waves in the inner foreshock region. +• The phase and group velocities of the whistlers are +much less than that of the upstream bulk speed +vup, so they approximately co-move with the up- +stream plasma and also grow in size to about five +ion inertial lengths as they approach the shock. +• Over time, the internal structures arising from the +whistler waves become highly complex and non- +linear, and they carve out density cavities. The +magnetic energy density is the highest at the pe- +riphery of each structure, and they are filled with +strong electric field that varies on small scales on +the order of the ion inertial length. +• Upstream electrons co-moving with the non-linear +structures in general experience a Lorentz force di- +rected away from them. This means that upstream +electrons in general do not resonate with the struc- +tures, thus are unlikely to be confined within them. +• For reflected electrons encountering the non-linear +structures, the Lorentz force is now directed to- +wards the structure center, so they can become +trapped if their gyroradii are small enough. This +enables, and indeed leads to, the return of a sub- +set of electrons that are initially reflected at the + +α' = π/3 +Probability Densit +0.75 +0.50 +0.25 +0.00 +0 +π/4 +T/2 +3π/4 +α' relative to B°Electron-Whistler Interactions in Oblique Shocks +15 +shock being trapped and returned to it, where ad- +ditional pre-acceleration is possible. +• The acceleration mechanism is directly analo- +gous to stochastic shock drift acceleration, but +is around 3 times more efficient. This is because +other components of the electric field, besides the +coherent motional electric field, can contribute to +the acceleration. +• By considering their pitch angles, at any given +time, we estimate that 20% of reflected electrons +have been redirected back towards the shock. This +corresponds to around 0.8% of the total upstream +electrons. +M.P. acknowledges support by DFG through grant +PO 1508/10-1. +A.B. and M.P. thank the Interna- +tional Space Science Institute (ISSI) for their hospitality +and the ISSI Team Energy Partition across collisionless +shocks for invaluable discussions. The numerical sim- +ulations were conducted on resources provided by the +North-German Supercomputing Alliance (HLRN) under +project bbp00033. +REFERENCES +Amano, T., & Hoshino, M. 2007, ApJ, 661, 190 +—. 2009, ApJ, 690, 244 +Amano, T., et al. 2022, Reviews of Modern Plasma Physics, +6, 29 +Axford, W. I., Leer, E., & Skadron, G. 1977, International +Cosmic Ray Conference, 11, 132 +Ball, L., & Melrose, D. B. 2001, PASA, 18, 361 +Bell, A. R. 1978, MNRAS, 182, 147 +Blandford, R. D., & Ostriker, J. 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Res., 89, 8857 +Xu, R., Spitkovsky, A., & Caprioli, D. 2020, ApJL, 897, L41 + diff --git a/XtAyT4oBgHgl3EQf9PrN/content/tmp_files/load_file.txt b/XtAyT4oBgHgl3EQf9PrN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fdef184a7adac4226c79e84057ce1ae9c1a62b4e --- /dev/null +++ b/XtAyT4oBgHgl3EQf9PrN/content/tmp_files/load_file.txt @@ -0,0 +1,881 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf,len=880 +page_content='Draft version January 4, 2023 Typeset using LATEX twocolumn style in AASTeX62 Pre-acceleration in the Electron Foreshock II: Oblique Whistler Waves Paul J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Morris,1 Artem Bohdan,1, 2 Martin S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Weidl,2 Michelle Tsirou,1 Karol Fulat,3 and Martin Pohl1, 3 1Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany 2Max-Planck-Institut für Plasmaphysik, Boltzmannstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2, DE-85748 Garching, Germany 3Institute of Physics and Astronomy, University of Potsdam, D-14476 Potsdam, Germany ABSTRACT Thermal electrons have gyroradii many orders of magnitude smaller than the finite width of a shock, thus need to be pre-accelerated before they can cross it and be accelerated by diffusive shock acceler- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' One region where pre-acceleration may occur is the inner foreshock, which upstream electrons must pass through before any potential downstream crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In this paper, we perform a large scale particle-in-cell simulation that generates a single shock with parameters motivated from supernova remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Within the foreshock, reflected electrons excite the oblique whistler instability and produce electromagnetic whistler waves, which co-move with the upstream flow and as non-linear structures eventually reach radii of up to 5 ion-gyroradii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We show that the inner electromagnetic configuration of the whistlers evolves into complex non-linear structures bound by a strong magnetic field around 4 times the upstream value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Although these non-linear structures do not in general interact with co-spatial upstream electrons, they resonate with electrons that have been reflected at the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We show that they can scatter, or even trap, reflected electrons, confining around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='8% of the total upstream electron population to the region close to the shock where they can undergo substantial pre-acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This acceleration process is similar to, yet approximately 3 times more efficient than, stochastic shock drift acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Keywords: acceleration of particles, instabilities, ISM – supernova remnants, methods – numerical, plasmas, shock waves 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' INTRODUCTION It was suggested by Fermi (1949) that hadrons could be accelerated by magnetic mirrors and give rise to the well-documented observed cosmic-ray power-law spec- trum (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Hillas 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Nagano 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The original Fermi acceleration assumes interactions between parti- cles and magnetic mirrors, which move at speed U, oc- cur with an isotropic distribution of incident angles be- tween the two (as measured by a stationary observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The relative orientation of these interactions can either cause particles to gain (when head-on) or lose (if head- tail) energy, with a slight preference (of order U/c) for the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The expected angle-averaged fractional en- ergy gain per collision is ∝ (U/c)2, where c is the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Astrophysical shocks occurring in nature are more accurately described by diffusive shock accelera- tion (DSA) (Krymskii 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Axford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Bell Corresponding author: Paul J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Morris paul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='morris@desy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='de 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Blandford & Ostriker 1978), where the interac- tions always occur head-on in the rest-frames both up- stream (unshocked plasma) and downstream (shocked plasma) of the shock front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' DSA analytically predicts a more efficient fractional energy-gain-per-crossing of ∝ (U/c) and power law spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' DSA has been gener- ally successful in explaining a wide variety of astrophys- ical sources such as active galactic nuclei (Marchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2017) and supernova remnants (SNRs) (Reynolds 2008), where we observe non-thermal emission that is often characterized by a power-law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The radiative prop- erties of these objects can be interpreted as originating from an underlying population of high-energy particles (protons, electrons etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' ), providing evidence to support DSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Despite its numerous successes, aspects of the under- lying micro-physics necessary for DSA to work are yet to be conclusively determined (Amano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This is because in DSA it is assumed that the shock is a perfect discontinuity, when in reality it has a finite width of the order of the gyroradius of a proton traveling with the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='00872v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='plasm-ph] 2 Jan 2023 2 Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' shock speed, rgi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' While this does not pose challenging for thermal ions to cross into the downstream, thermal electrons require a significant amount of pre-acceleration before their gyroradii are sufficiently enlarged that they can easily cross the shock transition from upstream to downstream, or vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' To accelerate particles to cosmic-ray energies, DSA requires them to cross the shock multiple times, thus only electrons that have al- ready undergone sufficient pre-acceleration can “be in- jected" into DSA and undergo further acceleration by this mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Particle-in-cell (PIC) simulations are an excellent tool with an eminent track record when it comes to investi- gating electron pre-acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' These simulations are fully kinetic, containing individual electrons and ions, thus allow for a self-consistent treatment when these particles move in their self-generated electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' From these, we can obtain time- and spatially dependent information concerning both individual par- ticles and the fields they experience which allow us to un- veil the underlying physical processes (Pohl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In this work, we use physical parameters appropri- ate for supernova remnants, which are characterized by non-relativistic outflows with sonic and Alfvénic Mach numbers of MS, MA ≈ 20 − 2000 (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In contrast, the low Mach number regime is associated with the Earth’s bow shock (Ms, MA < 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Focusing on SNR parameters is advantageous for many reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' First and foremost, it has been established for almost 50 years that cosmic rays (CRs) can be accelerated by SNRs, with the majority of Galactic CRs believed to originate from these objects (Axford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Krym- skii 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Drury 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Bell 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Blandford & Ostriker 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Additionally the close proximity of SNRs permits the study of non-thermal radiation in radio, X-, and γ- rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This radiation is often attributed to a population of accelerated electrons, thus understanding their accel- eration is essential to comprehend the radiative proper- ties of SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' A crucial parameter in governing the behavior of a shock is the obliquity angle, θBn, which subtends the upstream magnetic field with the shock normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Typ- ically perpendicular shocks, where θBn = 90◦, have well defined shock transitions, with a small (of order ∼ rgi) shock foot region leading up to the ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' These shocks have been thoroughly studied over the last decade using 2D PIC simulations (Amano & Hoshino 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Kato & Takabe 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2012, 2013, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Wieland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Bohdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2017, 2019a,b, 2020b,a, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Conversely, decreasing the shock obliq- uity angle more freely permits the escape of energetic particles back upstream as their trajectories are tied to the magnetic field lines, allowing them to outrun the shock if sufficient energization has taken place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The extended regions containing the reflected particles are known as the electron and ion foreshocks (depending on the particle species) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Burgess 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Fitzenreiter 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Treumann 2009) where the energy transported up- stream by these reflected particles can excite instabili- ties and generate turbulence which can in turn influence, and possibly pre-accelerate upstream electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' All up- stream electrons that eventually encounter the shock must first pass through the foreshock, thus a physical description of these regions is essential to fully compre- hend the overall description of electron pre-acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Prior work has demonstrated that energetic electrons that have been pre-energized by shock surfing accel- eration are more likely to be reflected back upstream (Amano & Hoshino 2007), where mirror reflection (also called shock drift acceleration, SDA) (Wu 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Leroy & Mangeney 1984) is the mechanism responsible for the reflection (Honda & Honda 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This latter mecha- nism operates on electrons gyrating close enough to the shock ramp so that part of their gyrational orbit en- close the region with enhanced magnetic field, causing a temporary orbital tightening and causing them to drift along the shock (Wu 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Leroy & Mangeney 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Results obtained from using 1-dimensional PIC simu- lations demonstrated that the energy content of these reflected electrons was sufficient to power electrostatic and electromagnetic waves in the shock foot, which ef- fectively trap electrons allowing them to undergo more cycles of shock drift acceleration (SDA), gaining more energy and eventually cross into the downstream region (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Kumar & Reville 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Bohdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' (2022) used 2D3V (2 spatial dimensions and all three velocity and field components) PIC simu- lations, with a combination of large scale and periodic boundary condition simulations, accompanied by ana- lytically solving the dispersion to elucidate the exact in- stabilities excited by reflected electrons in the electron foreshock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The first of these are electrostatic electron acoustic waves (EAW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In paper I of this series, Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' (2022, hereafter Paper I) investigated the effect of changing the orientation of the upstream magnetic field on the foreshock structure by performing a series of narrow box simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' It was found that EAWs are quickly excited within a few ion gyro-radii, and the EAWs are stronger for decreasing θBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' It was shown that these waves can interact with, and in ∼ 1% of cases, di- vert upstream electrons away from the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Bohdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' (2022) further identified electromagnetic whistler waves in the inner foreshock region, which require a com- paratively larger energy density of reflected electrons rel- Electron-Whistler Interactions in Oblique Shocks 3 ative to EAWs, subsequently excited at later times than EAWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' These whistler waves occur on spatial scales ap- proaching ion length scales, as opposed to the EAWs where the characteristic size is similar to the electron inertial length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In this paper we focus on the micro- physics of individual electrons which encounter these whistler waves, and interpret their behavior in the con- text of electron pre-acceleration in astrophysical shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In Section 2 we outline our simulation setup before pro- viding an overview of the shock structure in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Section 4 contains the main discussion, where we outline the properties and development of whistler waves before detailing their evolution and interaction with electrons present in the foreshock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' SIMULATION SETUP In Paper I, we investigated the effect of changing the obliquity angle, θBn and the plane-angle, φ, which characterize the orientation of the initial large-scale upstream magnetic field, on the electron foreshock at short times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The run-time of these simulations can be quantified in terms of the ion-gyrofrequency, defined as Ωci = |e|B0/mi, for electron charge magnitude |e|, mag- netic field amplitude B0 and ion mass mi, with the total run-time tsim = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='8Ω−1 ci .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' They further employed a nar- row box, spanning 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='8λsi transversely, where λsi is the ion skin length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This relatively small transverse size reduced the computational expense of a single simula- tion, and therefore permitted multiple simulations to be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The chosen parameters also additionally al- lowed for a comparison to the 3D simulations of Mat- sumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Conversely, in this paper, we investigate electron pre- acceleration as a consequence of the electromagnetic foreshock, which begins to emerge at around tsim ∼ 10Ω−1 ci .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This is characterized by the presence of whistler waves, which at late times develop into non-linear struc- tures that can reach up to approximately 1 − 10λsi in diameter, which are better captured by our 2D3V simu- lations in the out-of-plane (φ = 90◦) case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Accordingly, to adequately resolve these structures we perform a sim- ulation with a wider box, of transverse size 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='6λsi and with a longer total run-time of tsim = 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5Ω−1 ci , allowing us to follow their long-term evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This computa- tionally expensive simulation featured in Bohdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' (2022), and the setup will be briefly outlined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Our code is a modified version of TRISTAN (Buneman 1993), which cyclically solves Maxwell’s equations for fields defined on a simulation grid and updates the posi- tions of individual particles located in the grid cells ac- cording to Lorentz forces via the Vay solver (Vay 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This version allows us to track the progression and prop- erties of individual particles and measure the local field strengths they encounter to better elucidate the physical processes they experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' At the beginning of the simulation, we initialize a plasma slab by injecting ions and electrons, where mi and me are the ion and electron mass and mi/me = 50, co-spatially into the simulation box with the number of particles per cell per species given as n0 = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The two particle species are initialized in thermal equilibrium, such that kBTe = kBTi = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='86 · 10−4 mec2, where kB is the Boltzmann constant and c the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This defines the sound speed as cs = � 2ΓkBTi/mi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='0081c, for adiabatic index Γ = 5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Across this plasma slab, we apply a large-scale, uniform, magnetic field accord- ing to ⃗B0 = B0(cos θBn, sin θBn cos ϕ, sin θBn sin ϕ) = B0(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5, 0, √ 3/2), where θBn = 60◦ and φ = 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Such a setup defines the important temporal scales in our simulation, such as the electron plasma and gy- rofrequencies, ωpe and Ωce respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Their ratio is quantified by Ωce = |e|B0/me = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='06 ωpe, where ωpe is the electron plasma frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' From this, we can define the electron skin length, λse = c/ωpe = 8∆, that is re- solved by 8 grid cells (denoted by ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' For ions, their inertial length scales as λsi = � mi/me λse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We ensure that these relevant frequencies are sufficiently resolved in our simulation by advancing it in time-step units of δt = 1/16 ω−1 pe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We move our plasma slab with a bulk velocity as mea- sured in the simulation frame of ⃗vup/c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='20ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Con- sequentially, a motional electric field is produced and de- fined by ⃗E0 = −⃗vup × ⃗B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Such a setup leads to a large value of ∇ × ⃗E, and a correspondingly large ∂ ⃗B/∂t, at x = 0 which can induce a large initial transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We mit- igate this by tapering the initial upstream field values to zero over the region x ≤ 50∆ (Wieland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The nonzero value of ∇ × ⃗B is exactly compensated by a drift current carried by the ions, which is removed at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Constituent particles within the plasma that reach the boundary at x = 0 encounter a reflecting wall, which performs the transformation vx → −vx (vy and vz are unaffected) (Quest 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 1989), and so they propagate back upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The mag- netic field behind this upstream-moving plasma com- pletely isotropizes after approximately a few ion gyro times, with a compression ratio of nd/n0 ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='0 rela- tive to the undisturbed upstream plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In the sim- ulation frame, a quasi-stationary shock with velocity ⃗v∗ sh/c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='067 ˆx propagates upstream, which is equiv- alent to a shock velocity of ⃗vsh/c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='263 when mea- sured in the upstream rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The properties of the shock can be further quantified by the Alfvén velocity, 4 Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Reflecting wall setup of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' φ is the angle B0 makes relative to the simulation plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' θBn is the angle subtended between the shock normal, ˆn, and the magnetic field vector, B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Here we use θBn = 60◦ and φ = 90◦, which define the upstream magnetic field as B0 = B0(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5, 0, √ 3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The dotted line indicates the trans- verse size of the simulation box, which is 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='6 ion skin lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' vA = B0/ � µ0(neme + nimi), for vacuum permeabil- ity µ0 and ni = ne = n0 are the ion and the electron number densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This gives rise to the Alfvénic Mach number, MA = vsh/vA = 30, whereas the sonic Mach number is MS = vsh/cs = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The plasma beta value, which denotes the thermal-to-magnetic energy density ratio in the upstream region is β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Our simulation setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' As the shock propagates up- stream, the domain length in the x-direction increases, with new plasma injected into the new regions with the same properties outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The extension of the simulation box is essential to ensure that that we main- tain all reflected electrons within the boundaries of our simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' SHOCK STRUCTURE AND SUMMARY OF FORESHOCK CHARACTERISTICS The late-time (Ωcit = 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='9) shock structure is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' For illustrative purposes, we show only the inner electromagnetic foreshock, containing the whistler waves, and the beginning of the outer electrostatic fore- shock containing the EAWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The full simulation box at this timestep extends to ∼ 2000λsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' As explained in Bo- hdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' (2022) and Paper I, the EAWs here are not well captured because the simulation setup employs an out-of-plane field angle (φ = 90◦), with EAWs propa- gating in the direction of the upstream magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We therefore cannot see them so easily because they do not lie in the simulation plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In contrast to Paper I, slightly ahead of the shock transition at x/λsi ⪆ 410, we see large electromagnetic irregularities, which have developed from the oblique whistler instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Their onset begins here at Ωcit ∼ 10, beyond the simulation time in Paper I, and they are associated with under- dense electron cavities (panel (a), also present in ions ) as well as magnetic- and electric-field turbulence (pan- els (b) and (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The physical scales of these cavities is of order λsi, justifying the use of a larger transverse simulation box to allow a robust investigation of these phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We further note that these magnetic and electric field inhomogeneities extend into the upstream beyond the shock ramp (for x > 410λsi) in the field profiles in panels (d) and (e), but decrease in strength with increasing distance from the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' From these panels, where the field is averaged across the transverse direction of the simulation box, we see that (B/B0)2 approaches unity more quickly than (E/E0)2, indicat- ing the end of the inner electromagnetic foreshock and the beginning of the outer electrostatic foreshock, the latter of which is the subject of Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In Bohdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' (2022), it was demonstrated via means of periodic boundary condition simulations that the differences in the inner and outer foreshocks can be explained by differences in the reflected electron pop- ulations that excite them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The latter, which leads to the excitation of electrostatic EAWs, has in comparison to the inner foreshock a reflected electron beam density a factor of 10 lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Furthermore, the thermal spread of this electron beam is approximately 25% lower than that of the inner electromagnetic foreshock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' These dis- crepancies are enough such that in the inner foreshock the electromagnetic oblique whistler instability is the dominant excited instability, as opposed to the electron acoustic instability which is prevalent in the outer re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The analysis presented in Paper I focused on the be- havior of upstream electrons within the outer electro- static foreshock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In the remainder of this paper, we fo- cus on the inner electromagnetic foreshock, focusing on the properties of the waves, how they affect upstream electrons, and whether they can lead to electron pre- acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Shock Reflection Rate We first quantify the energy content in the reflected electrons which are essential to excite the electromag- netic oblique whistler instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We do so by first es- timating the shock location, xsh, taken as the location where ni/n0 = 2 for ion number density ni and the subscript 0 denotes the far upstream value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We define this based on ion density as that of the ions reflected at the shock is around 10−4n0 and is more stable in the foreshock relative to the electron number densities (due Reflecting wall Bo n 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='6入si Ba Simulation PlaneElectron-Whistler Interactions in Oblique Shocks 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Late time shock structure for Ωcit = 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Panel (a) shows the electron density map, (b) and (c) show fluctuations in the Bz and Ey components of the magnetic- and electric-fields, respectively, and (d) and (e) the magnetic and electric field profiles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We see the transition from the downstream to upstream regions at x/λsi ∼ 400, with the immediate upstream characterized with electromagnetic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The reflection rate, R = Ne,ref/(Ne,ref + Ne,ups) where Ne,ref is the number of reflected electrons and Ne,ups the number of upstream electrons that have not yet reached the shock, (blue dashed line), mean value of γ − 1 for the reflected electrons (red dotted line, simulation frame) and normalized kinetic energy contained within the reflected electron beam (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The latter quantity is nor- malized relative to the upstream kinetic energy such that UK,ref/UK0 = Rγref − 1/(γups − 1) for reflection rate R and γups = (1 − (vups/c)2)−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Quantities are calculated im- mediately ahead of the shock ramp between xsh + 2λsi ≤ x ≤ xsh + 10λsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Although the reflection rate falls with the onset of whistler waves at Ωcit ∼ 10, the overall energy den- sity increases as the reflected electrons are more energetic on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' to higher electron reflection rates) and electromagnetic field amplitudes, which are disturbed by the whistler waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In practice, this definition places xsh on the shock ramp, so we measure the reflection rate in the region defined by xsh + 2λsi ≤ x ≤ xsh + 10λsi to ensure we are measuring it for a region within the electron fore- shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Note that the parameters of the region of interest do not affect the presented results, so long as it resides within the electron foreshock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The chosen fixed region nevertheless lies close to the shock, and at late times is completely occupied by whistler waves, thus enables us to measure if they have any tangible effect on the reflection rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' As measured in the simulation frame, the electron spectrum in the defined upstream region has a double peak structure in Γ − 1, where the low energy peak cor- responds to the thermal population moving with the upstream bulk flow and the high energy peak corre- sponds to reflected electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' As in Paper I, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4, we use the local minimum between these peaks to dis- tinguish between reflected and upstream electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' At each timestep, reflected electrons are those within the defined region where Γ − 1 exceeds the value for which d(Ne(Γ−1))/d(Γ−1) = 0 and d2(Ne(Γ−1))/d(Γ−1)2 > 0 in the electron spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Those with Γ − 1 lower than this threshold are considered to be upstream electrons traveling with the bulk flow of the incoming plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Note that this definition has no dependency on the di- rection in which the electron is traveling, thus an ener- getic reflected electron that is re-directed towards the shock is still considered reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' (a) 1 25 log10 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='8 10 25 (b) (Bz Boz) 0 [Bo 0 10 10 (Ey Eoy) 0 [Eo 07 10 (E/Eo)2 (B/Bo)2 (d) (e) 400 500 600 700 800 x/Λsi14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='10 6 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='08 5 Fraction of reflected electrons, 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' ref/Uko 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='06 Uk, 6 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='04 4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='02 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='00 0 10 20 30 40 50 time (Qcit)6 Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 3 indicates that the onset of whistlers may indeed affect the reflection rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The blue dashed line shows that the reflection rate falls after the onset of whistler waves at around Ωcit ∼ 10, but is relatively stable at around 4% for Ωcit > 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' During the simulation, de- spite variations and a slow decline in the reflection rate, the energy density of the reflected beam increases at a roughly linear rate as indicated by the black solid line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Possible causes of this include more efficient acceleration of reflected particles or an acceleration re- gion which has a size that increases with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The red-dotted line shows that this can be explained by the fact that the mean Lorentz factor of reflected electrons also increases approximately linearly with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We have verified that the reflection rate calculation is robust to our choice of region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The changes in reflection rate for the region shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 3 occur further up- stream in the same manner, although with a time-lag as it takes the reflected electrons longer to travel upstream along ⃗B0 and reach those regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Accordingly, the en- ergy density of the reflected electron beam increases throughout the simulation and reaches the threshold value to excite whistlers further from xsh as the sim- ulation progresses, hence explaining why the size of the whistler region increases with simulation run-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' WHISTLER WAVES 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Wave Properties/Summary of Bohdan 2022 A study of the instabilities driving the waves that arise in the electron foreshock was undertaken in Bohdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' (2022), with results based on the same large-scale simulation that is presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In this earlier work, the electromagnetic waves present in the inner foreshock that we focus on in this paper were subject to a linear dispersion analysis, where it was established that they arise as a result of the oblique whistler instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Ev- idence for this came from the fact that the waves in question have approximately the same parallel and per- pendicular wavenumbers as well as growth rate as pre- dicted by linear theory for the fastest-growing oblique whistler mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Defining the wavevectors parallel and perpendicular to the upstream magnetic field as k∥ and k⊥, respectively, and noting that the oblique whistler instability is excited by a beam of electrons moving par- allel to the upstream flow with velocity vb, we summarize the results of Bohdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' (2022) as: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The dependence of the perpendicular wave num- ber, k⊥, of the fastest growing oblique whistler mode on the parallel velocity of the reflected elec- tron beam is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' For the parameters used here, the peak growth rate occurs at k⊥λse ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The excited waves are in resonance with electrons reflected at the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Hence, k∥ is extremely sen- sitive to v∥, with the lth order gyroresonance given by, ϖres = k∥vb − ℓ |Ωce| γb , l ≥ 1 (1) for beam Lorentz factor γb and electron gyrofre- quency Ωce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The beam can only excite fluctuations with suffi- ciently small phase speed (O(10−3c)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' waves with angular frequencies satisfying ϖW < ϖres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Decreasing the beam number density of the re- flected electrons results in a smaller growth rate of the whistler mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' From this latter point, and from Bohdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' (2022, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 9), we note that the dependence of ω on k for the whistlers is approximately linear, and hence the group velocity, vg, is of the same order of magnitude as the phase velocity, meaning vg << vup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Accordingly, we see the linear structures co-move with the upstream bulk flow when viewed in the frame of reference of our simu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Non-linear Structures Although the behavior of the initial whistler wave structure can be appropriately described by linear anal- ysis, at later times in the simulations they develop into non-linear wave packets, with a complex internal struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In this section we outline the structural evolution of the whistler waves into non-linear wave packets as they propagate towards the shock from the upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4 shows the electromagnetic field structure of a particularly prominent non-linear structure (developing from a whistler wave) occurring in the inner electro- magnetic foreshock of the simulation at Ωcit = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Quantities have been measured in the upstream rest frame, which removes the large scale motional electric field, thus the electric field structure displayed is domi- nated by that associated with the non-linear structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The top row illustrates the magnetic field structure, with the middle row showing the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' From left to right the figure shows the x-, y-, and z- components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We note that while all three magnetic field components have roughly similar peak magnitudes, the maximum abso- lute values of the Ex and Ey components are around an order of magnitude higher than for Ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We further note that the characteristic size of the whistlers increases as they approach the shock, which is consistent with our previous studies in Bohdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Electron-Whistler Interactions in Oblique Shocks 7 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The structure of a prominent whistler wave packet present in the electromagnetic foreshock at Ωcit = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='1 is shown as measured in the upstream rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' For vi- sual clarity, quantities are normalized relative to background simulation-frame field values (note there is no motional E- field in this frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Panels (a), (b) and (c) show Bx, By and Bz, panels (d), (e) and (f) show Ex, Ey and Ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Panels (g), (h) and (i) show the Lorentz force for a test particle moving at ⃗v = (−ve,th/ √ 3) · (1, 1, 1), for electron thermal velocity ve,th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Here, the Lorentz force components are normalized to the modulus of F0 = eve,thB0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' It is from within these waves at late times that highly non-linear structures develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' From panels (a) (c) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4, we see that the B-fields are in general strongest in magnitude at the edge of the whistlers, and progressively weaker towards the central region, such that in 2D space the value of |B/B0| reaches a maximum in a ring shape encircling the nonlinear wave structure, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The range of spatial radii from 1 − 5λsi of these structures is similar to that of short large amplitude magnetic structures (SLAMS) (Mann & Classen 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2020), which have been detected in the bow shocks of the Earth (Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 1994), Venus (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2020), and Jupiter (Tsu- rutani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' However, in contrast to SLAMs, where the density is amplified by a factor of a few rel- ative to the upstream plasma, we see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2 that the non-linear structures discussed here are associated with under-dense cavities, with under-densities as low as n/n0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We further note that observational evidence for whistler-mode induced structures has been provided by analysing satellite data from the Magne- tospheric Multiscale mission (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' These observational data apply to the Earth’s Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The structure of |B|/|B0| for the non-linear struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The magnetic energy density is strongest in a ring-like shape encircling the structure, with the field strength declin- ing towards the center and radially outside of the area of maximum strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' bow shock, thus the parameters used here are not con- sistent with or supernova remnant based simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' To comprehend any pre-acceleration that arises con- sequentially from the interactions of electrons with whistler waves, we must understand the forces they experience when interacting with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In general, an electron with charge qe immersed in both electric (E) and magnetic (B) fields experiences a Lorentz force, F , which is defined as, F = qe(E + v × B) (2) where bold quantities represent vectors with x-, y- and z- components in Cartesian space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' From Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2, the velocity of an electron, both in terms of direction and magnitude, can influence the resulting behavior when in- teracting with a region of strong electromagnetic fields, such as those within the non-linear structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We trace a sample of 10,000 upstream electrons to probe any interactions with the whistler waves and non- linear structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' These electrons are selected randomly from the upstream population at Ωcit = 30 from a re- gion between 360λsi and 410λsi ahead of the shock, and traced for the remainder of the simulation (over 20 Ω−1 ci ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Such a sample is representative of the global upstream population, and our sampled region permits an adequate duration for them to pass through the foreshock and in- teract with the shock itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Interaction of upstream (bulk flow) electrons with whistlers We have already established that the non-linear struc- tures have phase and group velocities of O(10−3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Be- 10 (b) (c) (a) 20 1 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' B-Bo 0 [Bol 10 1 10 40 (d) e 20 2 CC S E 0 [Eo] 10 2 40 (g) 101 20 100 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' O F/\\Fol 100 10 101 370 380 370 380 370 380 x/Λsi x/Λsi x/^si4 20 2 [B/Bol 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 15 1 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='2 370 380 x/Λsi8 Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' cause of this, they are quasi-stationary when viewed in the upstream rest frame, and any force components arising from their motion relative to the upstream bulk plasma is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' For these reasons, the upstream rest frame is appropriate to analysis interactions be- tween the structures and upstream electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In this frame, the upstream electron population is thermal, with a most probable speed of ve,th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='044c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This corresponds to an electron gyroradius of rge = ve,thΩ−1 ce = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='7∆ ≈ λsi/10, which is around two or- ders of magnitude smaller than the size of the largest non-linear structures such as that shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' As By = 0 and Bz > Bx, the By terms in the Lorentz force (see Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2) can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The colormap of panels (g), (h) and (i) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4 show the Cartesian compo- nent of the Lorentz force as measured in the upstream rest frame that would be experienced by a test electron moving with the upstream bulk flow such that its ve- locity is given by ⃗vtest = −ve,th/ √ 3(1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We choose Cartesian velocities of vtest such that the thermal veloc- ity magnitude is divided equally between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Thermal upstream electrons that are spatially coin- cident with the growing non-linear structure will expe- rience forces according to Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In the x-direction, the direction of Ex oscillates and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4 shows that this oscillation dominates the structure of Fx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This in com- bination with gyroradii typically much smaller than the radial extend of the structure ensures that the electrons will in general remain co-moving with the non-linear structure in the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Additionally, the force in the z-direction is comparatively weak relative to other components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The interaction of upstream co-spatial electrons with the non-linear structures is much more interesting in the y-component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' From Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2, Fy = qe(Ey+vzBx−vxBz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Firstly, when considering cool thermal upstream elec- trons, the qeEy term dominates, thus the overall Lorentz force directs them outwards and away from the non- linear structure, helping to carve out low density cav- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' To understand the behavior of hotter upstream electrons we need to account for the signs of the three terms that constitute Fy and the field geometry shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Here, we note that the cross terms are in the same direction as the qeEy term if they are both nega- tive, as is the case for our test particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In fact, this sce- nario is both plausible and likely as from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4 Bx and Bz are generally diametrically opposed (thus vx and vz share the same sign, and are positive and negative each for half of one gyration period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In the negative case, as for cool electrons the Lorentz force is again directed outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' However, when vx and vz are both positive, the magnitude of this outwards force reduces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Despite Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Probability distribution along the transverse direction before and after the wave shown in Figs 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Due to the shape and strength of the Ey component of the whistlers, electrons spatially coincident with the top half ex- perience an upwards force, while those approaching the lower half an downwards force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In these plots, we see the initial dis- tribution (top, at Ωcit = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='3) and distribution after (lower panel, at Ωcit = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='4), which is now bimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The vertical dashed red lines indicate the radial extent of the non-linear structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' this, the overwhelming majority of upstream electrons are too cool for the force to ever point inwards, with the overall force away from the structure center when averaged over the electron gyro-period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In general, as shown in panel (h) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4, there is a net force away from the center of the non-linear struc- tures on co-spatial upstream electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Using a sub-set of our traced electrons that are located within 2λsi of the radial extent of the structure, we see that the direction of the Lorentz force leads to a bi-modal distribution of these electrons, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Here, we see a slight preference for the electrons to be present beyond the radial extent of the structure, as opposed to centrally within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This effect is only noticeable for non-linear waves with particularly large amplitudes, and in general it preferentially expels relatively cooler electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Crucially, we note that for upstream electrons the force associated with the non-linear structures is small, and not generally towards the wave center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This means that upstream electrons are not likely to resonate with these waves, making the scenario where upstream elec- trons are trapped in the non-linear structures that have developed within the whistler potential highly unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Interaction of Reflected Electrons with Non-linear Structures 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='10 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='05 ity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='00 probabil (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='05 0 5 10 15 20 y/ΛsiElectron-Whistler Interactions in Oblique Shocks 9 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Panels (a) - (d) show the trajectory of a reflected electron though the simulation, with the current location at Ωcit = 49 marked by the circle center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Its path over the previous Ω−1 ci is shown by the black and white line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The background in panels (a) - (b) show fluctuations in Ex and Ey, while (c) and (d) show the Lorentz force components Fx, and Fy, using the particle velocity at Ωcit = 49 of v/c = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='47, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='86) normalized relative to F1 = evupB0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We see that these force components are now directed towards the wave center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Panel (e) shows the work done on the electron by all Cartesian components of the electric field in its own rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The interval shown here corresponds to the cyan-shaded area of panel (f) which shows the electron Lorentz factor as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Here, the dashed green line shows the current time of Ωcit = 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The solid lines within the cyan and salmon colored panels show the change in Lorentz factor vs ∆y for the color-indicated regions, and the analytically expected ∆γ vs ∆y from SDA is plotted with a black dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' For aesthetic purposes, the color bar label has been moved inside the figure for panels (a)-(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We see two prominent kinks in the particle trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The first of these occurs at Ωcit = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='44, which is caused by the electron deflecting around the lower half of a non-linear structure (note that the structure centered at (415, 12) here is not responsible, as it was further upstream at this timestep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The second kink occurs at Ωcit = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='75, at which point the electron is trapped by the non-linear structure we see it co-move with afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In general, as measured in the upstream reference frame, reflected electrons have positive values for at least two Cartesian velocity components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' A positive vz is needed as they travel along the upstream mag- netic field lines, of which the strongest component lies along ˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' An additional magnetic field component in ˆx, in combination with the fact that a positive vx is re- quired so the electron can outrun the shock (such that vx > vsh cos θBn = vsh/2) ensures vx is also positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' If this latter criterion is not met, it cannot be reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' When considering the vy component, we first note that reflected electrons require some pre-acceleration to be reflected from the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' These mechanism tend to provide acceleration due to work done by the motional electric field, which here lies in the −ˆy direction, lead- ing to electron acceleration in the ˆy direction by virtue of their negative charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This velocity component is purely gyrational, so oscillates around zero at the elec- tron gyrofrequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' From Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2, we see that this changes the forces a reflected electron experiences during an encounter with a whistler wave relative to an upstream electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7 shows such an interaction in the simulation frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Pan- els (a) and (b) show fluctuations in Ex and Ey (to visu- alize the other field components of the whistler see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4), while (c) and (d) show Fx and Fy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Here, they are normalized relative to modulus of F1 = evupB0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Panel (e) shows the Cartesian components of accelera- tion felt by the electron in its own rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' (f) shows the Lorentz factor of the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The red dashed line in this panel indicates the timestep corresponding to the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' For panels (a) - (d) the trajectory of the elec- tron during the previous ion gyroperiod is shown by the black-and-white lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Immediately we see that the trajectory of the reflected electron is deflected away from the direction of ⃗B0 and is influenced by the presence of the developed non-linear structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Furthermore, the electron appears to have been trapped by the (prominent) upper structure lo- cated at (x/λsi, y/λsi) = (410, 27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Crucially we can determine from Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2 that, in contrast to upstream electrons, the Fx and Fy forces are now directed towards the center of the non-linear structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' If we consider the y-direction in the simulation frame, in comparison to an interaction with an upstream electron (where vx < 0), the vxBz term is now aligned with the Ey term, with 10 400 400 40 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='002 e) Ex 32 30 Eo 30||Foul 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='001 1 16 W S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='000 M20 0 0 y f-16-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='001 1 10 10 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='002 Wx Wy Wz sum 0 10 0 400 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5 380 400 420 380 400 420 x/Λsi x/Λsi 10 400 40 40F 10 f) Ey - Eyo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5 32 S M 30 Eo 301 |Fou 8 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='0 6 5 0 5 0 0 Ay/si 2 16 4 1 10 10 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 32 25 2 0 ww Ay/Asi 10 0 400 380 400 420 380 400 420 35 40 45 50 x/Λsi x/si timestep, Qcit10 Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' each of these pointing inwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' If all three velocity components are positive, no matter how the electron approaches the whistler, all three terms in Fy (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2) point inwards, enabling trapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The three Cartesian components of the Lorentz force generally point inwards for reflected electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' As measured in the upstream rest frame and relative to the upstream magnetic field, typ- ical Larmor radii for reflected electrons are around 1-5 λsi, but can be up to ≈ 10λsi, which from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 5 are compressed by a factor of around 4 when encountering the strong magnetic field associated with a non-linear structure, thus reflected electrons can typically be con- tained within them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Interaction Probability We can estimate the probability that a reflected elec- tron will interact with a non-linear structure by consid- ering the path taken by such a particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The whistler waves and resulting non-linear structures move with group velocity vg << vup, hence we perform this calcu- lation in the upstream rest frame using the approxima- tion that the non-linear structures are stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We use primed quantities here to represent the upstream rest frame, and further assume the size of the whistler- containing electromagnetic foreshock from which the non-linear structures derive to be a constant size of x′ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Primed quantities with the subscript ∥ and ⊥ refer to components measured parallel or perpendicular to the magnetic field vector in the upstream rest frame, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Since the reflected electrons are gyrating, we can con- sider the path length to be a sum of 2 components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' These consist of a linear component, R′ lin as a result of the path of reflected electrons following the large-scale upstream magnetic field structure, and an oscillatory component, R′ osc, as a result of the gyration around the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The total path length can be considered to be, R′ path = � R′2 lin + R′2 osc (3) where R′ osc dominates if the electron Larmor radius is significantly larger than x′ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Otheriwse, R′ path ≈ x′ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' R′ lin is simply the size of the whistler-containing re- gion, such that R′ lin = x′ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' To compute the time an elec- tron takes to traverse it, we consider that they travel in the direction of the upstream rest frame magnetic field, ⃗B0 ′ which now makes an angle θ′ Bn with the simulation plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We therefore only see their propagation projected onto the simulation (xy) plane, with the time taken to traverse R′ lin = x′ w defined as, t′ lin = x′ w v′ ∥ cos θ′ Bn , (4) where the cos θ′ Bn term provides the necessary path cor- rection to compensate for the inclination of the upstream magnetic field with respect to the simulation plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' During time t′ lin, assuming it does not interact with any non-linear structures, a reflected electron completes t′ lin/τge oscillations, where τge = 2πrge/v⊥ is the elec- tron gyro-period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' As a result of our magnetic field orien- tation with respect to simulation plane, the shape made by the gyrational orbit of the electron on the simulation plane is elliptical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' On account of B′ y = 0, the size of the semi-major axis is a = rge, where rge is the electron gyroradius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The semi-minor axis appears contracted in the direction of motion, such that its value is given by b = rge sin θ′ Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' For the sake of presenting a more easily interpretable solution, we approximate the perimeter of the elliptical path, p, as, p ≈ 2π � a2 + b2 2 = √ 2πrge � 1 + sin2 θ′ Bn, (5) which is typically accurate to better than 5% assum- ing the ellipse is not too elongated (Muir 1902).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Other, more accurate, approximations can be found in the lit- erature (Ramanujan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The total gyrational path length is therefore given by R′ osc = pt′ lin/τge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Combining these, we can rewrite Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 3 as, R′ path = x′ w � 1 + v′ ⊥ √ 2v′ ∥ � 1 + sin2 θ′ Bn 1 − sin2 θ′ Bn � , (6) where this equation must also satisfy the reflection con- straint that v′ ∥ cos θ′ Bn ≥ vsh to remain valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The probability of interaction with a non-linear struc- ture follows, Pinteract ≈ nnlsσnlsR′ path = nnlsπr′2 nlsx′ w � �1 + 1 2 � v′ ⊥ v′ ∥ �2 �1 + sin2 θ′ Bn 1 − sin2 θ′ Bn �� � 1/2 , (7) for non-linear structure number density nnls and cross- sectional area σnls = πrnls′2 as measured in the upstream rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We can interpret our results within the con- text of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Firstly, we recover the intuitively expected results that both a larger number density of non-linear structures and their cross-sectional area linearly increase the in- teraction probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In the limit v′ ∥ → c, v′ ⊥ → 0 to prevent the electron becoming superluminal, and we re- cover the expected solution that R′ path = x′ w (the same is true if v′ ⊥ is small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7 also recovers the expected solution that the path length approaches ∞ in the case Electron-Whistler Interactions in Oblique Shocks 11 of a perpendicular shock where θ′ Bn → 90◦ because the reflected particles are unable to escape upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' More significantly, we note that the second term in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7 indicates that the probability of interaction is proportional to v⊥, but inversely proportional to v∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' One may conclude that this may favor reflected elec- trons with v′ ⊥ >> v′ ∥, however this is not accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' For a more realistic picture, we must again consider that for a reflected electron to outrun the shock it must sat- isfy v′ ∥ cos θ′ Bn ≥ vsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This, in addition to the constraint that v′2 = v′2 ⊥ + v′2 ∥ ≤ c, restricts the value of v′ ⊥ to be v′ ⊥ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='848c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The electron shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7 interacts with a non- linear structure, and becomes trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' During this time, it is carried towards the shock as it is unable to escape, until the wave ‘breaks’ when encountering the shock ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We note that the Larmor radius of re- flected electrons that are trapped by whistler waves need to be smaller in size relative to the whistler wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Typi- cally, when measured relative to the upstream magnetic field, reflected electrons have gyroradii in the range of 1 − 5λsi, though can approach 10λsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This comparison means the ratio of reflected electron gyroradii to the ra- dius of a non-linear structure lies in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='1 − 1, as the non-linear structures measure ≈ 5λsi radially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Al- though the magnetic field amplification associated with these structures will contract electron gyroradii that en- counter them, this may not be sufficient to allow the non-linear structures to trap the most energetic reflected electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In reality, we might expect the probability of trapping to fall off exponentially as the gyroradius ap- proaches the radius of the non-linear structure, rnls, such that Ptrap ∝ exp(−rge/rnls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' With a simplifying assumption that reflected electrons would move with a constant v∥ in a calm upstream re- gion, we can calculate the most probable value by as- suming Ptrap ∝ Pinteract · exp(−rge/rnls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Numerically solving this equation and accounting for the aforemen- tioned constraints, under our simulation setup we ob- tain a peak trapping probability for v′ ⊥ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='40c and v′ ∥ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='54c, which corresponds to reflected electrons that are around 18 times as energetic as thermal electrons in the far upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' While this analysis has allowed us to estimate trap- ping conditions for reflected electrons, it only provides a snapshot over a small time-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' More realistically, Bohdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' (2022) show that the size of the whistler region from which the non-linear structures arise from grows with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' By integrating the energy density of the reflected electrons up to the point that the whistler waves become detectable and extrapolating this to late times, Bohdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' (2022) estimate that the size of this Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Plot showing perpendicular vs parallel momen- tum for the electron in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7, as measured in the upstream rest frame relative to the local magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Arcs of con- stant momentum indicate pitch angle scattering, while in- creases in p′ ⊥ are suggestive of SDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' region would reach a steady state at Ωcit ≈ 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' At this time, the size of the whistler containing region would ex- tent to around 2000λsi ahead of the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Within the context of our analysis, from Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7 we would expect the trapping probability to increase up until Ωcit ≈ 125 for all reflected electrons with gyroradii small enough to be contained by the non-linear structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Further upstream beyond this region, the energy density of the electron acoustic waves (which are not well captured in this out-of-plane simulation) that are the subject of Pa- per I would dominate, hence the non-linear structures discussed here would cease to be important beyond this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Stochastic Shock Drift Acceleration From Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7, electrons with some perpendicular accel- eration are more likely to be trapped by the non-linear structures, assuming that they have enough parallel ac- celeration to escape the shock front and under the condi- tion that their gyroradii are smaller than the character- istic radius of the non-linear structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Trapped elec- trons will be returned to shock where they may undergo further pre-acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Identifying a mechanism that provided perpendicular acceleration can explain much about their behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Known mechanisms in this region that increase v⊥ include shock surfing acceleration (SSA) and stochas- tic shock drift acceleration (SSDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The former of these processes is dependent on the presence of electrostatic Buneman waves, which are excited by a velocity dif- ference between incoming electrons and reflected ions, which results in their production near the shock ramp (Buneman 1958;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Gary 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In perpendicular shocks, ion gyration at the shock is sufficient to excite them in the shock foot (Bohdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In oblique shocks, the Buneman instability is strongly modified due to the 10 50 pl/mec 45 5 40 35 0 5 0 5 10 p /mec12 Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' presence of whistler waves and so the overall efficiency of SSA might be different compared to perpendicular shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Note that ions propagating back upstream can- not drive Buneman waves in the foreshock region since the ion reflection rate is too small, nion,ref/n0 = 10−4, and the growth rate predicted for Buneman waves is over two orders of magnitude smaller than that for the whistler waves (Bohdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Another candidate acceleration mechanism is shock drift acceleration (SDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The original theory of SDA in- dicated that it could efficiently accelerate charged par- ticles (Wu 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Leroy & Mangeney 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' It occurs if the electron gyrates close enough to the shock ramp such that part of its orbit overlaps the region with en- hanced magnetic field, tightening its gyro-radius during these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The gradient in the magnetic field results in a drift analogous to ∇B drift with work done by, and in the direction of, the motional electric field, which is perpendicular to ⃗B0 by definition (Ball & Melrose 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Despite the efficient energization, Vandas (2001) demonstrated that SDA alone is not efficient enough to account for the observed power-law spectrum and fluxes of accelerated electrons in astrophysical sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Phys- ically, this occurs because in the original SDA theory, candidate electrons are not confined to the shock transi- tion region where acceleration occurs, thus limiting the efficiency of the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' One such way of overcom- ing this impediment is to add pitch angle scattering, with electrons scattering off whistlers being observed in the Earth’s bow shock (Oka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Katou & Amano (2019) proposed the stochastic shock drift ac- celeration (SSDA) mechanism which incorporates pitch angle scattering into the SDA model, increasing the time in the acceleration region and accordingly the en- ergy gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Additional evidence of SSDA has been found in both in 3D PIC simulations (Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2017) and in observations which support electron scattering by whistler waves (Oka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This latter work con- cludes that the energization directly via the whistlers is low, but the electrons, as here, are confined within the acceleration region and become more energetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This picture is consistent with our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 8 plots p′ ⊥ vs p′ ∥ (where the primes again represent the upstream rest frame) for the electron shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We see increases in p′ ⊥, supporting SDA as the mechanism that provided the acceleration, and variations in pitch an- gle for constant p′, which are indicative of scattering (Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Ha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The vertical red line at around (p′ ∥, p′ ⊥) = (5, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5) corresponds to the salmon pink region of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7 panel (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This occurs when Ωcit ≈ 49, when the electron has been returned to the shock by the non-linear structure that had trapped it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Another signature of SSDA is that the change in energy is directly proportional to the motional electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' For our magnetic field configuration electrons will drift in the +ˆy direction (Krauss-Varban & Wu 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We see from panel (f) that for the salmon-pink region, the predicted ∆γ (dashed black line) agrees closely with measured values (solid red line), verifying that SSDA is observed in our simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' However, the presence of the non-linear structures fur- ther complicates this picture, and the acceleration of the most energetic electrons cannot be fully described by SSDA alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Panel (f) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7 indicates that this par- ticular electron undergoes two periods of rapid and effi- cient acceleration, with relative quiescence in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' At around Ωcit = 41, the electron first encounters the shock and is accelerated, with this region indicated by the cyan panel and corresponding to the color-matched sub-panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The phase of constant energy corresponds to the electron traveling upstream, and includes the time- period when it is trapped by the non-linear structure, indicating the primary role of such structures is to keep electrons confined to the acceleration region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The cyan panel shows the measured ∆γ (blue solid line) and ana- lytical ∆γ (black dashed line) as a function of ∆y for the first acceleration period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Although there are incidences where the energization rate can be attributed to SSDA, ∆γ is shown to also increase for ∆y ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We see that this behavior is also evident in the subset of the most energetic traced electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 9 indicates that they generally have a linear relationship between ∆γ and ∆y, as for the trace electron shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7 is shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 9 by the red circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' However, on average, the accelera- tion is between three and four times more efficient than can be accounted from SSDA via the motional electric field alone, with this prediction indicated by the purple dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This requires further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' To explain this, we compute the total work done, W, by electric field on the electron in its own rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This is a sum of the Cartesian components, such that W = Wx + Wy + Wz, with these displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7 panel (e) for the interval Ωcet = 41 − 43 which cor- responds to the cyan area in panel (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Noting that (∆y/λsi, ∆γ) = (0, 0) on the latter plot corresponds to Ωcit = 41 on panel (e), we see that initially when Wy ≤ 0 the change in y-coordinate is ∆y ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Closer to Ωcit = 43, Wy becomes positive, in which regions we see acceleration consistent with SSDA (the blue solid line is quasi parallel to the black dashed line in panel (f), cyan background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' However, the important distinction to SSDA is that the overall acceleration during this pe- riod is typically greater than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We assess the relative importance of each component by computing the mean Electron-Whistler Interactions in Oblique Shocks 13 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The change in Lorentz factor as a function of ∆y/λsi at Ωcit = 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5 for the traced electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The neg- ative average ∆y typically occurs for electrons that have passed into the downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The location of the electron shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7 is indicated by the red circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Under the assumption that the work done to cause the accelera- tion comes from the motional electric field, SSDA predicts ∆γSSDA ≈ qeE0y∆y/(mec2), thus the energy gain for the most energetic electrons is 3-4 times greater than can be ac- counted for by SSDA from the motional electric field alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' work done across this time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' For this electron, the Wz component is particularly strong, and has a rel- ative contribution to the overall particle energy change that is comparable to that from Wy, with Wz ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='1Wy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Wx is weaker at Wx ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='17Wy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Typically for reflected electrons, we see similar levels of work from Wz and Wy with a smaller Wx, unlike in SSDA where we would expect Wy to drive the acceleration exclusively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The reason we see more efficient acceleration overall rela- tive to SSDA is therefore because all Cartesian compo- nents can in principle contribute, thus we would expect ∆γ/∆γSSDA ≥ 2 on account of the comparable Wy and Wz, which is consistent with what we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The precise details of the underlying microphysics of the acceleration are likely a consequence of the complex non-linear structures arriving at the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Indeed, pre- vious studies have shown that changes in local conditions can lead to more efficient electron acceleration (Kobzar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' For the electron considered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7, we compute the dot-product of ⃗B0 and the time averaged lo- cal magnetic field from Ωcit = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='4 to Ωcit = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5, when Wz dominates the work done by the electric field in the particle rest frame (panel (f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The cosine of the angle between then is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='9, whereas for the pure SDA consistent region (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' pink shaded region of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7(f) ) this number is ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This means that the local magnetic field in these regions on average subtends an angle with the upstream field of around 26◦ during this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' As the work done during acceleration via SDA occurs via purely perpendicular electric fields, this changing of the local field orientation permits acceleration in additional directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Indeed, the non-linear structures arriving at the shock also perturbate the local electric fields, with Ey/E0y ranging from ±15 in the cyan region of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 7f, yet averaging at Ey/E0y ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The overall acceleration is therefore highly sensitive on local values and orienta- tions of the electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Pitch Angle Distribution On the one hand, an electron with a larger v′ ⊥ will have a longer path length via Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 6, yet if v′ ⊥ is too large its gyroradius becomes too large for trapping to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Similarly, v′ ∥ is constrained by the necessity for reflected particles to be able to outrun the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The pitch angle, α′ = arctan(v′ ⊥/v′ ∥), is a useful quantity that can provide important information about the sub- sample of reflected electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We select all reflected electrons that reach γ ≥ 5 at the end of the simulation at Ωcit = 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5 and compute their pitch angles as measured in the upstream reference frame relative to B′ 0 over the final two Ω−1 ci in the simula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We show the probability distribution of these pitch angles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' By definition, electrons with pitch an- gles α > π/2 are heading back the shock, however, in this frame recall that v′ ∥ cos θ′ Bn > vsh is required for elec- trons to outrun the shock and travel upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This, in addition to the speed limit of c, sets v′ ∥,min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='53c and v′ ⊥,max ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='85c, from which we can calculate a maximum permitted pitch angle of α′ max = arctan(v′ ⊥,max/v′ ∥,min) for reflected electrons that are traveling towards the up- stream in the upstream rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Accordingly, all elec- trons with pitch angles α′ ≳ π/3 will be caught up to by the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Integrating the probability density shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 10 gives the probability of α′ > π/3 as P(α′ > π/3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='20, where this probability represents an estimate of the frac- tion of reflected electrons that are re-directed back to the shock as a consequence of interactions with the non- linear structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Considering the stable late-time elec- tron reflection rate of ≈ 4%, we estimate that around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='8% of all the incoming upstream electrons can be ener- gized by the enhanced SSDA mechanism discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This argument assumes that the reflected electrons con- sidered here are located within the upstream region con- taining the non-linear structures, which is true for the majority of reflected electrons within this sample, but the numbers presented here should nevertheless be con- sidered as an approximate upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The electron downstream distribution Figure 11 shows the electron downstream distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The low energy part of the spectra is represented by a △YsSDA 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='0 102 3△YsSDA 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5 4△YsSDA 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='0 100 50 25 0 25 50 AylAsi (from injection)14 Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Pitch angle distribution for the most energetic 70 traced particles which have γ ≥ 5 over the final 2Ω−1 ci of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' In this frame, all electrons with pitch angle α ≳ π/3 will be caught up by, or are heading towards, the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Maxwellian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The electron to ion tempera- ture ratio is about Te/Ti = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='17 which is close to that in a perpendicular shock simulation with similar shock pa- rameters (Bohdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' High energy electrons follow a power-law with an approximate index up to -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The fraction of electrons not covered by the thermal dis- tribution is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='8% and they hold about 7% of the downstream electron energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 11, at the end of the simulation, the most energetic downstream electrons present at the very tail of the cut-off already reach gyroradii that are comparable to the shock width or the ion upstream gyroradius, γinj,e ≈ mi me vsh c ≈ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The threshold for injection into DSA will be a factor of a few higher than that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Unfortunately, the simula- tion time is still too short to properly capture the DSA phase of acceleration both for electrons and ions, as well as formation of a nonthermal tail in the ion distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' CONCLUSIONS In this paper, we have presented results on the elec- tron foreshock of an oblique collision-less shock, based on a large scale PIC simulation with a total run-time characterized by Ωcit = 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The physical properties of this simulation are such that a shock front is gener- ated in the high Mach number regime, and thus consis- tent with the environments of supernova remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' We have discussed the onset and characteristic properties of whistler waves, and outlined their development into complex non-linear structures which carve out density cavities in the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' These non-linear structures are capable of trapping and scattering reflected electrons, confining them to the region close to the shock where Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The electron energy distribution in the shock downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The dotted line is the Maxwellian fit to the low-energy part of the electron distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The dashed line represents the slope of the nonthermal electron tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' they can be efficiently pre-accelerated, which increases the likelihood of being injected into DSA and being fur- ther energized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Our main conclusions can be summa- rized as follows: Reflected electrons at the shock excite the oblique whistler instability, generating electromagnetic whistlers waves in the inner foreshock region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The phase and group velocities of the whistlers are much less than that of the upstream bulk speed vup, so they approximately co-move with the up- stream plasma and also grow in size to about five ion inertial lengths as they approach the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Over time, the internal structures arising from the whistler waves become highly complex and non- linear, and they carve out density cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The magnetic energy density is the highest at the pe- riphery of each structure, and they are filled with strong electric field that varies on small scales on the order of the ion inertial length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' Upstream electrons co-moving with the non-linear structures in general experience a Lorentz force di- rected away from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This means that upstream electrons in general do not resonate with the struc- tures, thus are unlikely to be confined within them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' For reflected electrons encountering the non-linear structures, the Lorentz force is now directed to- wards the structure center, so they can become trapped if their gyroradii are small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=" This enables, and indeed leads to, the return of a sub- set of electrons that are initially reflected at the α' = π/3 Probability Densit 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content="00 0 π/4 T/2 3π/4 α' relative to B°Electron-Whistler Interactions in Oblique Shocks 15 shock being trapped and returned to it, where ad- ditional pre-acceleration is possible." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' The acceleration mechanism is directly analo- gous to stochastic shock drift acceleration, but is around 3 times more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This is because other components of the electric field, besides the coherent motional electric field, can contribute to the acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' By considering their pitch angles, at any given time, we estimate that 20% of reflected electrons have been redirected back towards the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' This corresponds to around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='8% of the total upstream electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' acknowledges support by DFG through grant PO 1508/10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQf9PrN/content/2301.00872v1.pdf'} +page_content=' thank the Interna- tional Space Science Institute (ISSI) for their hospitality and the ISSI Team Energy Partition across collisionless shocks for invaluable discussions.' metadata={'source': 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sha256:f766669f5e520f99e8dc3d3ebcc3742eaac7c32dea67a586c7de42777dce4a96 +size 116689 diff --git a/Z9FIT4oBgHgl3EQflCtt/content/tmp_files/2301.11303v1.pdf.txt b/Z9FIT4oBgHgl3EQflCtt/content/tmp_files/2301.11303v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4f03ea7742d250ccbd4d9fb4349781002fa9731b --- /dev/null +++ b/Z9FIT4oBgHgl3EQflCtt/content/tmp_files/2301.11303v1.pdf.txt @@ -0,0 +1,350 @@ +Proof to count bound state nodes +in supersymmetric quantum mechanics +A.Aynbund1, ∗ and V.V.Kiselev1, 2, † +1Landau Phystech-School, Moscow Institute of Physics and Technology, +Russia, 141701, Moscow Region, Dolgoprudny, Institutsky 9 +2State Research Center of the Russian Federation “Institute for High +Energy Physics” of National Research Centre “Kurchatov Institute”, +Russia, 142281, Moscow Region, Protvino, Nauki 1 +A normalizable static supersymmetric bound ground state annihilated by the +super-generators has got zero number of internal nodes in the framework of one- +dimensional supersymmetric quantum mechanics. The super-generator transforma- +tions between excited super-partner bound states as combined with the standard +technique of Wronskian provides an elegant and self-sufficient way to derive the +equality of internal nodes amount to the number of consequent excitation. +I. +INTRODUCTION +Counting an amount of internal nodes in wave functions of bound states is known as an +oscillation theorem. The theorem relates a sequence number of excited state in the order of +energy enlarge to number of its nodes. An ordinary proof, say, in classical book by F. Berezin +and M. Shubin [1] is a set of theorems, lemmas, consequences and prepositions in Sturm– +Liouville theory of differential equations. Such the way is rigorous and it looks like just +straightforwardly mathematical, while physical explanations merely give some qualitative +arguments but not enough to be provements (see, for instance, [2]). In occasion there is +a method based on coherent quantum structure, that provides an elegant and physically +clear way to derive the oscillation theorem by means of supersymmetry in one dimensional +quantum mechanics. This method is built on a construction introduced by M. Crum [3] and +∗Electronic address: aynbund.asya@phystech.edu +†Electronic address: Valery.Kiselev@ihep.ru; kiselev.vv@mipt.ru +arXiv:2301.11303v1 [hep-th] 26 Jan 2023 + +2 +incorporated to the supersymmetry by E. Witten [4] as reviewed in [5–7]. +In Section II basic notions and constructions are described to give the most important +properties of supersymmetric Hamiltonian. Section III deals with a ground state, and we +derive that the ground state has got no internal nodes. Section IV is devoted to a step to +the level next to the ground state to prove that the number of nodes increases to the unit +exactly. In Conclusion we summarize the results. +II. +MACHINERY OF SUPERSYMMETRIC QUANTUM MECHANICS +The theory under study is one-dimensional stationary Schr¨odinger equation. Suppose +that Ψ0 is a ground state wave function normalised in a Hilbert space of quantum states +for a Hamiltonian ˆH1. As it is well known, function Ψ0 can be set to be real since it has +got nodes at borders. Let the Hamiltonian have got several excitations of the ground state, +and the excited levels are counted in the order of energy enlargement as Ψ(1) +k . Following +M. Crum [3] and M. Darboux [8], one constructs a partner Hamiltonian ˆH2, that possesses +a set of bound states wave functions Ψ(2) +k . +Define two linear differential operators A− and A+, which are hermitian conjugated each +to other: +A− = − d +dx + Ψ′ +0 +Ψ0 +, +A+ = d +dx + Ψ′ +0 +Ψ0 +. +Introduce operators of super-generators Q and ¯Q ≡ Q† by E.Witten [4] +Q = +¯h +√ +2m +� +� 0 +0 +A− 0 +� +� and ¯Q = +¯h +√ +2m +� +�0 A+ +0 +0 +� +� acting on +� +�Ψ(1) +Ψ(2) +� +� +where the column refers to wave functions of super-partner Hamiltonians ˆH1 and ˆH2, re- +spectively. +A supersymmetric Hamiltonian H is constructed as +H = +� +Q, ¯Q +� +. +Consider properties of Hamiltonian H and supercharge operators. So, the super-generators +are nilpotent, i. e. Q2 = ¯Q2 = 0 and they commute with Hamiltonian +[Q, H] = 0. + +3 +Then the definition of super-partner Hamiltonians in terms of matrix H reads off +H = ¯h2 +2m +� +�A+A− +0 +0 +A−A+ +� +� = +� +� +ˆH1 − E0 +0 +0 +ˆH2 − E0 +� +� . +Two Schr¨odinger equations under study are +ˆ +H1Ψ(1) +k += EkΨ(1) +k , +ˆ +H2Ψ(2) +k += EkΨ(2) +k , +where the eigenvalues of Hamiltonians coincide. Indeed, according to the connection between +ˆ +H1 and operators A− and A+ the action of A+A− to the wave function Ψ(1) +k +gives +¯h2 +2mA+A−Ψ(1) +k += ( ˆ +H1 − E0)Ψ(1) +k += (Ek − E0)Ψ(1) +k . +Acting by A− results in the eigen state equation +A−A+(A−Ψ(1) +k ) = (Ek − E0)(A−Ψ(1) +k ) +⇒ +( ˆH2 − E0)(A−Ψ(1) +k ) = (Ek − E0)(A−Ψ(1) +k ), +determining the wave function Ψ(2) +k +def += A−Ψ(1) +k +as the bound state solution for Hamiltonian +ˆH2. Note that the superscripts of wave functions point to means the super-partner Hamil- +tonians, while the subscript denotes the number of excitation with the same energy Ek for +both Hamiltonians. +III. +GROUND STATE IN THE DISCRETE SPECTRUM +The ground state wave function Ψ0 satisfies the equation of ground state: +A−Ψ0 = 0. +Hence, any other function satisfying the same equation is a normalised solution for the +ground state and it is proportional to Ψ0, since the bound state energies are not degenerate. +Let us show that A−|Ψ0| = 0. Therefore, Ψ0 = const · |Ψ0|, and generically one can put +the constant to be positive, while Ψ0 > 0 at any internal point except borders. It means +that Ψ0 is nodeless. +Indeed, +˜Ψ +def += |Ψ0| = Ψ0 sign(Ψ0), + +4 +while the action by A− straightforwardly gives +A− ˜Ψ = +� +− d +dx + Ψ′ +0 +Ψ0 +� +Ψ0 sign(Ψ0) = − d +dx +� +Ψ0 sign(Ψ0) +� ++ Ψ′ +0 sign(Ψ0) = += ((((((( +( +− sign(Ψ0)Ψ′ +0 − Ψ0 +d +dx +� +sign(Ψ0) +� ++  + +Ψ′ +0 sign(Ψ0) = −Ψ0 +d +dx +� +sign(Ψ0) +� += += − +� +j +Ψ0(xj) 2 δ(xj) sign +� +Ψ′ +0(xj) +� +≡ 0, +(1) +where xj are positions of possible nodes of Ψ0, if exist, while Ψ′ +0(xj) ̸= 0 because +Ψ′(xj) = Ψ(xj) = 0 +would lead to Ψ(x) ≡ 0 as a solution of Schr¨odinger equation. +Thus, the super-generators provide us with the short proof of statement that the ground +state wave function Ψ0 is nodeless, so we consider it to be positive anywhere except border +points. +IV. +COUNTING THE NODES +As we plot in Fig. 1, there are two sets of states for the super-partner Hamiltonians with +energy E1. We show that one can transform one column of states to another by making use +the linear operator A− for moving from left to right (for example, if one acts on Ψ0, we will +get 0) and the operator A+ for moving back. These operators conserve the energy level. +nodeless +ground +ground +nodeless +A− ↔ Q +A− ↔ Q +A+ ↔ ¯Q +E1 +E0 +Ψ(1) +1 +Ψ0 +Ψ(2) +1 +0 +ˆ +H1 +ˆ +H2 +FIG. 1: The ground state and the exited level of Hamiltonian ˆH1 (left), the ground state of +Hamiltonian ˆH2 (right) with energies E0 and E1, respectively. + +5 +Ψ(1) +1 (x) +x +x1 +x′ +1 +x′′ +1 +FIG. 2: Node of the exited state and the direction of the derivative in it. +The figure points that the ground states for both super-partner Hamiltonians are node- +less, while more energy means that the wave function has more nodes as well known fact. +So, with this explicit scheme for the super-partner system we are going to prove that the +first excitation of ground state for ˆH1 has got exactly single internal node, while further +excitations have got numbers of nodes equal to its excitation numbers. +In previous section we proved so-called base case of mathematical induction method. +The next point is to prove the induction step. Acting on the first excited wave function of +Hamiltonian ˆH2 and by revealing the operator by definition we get +A+Ψ(2) +1 += Ψ +′(2) +1 ++ Ψ′ +0 +Ψ0 +Ψ(2) +1 += +� +Ψ0Ψ(2) +1 +�′ +Ψ0 += ⟨ or ⟩ = A+(A−Ψ(1) +1 ) = (E1 − E0)Ψ(1) +1 . +Then, multiplying by Ψ0 one can see the derivative of Wronskian for Ψ0 and Ψ(1) +1 : +� +Ψ0Ψ(2) +1 +�′ += (E1 − E0)Ψ(1) +1 Ψ0 = −W ′ +Ψ0Ψ(1) +1 +⇒ −WΨ0Ψ(1) +1 += Ψ0Ψ(2) +1 . +Therefore, we come to the following expression: +Ψ′ +0Ψ(1) +1 +− Ψ0Ψ′(1) +1 += Ψ0Ψ(2) +1 . +Consider the first term on the left side. We can see that as E1 is grater than E0 there are +strictly more nodes for the function of excited level. Hence, there are at least one node +which we will denote x1. Rewriting the previous equation at the node we get +Ψ0(x1)Ψ(2) +1 (x1) = −Ψ0(x1)Ψ′(1) +1 (x1), +where all of Ψ0(x1) and Ψ(2) +1 (x1) are positive by construction for the ground states (remem- +ber, that Ψ(2) +1 +is the ground state for Hamiltonian ˆH2). +Therefore, Ψ′(1) +1 (x1) < 0, as it is pictured in Fig. 2, for any node x1. It means that in +all of nodes the derivative has to be negative. Taking into account the continuity of wave + +6 +function, we have to conclude that there is no more than one change of sign Ψ(1) +1 , and the +wave function of the first excitation Ψ(1) +1 +has got the single internal node, exactly. +Further steps of mathematical induction are quite transparent. +V. +CONCLUSION +In the present paper we have presented the new original modern proof for counting the +internal nodes of bound states wave functions instead of Sturm–Liouville theory by making +use the elegant mathematical formalism of Supersymmetric Quantum Mechanics. It could +be incorporated into textbooks on modern quantum mechanics in order to provide some +acquaintance with up-to-date supersymmetric concepts and mathematical methods essential +in nowadays science. +[1] F. A. Berezin, M. Shubin, The Schr¨odinger Equation. (Springer Science & Business Media, +2012). +[2] M. Moriconi, American Journal of Physics 75(3), 284 (2007) [arXiv:quant-ph/0702260]. +[3] M. M. Crum, Quart. J. Math. Oxford Ser. 2 6, 1216 (1955) [arXiv:physics/9908019]. +[4] E. Witten, Nucl. Phys. B 188, 513 (1981). +[5] C. V. Sukumar, Phys. A: Math. Gen. 18, 2917 (1985). +[6] L. E. Gendenshtein, I. V Krive, Soviet Physics Uspekhi. 28, 645 (1985). +[7] F. Cooper, F. Khare, U. Sukhatme, Physics Reports 25, 267 (1995) [arXiv:hep-th/9405029]. +[8] M. G. Darboux, Comptes Rendus Acad. Sci. Paris 94, 1456 (1882) [arXiv:physics/9908003]. + diff --git a/Z9FIT4oBgHgl3EQflCtt/content/tmp_files/load_file.txt b/Z9FIT4oBgHgl3EQflCtt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5e438fdfb834f25874d3f63a9797bfceffb6fd9a --- /dev/null +++ b/Z9FIT4oBgHgl3EQflCtt/content/tmp_files/load_file.txt @@ -0,0 +1,142 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf,len=141 +page_content='Proof to count bound state nodes in supersymmetric quantum mechanics A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content='Aynbund1, ∗ and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content='Kiselev1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' † 1Landau Phystech-School,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Moscow Institute of Physics and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Russia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' 141701,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Moscow Region,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Dolgoprudny,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Institutsky 9 2State Research Center of the Russian Federation “Institute for High Energy Physics” of National Research Centre “Kurchatov Institute”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Russia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' 142281,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Moscow Region,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Protvino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Nauki 1 A normalizable static supersymmetric bound ground state annihilated by the super-generators has got zero number of internal nodes in the framework of one- dimensional supersymmetric quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' The super-generator transforma- tions between excited super-partner bound states as combined with the standard technique of Wronskian provides an elegant and self-sufficient way to derive the equality of internal nodes amount to the number of consequent excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' INTRODUCTION Counting an amount of internal nodes in wave functions of bound states is known as an oscillation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' The theorem relates a sequence number of excited state in the order of energy enlarge to number of its nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' An ordinary proof, say, in classical book by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Berezin and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Shubin [1] is a set of theorems, lemmas, consequences and prepositions in Sturm– Liouville theory of differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Such the way is rigorous and it looks like just straightforwardly mathematical, while physical explanations merely give some qualitative arguments but not enough to be provements (see, for instance, [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' In occasion there is a method based on coherent quantum structure, that provides an elegant and physically clear way to derive the oscillation theorem by means of supersymmetry in one dimensional quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' This method is built on a construction introduced by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Crum [3] and ∗Electronic address: aynbund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content='asya@phystech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content='edu †Electronic address: Valery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content='Kiselev@ihep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content='ru;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' kiselev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content='vv@mipt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content='ru arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content='11303v1 [hep-th] 26 Jan 2023 2 incorporated to the supersymmetry by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Witten [4] as reviewed in [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' In Section II basic notions and constructions are described to give the most important properties of supersymmetric Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Section III deals with a ground state, and we derive that the ground state has got no internal nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Section IV is devoted to a step to the level next to the ground state to prove that the number of nodes increases to the unit exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' In Conclusion we summarize the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' MACHINERY OF SUPERSYMMETRIC QUANTUM MECHANICS The theory under study is one-dimensional stationary Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Suppose that Ψ0 is a ground state wave function normalised in a Hilbert space of quantum states for a Hamiltonian ˆH1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' As it is well known, function Ψ0 can be set to be real since it has got nodes at borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Let the Hamiltonian have got several excitations of the ground state, and the excited levels are counted in the order of energy enlargement as Ψ(1) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Following M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Crum [3] and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Darboux [8], one constructs a partner Hamiltonian ˆH2, that possesses a set of bound states wave functions Ψ(2) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Define two linear differential operators A− and A+, which are hermitian conjugated each to other: A− = − d dx + Ψ′ 0 Ψ0 , A+ = d dx + Ψ′ 0 Ψ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Introduce operators of super-generators Q and ¯Q ≡ Q† by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content='Witten [4] Q = ¯h √ 2m � � 0 0 A− 0 � � and ¯Q = ¯h √ 2m � �0 A+ 0 0 � � acting on � �Ψ(1) Ψ(2) � � where the column refers to wave functions of super-partner Hamiltonians ˆH1 and ˆH2, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' A supersymmetric Hamiltonian H is constructed as H = � Q, ¯Q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Consider properties of Hamiltonian H and supercharge operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' So, the super-generators are nilpotent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Q2 = ¯Q2 = 0 and they commute with Hamiltonian [Q, H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' 3 Then the definition of super-partner Hamiltonians in terms of matrix H reads off H = ¯h2 2m � �A+A− 0 0 A−A+ � � = � � ˆH1 − E0 0 0 ˆH2 − E0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Two Schr¨odinger equations under study are ˆ H1Ψ(1) k = EkΨ(1) k , ˆ H2Ψ(2) k = EkΨ(2) k , where the eigenvalues of Hamiltonians coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Indeed, according to the connection between ˆ H1 and operators A− and A+ the action of A+A− to the wave function Ψ(1) k gives ¯h2 2mA+A−Ψ(1) k = ( ˆ H1 − E0)Ψ(1) k = (Ek − E0)Ψ(1) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Acting by A− results in the eigen state equation A−A+(A−Ψ(1) k ) = (Ek − E0)(A−Ψ(1) k ) ⇒ ( ˆH2 − E0)(A−Ψ(1) k ) = (Ek − E0)(A−Ψ(1) k ), determining the wave function Ψ(2) k def = A−Ψ(1) k as the bound state solution for Hamiltonian ˆH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Note that the superscripts of wave functions point to means the super-partner Hamil- tonians, while the subscript denotes the number of excitation with the same energy Ek for both Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' GROUND STATE IN THE DISCRETE SPECTRUM The ground state wave function Ψ0 satisfies the equation of ground state: A−Ψ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Hence, any other function satisfying the same equation is a normalised solution for the ground state and it is proportional to Ψ0, since the bound state energies are not degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Let us show that A−|Ψ0| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Therefore, Ψ0 = const · |Ψ0|, and generically one can put the constant to be positive, while Ψ0 > 0 at any internal point except borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' It means that Ψ0 is nodeless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' ˜Ψ def = |Ψ0| = Ψ0 sign(Ψ0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' 4 while the action by A− straightforwardly gives A− ˜Ψ = � − d dx + Ψ′ 0 Ψ0 � Ψ0 sign(Ψ0) = − d dx � Ψ0 sign(Ψ0) � + Ψ′ 0 sign(Ψ0) = = ((((((( ( − sign(Ψ0)Ψ′ 0 − Ψ0 d dx � sign(Ψ0) � + \x18\x18\x18\x18\x18\x18 \x18 Ψ′ 0 sign(Ψ0) = −Ψ0 d dx � sign(Ψ0) � = = − � j Ψ0(xj) 2 δ(xj) sign � Ψ′ 0(xj) � ≡ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' (1) where xj are positions of possible nodes of Ψ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' if exist,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' while Ψ′ 0(xj) ̸= 0 because Ψ′(xj) = Ψ(xj) = 0 would lead to Ψ(x) ≡ 0 as a solution of Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Thus, the super-generators provide us with the short proof of statement that the ground state wave function Ψ0 is nodeless, so we consider it to be positive anywhere except border points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' COUNTING THE NODES As we plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' 1, there are two sets of states for the super-partner Hamiltonians with energy E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' We show that one can transform one column of states to another by making use the linear operator A− for moving from left to right (for example, if one acts on Ψ0, we will get 0) and the operator A+ for moving back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' These operators conserve the energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' nodeless ground ground nodeless A− ↔ Q A− ↔ Q A+ ↔ ¯Q E1 E0 Ψ(1) 1 Ψ0 Ψ(2) 1 0 ˆ H1 ˆ H2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' 1: The ground state and the exited level of Hamiltonian ˆH1 (left), the ground state of Hamiltonian ˆH2 (right) with energies E0 and E1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' 5 Ψ(1) 1 (x) x x1 x′ 1 x′′ 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' 2: Node of the exited state and the direction of the derivative in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' The figure points that the ground states for both super-partner Hamiltonians are node- less, while more energy means that the wave function has more nodes as well known fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' So, with this explicit scheme for the super-partner system we are going to prove that the first excitation of ground state for ˆH1 has got exactly single internal node, while further excitations have got numbers of nodes equal to its excitation numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' In previous section we proved so-called base case of mathematical induction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' The next point is to prove the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Acting on the first excited wave function of Hamiltonian ˆH2 and by revealing the operator by definition we get A+Ψ(2) 1 = Ψ ′(2) 1 + Ψ′ 0 Ψ0 Ψ(2) 1 = � Ψ0Ψ(2) 1 �′ Ψ0 = ⟨ or ⟩ = A+(A−Ψ(1) 1 ) = (E1 − E0)Ψ(1) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Then, multiplying by Ψ0 one can see the derivative of Wronskian for Ψ0 and Ψ(1) 1 : � Ψ0Ψ(2) 1 �′ = (E1 − E0)Ψ(1) 1 Ψ0 = −W ′ Ψ0Ψ(1) 1 ⇒ −WΨ0Ψ(1) 1 = Ψ0Ψ(2) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Therefore, we come to the following expression: Ψ′ 0Ψ(1) 1 − Ψ0Ψ′(1) 1 = Ψ0Ψ(2) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Consider the first term on the left side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' We can see that as E1 is grater than E0 there are strictly more nodes for the function of excited level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Hence, there are at least one node which we will denote x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Rewriting the previous equation at the node we get Ψ0(x1)Ψ(2) 1 (x1) = −Ψ0(x1)Ψ′(1) 1 (x1), where all of Ψ0(x1) and Ψ(2) 1 (x1) are positive by construction for the ground states (remem- ber, that Ψ(2) 1 is the ground state for Hamiltonian ˆH2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Therefore, Ψ′(1) 1 (x1) < 0, as it is pictured in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' 2, for any node x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' It means that in all of nodes the derivative has to be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Taking into account the continuity of wave 6 function, we have to conclude that there is no more than one change of sign Ψ(1) 1 , and the wave function of the first excitation Ψ(1) 1 has got the single internal node, exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Further steps of mathematical induction are quite transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' CONCLUSION In the present paper we have presented the new original modern proof for counting the internal nodes of bound states wave functions instead of Sturm–Liouville theory by making use the elegant mathematical formalism of Supersymmetric Quantum Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' It could be incorporated into textbooks on modern quantum mechanics in order to provide some acquaintance with up-to-date supersymmetric concepts and mathematical methods essential in nowadays science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Berezin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Shubin, The Schr¨odinger Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' (Springer Science & Business Media, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' Moriconi, American Journal of Physics 75(3), 284 (2007) [arXiv:quant-ph/0702260].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FIT4oBgHgl3EQflCtt/content/2301.11303v1.pdf'} +page_content=' [3] M.' metadata={'source': 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b/ZtE5T4oBgHgl3EQfDQ6F/content/tmp_files/2301.05404v1.pdf.txt @@ -0,0 +1,1480 @@ +arXiv:2301.05404v1 [physics.optics] 13 Jan 2023 +Tunable virtual gain in resonantly absorbing media +Denis V. Novitsky∗ +B. I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Nezavisimosti Avenue 68, 220072 Minsk, Belarus +(Dated: January 16, 2023) +Virtual gain refers to the simulation of real light amplification using radiation with exponentially decaying +amplitude, so that its complex frequency corresponds to the scattering pole. We theoretically study virtual +gain in a two-level resonant medium for different regimes of light-matter interaction depending on the radiation +intensity. We show that virtual gain at the pole can be most clearly observed for low intensities, when the +medium is absorbing, in contrast to the saturated medium at high intensities. The efficiency of virtual gain +can be tuned with the light intensity and can be controlled dynamically through the population inversion of the +medium. Our results show that resonantly absorbing media paradoxically mimics gain-like response, which +admit a number of related phenomena and methods to mold both optical signals and material properties without +relying on instability-prone gain media. +I. +INTRODUCTION +Recent years have evidenced the rise of nanophotonics +studying the methods for controlling light with small-scale +(nanostructured) objects [1]. This approach has breathed new +life into such classic subjects of optics as scattering. In par- +ticular, subtle tuning of system parameters has made possible +to observe different anomalies in light scattering [2]. These +anomalies can be often associated with scattering poles and +zeros, which can be conveniently represented on the complex- +frequency plane. Generally, the poles and zeroes appear at +the complex frequencies, but their position can be regulated +by introducing gain or loss, thus, making the system non- +Hermitian. When a pole or zero goes to the real-frequency +axis, one observes lasing or antilasing (coherent perfect ab- +sorption, or CPA [3]), respectively. Specific non-Hermitian +systems obeying parity-time (PT ) symmetry demonstrate +both lasing and antilasing simultaneously [4, 5]. This is the +so-called CPA-lasing effect obtained under the coalescence of +a pole and a zero at the real-frequency axis. Analogous coa- +lescence in a Hermitian system corresponds to a bound state +in the continuum (BIC) – nonradiating mode having infinite +quality factor despite the system openness [6]. Sometimes, +one can transform the BIC into CPA-lasing point by using +geometric or non-Hermitian perturbations [7]. Among other +scattering anomalies, we mention exceptional points (non- +Hermitian degeneracies) [8], anapoles [9], and superscattering +[10, 11]. +Recently, dynamic effects have come under the spotlight +strongly widening the range of optical systems and applica- +tions under study. In particular, materials with time-varying +parameters (such as refractive index) attract much attention +[12] as a counterpart to the familiar space-varying systems +such as interfaces [13, 14], layers [15], photonic crystals [16], +metamaterials [17], and metasurfaces [18]. +A number of +temporal analogs of usual optical phenomena were reported, +including Brewster angle [19], Anderson localization [20], +topological protection [21], PT symmetry [22], negative +refraction [23], etc. What is most important for our discus- +∗ dvnovitsky@gmail.com +sion, variation of media in time serves as a source of energy +flowing into the system and allowing to circumvent the law +of energy conservation. As a result, frequency and momen- +tum switch their roles making possible such effects as the fre- +quency conversion under time modulation of materials [24]. +In the temporal analogs of photonic crystals, one can observe +the momentum bandgaps inside which one can observe ampli- +fication of waves [25–28]. However, practical possibilities of +time-varying photonics are strongly limited by the difficulties +with fast and spatially uniform modulations of materials and +by interference with common nonlinear effects [29]. +Another possibility is to harness time variation not for the +medium, but for light signal itself. As a result, not medium, +but radiation should be described with the complex frequency +making it “non-Hermitian”. This possibility was implemented +in the idea of virtual loss and gain, which attracted a lot of at- +tention recently. Virtual loss (or virtual perfect absorption, +VPA) can be considered as an imitation of CPA by taking ra- +diation with exponentially growing amplitude. If the rate of +exponential growth (i.e., the imaginary part of complex fre- +quency of radiation) is equal to the imaginary part of scatter- +ing zero frequency, radiation seems to be absorbed [30]. In +fact, it is stored inside the medium and released after the ex- +ponential signal is switched off. Theoretical idea of virtual +loss was initially demonstrated with the examples of dielec- +tric slab and cylinder [30]. Subsequently, it was expanded to +the discrete array of resonators or waveguides [31], ring mi- +crocavity coupled to a waveguide [32], and open metasurface- +based cavity [33]. The effect was experimentally proved with +elastic waves [34]. Virtual gain is a counterpart of virtual loss +engaging a scattering pole under exponentially decaying irra- +diation [35]. As further generalizations, we mention the vir- +tual PT symmetry with balanced loss and gain [35] and the +certain radiation waveforms absorbed at the exceptional points +[36]. The virtual processes based on manipulations with the +time-varying light signals are finding such applications as crit- +ical coupling for transformation of electromagnetic energy +[37], virtual pulling forces for particles transportation [38], +enhancement of light scattering [39, 40], and topological light +localization via non-Hermitian skin effect [41]. It should be +emphasized that the total energy of radiation does not grow +under virtual gain, so that there is no risk of instability usual +for systems with light amplification (such as PT -symmetric + +2 +e +-t/� +x +y +z +absorbing medium +L +Figure 1. Schematic representation of the situation studied: An expo- +nentially decaying radiation is impinging on the layer of resonantly +absorbing medium. The color gradient inside the medium shows the +growing intensity of light corresponding to virtual gain. +ones). +In this paper, we apply the idea of virtual gain to the res- +onantly absorbing media. The two-level model of resonant +quantum media have attracted much attention since the be- +ginning of laser era [42]. Pulsed-field dynamics in such me- +dia was studied in much detail on the basis of Maxwell- +Bloch equations [43, 44]. In particular, coherent pulses (much +shorter than relaxation times) evidently violate Beer’s law +propagating almost without attenuation due to such phenom- +ena as self-induced transparency [45, 46] and zero-area pulse +formation [47]. The further advances include studies of the +ever more shorter – few-cycle [48–50], subcycle [51], and +unipolar [52] – pulses as well as interactions between pulses in +resonant media [53, 54] resulting in such phenomena as pop- +ulation gratings [55] and diode-like effect [56], among oth- +ers. On the other hand, the incoherent waveforms with longer +characteristic timescales such as incoherent solitons [57] and +optical kinks [58, 59] can also withstand absorption in reso- +nant media. We also mention the studies of ultrashort pulses +[60] and wavefront propagation [61] in disordered resonantly +absorbing and amplifying media. The role of driving-field de- +cay rate in superradiance and subradiance dynamics was re- +vealed recently [62]. However, the virtual-gain dynamics of +quasi-continuous patterns without evident propagation effects +connected with pulses or wavefronts have not been studied +earlier. +Using +numerical +simulations +of +quasi-continuous +monochromatic +and +then +exponentially-decaying +radia- +tion in a two-level medium, we analyze the different regimes +depending on light intensity. +In the low-intensity regime, +we have the classic virtual gain, when the signal attenuating +inside the medium is replaced by the gradually growing +waveform. +In the high-intensity regime, the medium gets +saturated with the step-like patterns seen in the system’s +response. Moreover, one can switch between these different +types of response by simply choosing the time instant to start +the signal decay. Our results clearly point to the connection +of the virtual gain effect with the position of pole in the +complex-frequency plane and can be used for controlling +both the radiation patterns and quantum-medium state with +time-varying light signals. +II. +RESONANT MEDIUM DESCRIPTION +We consider the two-level resonant medium of thickness L +illuminated by the normally incident radiation of wavelength +λ (Fig. 1). Interaction of light with the medium is described +with the well-known Bloch equations [49], +dρ12 +dt += iω0ρ12 + iµ +¯h Ew− ρ12 +T2 +, +(1) +dw +dt = −4 µ +¯h EImρ12 − w+ 1 +T1 +, +(2) +where E is the electric field, ρ12 is the off-diagonal density +matrix element (atomic polarization), w = ρ22 − ρ11 is the +population inversion (difference between populations of the +excited and ground levels), ω0 is the resonance frequency, µ +is the dipole moment corresponding to the transition between +levels, T1 and T2 are the relaxation times, c is the speed of +light, and ¯h is the Planck constant. These equations should be +completed with the wave equation accounting for light propa- +gation, +∂ 2E +∂z2 − n2 +c2 +∂ 2E +∂t2 = 4π +c2 +∂ 2P +∂t2 , +(3) +where P = 2µCRe(ρ12) is the macroscopic polarization, n is +the background refractive index, and C is the concentration +of two-level atoms. Note that these equations do not employ +the popular rotating-wave approximation (RWA) and slowly- +varying-envelope approximation (SVEA). The reason is that +the fast change of light field can break these approximations +[48–51]. We address the applicability of RWA in Appendix +A with the conclusion of qualitative agreement between the +general and RWA cases. The Maxwell-Bloch equations (1)- +(3) are numerically solved using the approach described ear- +lier [63]. Further, we adopt the parameters of the medium as +follows: the thickness L = λ = 1 µm; the background refrac- +tive index n = 3; the relaxation rates T1 = 1 ns and T2 = 1 ps; +the density parameter governing the strength of light-matter +interaction g = 4πµ2C/3¯h = 3.7 · 1011 s−1; these values can +be reached, e.g., for quantum-dot media. Further, we also as- +sume that radiation is tuned exactly to the resonance with the +quantum transition, i.e., ω = 2πc/λ = ω0 with λ = 1 µm. +Transmission of light through the resonant medium +strongly depends on its intensity, which can be conveniently +characterized with the maximal Rabi frequency, Ωmax = +µA/¯h, where A is the wave amplitude. Figure 2(a) shows +transmission dynamics for the continuous wave instanta- +neously switched on at t = 0. +One can see that the +low-intensity radiation (ΩmaxT2 ≪ 1) is weakly transmitted +through the layer of thickness L = λ = 1 µm and is mostly ab- +sorbed inside the medium. On the contrast, the high-intensity +radiation (ΩmaxT2 ∼ 1) is almost perfectly transmitted as it + +3 +0 +5 +10 +15 +20 +0.0 +0.5 +1.0 +1.5 +max +=0.1/T +2 +max +=1/T +2 +max +=3/T +2 +(a) +Normalized intensity +t (ps) +0 +5 +10 +15 +20 +-0.8 +-0.4 +0.0 +0.4 +0.8 +max +=0.1/T +2 +max +=1/T +2 +max +=3/T +2 +Population inversion +t (ps) +(b) +Figure 2. (a) Transmitted light intensity and (b) inversion dynamics +at the entrance of the medium for different values of Rabi frequency. +Parameters: T1 = 1 ns, T2 = 1 ps, ωL = 3.7 · 1011 s−1, L = λ = 1 +µm, n = 3. +would be for the passive medium without absorbing particles +at all. These two situations correspond to the steady-state in- +versions [Fig. 2(b)] equal to wst ≈ −1 (most two-level emit- +ters remain in the ground state) and wst ≈ 0 (the equal num- +ber of emitters in the ground and excited states) and, there- +fore, will be referred to as “absorbing medium” and “saturated +medium”, respectively. Moreover, for high enough intensity +(ΩmaxT2 > 1), one can clearly observe the oscillations of trans- +mission and the corresponding Rabi oscillations of population +inversion. The inversion is given in Fig. 2(b) at the entrance of +the medium, but it has essentially the same behavior at other +distances due to small thickness of the layer. As an estimate, +the condition ΩmaxT2 = 1 requires the electric-field strength +to be E = ¯h/µT2, which for T2 = 1 ps and µ = 30 D (semi- +conductor quantum dots) gives E = 106 V/m corresponding to +the intensity ∼ 5 GW/m2. This value can be further decreased +for larger dipole moments and slower relaxation. +III. +VIRTUAL GAIN CONDITION +In order to study the possibility of virtual gain, we consider +the incident wave having constant intensity from t = 0 to tmax +and exponentially decaying after that, +E ∼ A +� +θ(tmax −t)+ e−(t−tmax)/τθ(t −tmax) +� +, +(4) +where θ(t) is the Heaviside step function, τ is the decay time. +The parameters of the system are chosen close to the pole, po- +sition of which in the complex-frequency plane can be evalu- +ated analytically for a layer of thickness L as follows +ωp +c L = 1 +n +� +πl + ilnn − 1 +n + 1 +� +, +(5) +where l is the integer number. The frequency of the wave +(4) is tuned to coincide with the real part of pole frequency, +ω = ω′ +p = 2πc/λp. The decay time is, in turn, limited by the +imaginary part of pole frequency, τ ≲ τp = 1/ω′′ +p. One can +see that for real n, λp can be kept constant by simply changing +simultaneously the thickness L and the pole number l; further, +we take λp = 1 µm for L = 1 µm, l = 6, and n = 3. On +the contrary, ω′′ +p can be varied by changing L, so that for the +above parameters we have τp ≈ 0.014 ps. This rather short +time can be problematic to realize in experiment. However, it +can be increased by taking thicker layer as shown further. As +a proof of principle, we consider the thin layer, because it is +easier and faster to calculate without significant influence of +radiation inhomogeneity which can smear the effect in thick +layers. +To give a simple explanation of the virtual gain in resonant +media, it is convenient to use the simplified Bloch equations +under RWA by adopting s = 0 in Eqs. (A2) and (A3). We +also assume the exact resonance, so that ω = ω0. First, we +consider the stationary limit, when the radiation amplitude Ω +is constant, whereas the polarization amplitude and popula- +tion inversion have reached the steady state, dp/dt = 0 and +dw/dt = 0. The steady-state values are as follows, +pst = i +2ΩT2wst, +(6) +wst = − +1 +1 + |Ω|2T1T2 +. +(7) +One can readily see that for low incident intensities (|Ω|2 ≪ +(T1T2)−1), the medium remains almost entirely unexcited, +wst ≈ −1. +On the contrary, for high intensity (|Ω|2 ≫ +(T1T2)−1), it is saturated, wst ≈ 0. +If the radiation is switched off after the medium has reached +steady state and decays exponentially (Ω = Ω0e−t/τ), we can +assume that the polarization follows the decay of radiation +as p = pste−t/τ. The population inversion, on the contrary, +undergoes rapid decay due to the first term (∼ e−2t/τ) in the +right-hand side of Eq. (A3) and then changes slowly. So, +we can assume that the inversion takes on a constant effective +value, which can be found after substituting the exponential +expressions for Ω and p in Eq. (A2). Thus, we have +wef f = +� +1 − T2 +τ +� +wst. +(8) + +4 +19.5 +20.0 +20.5 +0.0 +0.5 +1.0 +=0.1 ps +(b) +transmitted +reflected +incident +Normalized intensity +t (ps) +19.6 +19.8 +20.0 +20.2 +20.4 +0.0 +0.5 +1.0 +transmitted +reflected +incident +(a) +=0.0001 ps +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.1 +0.2 +0.3 +(d) +t=20 ps +t=20.01 ps +=0.1 ps +Normalized intensity +z / L +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.1 +0.2 +0.3 +(c) +t=20.01 ps +=0.0001 ps +t=20 ps +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +R +t (ps) + +=0.1 ps + +=0.01 ps + +=0.001 ps + +=0.0001 ps +(e) +10 +-5 +10 +-4 +10 +-3 +10 +-2 +10 +-1 +0.4 +0.8 +1.2 +1.6 +2.0 +(f) +=0 + t=20 ps + t=20.01 ps +R + (ps) +=0 +virtual +gain +p +Figure 3. Virtual gain in the absorbing medium under low-intensity radiation, Ωmax = 0.01/T2. (a) and (b) Light intensity profiles for different +values of decay time τ. (c) and (d) Radiation distributions at the switching-off moment (tmax = 20 ps) and a short time later (tmax +∆t = 20.01 +ps). (e) The ratio R as a function of ∆t for different τ. (f) The ratio R at different time instants as a function of the decay time τ. +Obviously, wef f will have the sign opposite to the steady- +state value wst, if the decay time is short enough, τ < T2. +This effect is especially pronounced in the absorbing medium +regime, when wst ≈ −1 and wef f > 0, i.e., the medium acts +effectively as if it would have gain. In the saturated medium +regime, when wst ≈ 0, this effect should be negligible. Al- +though this simplified explanation allows us to illustrate the +possibility of virtual gain in the resonantly absorbing media, +it cannot take into account the quantitative features due to the +radiation propagation in the medium of finite length. +Thus, we have several relations between the parameters of +medium and radiation, which should be justified to observe +virtual gain. Equation (5) governs the position of pole and the +corresponding limiting decay time τp as a function of thick- +ness L and refractive index n. Equation (8) requires τ < T2, so +that one can expect virtual gain for τ ≲ τp < T2 in the absorb- +ing medium regime governed by the condition ΩmaxT2 ≪ 1. +One can easily assess the possible material parameters from +these relations. Further, we justify this general reasoning with +full numerical simulations of the Maxwell-Bloch equations in +both the absorbing medium and saturated medium regimes. +IV. +ABSORBING MEDIUM REGIME +We start with the regime of absorbing medium, when ra- +diation is weak enough to saturate the medium. +Figure 3 +shows the results of calculations for Ωmax = 0.01/T2 and the +incident wave given by Eq. (4) with the switching-off mo- +ment tmax = 20 ps, i.e., the signal decays after the steady- +state transmission and reflection are established. +Behavior +of radiation strongly depends on the decay time τ: for slow +decay with τ = 0.1 ps, transmitted and reflected intensities +smoothly follow the profile of incident radiation [Fig. 3(b)], +whereas for abruptly switched-off radiation with τ = 10−4 ps, +transmission and reflection demonstrate sharp splash in in- +tensity [Fig. 3(a)]. The corresponding distributions of radi- +ation along the medium just after the switching-off moment +(at t = tmax + ∆t = 20.01 ps) are fundamentally different as +compared in Figs. 3(c) and 3(d). For τ = 0.1 ps, the pattern + +5 +of attenuating intensity characteristic for absorbing medium +is kept after switching off the radiation. On the contrary, for +τ = 10−4 ps, the reversal of the pattern is seen with attenuation +replaced by gradual growth of intensity despite the absorption +in the material. This is exactly what we mean by the virtual +gain in accordance with Ref. [35]. +We should emphasize that the virtual gain does not change +the fact that the total amount of energy decreases with time +as one would expect for a passive (absorbing) system [41]. +Moreover, the virtual gain is a transient phenomenon, i.e., +the reversed pattern of intensity growing along the medium +length exists only for a finite interval of time after radiation +was switched off. Therefore, we need an integral value to il- +lustrate the efficiency of virtual gain depending on decay time +τ and how it changes with time. To this end, we calculate the +ratio of average intensities in the second and the first halves of +the layer, +R(t) = +� L +L/2 I(t,z)dz/ +� L/2 +0 +I(t,z)dz. +(9) +Obviously, virtual gain corresponds to R > 1, although this +condition is not sufficient to make unequivocal conclusion on +the regime of light propagation as will be clear further. Figure +3(e) shows how the ratio R changes as a function of instant +after the switching-off moment ∆t for different values of τ. +It is seen that for large τ (slow decay), R stays lower than +unity. Only for short enough τ, the range of ∆t appears where +R(t) > 1 and virtual gain is possible. The transition to virtual +gain agrees well with the condition of the pole position, τ ≪ +τp. Note that the effect is observed very soon after tmax and R +rapidly (up to 0.1 ps) comes to unity. The reasons are the fast +redistribution of radiation leveling intensity along the medium +and rapid exit of radiation from the medium. This uniformly +distributed intensity grows in time as long as transmitted and +reflected intensities increase according to Fig. 3(a). The ratio +loses its meaning at very large ∆t, because almost no radiation +remains stored in the medium. +In Fig. 3(f), we show the dependence of R on the decay time +τ at the specific instant of time, tmax +∆t = 20.01 ps. One can +see that the ratio becomes larger than unity, when τ is much +less than τp. For τ = 10−4 ps, it almost reaches the value ob- +tained for instantaneous switching-off (τ = 0). Compare with +the value of R at tmax = 20 ps which is the same for every τ. +Note that the value of τp is somewhat overestimated, because +it is obtained for the real refractive index n, i.e., not taking into +account medium absorption. Nevertheless, this simplified ap- +proach is good enough to give us the qualitative estimate of +virtual gain possibility at short decay times. +As mentioned above, the value of τp grows with the +medium thickness L, so that virtual gain can be reached at +much higher decay times for a thick layer and may be easier +observed in experiment. To give an example, we increase the +thickness by 10 times, up to L = 10λ = 10 µm. As shown +in Fig. 4(a), for fast enough decay (τ = 10−3 ps), the in- +tensity distribution soon after switching-off moment clearly +demonstrates the transition from absorption to virtual gain. +No such change is observed for slow decay [τ = 1 ps, Fig. +4(b)]. Corresponding behavior of the ratio R [Figs. 4(c) and +0 +2 +4 +6 +8 +10 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +(b) +t=20 ps +t=20.1 ps +=1 ps +Normalized intensity +z / +0 +2 +4 +6 +8 +10 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +(a) +t=20.1 ps +=0.001 ps +t=20 ps +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0 +1 +2 +3 +4 +R +t (ps) + +=1 ps + +=0.1 ps + +=0.01 ps + +=0.001 ps +(c) +10 +-3 +10 +-2 +10 +-1 +10 +0 +0.0 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +(d) +=0 + t=20 ps + t=20.1 ps +R + (ps) +=0 +virtual +gain +p +Figure 4. Virtual gain in the thick layer (L = 10λ = 10 µm) of ab- +sorbing medium under low-intensity radiation, Ωmax = 0.01/T2. (a) +and (b) Radiation distributions at the switching-off moment (tmax = +20 ps) and a short time later (tmax +∆t = 20.1 ps). (c) The ratio R as a +function of ∆t for different decay times τ. (d) The ratio R at different +time instants as a function of τ. +4(d)] also corroborates the validity of observations made for + +6 +19.5 +20.0 +20.5 +0.0 +0.5 +1.0 +=0.1 ps +(b) +transmitted +reflected +incident +Normalized intensity +t (ps) +19.6 +19.8 +20.0 +20.2 +20.4 +0.0 +0.5 +1.0 +transmitted +reflected +incident +(a) +=0.0001 ps +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.5 +1.0 +(d) +t=20 ps +t=20.006 ps +=0.1 ps +Normalized intensity +z / L +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.5 +1.0 +(c) +t=20.006 ps +=0.0001 ps +t=20 ps +Figure 5. The case of saturated medium under high-intensity ra- +diation, Ωmax = 3/T2. (a) and (b) Light intensity profiles for dif- +ferent values of decay time. +(c) and (d) Radiation distributions +at the switching-off moment (tmax = 20 ps) and a short time later +(tmax +∆t = 20.006 ps). +the thin layer. All the time values such as ∆t corresponding +to the R peak in Fig. 4(c) increase by an order of magnitude. +The same can be said about the decay time, so that τ = 0.01 +ps ≪ τp = 0.14 ps is enough for virtual gain. One can ex- +pect that further increase of the layer thickness can relax the +requirements on the decay rapidity even stronger. +V. +SATURATED MEDIUM REGIME +Let us now consider the case of radiation powerful enough +to saturate the medium. In particular, we take the amplitude +Ωmax = 3/T2. Radiation is switched off at tmax = 20 ps, when +the steady state is already reached, i.e., the population inver- +sion is close to zero and transmission is close to unity (see Fig. +2). The key difference in this case is the absence of absorption +in the medium. As a result, light at the switching-off moment +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(b) +t=20 ps +t=20.006 ps +=0.1 ps +Normalized intensity +z / L +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(a) +t=20.006 ps +=0.0001 ps +t=20 ps +0.00 +0.02 +0.04 +0.06 +0.08 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +R +t (ps) + +=0.1 ps + +=0.01 ps + +=0.001 ps + +=0.0001 ps +(c) +Figure 6. The case of passive medium under low-intensity radiation, +Ωmax = 0.01/T2. (a) and (b) Radiation distributions at the switching- +off moment (tmax = 20 ps) and a short time later (tmax +∆t = 20.006 +ps). (c) The ratio R as a function of ∆t for different τ. +is evenly distributed along the layer, without any attenuation +(Fig. 5). If radiation is decaying slowly, this uniform distribu- +tion is saved [Fig. 5(d)] as well as unity transmission, i.e., ra- +diation gradually leaves the medium only in the forward direc- +tion. If radiation is decaying rapidly, the distribution takes the +form of a sharp step in intensity moving along the medium in +the forward direction [Fig. 5(c)]. The sharp steps are observed +also in the transmitted and reflected intensity profiles as shown +in Fig. 5(a). The profiles for transmitted radiation generally +follow the incident intensity. The difference between slow and +fast decays is that in the latter case, the profile forms a step due +to very sharp disappearance of impinging radiation [Fig. 5(a)]. +The corresponding step is observed also in reflection, unlike +almost total absence of reflected signal in the case of slow de- +cay [Fig. 5(b)]. The stepped distribution makes a fundamental +difference between the saturated medium regime and the case +of absorbing medium with its monotonously growing inten- +sity [compare Figs. 5(c) and 3(c)] and allows us to conclude +that the saturated medium does not support virtual gain. + +7 +0.0 +0.5 +1.0 +1.5 +2.0 +-1.0 +-0.5 +0.0 +0.5 +t +max +=0.8 ps +t +max +=0.6 ps +t +max +=0.4 ps +Inversion +t (ps) +t +max +=0.2 ps +(a) +0.00 +0.01 +0.02 +0.03 +0 +5 +10 +15 +R +t (ps) + t +max +=0.1 ps + t +max +=0.4 ps + t +max +=0.8 ps +(b) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +t=0.834 ps +(d) +t=0.8 ps +t=0.806 ps +t +max +=0.8 ps +Normalized intensity +z / +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.1 +0.2 +0.3 +0.4 +t=0.434 ps +(c) +t=0.408 ps +t +max +=0.4 ps +t=0.4 ps +0.6 +0.8 +1.0 +1.2 +0.0 +0.5 +1.0 +t +max +=0.8 ps +(f) +transmitted +reflected +incident +Normalized intensity +t (ps) +0.2 +0.4 +0.6 +0.8 +0.0 +0.5 +1.0 +transmitted +reflected +incident +(e) +t +max +=0.4 ps +-1.0 +-0.5 +0.0 +0.5 +0 +2 +4 +6 +8 +(g) +t=t +max +t=t +max ++ +t +virtual +gain +R +Inversion at t +max +Figure 7. Tunable virtual gain under high-intensity radiation, Ωmax = 3/T2. (a) Dynamics of population inversion for different switching-off +moments tmax. (b) The ratio R as a function of ∆t for different tmax. (c) and (d) Radiation distributions at the switching-off moment tmax and a +short time later (at t = tmax +∆t marked in the plot). (e) and (f) Light intensity profiles for different switching-off moments tmax. (g) The ratio +R as a function of inversion at tmax calculated at the instant t = 1.02tmax. The decay time is τ = 10−4 ps. +In fact, the saturated medium regime is equivalent to the +case of purely passive, lossless medium, which contains no +absorbing particles as illustrated by the characteristic step dis- +tribution [Fig. 6(a), compare with Fig. 5(c)]. In Fig. 6(c), +we plot the ratio R at different time instants showing the sharp +peak soon after the switching-off moment. This peak is clearly + +8 +associated with the step in the intensity profile [Fig. 5(a)] and +is due to radiation rapidly leaving the medium. The strong +fluctuations at ∆t > 0.2 ps illustrate the meaninglessness of R +at later times because there is almost no radiation inside the +layer. Note also that R remains around unity for slow decay +(τ = 0.1 ps) that corresponds to the even distribution of radi- +ation along the medium [Fig. 6(b)]. In the case of saturated +medium, the ratio demonstrates the same behavior as in Fig. +6(c). +In addition, in Appendix B, we discuss the system, which +does not satisfy the condition for pole (5), and show that it can +be considered as an intermediary case having features of both +absorbing and saturated medium regimes. +VI. +NONSTATIONARY REGIME +In the above consideration, we have assumed that radiation +is switched off only after the steady state of both transmission +and population inversion is established. Here, we turn to the +nonstationary case, when decay starts before the steady state +is reached. This opens a room for controlling the virtual re- +sponse by choosing the proper value of the switching-off mo- +ment tmax. To demonstrate this possibility, it is instructive to +take the high-intensity incident wave able to strongly change +the medium state in a short period of time. We concentrate +at the rising slope of the first Rabi oscillation seen Fig. 2. +In Fig. 7(a), we show the population-inversion dynamics for +Ωmax = 3/T2 and different tmax, but for the same decay time +τ = 10−4 ps. Changing tmax in the range of just 1 ps is enough +to get different values of inversion w(tmax) at the switching-off +moment and, as a result, different behavior of radiation inside +the medium. +The dynamics of ratio R as a function of time after +switching-off radiation demonstrate the sharp peaks with the +height depending on tmax [Fig. 7(b)]. It is maximal for tmax +corresponding to the population inversion close to zero and +diminishes for large absolute values of w(tmax). However, the +behavior hiding behind these peaks seems to be quite different +as evidenced by the comparison of radiation distributions for +tmax = 0.4 ps [Fig. 7(c)] and tmax = 0.8 ps [Fig. 7(d)], respec- +tively. In the first case, the nonperiodic (because of nonsta- +tionarity) distribution at t = tmax = 0.4 ps is replaced by grow- +ing intensity characteristic for virtual gain at t = 0.408 ps. In +the second case, the initial distribution at t = tmax = 0.8 ps +is more periodic (the system seems to be closer to the steady +state), whereas the distribution at t = 0.806 ps (maximal R) +is step-like as in the saturated-medium regime. At later times +[∆t = 0.034 ps in Figs. 7(c) and 7(d)], the distributions are +more uniform and lower in intensity. The corresponding in- +tensity profiles are shown in Figs. 7(e) and 7(f) for different +switching-off moments tmax at the same decay time τ = 10−4 +ps. For tmax = 0.4 ps, the population inversion is negative [see +Fig. 7(a)], so that transmission is decreasing as expected for +absorbing medium. For tmax = 0.8 ps, the population inver- +sion gets positive and leads to increasing transmission. As a +result, in the latter case, the switching-off moment is followed +by the clear step-like feature and subsequent very slow inten- +sity decay. These peculiarities of intensity profiles are linked +to the differences between the distributions shown in Fig. 7(c) +and 7(d). +The dependence of R calculated at the instant t = 1.02tmax +on the population inversion at the switching-off moment tmax +[Fig. 7(g)] clearly demonstrates that virtual gain is most effec- +tive for w(tmax) ≲ 0. For positive w(tmax), the ratio R rapidly +decreases and the response changes its character from virtual +gain to saturated-medium-like regime as discussed above. +Thus, for w(tmax) < 0, the medium can be considered as +absorbing and we obtain virtual gain. On the contrary, for +w(tmax) > 0, the situation is analogous to the case of saturated +medium and the peak in R is not connected to the virtual gain, +but is due to the step-like distribution. Regulating the value of +tmax allows one to control the regime of light-matter interac- +tion and switch virtual gain on and off. This idea can be used +as the tunability scheme allowing to obtain response corre- +sponding to different effective gains controlled simply by the +switching-off moment tmax of decaying radiation. +VII. +CONCLUSION +Our results corroborate the possibility to harness scattering +anomalies (such as poles) to obtain unusual response under +optical irradiation strongly varying in time. We demonstrate +that resonantly absorbing media illuminated with an exponen- +tially decaying light can be used to realize virtual gain with +the efficiency tuned dynamically using intensity-driven pop- +ulation inversion. Simultaneously, one can use this scheme +to control the medium state changing its variables (population +inversion and polarization) in a desired way. There is a certain +paradox that virtual gain is most pronounced in the presence +of absorption and not in the passive or saturated medium. It +would be intriguing to further generalize our approach to the +initially excited media or combination of several layers of dif- +ferent resonant media. +ACKNOWLEDGMENTS +The work has been supported by the Belarusian Repub- +lican Foundation for Fundamental Research (Project No. +F22TURC-001). The author is grateful to Viktoryia Kouhar +for help with preparation of the graphical materials. +Appendix A: Details on the Maxwell-Bloch equations and +applicability of the RWA +The applicability of the rotating-wave approximation +(RWA) is convenient to study with the approach described in +Ref. [51]. According to this approach, the Maxwell-Bloch +equations can be represented in the dimensionless form as fol- + +9 +19.6 +19.8 +20.0 +20.2 +20.4 +0.0 +0.5 +1.0 +g=3.7x10 +11 + s +-1 +(b) +transmitted +reflected +incident +Normalized intensity +t (ps) +19.6 +19.8 +20.0 +20.2 +20.4 +0.0 +0.5 +1.0 + + general + + RW A +transmitted +reflected +incident +(a) +g=3.7x10 +10 + s +-1 +Figure 8. Testing RWA applicability: Light intensity profiles at τ = +0.1 ps for different values of the Lorentz frequency: (a) ωL = 3.7 · +1010 s−1 and (b) ωL = 3.7·1011 s−1. +lows, +∂ 2Ω′ +∂ξ 2 − ∂ 2Ω′ +∂θ 2 − 2i∂Ω′ +∂ξ − 2i∂Ω′ +∂θ + (n2 − 1)Ω′ += 6ε +�∂ 2p +∂θ 2 + 2i∂ p +∂θ − p +� +, +(A1) +dp +dθ = iδ p + i +2(Ω′ + sΩ′∗e−2i(θ−ξ))w− γ′ +2p, +(A2) +dw +dθ = i(Ω′∗p − Ω′p∗)+ is +� +Ω′pe2i(θ−ξ) − Ω′∗p∗e−2i(θ−ξ)� +−γ′ +1(w+ 1), +(A3) +where we represented the electric field and atomic po- +larization as E = {Aexp[i(ωt − kz)] + c.c.}/2 and ρ12 = +pexp[i(ωt − kz)]. Here ω is the carrier frequency of radi- +ation, k = ω/c is the wavenumber, θ = ωt and ξ = kz are +dimensionless arguments, Ω′ = (µ/¯hω)A is the normalized +Rabi frequency, δ = ∆ω/ω = (ω0 − ω)/ω is the frequency +detuning, γ′ +1,2 = 1/T1,2ω are the normalized relaxation rates, +ε = g/ω = 4πµ2C/3¯hω is the strength of light-matter interac- +tion (normalized Lorentz frequency), and c.c. stands for com- +plex conjugate. The numerical method to solve these equa- +tions is essentially the same as in Ref. [63]. +The key factor governing account of RWA is the auxiliary +parameter s: when it is dropped out (s = 0), we have the usual +RWA; otherwise, the rapidly rotating terms are taken into ac- +count. The problem of RWA applicability is very well studied +for short pulses. It is justified for coherent pulses with dura- +tion much shorter than the relaxation times, but for very short, +few-cycle pulses it is violated [51]. For continuous or quasi- +continuous radiation as in our case, the RWA violation can be +regulated by the values of the relaxation times T1,2 and light- +matter-interaction strength, g = 4πµ2C/3¯h. It is known that +19.5 +20.0 +20.5 +0.0 +0.5 +1.0 +=0.1 ps +(b) +transmitted +reflected +incident +Normalized intensity +t (ps) +19.6 +19.8 +20.0 +20.2 +20.4 +0.0 +0.5 +1.0 +transmitted +reflected +incident +(a) +=0.001 ps +0.00 +0.02 +0.04 +0.06 +0.08 +0 +1 +2 +R +t (ps) + +=0.1 ps + +=0.01 ps + +=0.001 ps + +=0.0001 ps +(c) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.1 +0.2 +0.3 +(e) +t=20 ps +t=20.01 ps +=0.1 ps +Normalized intensity +z / +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.1 +0.2 +0.3 +t=20.03 ps +(d) +t=20.006 ps +=0.001 ps +t=20 ps +Figure 9. Virtual gain in the layer not satisfying the pole condition +(L = 1.1λ = 1.1 µm) under low-intensity radiation, Ωmax = 0.01/T2. +(a) and (b) Light intensity profiles for different values of decay time +τ. (c) The ratio R as a function of ∆t for different τ. (d) and (e) +Radiation distributions for different τ at the switching-off moment +(tmax = 20 ps) and a short time later (t = tmax +∆t). + +10 +the RWA can be violated in the strong-coupling regime, when +Rabi oscillations become possible [64]. An example of inten- +sity profile calculations for different g is shown in Fig. 8. One +can see that decreasing g makes the general and RWA cases +closer to each other. Simultaneously, transmission grows and +absorption decreases due to weaker light-matter interaction. +Since we deal with a rather thin layer (L = λ), the large value +of g is justified to observe the pronounced effects. Neverthe- +less, the qualitative features of virtual gain are observed both +in the general case and under the RWA. +Appendix B: Virtual gain in the system not satisfying the pole +condition +Here, we take the layer of slightly larger thickness as com- +pared to previous consideration, so that one cannot find the in- +teger l to match the pole condition (5). In particular, we take +L = 1.1λ = 1.1 µm. The results of calculations are shown +in Fig. 9. One can see that there is still strong dependence +on the decay time τ, although virtual gain can be observed in +not so sharp and clear form as at the pole. For large τ, trans- +mission and reflection follow the incident intensity [Fig. 9(b)] +and absorption is prevailing even after switching the signal off +[Fig. 9(e)], so that the ratio R remains lower than unity [Fig. +9(c), black line]. On the contrary, R for short τ demonstrates +several peaks at different time instants [see, e.g., blue line in +Fig. 9(c)]. These peaks have different origin: the first one cor- +responds to the step-like distribution of radiation, whereas the +second one more resembles virtual gain [compare blue and red +curves in Fig. 9(d)]. The profile shown in Fig. 9(a) corrob- +orates this interpretation containing both the sharp step-like +splash of intensity and the subsequent smooth decay of trans- +mitted signal. 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Novitsky, Disordered resonant media: Self-induced trans- +parency versus light localization, Phys. Rev. A 97, 013826 +(2018). +[61] D. V. Novitsky, D. Lyakhov, D. Michels, D. Redka, A. A. +Pavlov, and A. S. Shalin, Controlling wave fronts with tun- +able disordered non-Hermitian multilayers, Sci. Rep. 11, 4790 +(2021). +[62] S. Asselie, A. Cipris, and W. Guerin, Optical interpretation of +linear-optics superradiance and subradiance, Phys. Rev. A 106, +063712 (2022). +[63] D.V. Novitsky, Compression of an intensive light pulse in +photonic-band-gap structures with a dense resonant medium, +Phys. Rev. A 79, 023828 (2009). +[64] S. Hughes, Breakdown of the Area Theorem: Carrier-Wave +Rabi Flopping of Femtosecond Optical Pulses, Phys. Rev. Lett. +81, 3363 (1998). + diff --git a/ZtE5T4oBgHgl3EQfDQ6F/content/tmp_files/load_file.txt b/ZtE5T4oBgHgl3EQfDQ6F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6fda3655bf71e7e5d95ca156ccc875914d161b70 --- /dev/null +++ b/ZtE5T4oBgHgl3EQfDQ6F/content/tmp_files/load_file.txt @@ -0,0 +1,1191 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf,len=1190 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='05404v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='optics] 13 Jan 2023 Tunable virtual gain in resonantly absorbing media Denis V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Novitsky∗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Stepanov Institute of Physics, National Academy of Sciences of Belarus, Nezavisimosti Avenue 68, 220072 Minsk, Belarus (Dated: January 16, 2023) Virtual gain refers to the simulation of real light amplification using radiation with exponentially decaying amplitude, so that its complex frequency corresponds to the scattering pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' We theoretically study virtual gain in a two-level resonant medium for different regimes of light-matter interaction depending on the radiation intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' We show that virtual gain at the pole can be most clearly observed for low intensities, when the medium is absorbing, in contrast to the saturated medium at high intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The efficiency of virtual gain can be tuned with the light intensity and can be controlled dynamically through the population inversion of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Our results show that resonantly absorbing media paradoxically mimics gain-like response, which admit a number of related phenomena and methods to mold both optical signals and material properties without relying on instability-prone gain media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' INTRODUCTION Recent years have evidenced the rise of nanophotonics studying the methods for controlling light with small-scale (nanostructured) objects [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' This approach has breathed new life into such classic subjects of optics as scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In par- ticular, subtle tuning of system parameters has made possible to observe different anomalies in light scattering [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' These anomalies can be often associated with scattering poles and zeros, which can be conveniently represented on the complex- frequency plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Generally, the poles and zeroes appear at the complex frequencies, but their position can be regulated by introducing gain or loss, thus, making the system non- Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' When a pole or zero goes to the real-frequency axis, one observes lasing or antilasing (coherent perfect ab- sorption, or CPA [3]), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Specific non-Hermitian systems obeying parity-time (PT ) symmetry demonstrate both lasing and antilasing simultaneously [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' This is the so-called CPA-lasing effect obtained under the coalescence of a pole and a zero at the real-frequency axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Analogous coa- lescence in a Hermitian system corresponds to a bound state in the continuum (BIC) – nonradiating mode having infinite quality factor despite the system openness [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Sometimes, one can transform the BIC into CPA-lasing point by using geometric or non-Hermitian perturbations [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Among other scattering anomalies, we mention exceptional points (non- Hermitian degeneracies) [8], anapoles [9], and superscattering [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Recently, dynamic effects have come under the spotlight strongly widening the range of optical systems and applica- tions under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In particular, materials with time-varying parameters (such as refractive index) attract much attention [12] as a counterpart to the familiar space-varying systems such as interfaces [13, 14], layers [15], photonic crystals [16], metamaterials [17], and metasurfaces [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' A number of temporal analogs of usual optical phenomena were reported, including Brewster angle [19], Anderson localization [20], topological protection [21], PT symmetry [22], negative refraction [23], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' What is most important for our discus- ∗ dvnovitsky@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='com sion, variation of media in time serves as a source of energy flowing into the system and allowing to circumvent the law of energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' As a result, frequency and momen- tum switch their roles making possible such effects as the fre- quency conversion under time modulation of materials [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In the temporal analogs of photonic crystals, one can observe the momentum bandgaps inside which one can observe ampli- fication of waves [25–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' However, practical possibilities of time-varying photonics are strongly limited by the difficulties with fast and spatially uniform modulations of materials and by interference with common nonlinear effects [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Another possibility is to harness time variation not for the medium, but for light signal itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' As a result, not medium, but radiation should be described with the complex frequency making it “non-Hermitian”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' This possibility was implemented in the idea of virtual loss and gain, which attracted a lot of at- tention recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Virtual loss (or virtual perfect absorption, VPA) can be considered as an imitation of CPA by taking ra- diation with exponentially growing amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' If the rate of exponential growth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=', the imaginary part of complex fre- quency of radiation) is equal to the imaginary part of scatter- ing zero frequency, radiation seems to be absorbed [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In fact, it is stored inside the medium and released after the ex- ponential signal is switched off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Theoretical idea of virtual loss was initially demonstrated with the examples of dielec- tric slab and cylinder [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Subsequently, it was expanded to the discrete array of resonators or waveguides [31], ring mi- crocavity coupled to a waveguide [32], and open metasurface- based cavity [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The effect was experimentally proved with elastic waves [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Virtual gain is a counterpart of virtual loss engaging a scattering pole under exponentially decaying irra- diation [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' As further generalizations, we mention the vir- tual PT symmetry with balanced loss and gain [35] and the certain radiation waveforms absorbed at the exceptional points [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The virtual processes based on manipulations with the time-varying light signals are finding such applications as crit- ical coupling for transformation of electromagnetic energy [37], virtual pulling forces for particles transportation [38], enhancement of light scattering [39, 40], and topological light localization via non-Hermitian skin effect [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' It should be emphasized that the total energy of radiation does not grow under virtual gain, so that there is no risk of instability usual for systems with light amplification (such as PT -symmetric 2 e t/� x y z absorbing medium L Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Schematic representation of the situation studied: An expo- nentially decaying radiation is impinging on the layer of resonantly absorbing medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The color gradient inside the medium shows the growing intensity of light corresponding to virtual gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In this paper, we apply the idea of virtual gain to the res- onantly absorbing media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The two-level model of resonant quantum media have attracted much attention since the be- ginning of laser era [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Pulsed-field dynamics in such me- dia was studied in much detail on the basis of Maxwell- Bloch equations [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In particular, coherent pulses (much shorter than relaxation times) evidently violate Beer’s law propagating almost without attenuation due to such phenom- ena as self-induced transparency [45, 46] and zero-area pulse formation [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The further advances include studies of the ever more shorter – few-cycle [48–50], subcycle [51], and unipolar [52] – pulses as well as interactions between pulses in resonant media [53, 54] resulting in such phenomena as pop- ulation gratings [55] and diode-like effect [56], among oth- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' On the other hand, the incoherent waveforms with longer characteristic timescales such as incoherent solitons [57] and optical kinks [58, 59] can also withstand absorption in reso- nant media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' We also mention the studies of ultrashort pulses [60] and wavefront propagation [61] in disordered resonantly absorbing and amplifying media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The role of driving-field de- cay rate in superradiance and subradiance dynamics was re- vealed recently [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' However, the virtual-gain dynamics of quasi-continuous patterns without evident propagation effects connected with pulses or wavefronts have not been studied earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Using numerical simulations of quasi-continuous monochromatic and then exponentially-decaying radia- tion in a two-level medium, we analyze the different regimes depending on light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In the low-intensity regime, we have the classic virtual gain, when the signal attenuating inside the medium is replaced by the gradually growing waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In the high-intensity regime, the medium gets saturated with the step-like patterns seen in the system’s response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Moreover, one can switch between these different types of response by simply choosing the time instant to start the signal decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Our results clearly point to the connection of the virtual gain effect with the position of pole in the complex-frequency plane and can be used for controlling both the radiation patterns and quantum-medium state with time-varying light signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' RESONANT MEDIUM DESCRIPTION We consider the two-level resonant medium of thickness L illuminated by the normally incident radiation of wavelength λ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Interaction of light with the medium is described with the well-known Bloch equations [49],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' dρ12 dt = iω0ρ12 + iµ ¯h Ew− ρ12 T2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (1) dw dt = −4 µ ¯h EImρ12 − w+ 1 T1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (2) where E is the electric field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' ρ12 is the off-diagonal density matrix element (atomic polarization),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' w = ρ22 − ρ11 is the population inversion (difference between populations of the excited and ground levels),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' ω0 is the resonance frequency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' µ is the dipole moment corresponding to the transition between levels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' T1 and T2 are the relaxation times,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' c is the speed of light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' and ¯h is the Planck constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' These equations should be completed with the wave equation accounting for light propa- gation, ∂ 2E ∂z2 − n2 c2 ∂ 2E ∂t2 = 4π c2 ∂ 2P ∂t2 , (3) where P = 2µCRe(ρ12) is the macroscopic polarization, n is the background refractive index, and C is the concentration of two-level atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Note that these equations do not employ the popular rotating-wave approximation (RWA) and slowly- varying-envelope approximation (SVEA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The reason is that the fast change of light field can break these approximations [48–51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' We address the applicability of RWA in Appendix A with the conclusion of qualitative agreement between the general and RWA cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The Maxwell-Bloch equations (1)- (3) are numerically solved using the approach described ear- lier [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Further, we adopt the parameters of the medium as follows: the thickness L = λ = 1 µm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' the background refrac- tive index n = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' the relaxation rates T1 = 1 ns and T2 = 1 ps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' the density parameter governing the strength of light-matter interaction g = 4πµ2C/3¯h = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='7 · 1011 s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' these values can be reached, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=', for quantum-dot media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Further, we also as- sume that radiation is tuned exactly to the resonance with the quantum transition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=', ω = 2πc/λ = ω0 with λ = 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Transmission of light through the resonant medium strongly depends on its intensity, which can be conveniently characterized with the maximal Rabi frequency, Ωmax = µA/¯h, where A is the wave amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Figure 2(a) shows transmission dynamics for the continuous wave instanta- neously switched on at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' One can see that the low-intensity radiation (ΩmaxT2 ≪ 1) is weakly transmitted through the layer of thickness L = λ = 1 µm and is mostly ab- sorbed inside the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' On the contrast, the high-intensity radiation (ΩmaxT2 ∼ 1) is almost perfectly transmitted as it 3 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 max =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1/T 2 max =1/T 2 max =3/T 2 (a) Normalized intensity t (ps) 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 max =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1/T 2 max =1/T 2 max =3/T 2 Population inversion t (ps) (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (a) Transmitted light intensity and (b) inversion dynamics at the entrance of the medium for different values of Rabi frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Parameters: T1 = 1 ns, T2 = 1 ps, ωL = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='7 · 1011 s−1, L = λ = 1 µm, n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' would be for the passive medium without absorbing particles at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' These two situations correspond to the steady-state in- versions [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 2(b)] equal to wst ≈ −1 (most two-level emit- ters remain in the ground state) and wst ≈ 0 (the equal num- ber of emitters in the ground and excited states) and, there- fore, will be referred to as “absorbing medium” and “saturated medium”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Moreover, for high enough intensity (ΩmaxT2 > 1), one can clearly observe the oscillations of trans- mission and the corresponding Rabi oscillations of population inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The inversion is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 2(b) at the entrance of the medium, but it has essentially the same behavior at other distances due to small thickness of the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' As an estimate, the condition ΩmaxT2 = 1 requires the electric-field strength to be E = ¯h/µT2, which for T2 = 1 ps and µ = 30 D (semi- conductor quantum dots) gives E = 106 V/m corresponding to the intensity ∼ 5 GW/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' This value can be further decreased for larger dipole moments and slower relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' VIRTUAL GAIN CONDITION In order to study the possibility of virtual gain, we consider the incident wave having constant intensity from t = 0 to tmax and exponentially decaying after that, E ∼ A � θ(tmax −t)+ e−(t−tmax)/τθ(t −tmax) � , (4) where θ(t) is the Heaviside step function, τ is the decay time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The parameters of the system are chosen close to the pole, po- sition of which in the complex-frequency plane can be evalu- ated analytically for a layer of thickness L as follows ωp c L = 1 n � πl + ilnn − 1 n + 1 � , (5) where l is the integer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The frequency of the wave (4) is tuned to coincide with the real part of pole frequency, ω = ω′ p = 2πc/λp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The decay time is, in turn, limited by the imaginary part of pole frequency, τ ≲ τp = 1/ω′′ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' One can see that for real n, λp can be kept constant by simply changing simultaneously the thickness L and the pole number l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' further, we take λp = 1 µm for L = 1 µm, l = 6, and n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' On the contrary, ω′′ p can be varied by changing L, so that for the above parameters we have τp ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='014 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' This rather short time can be problematic to realize in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' However, it can be increased by taking thicker layer as shown further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' As a proof of principle, we consider the thin layer, because it is easier and faster to calculate without significant influence of radiation inhomogeneity which can smear the effect in thick layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' To give a simple explanation of the virtual gain in resonant media, it is convenient to use the simplified Bloch equations under RWA by adopting s = 0 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (A2) and (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' We also assume the exact resonance, so that ω = ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' First, we consider the stationary limit, when the radiation amplitude Ω is constant, whereas the polarization amplitude and popula- tion inversion have reached the steady state, dp/dt = 0 and dw/dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The steady-state values are as follows, pst = i 2ΩT2wst, (6) wst = − 1 1 + |Ω|2T1T2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (7) One can readily see that for low incident intensities (|Ω|2 ≪ (T1T2)−1), the medium remains almost entirely unexcited, wst ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' On the contrary, for high intensity (|Ω|2 ≫ (T1T2)−1), it is saturated, wst ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' If the radiation is switched off after the medium has reached steady state and decays exponentially (Ω = Ω0e−t/τ), we can assume that the polarization follows the decay of radiation as p = pste−t/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The population inversion, on the contrary, undergoes rapid decay due to the first term (∼ e−2t/τ) in the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (A3) and then changes slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' So, we can assume that the inversion takes on a constant effective value, which can be found after substituting the exponential expressions for Ω and p in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Thus, we have wef f = � 1 − T2 τ � wst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (8) 4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps (b) transmitted reflected incident Normalized intensity t (ps) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 transmitted reflected incident (a) =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0001 ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='3 (d) t=20 ps t=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps Normalized intensity z / L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='3 (c) t=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0001 ps t=20 ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 R t (ps) =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='001 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0001 ps (e) 10 5 10 4 10 3 10 2 10 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 (f) =0 t=20 ps t=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01 ps R (ps) =0 virtual gain p Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Virtual gain in the absorbing medium under low-intensity radiation, Ωmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01/T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (a) and (b) Light intensity profiles for different values of decay time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (c) and (d) Radiation distributions at the switching-off moment (tmax = 20 ps) and a short time later (tmax +∆t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01 ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (e) The ratio R as a function of ∆t for different τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (f) The ratio R at different time instants as a function of the decay time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Obviously, wef f will have the sign opposite to the steady- state value wst, if the decay time is short enough, τ < T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' This effect is especially pronounced in the absorbing medium regime, when wst ≈ −1 and wef f > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=', the medium acts effectively as if it would have gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In the saturated medium regime, when wst ≈ 0, this effect should be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Al- though this simplified explanation allows us to illustrate the possibility of virtual gain in the resonantly absorbing media, it cannot take into account the quantitative features due to the radiation propagation in the medium of finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Thus, we have several relations between the parameters of medium and radiation, which should be justified to observe virtual gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Equation (5) governs the position of pole and the corresponding limiting decay time τp as a function of thick- ness L and refractive index n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Equation (8) requires τ < T2, so that one can expect virtual gain for τ ≲ τp < T2 in the absorb- ing medium regime governed by the condition ΩmaxT2 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' One can easily assess the possible material parameters from these relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Further, we justify this general reasoning with full numerical simulations of the Maxwell-Bloch equations in both the absorbing medium and saturated medium regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' ABSORBING MEDIUM REGIME We start with the regime of absorbing medium, when ra- diation is weak enough to saturate the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Figure 3 shows the results of calculations for Ωmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01/T2 and the incident wave given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (4) with the switching-off mo- ment tmax = 20 ps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=', the signal decays after the steady- state transmission and reflection are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Behavior of radiation strongly depends on the decay time τ: for slow decay with τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps, transmitted and reflected intensities smoothly follow the profile of incident radiation [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 3(b)], whereas for abruptly switched-off radiation with τ = 10−4 ps, transmission and reflection demonstrate sharp splash in in- tensity [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 3(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The corresponding distributions of radi- ation along the medium just after the switching-off moment (at t = tmax + ∆t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01 ps) are fundamentally different as compared in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 3(c) and 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' For τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps, the pattern 5 of attenuating intensity characteristic for absorbing medium is kept after switching off the radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' On the contrary, for τ = 10−4 ps, the reversal of the pattern is seen with attenuation replaced by gradual growth of intensity despite the absorption in the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' This is exactly what we mean by the virtual gain in accordance with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' We should emphasize that the virtual gain does not change the fact that the total amount of energy decreases with time as one would expect for a passive (absorbing) system [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Moreover, the virtual gain is a transient phenomenon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=', the reversed pattern of intensity growing along the medium length exists only for a finite interval of time after radiation was switched off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Therefore, we need an integral value to il- lustrate the efficiency of virtual gain depending on decay time τ and how it changes with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' To this end, we calculate the ratio of average intensities in the second and the first halves of the layer, R(t) = � L L/2 I(t,z)dz/ � L/2 0 I(t,z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (9) Obviously, virtual gain corresponds to R > 1, although this condition is not sufficient to make unequivocal conclusion on the regime of light propagation as will be clear further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Figure 3(e) shows how the ratio R changes as a function of instant after the switching-off moment ∆t for different values of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' It is seen that for large τ (slow decay), R stays lower than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Only for short enough τ, the range of ∆t appears where R(t) > 1 and virtual gain is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The transition to virtual gain agrees well with the condition of the pole position, τ ≪ τp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Note that the effect is observed very soon after tmax and R rapidly (up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps) comes to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The reasons are the fast redistribution of radiation leveling intensity along the medium and rapid exit of radiation from the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' This uniformly distributed intensity grows in time as long as transmitted and reflected intensities increase according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The ratio loses its meaning at very large ∆t, because almost no radiation remains stored in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 3(f), we show the dependence of R on the decay time τ at the specific instant of time, tmax +∆t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' One can see that the ratio becomes larger than unity, when τ is much less than τp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' For τ = 10−4 ps, it almost reaches the value ob- tained for instantaneous switching-off (τ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Compare with the value of R at tmax = 20 ps which is the same for every τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Note that the value of τp is somewhat overestimated, because it is obtained for the real refractive index n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=', not taking into account medium absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Nevertheless, this simplified ap- proach is good enough to give us the qualitative estimate of virtual gain possibility at short decay times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' As mentioned above, the value of τp grows with the medium thickness L, so that virtual gain can be reached at much higher decay times for a thick layer and may be easier observed in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' To give an example, we increase the thickness by 10 times, up to L = 10λ = 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 4(a), for fast enough decay (τ = 10−3 ps), the in- tensity distribution soon after switching-off moment clearly demonstrates the transition from absorption to virtual gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' No such change is observed for slow decay [τ = 1 ps, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 4(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Corresponding behavior of the ratio R [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 4(c) and 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 (b) t=20 ps t=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps =1 ps Normalized intensity z / 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 (a) t=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='001 ps t=20 ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 0 1 2 3 4 R t (ps) =1 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='001 ps (c) 10 3 10 2 10 1 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 (d) =0 t=20 ps t=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps R (ps) =0 virtual gain p Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Virtual gain in the thick layer (L = 10λ = 10 µm) of ab- sorbing medium under low-intensity radiation, Ωmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01/T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (a) and (b) Radiation distributions at the switching-off moment (tmax = 20 ps) and a short time later (tmax +∆t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (c) The ratio R as a function of ∆t for different decay times τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (d) The ratio R at different time instants as a function of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 4(d)] also corroborates the validity of observations made for 6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps (b) transmitted reflected incident Normalized intensity t (ps) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 transmitted reflected incident (a) =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0001 ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 (d) t=20 ps t=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='006 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps Normalized intensity z / L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 (c) t=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='006 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0001 ps t=20 ps Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The case of saturated medium under high-intensity ra- diation, Ωmax = 3/T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (a) and (b) Light intensity profiles for dif- ferent values of decay time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (c) and (d) Radiation distributions at the switching-off moment (tmax = 20 ps) and a short time later (tmax +∆t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='006 ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' the thin layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' All the time values such as ∆t corresponding to the R peak in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 4(c) increase by an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The same can be said about the decay time, so that τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01 ps ≪ τp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='14 ps is enough for virtual gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' One can ex- pect that further increase of the layer thickness can relax the requirements on the decay rapidity even stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' SATURATED MEDIUM REGIME Let us now consider the case of radiation powerful enough to saturate the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In particular, we take the amplitude Ωmax = 3/T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Radiation is switched off at tmax = 20 ps, when the steady state is already reached, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=', the population inver- sion is close to zero and transmission is close to unity (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The key difference in this case is the absence of absorption in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' As a result, light at the switching-off moment 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 (b) t=20 ps t=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='006 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps Normalized intensity z / L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 (a) t=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='006 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0001 ps t=20 ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 R t (ps) =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='001 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0001 ps (c) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The case of passive medium under low-intensity radiation, Ωmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01/T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (a) and (b) Radiation distributions at the switching- off moment (tmax = 20 ps) and a short time later (tmax +∆t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='006 ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (c) The ratio R as a function of ∆t for different τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' is evenly distributed along the layer, without any attenuation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' If radiation is decaying slowly, this uniform distribu- tion is saved [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 5(d)] as well as unity transmission, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=', ra- diation gradually leaves the medium only in the forward direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' If radiation is decaying rapidly, the distribution takes the form of a sharp step in intensity moving along the medium in the forward direction [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 5(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The sharp steps are observed also in the transmitted and reflected intensity profiles as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The profiles for transmitted radiation generally follow the incident intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The difference between slow and fast decays is that in the latter case, the profile forms a step due to very sharp disappearance of impinging radiation [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 5(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The corresponding step is observed also in reflection, unlike almost total absence of reflected signal in the case of slow de- cay [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 5(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The stepped distribution makes a fundamental difference between the saturated medium regime and the case of absorbing medium with its monotonously growing inten- sity [compare Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 5(c) and 3(c)] and allows us to conclude that the saturated medium does not support virtual gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 t max =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 ps t max =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 ps t max =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 ps Inversion t (ps) t max =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 ps (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='03 0 5 10 15 R t (ps) t max =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps t max =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 ps t max =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 ps (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='834 ps (d) t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 ps t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='806 ps t max =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 ps Normalized intensity z / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='434 ps (c) t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='408 ps t max =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 ps t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 t max =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 ps (f) transmitted reflected incident Normalized intensity t (ps) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 transmitted reflected incident (e) t max =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 ps 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 0 2 4 6 8 (g) t=t max t=t max + t virtual gain R Inversion at t max Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Tunable virtual gain under high-intensity radiation, Ωmax = 3/T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (a) Dynamics of population inversion for different switching-off moments tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (b) The ratio R as a function of ∆t for different tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (c) and (d) Radiation distributions at the switching-off moment tmax and a short time later (at t = tmax +∆t marked in the plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (e) and (f) Light intensity profiles for different switching-off moments tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (g) The ratio R as a function of inversion at tmax calculated at the instant t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='02tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The decay time is τ = 10−4 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In fact, the saturated medium regime is equivalent to the case of purely passive, lossless medium, which contains no absorbing particles as illustrated by the characteristic step dis- tribution [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 6(a), compare with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 5(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 6(c), we plot the ratio R at different time instants showing the sharp peak soon after the switching-off moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' This peak is clearly 8 associated with the step in the intensity profile [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 5(a)] and is due to radiation rapidly leaving the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The strong fluctuations at ∆t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 ps illustrate the meaninglessness of R at later times because there is almost no radiation inside the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Note also that R remains around unity for slow decay (τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps) that corresponds to the even distribution of radi- ation along the medium [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 6(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In the case of saturated medium, the ratio demonstrates the same behavior as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In addition, in Appendix B, we discuss the system, which does not satisfy the condition for pole (5), and show that it can be considered as an intermediary case having features of both absorbing and saturated medium regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' NONSTATIONARY REGIME In the above consideration, we have assumed that radiation is switched off only after the steady state of both transmission and population inversion is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Here, we turn to the nonstationary case, when decay starts before the steady state is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' This opens a room for controlling the virtual re- sponse by choosing the proper value of the switching-off mo- ment tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' To demonstrate this possibility, it is instructive to take the high-intensity incident wave able to strongly change the medium state in a short period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' We concentrate at the rising slope of the first Rabi oscillation seen Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 7(a), we show the population-inversion dynamics for Ωmax = 3/T2 and different tmax, but for the same decay time τ = 10−4 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Changing tmax in the range of just 1 ps is enough to get different values of inversion w(tmax) at the switching-off moment and, as a result, different behavior of radiation inside the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The dynamics of ratio R as a function of time after switching-off radiation demonstrate the sharp peaks with the height depending on tmax [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 7(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' It is maximal for tmax corresponding to the population inversion close to zero and diminishes for large absolute values of w(tmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' However, the behavior hiding behind these peaks seems to be quite different as evidenced by the comparison of radiation distributions for tmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 ps [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 7(c)] and tmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 ps [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 7(d)], respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In the first case, the nonperiodic (because of nonsta- tionarity) distribution at t = tmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 ps is replaced by grow- ing intensity characteristic for virtual gain at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='408 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In the second case, the initial distribution at t = tmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 ps is more periodic (the system seems to be closer to the steady state), whereas the distribution at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='806 ps (maximal R) is step-like as in the saturated-medium regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' At later times [∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='034 ps in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 7(c) and 7(d)], the distributions are more uniform and lower in intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The corresponding in- tensity profiles are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 7(e) and 7(f) for different switching-off moments tmax at the same decay time τ = 10−4 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' For tmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 ps, the population inversion is negative [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 7(a)], so that transmission is decreasing as expected for absorbing medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' For tmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 ps, the population inver- sion gets positive and leads to increasing transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' As a result, in the latter case, the switching-off moment is followed by the clear step-like feature and subsequent very slow inten- sity decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' These peculiarities of intensity profiles are linked to the differences between the distributions shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 7(c) and 7(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The dependence of R calculated at the instant t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='02tmax on the population inversion at the switching-off moment tmax [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 7(g)] clearly demonstrates that virtual gain is most effec- tive for w(tmax) ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' For positive w(tmax), the ratio R rapidly decreases and the response changes its character from virtual gain to saturated-medium-like regime as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Thus, for w(tmax) < 0, the medium can be considered as absorbing and we obtain virtual gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' On the contrary, for w(tmax) > 0, the situation is analogous to the case of saturated medium and the peak in R is not connected to the virtual gain, but is due to the step-like distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Regulating the value of tmax allows one to control the regime of light-matter interac- tion and switch virtual gain on and off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' This idea can be used as the tunability scheme allowing to obtain response corre- sponding to different effective gains controlled simply by the switching-off moment tmax of decaying radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' CONCLUSION Our results corroborate the possibility to harness scattering anomalies (such as poles) to obtain unusual response under optical irradiation strongly varying in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' We demonstrate that resonantly absorbing media illuminated with an exponen- tially decaying light can be used to realize virtual gain with the efficiency tuned dynamically using intensity-driven pop- ulation inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Simultaneously, one can use this scheme to control the medium state changing its variables (population inversion and polarization) in a desired way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' There is a certain paradox that virtual gain is most pronounced in the presence of absorption and not in the passive or saturated medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' It would be intriguing to further generalize our approach to the initially excited media or combination of several layers of dif- ferent resonant media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' ACKNOWLEDGMENTS The work has been supported by the Belarusian Repub- lican Foundation for Fundamental Research (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' F22TURC-001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The author is grateful to Viktoryia Kouhar for help with preparation of the graphical materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Appendix A: Details on the Maxwell-Bloch equations and applicability of the RWA The applicability of the rotating-wave approximation (RWA) is convenient to study with the approach described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' According to this approach, the Maxwell-Bloch equations can be represented in the dimensionless form as fol- 9 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 g=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='7x10 11 s 1 (b) transmitted reflected incident Normalized intensity t (ps) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 general RW A transmitted reflected incident (a) g=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='7x10 10 s 1 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Testing RWA applicability: Light intensity profiles at τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps for different values of the Lorentz frequency: (a) ωL = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='7 · 1010 s−1 and (b) ωL = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='7·1011 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' lows, ∂ 2Ω′ ∂ξ 2 − ∂ 2Ω′ ∂θ 2 − 2i∂Ω′ ∂ξ − 2i∂Ω′ ∂θ + (n2 − 1)Ω′ = 6ε �∂ 2p ∂θ 2 + 2i∂ p ∂θ − p � , (A1) dp dθ = iδ p + i 2(Ω′ + sΩ′∗e−2i(θ−ξ))w− γ′ 2p, (A2) dw dθ = i(Ω′∗p − Ω′p∗)+ is � Ω′pe2i(θ−ξ) − Ω′∗p∗e−2i(θ−ξ)� −γ′ 1(w+ 1), (A3) where we represented the electric field and atomic po- larization as E = {Aexp[i(ωt − kz)] + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' }/2 and ρ12 = pexp[i(ωt − kz)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Here ω is the carrier frequency of radi- ation, k = ω/c is the wavenumber, θ = ωt and ξ = kz are dimensionless arguments, Ω′ = (µ/¯hω)A is the normalized Rabi frequency, δ = ∆ω/ω = (ω0 − ω)/ω is the frequency detuning, γ′ 1,2 = 1/T1,2ω are the normalized relaxation rates, ε = g/ω = 4πµ2C/3¯hω is the strength of light-matter interac- tion (normalized Lorentz frequency), and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' stands for com- plex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The numerical method to solve these equa- tions is essentially the same as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The key factor governing account of RWA is the auxiliary parameter s: when it is dropped out (s = 0), we have the usual RWA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' otherwise, the rapidly rotating terms are taken into ac- count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The problem of RWA applicability is very well studied for short pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' It is justified for coherent pulses with dura- tion much shorter than the relaxation times, but for very short, few-cycle pulses it is violated [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' For continuous or quasi- continuous radiation as in our case, the RWA violation can be regulated by the values of the relaxation times T1,2 and light- matter-interaction strength, g = 4πµ2C/3¯h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' It is known that 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps (b) transmitted reflected incident Normalized intensity t (ps) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 transmitted reflected incident (a) =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='001 ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='08 0 1 2 R t (ps) =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='001 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0001 ps (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='3 (e) t=20 ps t=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 ps Normalized intensity z / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='3 t=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='03 ps (d) t=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='006 ps =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='001 ps t=20 ps Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Virtual gain in the layer not satisfying the pole condition (L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 µm) under low-intensity radiation, Ωmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='01/T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (a) and (b) Light intensity profiles for different values of decay time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (c) The ratio R as a function of ∆t for different τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' (d) and (e) Radiation distributions for different τ at the switching-off moment (tmax = 20 ps) and a short time later (t = tmax +∆t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 10 the RWA can be violated in the strong-coupling regime, when Rabi oscillations become possible [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' An example of inten- sity profile calculations for different g is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' One can see that decreasing g makes the general and RWA cases closer to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Simultaneously, transmission grows and absorption decreases due to weaker light-matter interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Since we deal with a rather thin layer (L = λ), the large value of g is justified to observe the pronounced effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Neverthe- less, the qualitative features of virtual gain are observed both in the general case and under the RWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Appendix B: Virtual gain in the system not satisfying the pole condition Here, we take the layer of slightly larger thickness as com- pared to previous consideration, so that one cannot find the in- teger l to match the pole condition (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' In particular, we take L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The results of calculations are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' One can see that there is still strong dependence on the decay time τ, although virtual gain can be observed in not so sharp and clear form as at the pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' For large τ, trans- mission and reflection follow the incident intensity [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 9(b)] and absorption is prevailing even after switching the signal off [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 9(e)], so that the ratio R remains lower than unity [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 9(c), black line].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' On the contrary, R for short τ demonstrates several peaks at different time instants [see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=', blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 9(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' These peaks have different origin: the first one cor- responds to the step-like distribution of radiation, whereas the second one more resembles virtual gain [compare blue and red curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 9(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' The profile shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' 9(a) corrob- orates this interpretation containing both the sharp step-like splash of intensity and the subsequent smooth decay of trans- mitted signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Thus, we can still have the virtual gain outside the pole, but not so effective as exactly at the pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} +page_content=' Gaponenko, Introduction to Nanophotonics (Cambridge University Press, Cambridge, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfDQ6F/content/2301.05404v1.pdf'} diff --git a/_dAyT4oBgHgl3EQfqvhr/content/tmp_files/2301.00548v1.pdf.txt b/_dAyT4oBgHgl3EQfqvhr/content/tmp_files/2301.00548v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b3861d6b353c38398feca3fde26491beeee472cd --- /dev/null +++ b/_dAyT4oBgHgl3EQfqvhr/content/tmp_files/2301.00548v1.pdf.txt @@ -0,0 +1,896 @@ +1 + +RESEARCH PAPER + +Density-dependent and independent mechanisms jointly reduce species +performance under nitrogen enrichment + + +David Sampson Issaka1, Or Gross1, Itunuoluwa Ayilara1, Talia Schabes1, Niv DeMalach1*, + +1Institute of Plant Sciences and Genetics in Agriculture, The Hebrew University of Jerusalem, +Rehovot, Israel + +* Corresponding author: Niv.demalach@mail.huji.ac.il + + + +Keywords: competition, nutrient addition, nitrogen deposition, annual plants, species +diversity, gradient, grassland + + + + + +2 + +ABSTRACT +Nitrogen (N) deposition is a primary driver of species loss in plant communities globally. +However, the mechanisms by which high N availability causes species loss remain unclear. +Many hypotheses for species loss with increasing N availability highlight density- +dependent mechanisms, i.e., changes in species interactions. However, an alternative set of +hypotheses highlights density-independent detrimental effects of nitrogen (e.g., N +toxicity). We tested the role of density-dependent and density-independent mechanisms in +reducing species performance. For this aim, we used 120 experimental plant communities +(mesocosms) comprised of annual species growing together in containers under four +fertilization treatments: (1) no nutrient addition (control), (2) all nutrients except N (P, K, +and micronutrients), (3) Low N (3gN m-2) + other nutrients, and (4) high N (15gN m-2) + +other nutrients. Each fertilization treatment included two sowing densities to differentiate +between the effects of competition (N × density interactions) and other detrimental effects +of N. We focused on three performance attributes: the probability of reaching the +reproduction period, biomass growth, and population growth. We found that individual +biomass and population growth rates decreased with increasing sowing density in all +nutrient treatments, implying that species interactions were predominantly negative. The +common grass (Avena barbata) had a higher biomass and population growth under N +enrichment, regardless of sowing density. In contrast, the legume (Trifolium purpureum) +showed a density-independent reduction in biomass growth with increasing N. Lastly, the +small forb (Silene palaestina) showed a density-dependent reduction in population growth, +i.e., the decline occurred only under high density. Our results demonstrate that density- +dependent and density-independent mechanisms operate simultaneously to reduce species +performance under high N availability. Yet, their relative importance varies among species +and life stages. + + + + +3 + + +ACKNOWLEDGEMENTS +The study was supported by the Israel Science Foundation Grants no. 2403/22 and 672/22 . DSI +was supported by the Pears Foundation. TS was supported by the School of Environmental studies +of the Hebrew University. We thank Joanna Ouaknine for her help in the fieldwork. Tyler +Poppenwimer, Nir Band, and two anonymous reviewers have provided constructive on earlier +versions of the manuscript. + +AUTHOR CONTRIBUTIONS +ND and OG designed the experiment. DSI, IA, TS, and OG performed the experiment and +collected the data. DSI analyzed the data. DSI and ND wrote the first draft, and all other authors +substantially contributed to the writing of the manuscript. + +DATA AVAILABILITY +The data and codes for this manuscript are available in Figshare, +https://figshare.com/account/home#/collections/6183493 + + + + + +4 + + +1. INTRODUCTION +Human activities have elevated Nitrogen (N) deposition in various ecosystems (Simkin et al. 2016, +Ackerman et al. 2019, Borer and Stevens 2022). This increase in N availability is associated with +a decline in species diversity, especially in grassland communities (Stevens et al. 2004, Harpole et +al. 2016, Midolo et al. 2019, Band et al. 2022, Eskelinen et al. 2022). Moreover, the magnitude of +diversity loss following N addition often increases with spatial scale (Seabloom et al. 2021, but +see Lan et al. 2015) and can be irreversible (Isbell et al. 2013). Currently, N deposition is +considered the third greatest driver of biodiversity loss worldwide after climate and land-use +changes (Sala et al. 2000, Bobbink et al. 2010, Borer and Stevens 2022) +So far, numerous N addition experiments have been conducted worldwide, showing that elevated +N levels lead to species loss and decline of forbs and legumes (Bobbink et al. 2010, Soons et al. +2017, Midolo et al. 2019, Tognetti et al. 2021, Band et al. 2022). Two mechanisms can drive such +species loss within the framework of the classical niche theory (Hutchinson 1957). First, high N +levels can fall outside the fundamental niche of some species, i.e., they cannot persist regardless +of the density of other species. Alternatively, elevated N levels can fall outside the realized niche +of some species (extinction due to a competitive exclusion). Therefore, a better understanding of +the effects of elevated N requires separating density-dependent effects (competition) vs. density- +independent detrimental effects (e.g., Chase & Leibold, 2003, Thompson et al., 2020). +Many studies investigating community response to N enrichments have highlighted density- +dependent mechanisms, i.e., changes in competitive interactions with increasing N availability. +First, an increase in standing biomass enhances light competition, thereby reducing short species' +competitive ability (Eek and Zobel 2001, Hautier et al. 2009, Lamb et al. 2009, Borer et al. 2014a, +DeMalach et al. 2016, Eskelinen et al. 2022). Nonetheless, N addition can also intensify +belowground competition (Rajaniemi 2003a, Dickson and Foster 2011). Furthermore, eliminating +the N limitation can decrease the potential for niche partitioning among species (niche +dimensionality, sensu Harpole et al., 2016). Additionally, N addition can increase litter production, +thereby reducing seedlings' establishment (Tilman, 1993, Foster & Gross, 1998). While there is no +consensus on the relative importance of the above competitive exclusion mechanisms (Rajaniemi + +5 + +2003b, Dickson and Foster 2011, DeMalach and Kadmon 2017, Harpole et al. 2017) all these +mechanisms have one shared attribute of being density-dependent. +An alternative set of explanations does not invoke resource competition. Instead, it highlights the +detrimental effects of N, which do not necessarily depend on density ("the nitrogen detriment +hypotheses" sensu Band et al., 2022). For example, N addition can lead to ammonium toxicity +(Britto and Kronzucker 2002), soil acidification (Crawley et al. 2005) which prevents the +acquisition of nutrients (Tian et al. 2022), increased susceptibility to stress agents (Bobbink et al., +2010, Stevens et al., 2010), and an alternation of the soil microbiome (Farrer and Suding 2016, +Huang et al. 2021). Again, these specific mechanisms are difficult to disentangle and are highly +variable among systems. For example, differences in community sensitivity to N-induced +acidification depend on soil buffering capacities (Clark et al., 2007). Still, all these mechanisms +can potentially reduce performance independent of species interactions. +Recently, Band et al., (2022) conducted a global meta-analysis of ~600 experiments using biomass +as a proxy for density-dependent effects. They found that the biomass-mediated effect of N leads +to an average decline of ~2% in diversity, while the biomass-independent effect causes a more +substantial diversity decline of ~18%. These findings are surprising given that many competition +experiments have established a causal link between nitrogen addition, competition, and lower +species diversity(Gurevitch et al. 1990, Lamb et al. 2009, DeMalach et al. 2017b). +One possible explanation for the results of Band et al. (2022) is that biomass is not a good proxy +for density-dependent processes. Many studies have shown that biomass is affected by multiple +factors (resources, nutrients, herbivores, species composition, litter production) and that the +causality of the biomass-richness relationship is bidirectional (Grace et al. 2016). Alternatively, it +is possible that even within the same community, some species experience reduced performance +due to density-independent processes while other species experience reduced performance due to +competition. Surprisingly, despite the numerous N-addition experiments, we are unaware of any +experiment that separated the density-dependent and independent effects of N enrichment on the +performance of different species. +Here, we manipulated sowing density as the most direct approach to disentangle density-dependent +and independent processes. This simple manipulation has been widely used for monocultures but +rarely for communities (but see Goldberg & Estabrook, 1998, Rajaniemi, Turkington, & Goldberg, + +6 + +2009). We used this approach to investigate the effects of N availability on species performance +within an experimental community that included a small forb (Silene palaestina), a medium size +legume (Trifolium purpureum), and a tall competitive grass (Avena barbata). +We expected that the tall grass would benefit from N addition because tall grasses are often strong +competitors for light (Suding et al. 2005, Vojtech et al. 2007, Gough et al. 2012, DeMalach et al. +2017b) and are insensitive to the detrimental effects of high N availability (Tian et al. 2022). In +contrast, since the abundance of forbs and legumes often decreases with N availability (Suding et +al. 2005, Tognetti et al. 2021), we hypothesized that N addition would increase their sensitivity to +competition (competitive response sensu Miller & Werner, 1987). Additionally, we expected the +legume and forb species to experience density-independent detrimental effects of N addition (Band +et al. 2022). Still, we predicted that the density-independent effect would be minor compared to +competition within the scope of our short-term experiment (see also Eskelinen et al. 2022) because +toxic effects in the soil often accumulate over time (Tian et al. 2020). + +2. MATERIALS AND METHODS +2.2 Experimental design +The experiment was conducted at the experimental station of the Hebrew University of Jerusalem +in Rehovot, Israel (31°54'18.8 "N, 34°48'16.9 "E). The station is characterized by a Mediterranean +climate, with a mild, rainy winter and a hot, dry summer (annual rainfall of ~560mm yr-1). +The experiment mimicked the annual plant communities that grow on Hamra soils of Israel's +coastal region. These communities grow during winter, bloom in spring, and dry in the late spring. +Soil type is a primary driver of species composition in these communities. Communities growing +on a sandy-type Hamra are characterized by high diversity and abundance of forbs and legumes. +In contrast, loamy-type Hamra (with higher nutrients and water availability) has a lower diversity +and high dominance of tall grasses. +The experimental plant communities were sown in January 2021 in large plastic containers +(dimensions of ~1m × 1m × 1m) with holes for water drainage (Figure 1). The experiment included +120 experimental communities (3 soil types × 4 nutrient treatments × 2 sowing densities × 5 +replications) randomly located in an area of ~half a hectare. We filled the containers with soils + +7 + +(taken from at least 1m depth to avoid seed bank) from three nearby locations aiming for a gradient +of clay content (from sandy to loamy soil). However, two sources had almost identically low clay +content (and water-holding capacity). Thus, to increase statistical power, these two sources were +pooled into a single category (hereafter sandy soil ~7.5% clay content) and contrasted with the +other source, red soil (hereafter loamy soil ~ 15% clay content). +We chose four model species that were collected in Israel's coastal plain (by a commercial +company, 'Hila Pirchei Bar Ltd'): Silene palaestina (Caryophyllaceae, seed mass of 0.095mg), +Trifolium purpureum (Fabaceae, 1.47mg), Avena barbata (Poaceae, 15.12mg), and Lupinus +palaestinus (Fabaceae, 257.3mg). The species were chosen based on the following criteria: (i) +being common in Hamra communities such that a sufficient amount of seeds was available (1000g +per species). (ii) Representing different functional groups (forbs, legumes, and grasses) and a wide +range of seed masses. +We fertilized the plots a few days after sowing in January 2021. The four fertilization treatments +included: (i) no nutrient addition, (ii) the addition of all nutrients except N (P, K, and +micronutrients following NutNet protocol, Borer et al., 2014), (iii) Low N (3gN m-2, applied as +urea) together with all other nutrients, and (iv) high N (15gN m-2) with all other nutrients. Since +we focused on N, we removed the potential limitation of all other nutrients by adding them to all +plots and viewed treatment ii as another 'control'. However, to quantify the potential limitations of +other nutrients, we also included a standard control with no addition. Hereafter, we refer to the no- +nutrient treatment as the control to avoid confusion. +Under each nutrient treatment, the plant communities (all four species) were sown in two densities, +low (3g m-2) and high (16g m-2), separated equally across the four species (in terms of weight). +Sowing species in equal weight (rather than equal seed number), implies that the number of sown +seeds is higher for small-seeded species (and vice versa). The rationale of this choice is that small- +seeded species have a lower emergence probability (Table S1) and lower survival probability of +seedlings (Tables S2, S3). +We chose 16g m-2 as the high sowing density following previous studies of annual plant +communities using this amount as the highest end of a density gradient (Godoy et al. 2014, Pérez- +Ramos et al. 2019). The low-density level was chosen to reduce competitive interactions (although +we did not assume that interactions are entirely eliminated). At the same time, a major requirement + +8 + +of our experiment was that even under low density, there would be enough individuals to calculate +species' performance. Thus, we have sown only four species (having more species would reduce +each species' density, i.e., separating the 3g among more species). In the experiment, the densities +were sufficient for all species except Lupinus. Lupinus was missing in many (low-density) plots +and excluded from the analysis. +Throughout the growing season, we weeded all species that were not part of the experiment. +Additionally, the experiment was irrigated weekly during the spring (March-April). Otherwise, the +containers would dry faster than the surrounding vegetation because of their high drainage +(DeMalach, Ron, & Kadmon, 2019). +2.3 Sampling +We estimated population density by sampling seedlings during February 2021 and February 2022. +Sampling was conducted in four 20cm × 20cm quadrates in each plot in the first year. However, +due to logistic constraints, we sampled in two quadrates only for Silene and Trifolium and in four +quadrates for Avena. During the second-year sampling, we avoided spots where biomass was +collected in the first year. +To quantify individual biomass, we collected biomass samples from two 20cm × 20cm quadrates +in the center of each plot (at the end of the first growing season, May 2021). First, the biomass was +oven-dried at 600C for 48hrs and sorted into species. Then, we counted the number of individuals +of each species and weighed their biomass. In the analysis, all samples within a plot were +aggregated (by taking the mean). Lastly, density units were converted from individuals per sample +into individuals per square meter (i.e., multiplied by 25, the ratio between a quadrate area and a +square meter). +2.4 Statistical analyses +All statistical analyses were conducted with R (version 4.1.1). We tested the effects of the +treatments on the probability of reaching reproduction (the ratio between the number of sown seeds +and the number of adults in the first year), biomass growth (the ratio between adult and seed +masses), and population growth (the ratio between seedling densities in two consecutive years). +For each species in each soil type, we built the following linear regression model using dummy +variables to describe the four nutrient treatments and the two sowing density levels: + +9 + +𝑌 = 𝛽0 + 𝛽1𝐷𝑒𝑛𝑠𝑖𝑡𝑦 + 𝛽2𝑀𝑖𝑐𝑟𝑜 + 𝛽3𝑁3 + 𝛽4𝑁15 + 𝛽5(𝐷𝑒𝑛𝑠𝑖𝑡𝑦 × 𝑁15)𝑖𝑗 + 𝜀, 𝜀 ~ 𝑁 (0, 𝜎𝜀 +2) +Here, Y is a performance attribute (reproduction probability, biomass growth, or population +growth rate). Density represents the sowing density (low[0] vs. high[1]), and Micro indicates +whether or not PK and micronutrients were added. The variables N3 and N15 represent the addition +of low and high levels of N. Using dummy variables rather than a continuous variable of N level +was needed to avoid assuming a linear response to N addition. +The model includes an interaction term between sowing density and high levels of N (interactions +with other nutrient treatments were insignificant and therefore removed to increase the statistical +power and avoid overfitting). Notably, a significant interaction between N addition and sowing +density is interpreted as evidence of a density-dependent response to N (i.e., N addition modifies +competitive interactions). In contrast, when N has a significant effect, but there were no +interactions, we interpreted it as evidence that the response to N addition is primarily density- +independent. +To ensure normally distributed residuals in the linear regression (𝜀 ~ 𝑁 (0, 𝜎𝜀2), biomass and +population growth were loge-transformed while reproduction probability was logit-transformed +(𝑙𝑜𝑔𝑖𝑡(𝑝) = log𝑒( +𝑝 +1−𝑝) ). To avoid zeros in the logit transformation (leading to undefined values), +we added 0.01 to all observations. Additionally, for Avena, there were a few cases where +reproduction probability was equal to or greater than one (because the density of the whole plot +was extrapolated from quadrates and spatial distribution was non-homogenous), and those values +were converted to 0.99 (logit of one is also undefined). + + + + +10 + + + +Figure 1. (A) a photo of the experiment (B) a scheme of the experimental design. Two sowing +densities (low vs. high) were applied to quantify both density-independent and density- +dependent performance. Besides density manipulation, the experiment included a factorial +combination of two soil types (sandy vs. loamy) and four fertilization treatments: (1) no +nutrient addition (control), (2) the addition of all nutrients (P, K, and micronutrients) except +N (Micro), (3) Low N together with all other nutrients (N3), and (4) high N with all other +nutrients (N15). + +(A) +(B) +Sandy soil +Loamy soil +Lowdensity +Highdensity +Lowdensity +Highdensity +Control +Micro +N3 +N1511 + +3. RESULTS +Reproduction probability, the probability of the seed reaching the reproduction period, was low in +the small-seeded Silene, intermediate in Trifolium, and highest for the large-seeded Avena (Fig. 2, +Tables S2 and S3). The effect of sowing density was negative for Silene (Psandy < 0.001, Ploamy = +0.004), positive for Trifolium in the sandy soil (P < 0.001), and insignificant for Avena. The +effects of N on reproduction probability were insignificant except for a marginally significant +positive effect on Silene in the sandy soil (P N15 = 0.065). The addition of other nutrients reduced +the reproduction probability of Trifolium in the sandy soil (P = 0.003) and increased it for Silene +in the loamy soil (P = 0.04). +Biomass growth, the ratio between adult and seed biomass, was highest in the small-seeded Silene, +intermediate in Trifolium, and lowest for the large-seeded Avena (Fig 3). The effects of increasing +sowing density on biomass growth were strongly negative for all the species indicating the +dominance of competitive interactions in the system (Table S4, S5). High levels of nitrogen +addition increased the biomass growth of Silene (Ploamy = 0.04) and Avena (Ploamy = 0.06). However, +the other nutrients did not affect biomass growth except for a positive effect on Trifolium (Ploamy = +0.004). Importantly, the biomass growth of Trifolium was reduced in the sandy soil under the high +nitrogen addition (Psandy = 0.058) and in loamy soils under low and high N addition (Ploamy, N3 = +0.038, Ploamy, N15 = 0.019). Strikingly, in low-density plots where all nutrients were added, +Trifolium's biomass growth was reduced from ~510 to ~210 and ~140 under low and high nitrogen +levels, respectively. Moreover, we found no interactions between N and density, indicating that +the detrimental effects of N were density-independent. +Increasing sowing density reduced the population growth (the ratio between seedling densities in +the two years) of all species, demonstrating that competition strongly affects population dynamics +(Fig 4, Tables S6, S7). Notably, Silene had strong interactions between growth rate and density +(Ploamy, N15*Density = 0.0019). Under high nitrogen, its population growth rate was reduced from 2 +(population increase) under low density to 0.4 (population decline) under high density. In contrast, +the population growth of Trifolium was unaffected by N (but was reduced in the loamy soil). +Lastly, for Avena, the population growth increased under high N levels in the sandy soils regardless +of sowing density (Psandy= 0.012). Additionally, we found no effect of adding other nutrients on +species' population growth, except for a marginal increase for Trifolium (Ploamy = 0.06). + +12 + + +Figure 2. The effects of fertilization treatments and sowing density on the probability of +reaching reproduction (the ratio between density in the spring and sowing density). The blue +and red circles represent the means in the low and high sowing densities, respectively. The +error bars are standard errors. (Mi) - addition of all nutrients (P, K, and micro) except N, +(N3), and (N15) are treatments with N addition of 3g m-2 and 15g m-2 (together with all other +nutrients). The left and right panels show the results for sandy and loamy soil types. See table +S1 for statistical analyses. Note the different ranges of the y-axes. + +(a) +silene (sandy soil) +(b) +silene (Loamy soil +0.07 +.low density +0.07 +0.06 +. high density +0.06 +0.05 +0.05 +0.04 +0.03 +0.03 +0.02 +0.02 +10 +0.01 +0.00 +Control +Mi +N3 +N15 +Control +Mi +N3 +N15 +(c) +Trifolium (Sandy soil) +(p) +Trifolium (Loamy soil) +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +0.0 +Control +Mi +N3 +N15 +Control +Mi +N3 +N15 +(e) +Avena (Sandy soil) +(f) +Avena (Loamy soil) +1.0 +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +10 +0.0 +Control +Mi +N3 +N15 +Control +Mi +N3 +N15 +Fertization treatment +Fertilization treatment13 + + +Figure 3. The effects of fertilization treatments and sowing density on the biomass growth +rate (the ratio between seed and adult masses). The blue and red circles represent the means +in the low and high sowing densities. The error bars are standard errors. (Mi) - addition of +all nutrients (P, K, and micro) except N, (N3), and (N15) are treatments with N addition of +3g m-2 and 15g m-2 (together with all other nutrients). The left and right panels show the +results for sandy and loamy soil types. Note the logarithmic scale of the y-axes. + + + +(a) +silene (Sandy soil) +(b) +silene (Loamy soil) +rowth +5000 +5000 +2000 +2000 +1000 +1000 +iomass +500 +500 +200 +200 +100 +low density +100 +50 +high density +50 +B +20 +20 +Control +Mi +N3 +N15 +Control +Mi +N3 +N15 +(c) +Trifolium (Sandy soil) +(d) +Trifolium (Loamy soil) +5000 +5000 +2000 +2000 +0. +1000 +1000 +500 +500 +200 +200 +100 +100 +50 +50 +20 +20 +Control +Mi +N3 +N15 +Control +Mi +N3 +N15 +(e) +Avena (Sandy soil) +(f) +Avena (Loamy soil) +5000 +5000 +2000 +2000 +1000 +1000 +500 +500 +siomass +200 +200 +100 +100 +50 +50 +20 +B +20 +10 +10 +Control +Mi +N3 +N15 +Control +Mi +N3 +N15 +Fertilization treatment +Fertilization treatment14 + + +Figure 4. The effects of fertilization treatments and sowing density on the population growth +rate (the ratio between seedlings densities in year two and year one). The dashed line +represents the growth rate of one, i.e., no change in population size. The blue and red circles +represent the means in the low and high sowing densities. The error bars are standard errors. +(Mi) - addition of all nutrients (P, K, and micro) except N, (N3), and (N15) are treatments +with N addition of 3g m-2 and 15g m-2 (together with all other nutrients). The left and right +panels show the results for sandy and loamy soil types. Note the logarithmic scale and the +different ranges of the y-axes. + + + + +(a) +silene (Sandy soil) +(b) +silene (Loamy soil) +10.0 +10.0 +low density +5.0 +5.0 +high density +2.0 +2.0 +1.0 +1.0 +0.5 +0.5 +0.2 +0.2 +Control +Mi +N3 +N15 +Control +Mi +N3 +N15 +(c) +Trifolium (Sandy soil) +(d) +Trifolium (Loamy soil) +500 +500 +200 +200 +100 +100 +50 +50 +20 +20 +10 +10 +5 +5 +2 +2 +Control +Mi +N3 +N15 +Control +Mi +N3 +N15 +(e) +Avena (Sandy soil) +(f) +Avena (Loamy soil) +10.0 +10.0 +5.0 +5.0 +2.0 +2.0 +1.0 +1.0 +0.5 +0.5 +0.2 +0.2 +Control +Mi +N3 +N15 +Control +Mi +N3 +N15 +Fertilization treatment +Fertilization treatment15 + +4. DISCUSSION +We tested the role of density-dependent and density-independent mechanisms in reducing species +performance with increasing N availability. In our experimental system, biomass growth and +population growth decreased with sowing density for all species, implying that species interactions +were predominantly negative, a necessary condition for testing N-induced density-dependent +mechanisms of reduced performance. Additionally, we found that N limited growth for the forb +and grass species but not for the legume species that responded positively to the addition of other +resources (probably because of phosphorus limitations). +We demonstrated evidence for density-dependent and density-independent performance +reductions following N addition. The legume species (Trifolium) declined in biomass growth under +high N levels regardless of density, indicating a density-independent detrimental effect of N. In +contrast, the small forb (Silene) experienced lower population growth with N addition but only +under high density, demonstrating a density-dependent effect (competition). +Our results agree with previous empirical studies in the region (DeMalach et al., 2017b) as well as +places (Xia and Wan 2008, Bobbink et al. 2010, Maskell et al. 2010, Eskelinen and Harrison 2015, +DeMalach et al. 2017a, Seabloom et al. 2021, Vázquez et al. 2022), showing that tall competitive +grasses like Avena benefit from N enrichment while legumes and forbs decline. However, our +findings further suggest that this reduction is driven by different processes, density-dependent +mechanisms for forbs and density-independent mechanisms for legumes. Below, we discuss the +implications and interpretations of our findings. +4.1 Density-independent mechanisms +Many studies have shown that N enrichment affects soil chemical properties (Tian et al. 2020) and +microbiomes (Lekberg et al. 2021, Huang et al. 2021) with the potential to reduce species diversity +regardless of the competitive interactions. In accordance, global analyses have shown that the +negative effects of N on diversity are significant even after controlling for its effects on biomass +(Band et al. 2022) or light (DeMalach, 2018). Moreover, a causal link between N-induced +acidification and diversity decline has been established by adding calcium to buffer soil +acidification, thereby reducing species loss. However, all the above studies do not provide direct +evidence for density-independent mechanisms. For example, N-induced acidification can also +change competitive interactions (Yao and Feng 2022). The same reasoning applies to the soil + +16 + +microbiome, affecting plant performance in density-dependent and independent manners (Ke and +Wan 2020). Thus, as far as we know, this is the first study to quantify the density-independent +effects of N directly. +In our system, we expected density-independent mechanisms to be relatively weak because soil +acidification is unlikely (high limestone content, pH ~ 7.8), and the time scale is too short for long- +term changes in other soil properties (Tian et al. 2020). Nonetheless, we found a very strong +density-independent response of Trifolium, which experienced up to a three-fold reduction in +biomass growth. Furthermore, we found that Trifolium's density-independent biomass was higher +in the fertile loamy soil compared with the poor sandy soil. Still, the differences between the soils +could be explained by lower aeration rather than a nutrient-induced response. +A possible driver of reduced performance is N toxicity. While grass species have a rhizosheath (a +layer of adhering soil particles to the root surface) acting as a "biofilm-like shield" enhancing their +tolerance to N-induced stress, other species do not (Tian et al. 2022). Alternatively, the reduction +in performance of Trifolium could be related to changes in its microbiome. Such interpretation is +supported by the finding that Trifolium's performance is highly reduced without its microbiome +(Ayilara 2022). +4.2 Density-dependent mechanisms +Based on theoretical expectations, we predicted that the small forb would experience a density- +dependent reduction in survival and biomass growth with increasing nitrogen availability because +of asymmetric competition for light (DeMalach, Zaady, Weiner, & Kadmon, 2016). Nonetheless, +we found no evidence that sensitivity to competition in reproduction probability and biomass +growth increases with N availability (no interactions between density and nitrogen). Moreover, the +biomass of Silene increased rather than declined with N regardless of density (no interactions). +A strong interaction between N availability and density was found for the population growth rate +of Silene. While its population increased under low density, it declined under high density. We +attribute the decline in population growth rate with N to a low establishment probability of this +small-seeded species when there was high litter from the previous year (Thompson, 1987). This +finding highlights the importance of measuring the actual population growth rate rather than using +biomass growth rate or fecundity as proxies in competition experiments (Goldberg et al. 1999). +Currently, empirically-calibrated models of species interactions incorporate density-dependent + +17 + +effects on fecundity only (Stouffer 2022), but we hope that future studies will also integrate +density-dependent effects on the establishment. +4.3 Methodological issues +The experimental design of competition experiments along environmental gradients involves two +main tradeoffs: (i) realism vs. minimizing noise and (ii) resolution of environmental conditions +vs. resolution of species interactions. For the first tradeoff, we chose an intermediate setup, while +for the second, we focused on maximizing the number of environmental treatments (soil types and +fertilization). +The main advantage of growing plants in containers is reducing noise. We created homogenous +conditions and differentiated between two soil types (the natural landscape is patchy with many +intermediate levels of clay). Additionally, the containers allowed for minimizing the need to +remove weeds. However, these containers cannot fully mimic the natural soil. Thus, the next +challenge is testing for density-dependent and density-independent mechanisms of N enrichment +in even more realistic conditions. For example, when plants grow in the natural soil and nitrogen +is deposited in low quantities for a long time rather than in high quantities for a short time. +Importantly, our experiment is closer to the real world than many greenhouse experiments because +the plants experienced ambient climatic conditions. Furthermore, the plants grew in large +containers rather than in the few-liter pots used in most competition experiments. We believe that +large containers are crucial to quantify density-dependent performance because they enable niche +partitioning of species varying in root depth and sufficient shading to produce light competition. +Recently, Hart, Freckleton & Levine (2018) highlighted the need to separate conspecific and +heterospecific competition to calibrate a phenomenological model of community dynamics. +Assuming strictly pairwise interactions and accurate estimation of the model parameters, such a +model can predict long-term coexistence patterns. While we appreciate this approach's strength, it +requires many competition treatments, including monocultures, all pairwise combinations, and +several densities (Godoy et al. 2014). Therefore, as far as we know, it was never applied under +more than two environmental conditions (Wainwright et al. 2019, Pérez-Ramos et al. 2019, Van +Dyke et al. 2022). Similarly, experiments where a monoculture of each species is compared to a +mixture also require many combinations. To the best of our knowledge, the monoculture-mixtures + +18 + +approach has not been applied to an experiment like ours that include eight types of conditions +(two soil types and four nutrient treatments). +Following Goldberg and her colleagues (Goldberg & Estabrook, 1998; Rajaniemi et al., 2009), we +lumped together all density-dependent effects. Of course, such pooling cannot capture all the fine +details of the competition nor predict long-term coexistence. Yet, the strength of Goldberg's +approach is its simplicity, i.e., it requires only two treatments, low and high density. This simplicity +enabled us to investigate density-dependent and independent performance under eight +environmental conditions (two soil types and four fertilization treatments). Moreover, another +advantage of Goldberg's approach is that it does not require simplifying assumptions, such as +interactions being always negative and strictly pairwise (Kleinhesselink et al. 2022, Stouffer 2022) +or species establishment being density-independent. Indeed, despite the prevalence of negative +interactions in our system, positive interactions also occur (reproduction probability of Trifolium). +Similarly, the mounting evidence of the role of litter in the establishment of plants suggests that +density-dependent effects during the establishment should not be ignored (Facelli and Pickett +1991, Foster and Gross 1998). +4.4 Conclusion +We showed that density-independent and density-dependent mechanisms operate together to +reduce species performance under high levels of N. Specifically, we demonstrated a density- +independent effect for the legume species and a density-dependent effect for the forb. It remains +to be tested whether these results can be generalized to other legumes and forbs. Moreover, a +critical open question is how different environmental conditions affect density-dependent and +density-independent responses to N enrichment. This question is timely given the significant role +of N eutrophication in driving biodiversity loss worldwide. Hence, the simplicity of the approach +applied here makes it a promising way forward. +REFERENCES +Ackerman, D. et al. 2019. Global estimates of inorganic nitrogen deposition across four decades. +- Global Biogeochem. Cycles 33: 100–107. +Ayilara, I. 2022. Competition and plant-soil feedback following nitrogen addition: a greenhouse +experiment. + +19 + +Band, N. et al. 2022. Assessing the roles of nitrogen, biomass, and niche dimensionality as +drivers of species loss in grassland communities. - Proc. Natl. Acad. Sci. U. S. A. 119: 1– +11. +Bobbink, R. et al. 2010. 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Species existence and coexistence under nutrient enrichment in the +Park Grass. - bioRxiv in press. + + diff --git a/_dAyT4oBgHgl3EQfqvhr/content/tmp_files/load_file.txt b/_dAyT4oBgHgl3EQfqvhr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..17cc51779da93106f1fee3fdb35e07c2ee195592 --- /dev/null +++ b/_dAyT4oBgHgl3EQfqvhr/content/tmp_files/load_file.txt @@ -0,0 +1,935 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf,len=934 +page_content='1 RESEARCH PAPER Density-dependent and independent mechanisms jointly reduce species performance under nitrogen enrichment David Sampson Issaka1, Or Gross1, Itunuoluwa Ayilara1, Talia Schabes1, Niv DeMalach1*, 1Institute of Plant Sciences and Genetics in Agriculture, The Hebrew University of Jerusalem, Rehovot, Israel Corresponding author: Niv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='demalach@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='huji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='il Keywords: competition, nutrient addition, nitrogen deposition, annual plants, species diversity, gradient, grassland 2 ABSTRACT Nitrogen (N) deposition is a primary driver of species loss in plant communities globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' However, the mechanisms by which high N availability causes species loss remain unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Many hypotheses for species loss with increasing N availability highlight density- dependent mechanisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', changes in species interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' However, an alternative set of hypotheses highlights density-independent detrimental effects of nitrogen (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', N toxicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' We tested the role of density-dependent and density-independent mechanisms in reducing species performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' For this aim, we used 120 experimental plant communities (mesocosms) comprised of annual species growing together in containers under four fertilization treatments: (1) no nutrient addition (control), (2) all nutrients except N (P, K, and micronutrients), (3) Low N (3gN m-2) + other nutrients, and (4) high N (15gN m-2) + other nutrients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Each fertilization treatment included two sowing densities to differentiate between the effects of competition (N × density interactions) and other detrimental effects of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' We focused on three performance attributes: the probability of reaching the reproduction period, biomass growth, and population growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' We found that individual biomass and population growth rates decreased with increasing sowing density in all nutrient treatments, implying that species interactions were predominantly negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The common grass (Avena barbata) had a higher biomass and population growth under N enrichment, regardless of sowing density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' In contrast, the legume (Trifolium purpureum) showed a density-independent reduction in biomass growth with increasing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Lastly, the small forb (Silene palaestina) showed a density-dependent reduction in population growth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', the decline occurred only under high density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Our results demonstrate that density- dependent and density-independent mechanisms operate simultaneously to reduce species performance under high N availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Yet, their relative importance varies among species and life stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 3 ACKNOWLEDGEMENTS The study was supported by the Israel Science Foundation Grants no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2403/22 and 672/22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' DSI was supported by the Pears Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' TS was supported by the School of Environmental studies of the Hebrew University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' We thank Joanna Ouaknine for her help in the fieldwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Tyler Poppenwimer, Nir Band, and two anonymous reviewers have provided constructive on earlier versions of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' AUTHOR CONTRIBUTIONS ND and OG designed the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' DSI, IA, TS, and OG performed the experiment and collected the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' DSI analyzed the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' DSI and ND wrote the first draft, and all other authors substantially contributed to the writing of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' DATA AVAILABILITY The data and codes for this manuscript are available in Figshare, https://figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='com/account/home#/collections/6183493 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' INTRODUCTION Human activities have elevated Nitrogen (N) deposition in various ecosystems (Simkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2016, Ackerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2019, Borer and Stevens 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' This increase in N availability is associated with a decline in species diversity, especially in grassland communities (Stevens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2004, Harpole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2016, Midolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2019, Band et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2022, Eskelinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Moreover, the magnitude of diversity loss following N addition often increases with spatial scale (Seabloom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2021, but see Lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2015) and can be irreversible (Isbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Currently, N deposition is considered the third greatest driver of biodiversity loss worldwide after climate and land-use changes (Sala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2000, Bobbink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2010, Borer and Stevens 2022) So far, numerous N addition experiments have been conducted worldwide, showing that elevated N levels lead to species loss and decline of forbs and legumes (Bobbink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2010, Soons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2017, Midolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2019, Tognetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2021, Band et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Two mechanisms can drive such species loss within the framework of the classical niche theory (Hutchinson 1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' First, high N levels can fall outside the fundamental niche of some species, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', they cannot persist regardless of the density of other species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Alternatively, elevated N levels can fall outside the realized niche of some species (extinction due to a competitive exclusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Therefore, a better understanding of the effects of elevated N requires separating density-dependent effects (competition) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' density- independent detrimental effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', Chase & Leibold, 2003, Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Many studies investigating community response to N enrichments have highlighted density- dependent mechanisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', changes in competitive interactions with increasing N availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=" First, an increase in standing biomass enhances light competition, thereby reducing short species' competitive ability (Eek and Zobel 2001, Hautier et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2009, Lamb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2009, Borer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2014a, DeMalach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2016, Eskelinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Nonetheless, N addition can also intensify belowground competition (Rajaniemi 2003a, Dickson and Foster 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Furthermore, eliminating the N limitation can decrease the potential for niche partitioning among species (niche dimensionality, sensu Harpole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=" Additionally, N addition can increase litter production, thereby reducing seedlings' establishment (Tilman, 1993, Foster & Gross, 1998)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' While there is no consensus on the relative importance of the above competitive exclusion mechanisms (Rajaniemi 5 2003b, Dickson and Foster 2011, DeMalach and Kadmon 2017, Harpole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2017) all these mechanisms have one shared attribute of being density-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' An alternative set of explanations does not invoke resource competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Instead, it highlights the detrimental effects of N, which do not necessarily depend on density ("the nitrogen detriment hypotheses" sensu Band et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' For example, N addition can lead to ammonium toxicity (Britto and Kronzucker 2002), soil acidification (Crawley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2005) which prevents the acquisition of nutrients (Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2022), increased susceptibility to stress agents (Bobbink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', 2010, Stevens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', 2010), and an alternation of the soil microbiome (Farrer and Suding 2016, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Again, these specific mechanisms are difficult to disentangle and are highly variable among systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' For example, differences in community sensitivity to N-induced acidification depend on soil buffering capacities (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Still, all these mechanisms can potentially reduce performance independent of species interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Recently, Band et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', (2022) conducted a global meta-analysis of ~600 experiments using biomass as a proxy for density-dependent effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' They found that the biomass-mediated effect of N leads to an average decline of ~2% in diversity, while the biomass-independent effect causes a more substantial diversity decline of ~18%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' These findings are surprising given that many competition experiments have established a causal link between nitrogen addition, competition, and lower species diversity(Gurevitch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 1990, Lamb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2009, DeMalach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' One possible explanation for the results of Band et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' (2022) is that biomass is not a good proxy for density-dependent processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Many studies have shown that biomass is affected by multiple factors (resources, nutrients, herbivores, species composition, litter production) and that the causality of the biomass-richness relationship is bidirectional (Grace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Alternatively, it is possible that even within the same community, some species experience reduced performance due to density-independent processes while other species experience reduced performance due to competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Surprisingly, despite the numerous N-addition experiments, we are unaware of any experiment that separated the density-dependent and independent effects of N enrichment on the performance of different species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Here, we manipulated sowing density as the most direct approach to disentangle density-dependent and independent processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' This simple manipulation has been widely used for monocultures but rarely for communities (but see Goldberg & Estabrook, 1998, Rajaniemi, Turkington, & Goldberg, 6 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' We used this approach to investigate the effects of N availability on species performance within an experimental community that included a small forb (Silene palaestina), a medium size legume (Trifolium purpureum), and a tall competitive grass (Avena barbata).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' We expected that the tall grass would benefit from N addition because tall grasses are often strong competitors for light (Suding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2005, Vojtech et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2007, Gough et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2012, DeMalach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2017b) and are insensitive to the detrimental effects of high N availability (Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' In contrast, since the abundance of forbs and legumes often decreases with N availability (Suding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2005, Tognetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2021), we hypothesized that N addition would increase their sensitivity to competition (competitive response sensu Miller & Werner, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Additionally, we expected the legume and forb species to experience density-independent detrimental effects of N addition (Band et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Still, we predicted that the density-independent effect would be minor compared to competition within the scope of our short-term experiment (see also Eskelinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2022) because toxic effects in the soil often accumulate over time (Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' MATERIALS AND METHODS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content="2 Experimental design The experiment was conducted at the experimental station of the Hebrew University of Jerusalem in Rehovot, Israel (31°54'18." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='8 "N, 34°48\'16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='9 "E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The station is characterized by a Mediterranean climate, with a mild, rainy winter and a hot, dry summer (annual rainfall of ~560mm yr-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=" The experiment mimicked the annual plant communities that grow on Hamra soils of Israel's coastal region." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' These communities grow during winter, bloom in spring, and dry in the late spring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Soil type is a primary driver of species composition in these communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Communities growing on a sandy-type Hamra are characterized by high diversity and abundance of forbs and legumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' In contrast, loamy-type Hamra (with higher nutrients and water availability) has a lower diversity and high dominance of tall grasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The experimental plant communities were sown in January 2021 in large plastic containers (dimensions of ~1m × 1m × 1m) with holes for water drainage (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The experiment included 120 experimental communities (3 soil types × 4 nutrient treatments × 2 sowing densities × 5 replications) randomly located in an area of ~half a hectare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' We filled the containers with soils 7 (taken from at least 1m depth to avoid seed bank) from three nearby locations aiming for a gradient of clay content (from sandy to loamy soil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' However, two sources had almost identically low clay content (and water-holding capacity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Thus, to increase statistical power, these two sources were pooled into a single category (hereafter sandy soil ~7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='5% clay content) and contrasted with the other source, red soil (hereafter loamy soil ~ 15% clay content).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=" We chose four model species that were collected in Israel's coastal plain (by a commercial company, 'Hila Pirchei Bar Ltd'): Silene palaestina (Caryophyllaceae, seed mass of 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='095mg), Trifolium purpureum (Fabaceae, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='47mg), Avena barbata (Poaceae, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='12mg), and Lupinus palaestinus (Fabaceae, 257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='3mg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The species were chosen based on the following criteria: (i) being common in Hamra communities such that a sufficient amount of seeds was available (1000g per species).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' (ii) Representing different functional groups (forbs, legumes, and grasses) and a wide range of seed masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' We fertilized the plots a few days after sowing in January 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The four fertilization treatments included: (i) no nutrient addition, (ii) the addition of all nutrients except N (P, K, and micronutrients following NutNet protocol, Borer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', 2014), (iii) Low N (3gN m-2, applied as urea) together with all other nutrients, and (iv) high N (15gN m-2) with all other nutrients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=" Since we focused on N, we removed the potential limitation of all other nutrients by adding them to all plots and viewed treatment ii as another 'control'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' However, to quantify the potential limitations of other nutrients, we also included a standard control with no addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Hereafter, we refer to the no- nutrient treatment as the control to avoid confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Under each nutrient treatment, the plant communities (all four species) were sown in two densities, low (3g m-2) and high (16g m-2), separated equally across the four species (in terms of weight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Sowing species in equal weight (rather than equal seed number), implies that the number of sown seeds is higher for small-seeded species (and vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The rationale of this choice is that small- seeded species have a lower emergence probability (Table S1) and lower survival probability of seedlings (Tables S2, S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' We chose 16g m-2 as the high sowing density following previous studies of annual plant communities using this amount as the highest end of a density gradient (Godoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2014, Pérez- Ramos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The low-density level was chosen to reduce competitive interactions (although we did not assume that interactions are entirely eliminated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=" At the same time, a major requirement 8 of our experiment was that even under low density, there would be enough individuals to calculate species' performance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=" Thus, we have sown only four species (having more species would reduce each species' density, i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', separating the 3g among more species).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' In the experiment, the densities were sufficient for all species except Lupinus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Lupinus was missing in many (low-density) plots and excluded from the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Throughout the growing season, we weeded all species that were not part of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Additionally, the experiment was irrigated weekly during the spring (March-April).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Otherwise, the containers would dry faster than the surrounding vegetation because of their high drainage (DeMalach, Ron, & Kadmon, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='3 Sampling We estimated population density by sampling seedlings during February 2021 and February 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Sampling was conducted in four 20cm × 20cm quadrates in each plot in the first year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' However, due to logistic constraints, we sampled in two quadrates only for Silene and Trifolium and in four quadrates for Avena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' During the second-year sampling, we avoided spots where biomass was collected in the first year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' To quantify individual biomass, we collected biomass samples from two 20cm × 20cm quadrates in the center of each plot (at the end of the first growing season, May 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' First, the biomass was oven-dried at 600C for 48hrs and sorted into species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Then, we counted the number of individuals of each species and weighed their biomass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' In the analysis, all samples within a plot were aggregated (by taking the mean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Lastly, density units were converted from individuals per sample into individuals per square meter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', multiplied by 25, the ratio between a quadrate area and a square meter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='4 Statistical analyses All statistical analyses were conducted with R (version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' We tested the effects of the treatments on the probability of reaching reproduction (the ratio between the number of sown seeds and the number of adults in the first year), biomass growth (the ratio between adult and seed masses), and population growth (the ratio between seedling densities in two consecutive years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' For each species in each soil type, we built the following linear regression model using dummy variables to describe the four nutrient treatments and the two sowing density levels: 9 𝑌 = 𝛽0 + 𝛽1𝐷𝑒𝑛𝑠𝑖𝑡𝑦 + 𝛽2𝑀𝑖𝑐𝑟𝑜 + 𝛽3𝑁3 + 𝛽4𝑁15 + 𝛽5(𝐷𝑒𝑛𝑠𝑖𝑡𝑦 × 𝑁15)𝑖𝑗 + 𝜀, 𝜀 ~ 𝑁 (0, 𝜎𝜀 2) Here, Y is a performance attribute (reproduction probability, biomass growth, or population growth rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Density represents the sowing density (low[0] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' high[1]), and Micro indicates whether or not PK and micronutrients were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The variables N3 and N15 represent the addition of low and high levels of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Using dummy variables rather than a continuous variable of N level was needed to avoid assuming a linear response to N addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The model includes an interaction term between sowing density and high levels of N (interactions with other nutrient treatments were insignificant and therefore removed to increase the statistical power and avoid overfitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Notably, a significant interaction between N addition and sowing density is interpreted as evidence of a density-dependent response to N (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', N addition modifies competitive interactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' In contrast, when N has a significant effect, but there were no interactions, we interpreted it as evidence that the response to N addition is primarily density- independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' To ensure normally distributed residuals in the linear regression (𝜀 ~ 𝑁 (0, 𝜎𝜀2), biomass and population growth were loge-transformed while reproduction probability was logit-transformed (𝑙𝑜𝑔𝑖𝑡(𝑝) = log𝑒( 𝑝 1−𝑝) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' To avoid zeros in the logit transformation (leading to undefined values), we added 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='01 to all observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Additionally, for Avena, there were a few cases where reproduction probability was equal to or greater than one (because the density of the whole plot was extrapolated from quadrates and spatial distribution was non-homogenous), and those values were converted to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='99 (logit of one is also undefined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 10 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' (A) a photo of the experiment (B) a scheme of the experimental design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Two sowing densities (low vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' high) were applied to quantify both density-independent and density- dependent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Besides density manipulation, the experiment included a factorial combination of two soil types (sandy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' loamy) and four fertilization treatments: (1) no nutrient addition (control), (2) the addition of all nutrients (P, K, and micronutrients) except N (Micro), (3) Low N together with all other nutrients (N3), and (4) high N with all other nutrients (N15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' (A) (B) Sandy soil Loamy soil Lowdensity Highdensity Lowdensity Highdensity Control Micro N3 N1511 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' RESULTS Reproduction probability, the probability of the seed reaching the reproduction period, was low in the small-seeded Silene, intermediate in Trifolium, and highest for the large-seeded Avena (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2, Tables S2 and S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The effect of sowing density was negative for Silene (Psandy < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='001, Ploamy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='004), positive for Trifolium in the sandy soil (P < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='001), and insignificant for Avena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The effects of N on reproduction probability were insignificant except for a marginally significant positive effect on Silene in the sandy soil (P N15 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='065).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The addition of other nutrients reduced the reproduction probability of Trifolium in the sandy soil (P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='003) and increased it for Silene in the loamy soil (P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='04).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Biomass growth, the ratio between adult and seed biomass, was highest in the small-seeded Silene, intermediate in Trifolium, and lowest for the large-seeded Avena (Fig 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The effects of increasing sowing density on biomass growth were strongly negative for all the species indicating the dominance of competitive interactions in the system (Table S4, S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' High levels of nitrogen addition increased the biomass growth of Silene (Ploamy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='04) and Avena (Ploamy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' However, the other nutrients did not affect biomass growth except for a positive effect on Trifolium (Ploamy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Importantly, the biomass growth of Trifolium was reduced in the sandy soil under the high nitrogen addition (Psandy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='058) and in loamy soils under low and high N addition (Ploamy, N3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='038, Ploamy, N15 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=" Strikingly, in low-density plots where all nutrients were added, Trifolium's biomass growth was reduced from ~510 to ~210 and ~140 under low and high nitrogen levels, respectively." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Moreover, we found no interactions between N and density, indicating that the detrimental effects of N were density-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Increasing sowing density reduced the population growth (the ratio between seedling densities in the two years) of all species, demonstrating that competition strongly affects population dynamics (Fig 4, Tables S6, S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Notably, Silene had strong interactions between growth rate and density (Ploamy, N15*Density = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Under high nitrogen, its population growth rate was reduced from 2 (population increase) under low density to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='4 (population decline) under high density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' In contrast, the population growth of Trifolium was unaffected by N (but was reduced in the loamy soil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Lastly, for Avena, the population growth increased under high N levels in the sandy soils regardless of sowing density (Psandy= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=" Additionally, we found no effect of adding other nutrients on species' population growth, except for a marginal increase for Trifolium (Ploamy = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 12 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The effects of fertilization treatments and sowing density on the probability of reaching reproduction (the ratio between density in the spring and sowing density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The blue and red circles represent the means in the low and high sowing densities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The error bars are standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' (Mi) - addition of all nutrients (P, K, and micro) except N, (N3), and (N15) are treatments with N addition of 3g m-2 and 15g m-2 (together with all other nutrients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The left and right panels show the results for sandy and loamy soil types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' See table S1 for statistical analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Note the different ranges of the y-axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' (a) silene (sandy soil) (b) silene (Loamy soil 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='07 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='low density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='06 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' high density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='02 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='00 Control Mi N3 N15 Control Mi N3 N15 (c) Trifolium (Sandy soil) (p) Trifolium (Loamy soil) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 Control Mi N3 N15 Control Mi N3 N15 (e) Avena (Sandy soil) (f) Avena (Loamy soil) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 Control Mi N3 N15 Control Mi N3 N15 Fertization treatment Fertilization treatment13 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The effects of fertilization treatments and sowing density on the biomass growth rate (the ratio between seed and adult masses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The blue and red circles represent the means in the low and high sowing densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The error bars are standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' (Mi) - addition of all nutrients (P, K, and micro) except N, (N3), and (N15) are treatments with N addition of 3g m-2 and 15g m-2 (together with all other nutrients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The left and right panels show the results for sandy and loamy soil types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Note the logarithmic scale of the y-axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' (a) silene (Sandy soil) (b) silene (Loamy soil) rowth 5000 5000 2000 2000 1000 1000 iomass 500 500 200 200 100 low density 100 50 high density 50 B 20 20 Control Mi N3 N15 Control Mi N3 N15 (c) Trifolium (Sandy soil) (d) Trifolium (Loamy soil) 5000 5000 2000 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 1000 1000 500 500 200 200 100 100 50 50 20 20 Control Mi N3 N15 Control Mi N3 N15 (e) Avena (Sandy soil) (f) Avena (Loamy soil) 5000 5000 2000 2000 1000 1000 500 500 siomass 200 200 100 100 50 50 20 B 20 10 10 Control Mi N3 N15 Control Mi N3 N15 Fertilization treatment Fertilization treatment14 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The effects of fertilization treatments and sowing density on the population growth rate (the ratio between seedlings densities in year two and year one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The dashed line represents the growth rate of one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', no change in population size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The blue and red circles represent the means in the low and high sowing densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The error bars are standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' (Mi) - addition of all nutrients (P, K, and micro) except N, (N3), and (N15) are treatments with N addition of 3g m-2 and 15g m-2 (together with all other nutrients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The left and right panels show the results for sandy and loamy soil types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Note the logarithmic scale and the different ranges of the y-axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' (a) silene (Sandy soil) (b) silene (Loamy soil) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 low density 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 high density 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='2 Control Mi N3 N15 Control Mi N3 N15 (c) Trifolium (Sandy soil) (d) Trifolium (Loamy soil) 500 500 200 200 100 100 50 50 20 20 10 10 5 5 2 2 Control Mi N3 N15 Control Mi N3 N15 (e) Avena (Sandy soil) (f) Avena (Loamy soil) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='2 Control Mi N3 N15 Control Mi N3 N15 Fertilization treatment Fertilization treatment15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' DISCUSSION We tested the role of density-dependent and density-independent mechanisms in reducing species performance with increasing N availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' In our experimental system, biomass growth and population growth decreased with sowing density for all species, implying that species interactions were predominantly negative, a necessary condition for testing N-induced density-dependent mechanisms of reduced performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Additionally, we found that N limited growth for the forb and grass species but not for the legume species that responded positively to the addition of other resources (probably because of phosphorus limitations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' We demonstrated evidence for density-dependent and density-independent performance reductions following N addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The legume species (Trifolium) declined in biomass growth under high N levels regardless of density, indicating a density-independent detrimental effect of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' In contrast, the small forb (Silene) experienced lower population growth with N addition but only under high density, demonstrating a density-dependent effect (competition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Our results agree with previous empirical studies in the region (DeMalach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', 2017b) as well as places (Xia and Wan 2008, Bobbink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2010, Maskell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2010, Eskelinen and Harrison 2015, DeMalach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2017a, Seabloom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2021, Vázquez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2022), showing that tall competitive grasses like Avena benefit from N enrichment while legumes and forbs decline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' However, our findings further suggest that this reduction is driven by different processes, density-dependent mechanisms for forbs and density-independent mechanisms for legumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Below, we discuss the implications and interpretations of our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='1 Density-independent mechanisms Many studies have shown that N enrichment affects soil chemical properties (Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2020) and microbiomes (Lekberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2021, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2021) with the potential to reduce species diversity regardless of the competitive interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' In accordance, global analyses have shown that the negative effects of N on diversity are significant even after controlling for its effects on biomass (Band et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2022) or light (DeMalach, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Moreover, a causal link between N-induced acidification and diversity decline has been established by adding calcium to buffer soil acidification, thereby reducing species loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' However, all the above studies do not provide direct evidence for density-independent mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' For example, N-induced acidification can also change competitive interactions (Yao and Feng 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The same reasoning applies to the soil 16 microbiome, affecting plant performance in density-dependent and independent manners (Ke and Wan 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Thus, as far as we know, this is the first study to quantify the density-independent effects of N directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' In our system, we expected density-independent mechanisms to be relatively weak because soil acidification is unlikely (high limestone content, pH ~ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='8), and the time scale is too short for long- term changes in other soil properties (Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Nonetheless, we found a very strong density-independent response of Trifolium, which experienced up to a three-fold reduction in biomass growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=" Furthermore, we found that Trifolium's density-independent biomass was higher in the fertile loamy soil compared with the poor sandy soil." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Still, the differences between the soils could be explained by lower aeration rather than a nutrient-induced response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' A possible driver of reduced performance is N toxicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' While grass species have a rhizosheath (a layer of adhering soil particles to the root surface) acting as a "biofilm-like shield" enhancing their tolerance to N-induced stress, other species do not (Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Alternatively, the reduction in performance of Trifolium could be related to changes in its microbiome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=" Such interpretation is supported by the finding that Trifolium's performance is highly reduced without its microbiome (Ayilara 2022)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='2 Density-dependent mechanisms Based on theoretical expectations, we predicted that the small forb would experience a density- dependent reduction in survival and biomass growth with increasing nitrogen availability because of asymmetric competition for light (DeMalach, Zaady, Weiner, & Kadmon, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Nonetheless, we found no evidence that sensitivity to competition in reproduction probability and biomass growth increases with N availability (no interactions between density and nitrogen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Moreover, the biomass of Silene increased rather than declined with N regardless of density (no interactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' A strong interaction between N availability and density was found for the population growth rate of Silene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' While its population increased under low density, it declined under high density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' We attribute the decline in population growth rate with N to a low establishment probability of this small-seeded species when there was high litter from the previous year (Thompson, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' This finding highlights the importance of measuring the actual population growth rate rather than using biomass growth rate or fecundity as proxies in competition experiments (Goldberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Currently, empirically-calibrated models of species interactions incorporate density-dependent 17 effects on fecundity only (Stouffer 2022), but we hope that future studies will also integrate density-dependent effects on the establishment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='3 Methodological issues The experimental design of competition experiments along environmental gradients involves two main tradeoffs: (i) realism vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' minimizing noise and (ii) resolution of environmental conditions vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' resolution of species interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' For the first tradeoff, we chose an intermediate setup, while for the second, we focused on maximizing the number of environmental treatments (soil types and fertilization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' The main advantage of growing plants in containers is reducing noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' We created homogenous conditions and differentiated between two soil types (the natural landscape is patchy with many intermediate levels of clay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Additionally, the containers allowed for minimizing the need to remove weeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' However, these containers cannot fully mimic the natural soil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Thus, the next challenge is testing for density-dependent and density-independent mechanisms of N enrichment in even more realistic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' For example, when plants grow in the natural soil and nitrogen is deposited in low quantities for a long time rather than in high quantities for a short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Importantly, our experiment is closer to the real world than many greenhouse experiments because the plants experienced ambient climatic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Furthermore, the plants grew in large containers rather than in the few-liter pots used in most competition experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' We believe that large containers are crucial to quantify density-dependent performance because they enable niche partitioning of species varying in root depth and sufficient shading to produce light competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Recently, Hart, Freckleton & Levine (2018) highlighted the need to separate conspecific and heterospecific competition to calibrate a phenomenological model of community dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Assuming strictly pairwise interactions and accurate estimation of the model parameters, such a model can predict long-term coexistence patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=" While we appreciate this approach's strength, it requires many competition treatments, including monocultures, all pairwise combinations, and several densities (Godoy et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Therefore, as far as we know, it was never applied under more than two environmental conditions (Wainwright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2019, Pérez-Ramos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2019, Van Dyke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Similarly, experiments where a monoculture of each species is compared to a mixture also require many combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' To the best of our knowledge, the monoculture-mixtures 18 approach has not been applied to an experiment like ours that include eight types of conditions (two soil types and four nutrient treatments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Following Goldberg and her colleagues (Goldberg & Estabrook, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Rajaniemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', 2009), we lumped together all density-dependent effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Of course, such pooling cannot capture all the fine details of the competition nor predict long-term coexistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=" Yet, the strength of Goldberg's approach is its simplicity, i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=', it requires only two treatments, low and high density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' This simplicity enabled us to investigate density-dependent and independent performance under eight environmental conditions (two soil types and four fertilization treatments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=" Moreover, another advantage of Goldberg's approach is that it does not require simplifying assumptions, such as interactions being always negative and strictly pairwise (Kleinhesselink et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2022, Stouffer 2022) or species establishment being density-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Indeed, despite the prevalence of negative interactions in our system, positive interactions also occur (reproduction probability of Trifolium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Similarly, the mounting evidence of the role of litter in the establishment of plants suggests that density-dependent effects during the establishment should not be ignored (Facelli and Pickett 1991, Foster and Gross 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content='4 Conclusion We showed that density-independent and density-dependent mechanisms operate together to reduce species performance under high levels of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Specifically, we demonstrated a density- independent effect for the legume species and a density-dependent effect for the forb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' It remains to be tested whether these results can be generalized to other legumes and forbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Moreover, a critical open question is how different environmental conditions affect density-dependent and density-independent responses to N enrichment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' This question is timely given the significant role of N eutrophication in driving biodiversity loss worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Hence, the simplicity of the approach applied here makes it a promising way forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' REFERENCES Ackerman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Global estimates of inorganic nitrogen deposition across four decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Global Biogeochem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Cycles 33: 100–107.' metadata={'source': 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+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' - J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Ecol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 106: 1902–1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Hautier, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Competition for Light Causes Plant Biodiversity Loss After Eutrophication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' - Science 324: 636–638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Huang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Plant–soil biota interactions explain shifts in plant community composition under global change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' - Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Ecol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 35: 2778–2788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Hutchinson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Concluding Remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' - Cold Spring Harb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 22: 415– 427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Isbell, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Low biodiversity state persists two decades after cessation of nutrient enrichment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' - Ecol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 16: 454–460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Ke, P.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 90: e01391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Kleinhesselink, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Detecting and interpreting higher-order interactions in ecological communities.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Shoot, but not root, competition reduces community diversity in experimental mesocosms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' - J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' Ecol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 97: 155–163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' 22 Lan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfqvhr/content/2301.00548v1.pdf'} +page_content=' et al.' metadata={'source': 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sha256:17a24f1da9154f938d5d611bb596fbe8576b773d08190a75d92f5bd5c8e69fe0 +size 4128813 diff --git a/bNFIT4oBgHgl3EQfmCv-/vector_store/index.pkl b/bNFIT4oBgHgl3EQfmCv-/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..f5d57bc2e5650ce7f4e8f119fdb7f1eb6403be55 --- /dev/null +++ b/bNFIT4oBgHgl3EQfmCv-/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9cd0b295f140abaf6e98c008b618cd87828fa425552eb754201b587e25453088 +size 231265 diff --git a/cNE4T4oBgHgl3EQfPwwb/content/tmp_files/2301.04975v1.pdf.txt b/cNE4T4oBgHgl3EQfPwwb/content/tmp_files/2301.04975v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f1dd7a7335a6116e486df0a46999bea43294b0a4 --- /dev/null +++ b/cNE4T4oBgHgl3EQfPwwb/content/tmp_files/2301.04975v1.pdf.txt @@ -0,0 +1,2649 @@ +arXiv:2301.04975v1 [math.OA] 12 Jan 2023 +EQUIVARIANT COVERING SPACES OF QUANTUM +HOMOGENEOUS SPACES +MAO HOSHINO +Abstract. We develop a fundamental theory of compact quantum group +equivariant finite extensions of C*-algebras. +In particular we focus on the +case of quantum homogeneous spaces and give a Tannaka-Krein type result +for equivariant correspondences. As its application, we show that every Jones’ +value appears as the index of an equivariant conditional expectation. In the +latter half of this paper, we give an imprimitivity theorem in some cases: for +general compact quantum groups under a finiteness conditions, and for the +Drinfeld-Jimbo deformation Gq of a simply-connected compact Lie group G. +As an application, we give a complete classification of finite index discrete +quantum subgroups of � +Gq. +1. Introduction +The notion of a C*-tensor category has played an important role in recent +developments of theories of operator algebras. It firstly appeared in the subfactor +theory, as a tool of classification of subfactors: Actually every inclusion of factors +gives a C*-tensor category as an invariant, which is complete in the amenable case +([Po94, Remark 7.2.1]). Moreover we also can use it to understand V. F. Jones’ +celebrated work on the range of indices of subfactors ([Jo83, Theorem 4.3.1]). +C*-tensor categories also appear in the theory of compact quantum groups, as +their representation categories. Moreover the notion of a module category over +a C*-tensor category is also useful. By using this, the classical Tannaka-Krein +duality is generalized not only to a compact quantum group itself, but also to +its action on C*-algebra ([DY13, Theorem 6.4]). De Commer and Yamashita use +this duality theorem to obtain a one-to-one correspondence between quantum ho- +mogeneous spaces of SUq(2) and concrete combinatorial datum ([DY15, Theorem +2.4]). +The purpose of this paper is to connect these C*-tensor categorical approaches +and study an inclusion of C*-algebras with actions of a compact quantum group. +In this paper, we call it as an equivariant finite quantum covering spaces, since +it can be regarded as a genuine finite covering space in the case of commutative +C*-algebras. We also call its minimal indices as its covering degree. As in the +non-equivariant case, an equivariant finite quantum covering space can be un- +derstood as a Q-system in a suitable C*-tensor category, namely the category of +equivariant correspondences. The following theorem on this category, which is +a generalization of the observation due to De Commer and Yamashita ([DY13, +Theorem 7.1]), is fundamental throughout this paper. +2020 Mathematics Subject Classification. Primary 46L67, Secondary 46L08, 17B37. +Key words and phrases. operator algebra, quantum group, tensor category. +1 + +2 +MAO HOSHINO +Theorem 1 (Theorem 3.14). For quantum homogeneous spaces A and B of a +compact quantum group G, we have the following canonical equivalence of C*- +categories: +G- Corrrf +A,B ∼= [G- Modf +A, G- Modf +B]Repf G. +The first application of this result is the following existence theorem. +Theorem 2 (Theorem 3.15). For any d ∈ {4 cos2(π/n) | n ≥ 3} ∪ [4, ∞), we +have a compact quantum group G and a finite quantum G-covering space A ⊂ B +with its covering degree d. +In Section 4, we focus on the G-actions induced from the maximal Kac quantum +subgroup K of G. Instead of treating with them directly, we work on Repf G- +module categories and module functors between them. This enables us to trans- +late the imprimitivity theorem to the comparison theorem of Repf G-module func- +tors and Repf K-module functors. Then we develop a general theory of module +categories admitting a module trace, concluding the following theorem for actions +of compact quantum groups. +Theorem 3 (Theorem 4.29, Theorem 4.31, Theorem 4.33). Let A be a quantum +homogeneous space of K and �A be its induced G-C*algebra. +(i) If A has a tracial state and Irr K- Modf +A is finite, the induction func- +tor gives an equivalence of C*-tensor categories: K- Corrrf +A ∼= G- Corrrf +� +A. +Moreover both of them are rigid. +(ii) If A is induced from a quantum homogeneous space of a cocommutative +quantum subgroup of K, the induction functor gives group isomorphisms +PicK(A) ∼= PicG( �A) and AutK(A) ∼= AutG( �A). +The statement (i) contains the proceeding research [CKS, So05] on the quantum +Bohr compactification as a special case. +In Section 5, we deal with the Drinfeld-Jimbo deformation. By using the well- +developed representation theory of C(Gq) and C(T\Gq), we show the following +theorems. +Theorem 4 (Corollary 5.5). Let A, B be quantum homogeneous spaces of T +and �A, �B be its induced Gq-C*algebras. +Then the induction functor gives an +equivalence of C*-categories: T- Corrrf +A,B ∼= Gq- Corrrf +� +A, � +B +Theorem 5 (Corollary 5.6). Let �A be a quantum homogeneous space containing +C(T\Gq) and admitting a tracial state. Then it is of the form Ind Gq +T A, where A +is a quantum homogeneous space of T. +At the last of this paper, we use the result in Section 4 to give a complete +classification result of finite index discrete quantum subgroups of � +Gq. +Theorem 6 (Theorem 5.11). Let P (resp. Q) be the weight (resp. root) lattice +of G. There is a canonical one-to-one correspondece between finite index discrete +quantum subgroups of � +Gq and subgroups of P/Q. + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +3 +2. Preliminaries +In this paper, the symbol – ⊗ – denotes the spatial tensor product of C*- +algebras, the algebraic tensor product of C-vector spaces and the external tensor +product of Hilbert C*-modules. +For a Hilbert space H, its inner product ⟨–, –⟩ is C-linear with respect to the +second argument. An element ξ ∈ H is considered as an operator from C to H. +If H is finite dimensional, the unnormalized trace on B(H) is denoted by Tr. +2.1. Compact quantum groups and their dual. In this subsection, we give +a brief review on compact quantum groups. See [NT13] for detailed discussions. +A compact quantum group is a pair G = (A, ∆) of a unital C*-algebra A and +∗-homomorphism ∆: A −→ A ⊗ A which satisfies the following conditions: +• the coassociativity (∆ ⊗ id)∆ = (id ⊗∆)∆, +• the cancellation property (A ⊗ C)∆(A) = (C ⊗ A)∆(A) = A ⊗ A. +In this case A is denoted by C(G). A Haar state on G is a state h on C(G) +satisfying the bi-invariance property (h⊗id)∆(x) = (id ⊗h)∆(x) = h(x)1G. Such +a state always exists and is unique. It is said that G is reduced when h is faithful. +We only consider reduced compact quantum groups unless otherwise noted. +A unitary representation of G is a pair π = (Hπ, Uπ) of a Hilbert space Hπ +and a unitary Uπ ∈ M(K(Hπ) ⊗ C(G)) which satisfies (id ⊗∆)(Uπ) = Uπ,12Uπ,13. +The category of unitary representations of G is denoted by Rep G, and its full +subcategory of the finite dimensinal ones is denoted by Repf G. +Let π be a finite dimensional unitary representation of G and ξ, η be elements +of Hπ. +Then (ξ∗ ⊗ 1)Uπ(η ⊗ 1) defines an element of C(G), called a matrix +coefficient of π. The set of all matrix coefficients of all finite dimensional unitary +representations is called the algebraic core of G and denoted by O(G). We can +make O(G) into a Hopf ∗-algebra by using the product and the coproduct of +C(G). Its counit and antipode are denoted by ε and S, respectively. In general +S satisfies S(S(x)∗)∗ = x for any x ∈ O(G), but does not S2 = id. It is said that +G is of Kac type when it holds. +Let H be another compact quantum group. A homomorphism from H to G is +a homomorphism of Hopf ∗-algebras from O(G) to O(H). One should note that +ϕ need not be defined on C(G). On the other hand we can still define (ϕ ⊗ id)∆ +as a ∗-homomorphism from C(G) to C(H) ⊗ C(G). More generally, we have the +following lemma, which is a quantum analogue of the Fell’s absorption principle +for �G: +Lemma 2.1. Let A be a C*-algebra and ϕ: O(G) −→ A be a ∗-homomorphism. +Then (ϕ ⊗ id)∆ extends to a ∗-homomorphism from C(G) to A ⊗ C(G). +Proof. Let (π, L2(G)) be the GNS representation of C(G) with respect to the +Haar state of G. Since G is reduced, this representation is faithful. +It suffices to show in the case of A = B(H) for a Hilbert space H. Let V be +an operator on H ⊗ L2(G) satisfying +V (ξ ⊗ Λ(x)) = (ϕ ⊗ π)(∆(x))(ξ ⊗ Λ(1)). + +4 +MAO HOSHINO +Then V is unitary and satisfies V (1 ⊗ π(x))V ∗ = (ϕ ⊗ π)(∆(x)). +Then the +statement follows from the faithfulness of π. +□ +If we are given a Hopf ∗-algebra A generated by matrix coefficients of its finite +dimensional unitary representations, we can construct a compact quantum group +whose algebraic core coincides with A. +This fact allows us to construct the +maximal Kac quantum subgroup. +Definition 2.2. The maximal Kac quantum subgroup of G is the compact quan- +tum group corresponding to a Hopf ∗-algebra obtained as the quotient of O(G) +divided by a two-sided ideal generated with S2(x) − x for all x ∈ O(G). +Remark 2.3. Actually K is of Kac type and has the following universal property: +Let H be a compact quantum group of Kac type. Then any homorphism from +H to G factors through K. This follows from that any homomorphism of Hopf +∗-algebra preserves antipodes. +Remark 2.4. Let Gu be the universal form of G and �K be its canonical Kac sub- +group, introduced in [So05, Appendix]. Then we have a canonical ∗-homomorphism +�q: O(G) −→ O(�K). This map satisfies �q ◦ S2 = �q, hence �q factors through O(K). +On the other hand, the universal property of Gu implies that a canonical map +q: O(G) −→ O(K) extends to ∗-homomorphism from Cu(G) to Cr(K). Since +the Haar state of K is faithful on Cr(K), this map factors through C(�K) by its +construction. This implies that q factors through O(�K) and gives an isomorphism +from O(�K) to O(K). +Example 2.5. Let G be a semisimple compact Lie group and q ∈ (−1, 1). Then +there is a compact quantum group Gq called the Drinfeld-Jimbo q-deformation of +G, see [NT13, Definition 2.4.5.] for a precise definition. The representation theory +of Gq is quite similar to that of G, which means that they have the common set +of equivalence classes of irreducible representations and a common fusion rule. +But actually Repf Gq is not equivalent to Repf G as a C*-tensor category. +It is shown that the maximal Kac quantum subgroup of Gq coincides with the +maximal torus T of G ([To07, Lemma 4.10.]). +We end this subsection with the notion of discrete quantum group. Its definition +is given by the Pontrjagin dual of a compact quantum group, hence the reduced +C*-algebra of discrete quantum group Γ is nothing but C(G) when Γ is the +Pontrjagin dual of G. In this case Γ is denoted by �G. +For the notion of closed quantum subgroup, see [DKSS]. There are several +approaches, but all of them are equivalent in the discrete case ([DKSS, Theorem +6.2]). As a result, we can take the following definition: a closed quantum subgroup +of G is a pair of discrete quantum group �H and a coproduct-preserving unital +inclusion C(H) ⊂ C(G). +In this case a unitary representation of H gives a +unitary repsentation of G with the same intetwiner space. Combining this with +the Woronowicz’s Tannaka-Krein duality ([NT13, Theorem 2.3.2]), we have a one- +to-one correspondence between discrete quantum subgroups of �G and C*-tensor +full subcategories of Repf G. + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +5 +2.2. Actions of compact quantum groups. For detailed discussions and ref- +erences, see [DC16]. Let G be a compact quantum group. A (reduced) G-C*- +algebra or G-action on a C*-algebra is a pair (A, α) of a C*-algebra A and faithful +∗-homomorphism α: A −→ A ⊗ C(G) with the following conditions: +• (id ⊗∆)α = (α ⊗ id)α, +• span (C ⊗ C(G))α(A) = A ⊗ C(G). +Its fixed point subalgebra {x ∈ A | α(x) = x ⊗ 1} is denoted by AG or Aα. We +also define the algebraic core of A as A = {x ∈ A | α(x) ∈ A ⊗ O(G)}. Then we +have a right coaction of the Hopf ∗-algebra O(G) on A given by the restriction +of α to A. +We say that G-C*-algebra A is a quantum homogeneous space of G if it is unital +and AG = C1A. +Example 2.6. The C*-algebra C(G) has a canonical G-action defined by ∆. +Moreover, every G-invariant C*-subalgebra of C(G) also has a G-action and +actually is a quantum homogeneous space of G. We call it a right coideal of G. +Example 2.7. Let H be a compact quantum subgroup of G and q: O(G) −→ +O(H) be the canonical map. Then (q ⊗ id)∆ extends to a ∗-homomorphism ℓH +on C(G) by Lemma 2.1. Since ℓH is G-equivariant with respect to the canonical +G-action on C(G), we have a right coideal C(H\G) = {x ∈ C(G) | ℓH(x) = 1⊗x}. +More generally, for a H-C*-algebra, the induced G-C*-algebta of B is defined +as a pair of the following: +• Ind G +HA = {x ∈ A ⊗ C(G) | (α ⊗ id)(x) = (id ⊗ℓH)(x)}, +• �α = the restriction of id ⊗∆. +For example, the induced G-C*-algebra of the trivial H-C*-algebra C is isomor- +phic to C(H\G). +If A and B are G-C*-algebras with the algebraic cores A and B respectively, we +say that a linear map ϕ: A −→ B is G-equivariant when (ϕ ⊗ id)α = βϕ holds. +Similarly the equivariance for ψ: A −→ B is also defined when ψ ⊗ id: A ⊗ +C(G) −→ B ⊗ C(G) is defined. +The following generalization of Lemma 2.1 is useful to get a G-equivariant +c.p. map. +If A is ∗-algebra and B is C*-algebra, we say that a linear map +ϕ: A −→ B is completely positive if (ϕ ⊗ id)(X∗X) is positive in B ⊗ Mn(C) for +any X ∈ A ⊗ Mn(C). +Proposition 2.8. Let A and B be G-C*-algebras with the algebraic cores A and +B respectively. If ϕ: A −→ B is G-equivariant and completely positive, it extends +to a G-equivariant c.p. map from A to B. +For a proof of this proposition, see Subsection A.1. +Next we move on the notion of modules over G-C*-algebras. +Definition 2.9. Let A be a G-C*-algebra. A G-equivariant Hilbert A-module is +a pair (E, αE) of a Hilbert A-module E and a linear map αE : E −→ E ⊗ C(G) +with the following conditions: +• α(⟨ξ, η⟩A) = ⟨αE(ξ), αE(η)⟩A⊗C(G) for any ξ, η ∈ E, +• span αE(E)(C ⊗ C(G)) = E ⊗ C(G). + +6 +MAO HOSHINO +Here E ⊗ C(G) means the external tensor product of the Hilbert A-module E +and the Hilbert C(G)-module C(G). +If A is a quantum homogeneous space and E is finitely generated, E must be +projective ([DC16, Theorem 6.21.]). +The category of G-equivariant Hilbert A-module is denoted by G- ModA, and +its full subcategory of finitely generated ones is denoted by G- Modf +A. We use the +symbol LA(–, –) for the spaces of adjointable right A-module maps. We also use +LG +A(–, –) for the spaces of G-equvariant ones. +For E ∈ G- Modf +A, we have a G-action �α: G ↷ KA(E) on the algebra of com- +pact A-module maps. Moreover we can extend this action to a ∗-homomorphism +�α: LA(E) −→ LA⊗C(G)(E ⊗ C(G)), and this satisfies (�α ⊗ id)�α = (id ⊗∆)�α. For +any T ∈ LA(E), the operator �α(T) is the only operator on E ⊗ C(G) satisfying +�α(T)αE(ξ) = αE(Tξ). +We can also define the algebraic core and the induction of equivariant Hilbert +C*-modules as follows: +• E = {ξ ∈ E | αE(ξ) ∈ E ⊗ O(G)}, +• Ind G +HF = {x ∈ F ⊗ C(G) | (αF ⊗ id)(x) = (id ⊗ℓH)(x)}. +It can be easily seen that E has a right A-action and a right coaction of O(G). +2.3. C*-tensor categories and their module categories. For detailed de- +scriptions, see [NT13]. A C*-category is a C-linear category with a norm on each +Hom space and an anti-linear involution ∗: C(X, Y ) −→ C(Y, X) satisfying the +C*-identity: ∥T ∗T∥ = ∥T∥2. In this paper, we also assume that C*-categories +are closed under taking direct sums and subobjects. +Let C and D be C*-categories. A functor F : C −→ D is called a C*-functor if +it preserves the adjoints, i.e. F(T ∗) = F(T)∗ for any morphism T. For a natural +transformation between C*-functors, we can define its adjoint by taking the ad- +joint of each components. We use the symbol [C, D]b to denotes the category of +C*-functors from C to D and bounded natural transformations. Here the bound- +edness of a natural transformation is defined by the uniform norm-boudedness of +its components. Obviously [C, D]b also has a structure of C*-category. +A strict C*-multitensor category is a triple (C, ⊗, 1) of C*-category C, a C*- +bifunctor –⊗–: C ×C −→ C and an object 1 called the unit object which satisfies +U ⊗(V ⊗W) = (U ⊗V )⊗W and 1⊗U = U = U ⊗1 for any objects U, V, W ∈ C. +If C(1, 1) = C id1 holds, C is said to be a strict C*-tensor category. Let X be +an object of C. We say that X is rigid if there is a quadruple (X, X, R, R) with +an object X of C, R ∈ C(1, X ⊗ X) and R ∈ C(1, X ⊗ X) which satisfies the +conjugate equations +(idX ⊗R∗)(R ⊗ idX) = idX, +(idX ⊗R +∗)(R ⊗ idX) = idX . +This quadruple is called a solution of the conjugate equations. The full subcate- +gory of C consisting of all rigid objets of C is denoted by Cf, and C is said to be +a rigid when C = Cf, i.e. all objects of C are rigid. In such a C*-tensor category, +each Hom space is finite dimensional ([NT13, Proposition 2.2.8]). +Let X be a rigid object of a strict rigid C*-tensor category C. Then we can +minimize the value ∥R∥∥R∥ for a solution of conjugate equation. This minimum + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +7 +value is called the categorical dimension of X, denoted by d(X). +A solution +(X, X, R, R) is said to be standard when ∥R∥2 = ∥R∥2 = d(X). We remark +here that any two standard solutions with fixed X are unitary equivalent to +each other, which means that we can get all standard solutions by considering +(X, X +′, (u ⊗ id)R, (id ⊗u)R) with a fixed standard solution (X, X, R, R) and an +arbitrary unitary u ∈ C(X, X +′). This fact enables us to define the categorical +trace TrX : C(X, X) −→ C given by the formula TrX(T) id1 = R∗(id ⊗T)R using +a standard solution. This is independent of the choice of standard solutions and +equal to R +∗(T ⊗ id)R. In a similar way, we can also define the partial traces +TrX ⊗ id: C(X ⊗ Y, X ⊗ Z) −→ C(Y, Z) and id ⊗ TrX : C(Y ⊗ X, Z ⊗ X) −→ +C(Y, Z). The categorical traces are actually tracial in the following sense: For +any morphisms S, T ∈ C(X, Y ) we have TrX(S∗T) = TrY (TS∗). +Let C, D be strict C*-multitensor categories. A C*-tensor functor from C to D +is a pair (Θ, θ) of a C*-functor Θ: C −→ D and a unitary natural transformation +θ: Θ(–) ⊗ Θ(–) −→ Θ(– ⊗ –) with the conditions +• Θ(1C) = 1D, +• The following diagram commutes for U, V, W ∈ C: +Θ(U) ⊗ Θ(V ) ⊗ Θ(W) +id ⊗θV,W � +θU,V ⊗id +� +Θ(U) ⊗ Θ(V ⊗ W) +θU,V ⊗W +� +Θ(U ⊗ V ) ⊗ Θ(W) +θU⊗V,W +� Θ(U ⊗ V ⊗ W). +If (Θ′, θ′) is also a C*-tensor functor from C to D, then a natural transformation +η: F −→ G is said to be a monoidal if it satisfies θ′ +U,V ◦ (ηU ⊗ ηV ) = ηU⊗V ◦ θU,V +for U, V ∈ C. +Definition 2.10 ([DY13, Definition 2.14.]). Let C be a strict rigid C*-tensor +category. A C-module category is a triple (M, ⊗, a) of a C*-category M, a C*- +bifunctor – ⊗–: C ×M −→ M and a unitary natural transformation a: – ⊗(– ⊗ +–) −→ (– ⊗ –) ⊗ – with the following conditions. +• 1 ⊗ X = X for any X ∈ M, +• The following diagram commutes for any U, V, W ∈ C and X ∈ M: +U ⊗ (V ⊗ (W ⊗ X)) +aU,V,W ⊗X� +id ⊗aV,W,X +� +(U ⊗ V ) ⊗ (W ⊗ X) +aU⊗V,W,X +� +U ⊗ ((V ⊗ W) ⊗ X) aU,V ⊗W,X +� (U ⊗ V ⊗ W) ⊗ X. +We remark that a C-module structure on a fixed C*-category M gives rise to +a C*-tensor functor from C to [M, M]b and vice versa. It is said that M is +semisimple if M(X, Y ) is finite dimensional for any X, Y ∈ M. We assume that +all C-module categories in this paper are semisimple. A C-module category M is +said to be connected if, for any X, Y ∈ M, there is U ∈ C and an isometry from +X to U ⊗ Y . +Example 2.11. Let G be a compact quantum group. Then its representaiton +category Rep G is a C*-tensor category and Repf G is a rigid C*-tensor category. + +8 +MAO HOSHINO +Take a quantum homogeneous space A of G. Then G- Modf +A has a structure of +Repf G-module category: For π ∈ Repf G and E ∈ G- Modf +A, Hπ⊗E can be made +into a G-equivariant Hilbert A-module with a G-action v⊗ξ �−→ Uπ,13(v⊗αE(ξ)). +This RepG-module category is connected and semisimple ([DY13, Proposition +3.11]). +If N is another C-module category, a C-module functor from M to N is a pair of +a C*-functor F : M −→ N and a unitary natural transformation f : F(–⊗–) −→ +– ⊗ F(–) with the commutativity of the following diagram for any U, V ∈ C and +X ∈ M: +F(U ⊗ (V ⊗ X)) +fU,V ⊗X� +F (aU,V,X) +� +U ⊗ F(V ⊗ X) +id ⊗fV,X� U ⊗ (V ⊗ F(X)) +aU,V,F (X) +� +F((U ⊗ V ) ⊗ X) +fU⊗V,X +� (U ⊗ V ) ⊗ F(X). +The category of C-module functors from M to N is denoted by [M, N ]C +b, here +a morphism from (F, f) to (G, g) is given by a bounded natural transformation +η: F −→ G making the following diagram commutative for any U ∈ C and +X ∈ M: +F(U ⊗ X) +fU,X � +ηU⊗X +� +U ⊗ F(X) +id ⊗ηX +� +G(U ⊗ X) +gU,X +� U ⊗ G(X). +If we regard a C-module structure on M as a C*-tensor functor (Φ, ϕ): C −→ +[M, M]b, then [M, M]C +b can be thought as something like a Drinfeld center. +Actually, if we are given a C-module functor (F, f) from M to M, then f defines +a unitary natural transformation c from F ◦Φ(–) to Φ(–)◦F satisfying the braiding +equation cU⊗V = (ϕU,V ⊗ idF)(id ⊗cV )(cU ⊗ id)(id ⊗ϕU,V ∗). The converse also +holds, and then a morphism of C-module functor can be considered as a morphism +of unitary half-braiding. +2.4. Index of conditional expectation. In this subsection, we collect defini- +tions and facts on indices of conditional expectations, introduced in [Wa90]. +Let B be a unital C*-algebra and A be a unital C*-subalgebra of B with a +conditional expectation E : B −→ A. Then we say that E is with finite index if +it admits a quasi-basis i.e. a finite family (ui)n +i=1 ⊂ A satisfying +a = +n +� +i=1 +uiE(u∗ +i a) +for any a ∈ A. In this case we can define the index of E by the formula Index E = +�n +i=1 uiu∗ +i . This is independent of the choice of a quasi-basis, and Index E is an +element of Z(A)×. +On the other hand, we also have a more classical notion +of index based on the Pimsner-Popa inequality [PP86, Proposition 2.1.]. The +following value is called the probabilistic index of E ([Po95, Definition 1.1.1]): +Indexp E = min{c > 0 | cE − idA is positive.}. + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +9 +We also define the scalar index of E by replacing the positivity by the complete +positivity: +Indexs E = min{c > 0 | cE − idA is completely positive.}. +Obviously we have Indexp E ≤ Indexs E ≤ ∞. Moreover the following proposi- +tion holds. +Proposition 2.12 ([FK00, Theorem 1.]). Let E : B −→ A be a conditional +expectation. Then the probabilistic index of E is finite if and only if the scalar +index of E is finite. +We use the symbol BE for a Hilbert A-module obtained by taking the com- +pletion of A with respect to a B-valued inner product ⟨x, y⟩B = E(x∗y). The +following generalization of [Wa90, Proposition 2.1.5.] +may be well-known for +experts (c.f. [BDH, Th´eor`eme 3.5.]). For its proof, see Subsection A.2. +Proposition 2.13. Let E : B −→ A be a conditional expectation. Then E is +with finite index if and only if it satisfies both of the following two conditions: +(i) Indexp E < ∞. +(ii) The Hilbert A-module BE can be decomposed into a direct sum of finitely +generated projective Hilbert A-modules. +In this case we have Indexs E = ∥Index E∥. +3. Equivariant finite quantum covering spaces +3.1. Definition and characterizations. Let G be a compact quantum group. +If A ⊂ B is an inclusion of G-C*-algebras, we say that E : B −→ A is a G- +expectation when E is a G-equivariant conditional expectation. +Definition 3.1. Let A ⊂ B be a unital inclusion of unital G-C*-algebras. We +say that A ⊂ B is a finite quantum G-covering space over A if it admits a G- +expectation E : B −→ A with finite index. +In this paper, we only treat with finite quantum G-covering spaces over quan- +tum homogeneous spaces. In such cases, we can replace the finiteness of the index +by the finiteness of the probabilistic index. +Let A be a quantum homogeneous space of G. +Theorem 3.2. Let A ⊂ B be a unital inclusion of unital G-C*-algebras with a +G-expectation E. Then the following conditions are equivalent: +(i) The index of E is finite. +(ii) The probabilistic index of E is finite. +Moreover, under these conditions, we can take a quasi-basis of E in the algebraic +core B of B. +Proof. At first one should note that BE can be made into a G-equivariant Hilbert +A-module by the action of G on B. Then BE decomposes into a direct sum of +finitely generated projective Hilbert A-modules by [DC16, Theorem 6.21.], hence +the equivalence of (i) and (ii) follows from Proposition 2.13. +Next we show the last statement. Since BE is a finitely generated G-equivariant +Hilbert A-module, it has a irreducible decomposition. Hence we can take a finite + +10 +MAO HOSHINO +dimensional unitary representation π of G and a embedding V : BE −→ Hπ ⊗ +A of G-equivariant Hilbert A-module by [DC16, Theorem 6.23.]. Then V and +its adjoint V ∗ preserve their algebraic core, hence we can obtain a quasi-basis +(V ∗(ei ⊗ 1))n +i=1 in B where (ei)n +i=1 is an orthonormal basis of Hπ. +□ +Corollary 3.3. Let A ⊂ B be a finite quantum G-covering space. Then for any +intermediate unital G-C*-subalgebra C of A ⊂ B, A ⊂ C is also a finite quantum +G-covering space. +Proof. Fix a G-expectation E : B −→ A with finite index. Then we can easily +check the finiteness of the restriction of E on C by using the condition (ii) of +Theorem 3.2. +□ +If G is of Kac type, we can drop the equivariance of a conditional expectation +by the averaging procedure. Let A, B be G-C*-algebras and A, B be their al- +gebraic cores respectively. If we are given a C-linear map ϕ: B −→ A, then its +equivariantization is a C-linear map �ϕ: B −→ A given by the following: +�ϕ(x) = (id ⊗h)(α(ϕ(x(0)))(1 ⊗ S(x(1)))). +Here h and S denote the Haar state and the antipode of G respectively. By using +the strong bi-invariance of h ([KV00, Proposition 5.24, Corollary 5.35]), we can +see that �ϕ is a G-equivariant map from B to A. +Lemma 3.4. Let ϕ: B −→ A be a c.p. map. If G is of Kac type, the equivari- +antization of ϕ extends to a G-equivariant c.p. map from B to A. +Proof. Since G is of Kac type, we have +�ϕ(y∗x) = (id ⊗h)((1 ⊗ S(y(1))∗)α(ϕ(y∗ +(0)x(0)))(1 ⊗ S(x(1)))). +This equality implies the complete positivity of �ϕ, hence we can apply Proposition +2.8. +□ +Proposition 3.5. Assume G is of Kac type. For a unital inclusion A ⊂ B of +unital G-C*-algebras with Aα = C1A, the following conditions are equivalent: +(i) The inclusion A ⊂ B is a finite quantum G-covering space. +(ii) There is an expectation E : B −→ A with Indexp E < ∞. +Proof. Let E be a conditional expection from B to A. At first we have to show +that the equivariantization �E of E is again a conditional expectation. Since �E +is a u.c.p. map by the previous proposition, it suffices to show �E is A-bimodule +map on B. Take a ∈ A and x ∈ B. Then we have +�E(xa) = (id ⊗h)(α(E(x(0)a(0)))(1 ⊗ S(x(1)a(1)))) += (id ⊗h)(α(E(x(0)))α(a(0))(1 ⊗ S(a(1)))(1 ⊗ S(x(1)))) += (id ⊗h)(α(E(x(0)))(a ⊗ S(x(1)))) += �E(x)a. +The equality �E(ax) = a �E(x) can be seen as follows: +�E(ax) = �E(x∗a∗)∗ = ( �E(x)∗a∗)∗ = a �E(x). + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +11 +Now the probabilistic index of E is finite, hence the scalar index c of E is also +finite by Proposition 2.12. Then the equivariantization of cE − idB is a c.p. map +and coincides with c �E − idB. This implies that �E has the finite scalar index, +hence A ⊂ B is a finite quantum G-covering space. +□ +3.2. Covering degree. Next we consider a notion which is analogous to the +covering degree of covering space. +Definition 3.6. Let A ⊂ B be a finite quantum G-covering space. Then the cov- +ering degree of A ⊂ B is the infimum of the scalar indices over all G-expectations +from B to A. +Remark 3.7. If G is of Kac type, the covering degree coincides with the infimum +of the scalar indices of all conditional expectations of the inclusion. This fact can +be seen from the proof of Proposition 3.5. +To calculate the covering degree of a given finite quantum G-covering space, +it is usuful to regard it as the dimension of a rigid object of a suitable C*-tensor +category. +Let A and B be quantum homogeneous spaces of G. A G-equivariant (A, B)- +correspondence is a pair of G-equivariant Hilbert B-module E and a unital G- +equivariant ∗-homomorphism from A to LB(E). The category of G-equivariant +(A, B)-correspondences is denoted by G- CorrA,B, and its full subcategory con- +sisting of right-finitely generated correspondences is denoted by G- Corrrf +A,B. We +also use the symbols G- CorrA and G- Corrf +A when A = B. +Let A ⊂ B be an inclusion of unital G-C*-algebras with Aα = C1A. If we +are given a G-expectation E : B −→ A with finite index, then we have a G- +correspondence BE over A, on which the left action is given by the left multipli- +cation. Moreover BE can be made into a C*-Frobenius algebra ([BKLR, Section +3.1]) in G- Corrf +A with the following maps: +• m: BE ⊗A BE −→ BE, induced by the multiplication of m, +• ι: A −→ BE, induced by the inclusion A ⊂ B. +By using a quasi-basis (vi)n +i=1 of E, m∗ : BE −→ BE ⊗A BE can be calculated as +follows: +m∗(x) = +n +� +i=1 +vi ⊗ v∗ +i x +Hence we have mm∗(1B) = Index E and ι∗mm∗ι(a) = E(Index E)a. Since ι∗(b) = +E(b) for any b ∈ B, we also have +d(BE) ≤ ∥ι∗mm∗ι∥ = E(Index E) ≤ ∥Index E∥, +here the left-hand side is the dimension of BE as an object of G- Corrf +A. More- +over, if (BE, m, ι) is a Q-system, these inequalities turns out to be equalities. +Conversely any C*-Frobenius algebra in G- Corrf +A gives a pair of a finite quan- +tum G-covering space A ⊂ B and G-expectation E : B −→ A with finite index. +Hence we can get the following proposition since any Frobenius C*-algebra is +isomorphic to a Q-system, as shown in [NY18, Theorem 2.9] +Proposition 3.8. Let A be a quantum homogeneous space of G. + +12 +MAO HOSHINO +(i) There is a one-to-one correspondence between finite quantum G-covering +spaces of A and Q-systems in G- Corrf +A. +(ii) Let A ⊂ B be a finite quantum G-covering space. Its covering degree d +coincides with the dimension of the corresponding Q-system. Moreover, +there is a G-expectation E : B −→ A with Index E = d1B. +Example 3.9. When A = C, the trivial G-C*-algebra, then G- Corrf +C is nothing +but Repf G. +Hence the covering degree of a finite quantum G-space over C +coincides with the quantum dimension as a unitary representation. In particular, +the covering degree of C ⊂ B(Hπ) is (dimq π)2 if we consider the adjoint action +on B(Hπ) +As an application of Proposition 3.8, we show that the covering degree of a +finite quantum G-spaces over C(G) must be a positive integer. +Let Rep O(G) be the category of unital ∗-representations of O(G). By using +the coproduct on O(G), we can make Rep O(G) into a C*-tensor category. For +any (π, Hπ) ∈ Rep O(G), we can construct a G-equivariant correspondence Eπ +over C(G) as follows: +• As a Hilbert C(G)-module, Eπ = Hπ ⊗ C(G). +• The left action of G is given by id ⊗∆. +• The left action of C(G) is given by (π ⊗ λ)∆, where λ is the left multi- +plication of C(G) on C(G). +Here (π ⊗ λ)∆ can be defined on C(G) by Lemma 2.1. +The following proposition is a quantum analogue of [AV16, Proposition 2.2]. +Proposition 3.10. The functor (π, Hπ) �−→ Eπ gives an equivalence of C*- +tensor categories from Rep O(G) to G- CorrC(G). +Before proving this, we should prepare the following lemma. +Lemma 3.11. Let E be a G-equivariant Hilbert C(G)-module. The the following +formula gives an semi-inner product on the algebraic core E of E: +⟨ξ, η⟩ = ε(⟨ξ, η⟩C(G)) +Proof. The only non-trivial part is the positivity of ⟨ξ, ξ⟩ for each ξ ∈ E. By +replacing E with its G-equivariant Hilbert C(G) submodule generated by ξ and +η, we may assume E is finitely generated. +If we are given another F ∈ G- Modf +C(G) and an isometry V ∈ LG +C(G)(E, F), we +have +⟨V ξ, V η⟩ = ε(⟨V ξ, V η⟩C(G)) = ε(⟨ξ, η⟩C(G)) = ⟨ξ, η⟩ . +Since G- Modf +C(G) is connected as a Repf G-module category, the equality above +implies that it suffices to show the statement for Hπ ⊗ C(G) with an arbitrary +π ∈ Repf G But this is trivial since we have +⟨v ⊗ x, u ⊗ y⟩ = ⟨v, u⟩ ε(x∗y). +□ + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +13 +Proof of Proposition 3.10. At first we make the functor in the statement into a +C*-tensor functor. This can be completed by associating the following unitary: +(Hπ ⊗ C(G)) ⊗C(G) (Hρ ⊗ C(G)) +Uπ,ρ +� (Hπ ⊗ Hρ) ⊗ C(G). +(ξ ⊗ x) ⊗ (η ⊗ y) ✤ +� (ξ ⊗ ρ(x(1))η) ⊗ x(2)y. +It can be easily seen that this unitary satisfies the conditions in the definition +of C*-tensor functor. +To show that this functor gives the equivalence, we construct a quasi-inverse +functor. Let E be a G-equivariant corrrespondence over C(G). Set HE as a +completion of the algebraic core E of E, with respect the inner product in the +previous lemma. Then each element x ∈ O(G) acts on HE from the left, since +O(G) is a linear span of matrix coefficients of O(G)-valued unitary matrices. +Now we have a ∗-representation (πE, Hπ) of O(G). +To complete the proof, we will check that these functors give a quasi-inverse +of each other. For (π, Hπ) ∈ Repf O(G), the algebraic core of Eπ is precisely +Hπ ⊗ O(G). Hence we have a unitary from HEπ to Hπ given by ξ ⊗ x �−→ ε(x)ξ +and this is an intertwiner of representations of O(G). On the other hand, for any +E ∈ G- Corrrf +C(G), we have a map from E to HE ⊗ C(G) given as follows: +ξ ∈ E �−→ ξ(0) ⊗ ξ(1) ∈ HE ⊗ C(G). +This map is a G-equivariant isometry and intertwines with left C(G)-action. To +show that this is a right C(G)-module map, one should note the following: For +any x ∈ O(G) and ξ ∈ E, we have +∥ε(x)ξ − ξx∥2 +HE = ε(⟨ε(x)ξ − ξx, ε(x)ξ − ξx⟩C(G)) = 0. +Hence we have ξx = ε(x)ξ in HE and +(ξx)(0) ⊗ (ξx)(1) = ε(x(0))ξ(0) ⊗ ξ(1)x(1) = ξ(0) ⊗ ξ(1)x. +This implies that the map above is an embedding of G-equivariant correspondence +from E to HE ⊗ C(G). Hence the range of this map contains ξ(0) ⊗ ξ(1)x for any +ξ ∈ E and x ∈ C(G). Combining this with span αE(E)(C ⊗ C(G)) = E ⊗ C(G), +we can see that this map is unitary. Now we complete the proof. +□ +Remark 3.12. We also have an equivalence Repf O(G) ∼= G- Corrrf +C(G). +Let us recall the quantum Bohr compactification due to So�ltan [So05]. For +the dual discrete quantum group �G of G, its quantum Bohr compactification +b�G is of Kac type ([So05, Theorem 4.5]). Moreover every finite dimensional ∗- +representation of O(G) gives rise to a finite dimensional unitary representation of +b�G (c.f. [CKS, Theorem 2.3]). These facts imply that Repf O(G) is rigid and the +forgetful functor Repf O(G) �−→ Hilbf is dimension-preserving. Now Proposition +3.10 yields the desired result. +Corollary 3.13. For a finite quantum G-covering space over C(G), its covering +degree must be a positive integer. + +14 +MAO HOSHINO +3.3. A module categorical description. Let us recall the Tannaka-Krein type +result for quantum homogeneous spaces [DY13, Theorem 6.4], [Ne14, Theorem +3.3]. This suggests us the possibility to understand finite quantum G-covering +space by the language of module categories. +Let M be a right-finitely generated G-equivariant (A, B)-correspondence. Then +we have a Repf G-module functor from G- Modf +A to G- Modf +B given by the follow- +ing: +• As a C*-functor, it is given by taking the internal tensor product with E +over A i.e. the functor sends E to E⊗AM. Here E⊗AM is considered as +a G-equivariant Hilbert B-module by ξ ⊗ m �−→ αE(ξ)13βM(m)23. Since +E and M is finitely generated, this Hilbert B-module is also finitely +generated. +• The unitary from (Hπ ⊗ E) ⊗A M to Hπ ⊗ (E ⊗A M) is given by (v ⊗ +ξ) ⊗ m �−→ v ⊗ (ξ ⊗ m). +This Repf G-module functor is denoted by – ⊗A M. +The following is a main theorem of this subsection, which is a generalization +of [DY13, Theorem 7.1] +Theorem 3.14. Let A and B be quantum homogeneous spaces of G. Then the +C*-functor M �−→ – ⊗A M gives the following equivalence of C*-categories: +G- Corrrf +A,B ∼= [G- Modf +A, G- Modf +B]Repf G +b +We would like to use an argument used in [DY13, Theorem 7.1] for the proof +of the above theorem. +At first, we fix a complete set Irr G of representatives of all equivalence classes +of irreducible unitary representations of G. +For any π ∈ Irr G, π denotes an +element of Irr G unitary equivalent to the conjugate representation of π, which is +uniquely determined. +Let E be a G-equivariant Hilbert A-module and E be its algebraic core. For +each π ∈ Irr G, Eπ denotes the space LG +A(Hπ ⊗ A, E) ⊗ Hπ. Then we can regard +Eπ as a subspace of E via T ⊗ ξ �−→ T(ξ ⊗ 1) and have the spectral decompo- +sition E = � +π Eπ. By considering A as a G-equivariant Hilbert A-module, this +decomposition also can be applied to A. +Now we can present the right A-action, the A-valued inner product and the +G-action on E as follows: For T ⊗ ξ ∈ Eπ, S ⊗ η ∈ Eρ and s ⊗ v ∈ Aρ, we have +(T ⊗ ξ)(s ⊗ v) = +� +σ∈Irr G +� +V ∈(σ,πρ) +T(idπ ⊗s)(V ⊗ idA) ⊗ V ∗(ξ ⊗ v), +⟨T ⊗ ξ, S ⊗ η⟩A = +� +σ∈Irr G +� +V ∈(σ,πρ) +(R∗ +π ⊗ idA)(idπ ⊗T ∗S)(V ⊗ idA) +⊗ (ξ∗ ⊗ V ∗)(Rπ(1) ⊗ η), +αE(T ⊗ ξ) = T ⊗ Uπ(ξ ⊗ 1). +Here we use the following notations: +• The quadruple (π, π, Rπ, Rπ) denotes a standard solution in Repf G. + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +15 +• The symbol � +V ∈(σ,πρ) means the summation over all elements of a fixed +orthonormal basis of HomG(σ, π ⊗ ρ), equipped with an inner product +⟨V, W⟩ = V ∗W. +These formulae do not depend on the choices of the standard solutions and the +orthonormal bases. We also have the following presentation of the involution of +A under these notations: +(s ⊗ v)∗ = (R∗ +π ⊗ id)(idπ ⊗s∗) ⊗ (v∗ ⊗ id)Rπ(1). +Now we finishes the preparation for the proof of Theorem 3.14. +Proof of Theorem 3.14. At first we construct a functor from [G- Modf +A, G- Modf +B]Repf G +b +to G- Corrrf +A,B. Let (F, f) be a Repf G-module functor from G- Modf +A to G- Modf +B. +Then we can associate a left action of A on F(A) as follows: For s ⊗ v ∈ Aπ and +η ∈ F(A), the left multiplication by s ⊗ v is given by +(s ⊗ v)η = F(s)f ∗ +π,A(v ⊗ η). +Then some direct calculations show that this defines an adjointable B-module +map. Moreover we can see the following relations: +⟨η, (s ⊗ v)η′⟩B = ⟨(s ⊗ v)∗η, η′⟩B , +βF (A)((s ⊗ v)η) = α(s ⊗ v)βF (A)(η). +Hence Proposition 2.8 shows that this left A-action extends to a left A-action +on F(A) compatible with the G-actions. Now F(A) is equipped with a struc- +ture of G-equivariant (A, B)-correspondence. Moreover if we are given another +Repf G-module functor (G, g) and a morphism θ: (F, f) −→ (G, g), its compo- +nent θA : F(A) −→ G(A) is a morphism of G-equivariant (A, B)-correspondence. +These constructions give rise to a functor we want. +We will complete the proof by showing that these functors give a quasi-inverse +of each other. Fix an right-finitely generated G-equivariant (A, B)-correspondence +M. Then A ⊗A M is isomorphic to M as a G-equivariant Hilbert B-module map +via x ⊗ ξ �−→ xξ. Hence it suffices to show the compatibility with the left A- +actions. For s ⊗ v ∈ Aπ and a ⊗ ξ ∈ A ⊗A M, the action as above is calculated +as follows: +(s ⊗ v)(a ⊗ ξ) = (s ⊗ idM)((v ⊗ a) ⊗ ξ) = s(v ⊗ a) ⊗ ξ. +Since s ⊗v corresponds to s(v ⊗1) ∈ A, this equality shows the compatibility we +required. +Next we take a Repf G-module functor (F, f). For any finitely generated G- +equivariant Hilbert A-module E, we have a map from E ⊗ F(A) to F(E) which +sends (T ⊗ v) ⊗ ξ to F(T)f ∗ +π,A(v ⊗ ξ). Then we have +� +F(T)f ∗ +π,A(v ⊗ ξ), F(S)f ∗ +ρ,A(w ⊗ η) +� +B += +� +v ⊗ ξ, fπ,AF(T ∗S)f ∗ +ρ,A(w ⊗ η) +� +B += +� +v ⊗ ξ, (idπ ⊗(R∗ +π ⊗ idF (A))(idπ ⊗fπ,AF(T ∗S)f ∗ +ρ,A))(Rπ ⊗ idρ ⊗ idF (A))(w ⊗ η) +� +B + +16 +MAO HOSHINO +By using (R∗ +π ⊗ idF (A))fπ⊗π,A = F(R∗ +π ⊗ idA) and fπ⊗π,A = (idπ ⊗fπ,A)fπ,π⊗A, we +also have +(R∗ +π ⊗ idF (A))(idπ ⊗fπ,AF(T ∗S)f ∗ +ρ,A) = F((R∗ +π ⊗ idA)(idπ ⊗T ∗S))f ∗ +π⊗ρ,A. +Hence the map induces an isometry from E ⊗A F(A) to F(E). It can be easily +seen that this isometry is a G-equivariant Hilbert B-module map. To prove the +unitarity, one should note that F preserves the range projection i.e. +for any +T ∈ LG +A(Hπ ⊗ A, E) with the range projection p ∈ LG +A(E), the range projection +of F(T): F(Hπ ⊗ A) −→ F(E) is F(p). This fact implies that F(E) is contained +the sum space of the range of F(T) over all T ∈ � +π∈Irr G LG +A(Hπ ⊗ A, E). Hence +the map from E ⊗A F(A) to F(E) is unitary. Now the proof is completed. +□ +As an application of this theorem, we can see that each Jones’ value appears +as the covering degree of a finite quantum G-covering space for some G. +Theorem 3.15. The covering degree of a finite quantum G-covering space is +contained in {4 cos2(π/n) | n ≥ 3} ∪ [4, ∞). Conversely, for any element d of +this set, we have a compact quantum group G and a finite quantum G-covering +space A ⊂ B with its covering degree d. +In the following proof, we use the monoidal opposite Cop of a given C*-tensor +category C (c.f. [EGNO, Definition 2.1.5]). As a C*-category, Cop = C. But its +tensor product – ⊗ –: Cop × Cop −→ Cop is given by (X, Y ) �−→ Y ⊗ X. +Proof. The first half of the statement follows from a general theory on Q-system: +Take a finite quantum G-covering space A ⊂ B and let d be its covering degree. +Then we have a Q-system X in G- Corrf +A correspoinding to A ⊂ B with d(X) = d +(Proposition 3.8). Let C be a C*-tensor category generated by X. Then we can +find a finite factor N and a fully faithful embedding i: C −→ Bimodf +N by [Ya03, +Theorem 3.9]. Since a Q-system in Bimodf +N corresponds to a finite extension of +von Neumann algebra [Lo94, Theorem 6.1], we have a finite extension N ⊂ M +whose minimum index is d(i(X)) = d(X) = d. Now our assertion follows from +the Jones’ result on the value of index ([Jo83, Theorem 4.3.1]). +To prove the second half, take an arbitrary d from the set in the statement. +If d ≥ 4, we have a q ∈ (0, 1) such that the fundamental representaion π1/2 +of SUq(2) satisfies dimq π1/2 = +√ +d. Then Example 3.9 gives a finite quantum +SUq(2)-covering space with the covering degree d. +Now consider the case d < 4. Then we can take a fusion category C and its +object X with d(X) = +√ +d (For example, let C be the Temperley-Lieb-Jones +category [NT13, Section 2.5]). +We may assume C is generated by X and its +conjugate object by replacing C with its full subcateogry. +Take a q ∈ [−1, 1] \ {0} and a C*-tensor functor Φ: Repf SUq(2) −→ C with +Φ(π1/2) = X ⊕X. The existence of those is guaranteed by the universal property +of Repf SUq(2) ([NT13, Theorem 2.5.3]). Then C can be thought as a Repf SUq(2)- +module category by π ⊗Y = Φ(π)⊗Y for π ∈ Repf SUq(2) and Y ∈ C. Moreover +our assumption implies that this is connected. Hence we can use the duality +theorem [DY13, Theorem 6.4] to find a quantum homogeneous space A of SUq(2) + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +17 +for which there is an equivalence C ∼= SUq(2)- Modf +A of Repf SUq(2)-module cate- +gories. On the other hand, by using the right multiplication of C on C, we have a +C*-tensor functor from Cop to [SUq(2)- Modf +A, SUq(2)- Modf +A]Repf SUq(2) +b +. Since any +C*-tensor functor from a fusion category is dimension- preserving, this implies we +have an object Y in [SUq(2)- Modf +A, SUq(2)- Modf +A]Repf SUq(2) +b +with the dimension +√ +d. Hence we have a Q-system Y ⊗Y ∈ [SUq(2)- Modf +A, SUq(2)- Modf +A]Repf SUq(2) +b +, +and we have a finite quantum SUq(2)-covering space with the covering degree d +by Proposition 3.8. +□ +4. Induction from actions of the maximal Kac quantum subgroup +4.1. Module categorical interpretation of induced actions. Let G be a +compact quantum group and H be its quantum subgroup. For an H-C*algebra +A, we use the symbol �A to present its induced G-C*-algebra. +Take π ∈ Repf G and E ∈ H- Modf +A arbitrarily. Then the unitary U−1 +π,13 on +Hπ ⊗ E ⊗ C(G) is restricted to an unitary operator from Ind G +H(Hπ|H ⊗ E) to +Hπ ⊗ Ind G +HE. If we regard H- Modf +A as a Repf G-module category via the re- +striction functor Repf G −→ Repf H, the collection of such unitaries makes the +induction functor Ind G +H : H- Modf +A −→ G- Modf +� +A into a Repf G-module functor. +The following theorem is essentially proved in [Va05, Theorem 7.3.] +Theorem 4.1. The functor Ind G +H gives an equivalence H- ModA ∼= G- Mod � +A as +Repf G-module categories. This equivalence also gives an equivalence H- Modf +A ∼= +G- Modf +� +A. +Instead of relying on the Vaes’ generalization of Green imprimitivity, we will +give a quasi-inverse of Ind G +H. +The algebraic core of �A is denoted by � +A. The following is the explict presen- +tation of � +A: +� +A = {x ∈ A ⊗ O(G) | (α ⊗ id)(x) = (id ⊗ℓH)(x)}. +Here q is the canonical surjection from O(G) to O(H). +Lemma 4.2. Let �E be a G-equivariant Hilbert �A-module and �E be its algebraic +core. Then the following gives an A-valued semi-inner product on �E: +⟨ξ, η⟩A = (id ⊗ε)(⟨ξ, η⟩ � +A) +Proof. By replacing �E with its G-equivariant Hilbert �A-submodule generated by ξ +and η, we may assume that �E is countably generated. Then we can find a unitary +representation π of G such that there is an isometry V ∈ LG +� +A( �E, Hπ⊗ �A), by using +the equivariant Kasparov stabilization theorem ([Ve02, Th´eor`eme 3.2]). Hence it +suffices to show the statement for Hπ ⊗ �A. In this case we have E = Hπ ⊗ � +A and +⟨v ⊗ x, w ⊗ y⟩A = ⟨v, w⟩ (id ⊗ε)(x)∗(id ⊗y). +for v, w ∈ Hπ and x, y ∈ � +A. Now the statement follows from this. +□ + +18 +MAO HOSHINO +Let �E0 be a Hilbert A-module obtained as the completion of �E with respect +to this A-valued semi-inner product. Then the map (id ⊗q)�α � +E : �E −→ �E ⊗ O(H) +induces a H-action on �E0, making �E into an H-equivariant Hilbert A-module. +Lemma 4.3. Let E be an H-equivariant Hilbert A-module. Then the following +map defined PE on E ⊗O(G) is a surjection onto �E, the algebraic core of Ind G +HE: +PE(ξ ⊗ x) = (id ⊗hH)((id ⊗S)αE(ξ)(1 ⊗ q((x(1))))) ⊗ x(2). +Here hH is the Haar state of H. +Proof. Since we have +�E = {ξ ∈ E ⊗ O(G) | (αE ⊗ id)(ξ) = (id ⊗ℓH)(ξ)}, +it can be easily seen that PE is identical on �E. The remaining part is to show +PE(E ⊗ O(G)) ⊂ �E. Take arbitrary ξ ∈ E and x ∈ O(G). Then we have the +following: +(αE ⊗ id)(PE(ξ ⊗ x)) = (id ⊗ id ⊗hH)((id ⊗(id ⊗S)∆)(αE(ξ))(1 ⊗ 1 ⊗ q(x(1)))) ⊗ x(2) += (id ⊗hH ⊗ S−1)((id ⊗∆ ◦ S)(αE(ξ))(1 ⊗ q(x(1)) ⊗ 1)) ⊗ x(2) +(id ⊗ℓH)(PE(ξ ⊗ x)) = (id ⊗hH ⊗ id)(((id ⊗S)(αE(ξ)) ⊗ 1)(1 ⊗ ∆(q(x1)))) ⊗ x(2). +Hence the statement follows from the strong bi-invariance of hH. +□ +Remark 4.4. If H is of Kac type, PE extends to a G-equivariant c.c. map from +E ⊗ C(G) to Ind G +HG. This follows from Proposition 2.8 by using the corner trick +to E ⊂ KA(A ⊕ E) and �E ⊂ K � +A( �A ⊕ �E) ∼= Ind G +HKA(A ⊕ E). +Proof of Theorem 4.1. We have two functors: The first one is Ind G +H, and the +second one is EvH +G : G- Mod � +A −→ H- ModA given by �E �−→ �E0. The remaining is +to show that these functors are quasi-inverses of each other. +Take E ∈ H- ModA arbitrarily. Then we have a linear map (id ⊗ε)| �E : �E −→ E, +for which we have +⟨(id ⊗ε)(ξ), (id ⊗ε)(η)⟩A = ⟨ξ, η⟩A . +Hence this map induces an isometry VE ∈ LG +A((Ind G +HE)0, E). To see the surjec- +tivity of VE, we use PE as in the previous lemma. For any ξ ∈ E and x ∈ O(G), +we have +(id ⊗ε) ◦ PE(ξ ⊗ x) = (id ⊗hH)((id ⊗S)(αE(ξ))(1 ⊗ q(x))). +Since span αE(E)(C ⊗ O(H)) = E ⊗ O(H), the above equality implies the surjec- +tivity of VE. +Next take �E ∈ G- Mod � +A arbitrarily. Then �α � +E defines a linear map from �E to +�E0 ⊗ O(G), for which we have +� +�α � +E(ξ), �α � +E(η) +� +� +A = ⟨ξ, η⟩ � +A . +Moreover we also have +� +�α � +E(ξ), �α � +E(ηx) +� +� +A = (id ⊗ε ⊗ id)(�α(⟨ξ, η⟩ � +A x)) += (id ⊗ε ⊗ id)(�α(⟨ξ, η⟩ � +A))x = +� +�α � +E(ξ), �α � +E(η)x +� +� +A . + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +19 +Hence �α � +E induces an isometry U � +E ∈ LG +� +A( �E, �E0 ⊗ C(G)). It can be easily seen +that the image of U � +E is contained in Ind G +H �E0. To see that U � +E( �E) = Ind G +H �E0, we +can use Lemma 4.6 by considering the case of B = A and M = A. +□ +Remark 4.5. The functor EvH +G : G- Mod � +A −→ H- ModA also has a structure of +Repf G-module functor, namely the collection of unitaries v ⊗ ξ ∈ (Hπ ⊗ �E)0 �−→ +v ⊗ ξ ∈ Hπ|H ⊗ �E0. +Now let A, B be H-C*-algebras and �A, �B be their induced G-C*-algebra. Then +we also have an induction functor Ind G +H : H- Corr(A,B) −→ G- Corr � +A, � +B. +Lemma 4.6. Let �E ∈ G- Mod � +A and M ∈ H- CorrA,B. Then we have the following +unitary equivalence +�E ⊗ � +A Ind G +HM ∼= Ind G +H(EvH +G �E ⊗A M), +induced by ξ ⊗ η ⊗ x �−→ �α � +E,13(ξ)(1 ⊗ η ⊗ x). +Proof. The only non-trivial part is the surjectivity of the map in the statement. +Let �E, M and (EvH +G �E ⊗A M)alg be the algebraic cores of �E, M and EvH +G �E ⊗A M +respectively. Let us consider the following diagram: +�E ⊗ M ⊗ O(G) +� +id ⊗PM +� +�E ⊗ M ⊗ O(G) +� +(EvH +G �E ⊗A M)alg ⊗ O(G) +PEvH +G +� +E⊗AM +� +�E ⊗ � +A Ind G +HM +� Ind G +H(EvH +G �E ⊗A M). +Here the morphism at the top is given by ξ ⊗ η ⊗ x �−→ �α � +E,13(ξ)(1 ⊗ η ⊗ x), +and the morphism at the bottom is as in the statement. Then we can see that +this diagram is commutative and that the morphisms at the top and at the right +have dense ranges. By using these observation, we can see the surjectivity of the +morphism at the bottom. +□ +Remark 4.7. One should note that the collection of the unitary equivalences +over all �E gives rise to a unitary equivalence of Repf G-module functor from +– ⊗ � +A Ind G +HM to Ind G +H(EvH +G(–) ⊗A M) +Let C be another H-C*-algebra. For M ∈ H- CorrA,B and N ∈ H- CorrB,C, we +have a unitary equivalence Ind G +H(M ⊗B N) ∼= (Ind G +HM) ⊗ � +B (Ind G +HN), namely the +restriction of (ξ ⊗ η) ⊗ x �−→ (ξ ⊗ 1) ⊗ (η ⊗ x). Moreover the collection of these +unitaries makes Ind G +H : H- CorrA −→ G- Corr � +A into a C*-tensor functor. + +20 +MAO HOSHINO +Theorem 4.8. Let A, B be quantum homogeneous spaces of H and �A, �B be their +induced quantum homogeneous spaces of G. Then the following diagram of C*- +functors commutes up to a canonical unitary natural transformation: +H- Corrrf +A,B +Ind G +H +� +∼ += +� [H- Modf +A, H- Modf +B]Repf H +� +[H- Modf +A, H- Modf +B]Repf G +Ind G +H◦–◦EvH +G +∼ += +� +G- Corrrf +� +A, � +B +∼ += +� [G- Modf +� +A, G- Modf +� +B]Repf G. +Moreover, the canonical unitary natural transformation is monoidal if A = B. +Proof. The canonical unitary natural transformation is given by the collection +of unitaries as in Lemma 4.6. +Then the statement can be checked by direct +calculations. +□ +4.2. Tracial module category and its Plancherel weight. Theorem 4.8 en- +ables us to show an imprimitivity-type result by a comparison of Repf G-module +functors and Repf H-module functors. The purpose of this subsection is to build +a general theory of such a comparison for general module categories with traces. +At first we introduce the notion of a trace on a module category, which already +appears in [Sc13, Definition 3.7] for purely algebraic cases. +Let C be a rigid strict C*-tensor category and M be a C-module category. The +endomorphism algebra of X ∈ M is denoted by M(X). For U ∈ C and X ∈ M, +we can define the partial trace TrU ⊗ id: M(U ⊗ X) −→ M(X) as follows: +(TrU ⊗ id)(T) = (R∗ +U ⊗ idX)(idU ⊗X)(RU ⊗ idX) +Here (U, U, RU, RU) is a standard solution in C. +Definition 4.9. Let M be a C-module category. A C-module trace on M is a +family {TrX : M(X) −→ C}X∈M of positive linear maps satisfying the following +conditions: +(i) For any f, g ∈ M(X, Y ), TrX(g∗f) = TrY (fg∗). +(ii) TrU⊗X = TrX ◦(TrU ⊗ id). +We call a pair of C-module category and C-module trace on it as tracial C-module +category. In this case we define the dimension of X ∈ M as d(X) = TrX(idX). +By (ii), we have d(U ⊗ X) = d(U)d(X) for U ∈ C and X ∈ M. +Remark 4.10. If M is indecomposable, there exists at most one C-module trace +on M up to scalar multiplication. This can be seen by using the compatibility +with the left C-action. +Example 4.11. If we consider C as a C-module category in usual way, then +the family of categorical traces defines a C-module trace. Moreover, if another +rigid strict C*-tensor category D and dimension-preserving C*-tensor functor +F : C −→ D are given, D can be considered as a tracial C-module category by +U ⊗ X = F(U) ⊗ X and the family of categorical traces. + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +21 +Example 4.12. Let A be a Q-system in C. Then the category Mod -A of right +A-modules in C has a C-module trace. +Example 4.13. If C is a fusion category, every C-module category has a C-module +trace. This follows from the fact that every indecomposable C-module category +arises as a corner of a rigid C*-2-category with C in its diagonal. +Let M be a tracial C-module category. +Fix a set of representatives of all +equivalence classes of irreducible objects of M. It is denoted by Irr M. +For F ∈ [M, M]b, the algebra of bounded natural transformations from F to +F is denoted by Endb(F). Then Endb(idM) is isomorphic to ℓ∞(Irr M) via a +map a �−→ (ai)i∈Irr M. By using the dimension of objects in M, we can define a +weight ωM on Endb(idM) as follows: +ωM(a) = +� +i∈Irr M +d(i)2ai. +We call ωM the Plancherel weight of M. This defines a linear functional if and +only if Irr M is a finite set. +Recall we have a C*-tensor functor Φ: C −→ [M, M]b induced by the left +action of C on M. +Namely Φ(U) is a C*-fuctor from M to M which sends +X ∈ M to U ⊗ X ∈ M. We use the symbol ϕU,V to denote the unitary from +Φ(U) ⊗ Φ(V ) to Φ(U ⊗ V ). We can show the following nice property of ωM. +Proposition 4.14. Let M be a tracial C-module category and ωM be the Plancherel +weight. Then we have the following equalities for any U ∈ C and a positive natural +transformation η: Φ(U) −→ Φ(U): +ωM(Φ(RU)∗ϕU,U(id ⊗η)ϕ∗ +U,UΦ(RU)) = ωM(Φ(RU)∗ϕU,U(η ⊗ id)ϕ∗ +U,UΦ(RU)). +To prove this proposition, it is convenient to replace [M, M]b by the category +of column-finite bi-graded Hilbert spaces introduced in [DY13, Notation A.3.1]. +Set I = Irr M. Then the category HilbI×I of I × I-graded Hilbert spaces has +a canonical structure of a C*-multitensor category. Namely, for H = (Hij)ij and +K = (Kij)ij, their tensor product H ⊗ K is a I × I-graded Hilbert space whose +(i, j)-component is given by +� +k∈I +Hik ⊗ Kkj. +A I × I-graded Hilbert space H is said to be column-finite if, for each j ∈ I, +there are finitely many i ∈ I such that Hij ̸= 0. We also say that H is uniformly +finite if it satisfies the following: +sup +i∈I +� +j∈I +dim Hij < ∞, +sup +j∈I +� +i∈I +dim Hij < ∞. +The full subcategory consisting of column-finite (resp. uniformly finite) I × I- +graded Hilbert spaces is denoted by Hilbcf +I×I (resp. Hilbf +I×I). Then the rigid part +of HilbI×I is precisely Hilbf +I×I ([DY13, Lemma A.3.2]). +Now we compare [M, M]b and Hilbcf +I×I. For F ∈ [M, M]b, we have a column- +finite I×I-graded Hilbert space HF whose (i, j)-component is given by M(i, F(j)). +Here M(i, F(j)) is regarded as a Hilbert space by the inner product ⟨S, T⟩ = S∗T. + +22 +MAO HOSHINO +Then this construction gives an equivalence of C*-multitensor categories from +[M, M]b to Hilbcf +I×I ([DY13, Theorem A.2.1, Theorem A.2.2]). Moreover we also +have an equivalence [M, M]f +b ∼= Hilbf +I×I ([DY13, Proposition A.3.3]). From now +on, we identify these categories by this equivalence. +Let (U, U, R, R) be a standard solution in C. Then ϕ∗ +U,UΦ(R) gives a morphism +from 1 to HΦ(U) ⊗ HΦ(U). By the definition of a tensor product of I × I-graded +Hilbert spaces, this morphism can be presented as a direct sum of +Rji : C −→ M(i, U ⊗ j) ⊗ M(j, U ⊗ i), +where i, j ∈ I. Similarly ϕ∗ +U,UΦ(R) is presented as a direct sum of +Rji : C −→ M(j, U ⊗ i) ⊗ M(i, U ⊗ j). +In the following lemma, U ⊗ i is denoted by Ui. +Lemma 4.15. For any i, j ∈ I, (M(j, Ui), M(i, Uj), Rji, Rji) is a solution +of a conjugate equation in Hilbf. +Moreover, if we replace Rji and Rji with +d(i)1/2d(j)−1/2Rji, d(i)−1/2d(j)1/2Rji respectively, it gives a standard solution. +Proof. Since (Φ(U), Φ(U), ϕ∗ +U,UΦ(R), ϕU,UΦ(R)) is a solution of conjugate equa- +tion in [M, M]b, the following composition of maps is the identity: +� +j,i∈I +M(j, Ui) +� +i,k +� +j=l Rjk⊗id +−−−−−−−−−−→ +� +j,i∈I +� +k,l∈I +M(j, Uk) ⊗ M(k, Ul) ⊗ M(l, Ui) +� +j,l +� +i=k id ⊗R∗ +li +−−−−−−−−−−→ +� +j,i∈I +M(j, Ui). +Hence we have (id ⊗R∗ +ji)(Rji ⊗ id) = id for any i, j ∈ I. The same argument also +shows the other equality (id ⊗Rji∗)(Rji ⊗ id) = id. Now we complete the proof +of the former half of the statement. +To prove the latter half, we calculate ∥Rji∥ and ∥Rji∥. At first one should note +that M(i, Uj)⊗M(j, Ui) can be embedded in M(i, UUi) by T⊗S �−→ (idj ⊗S)T. +Moreove the projection pj onto its image is given by the following: +pj(X) = +� +V ∈(j,Ui) +(id ⊗V V ∗)X. +Then Rji(1) corresponds to the image by pj of the component of ϕ∗ +U,UΦ(R) at +i ∈ I. Hence we have +∥Rji∥2 = +� +V ∈(j,V i) +(R ⊗ idi)∗(id ⊗V V ∗)(R ⊗ idi) += +1 +d(i) +� +V ∈(j,V i) +TrUi(V V ∗) += +1 +d(i) +� +V ∈(j,V i) +Trj(V ∗V ) += d(j) +d(i) dim M(j, Ui). + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +23 +Similarly we also have +∥Rji∥2 = d(i) +d(j) dim M(i, Uj) = d(i) +d(j) dim M(j, Ui). +Hence we have ∥Rji∥∥Rji∥ = (dim M(j, Ui))2. By using the characterization of +standard solutions, we get the conclusion. +□ +Proof of Proposition 4.14. By the previous lemma, we have +Rji(id ⊗θ)R∗ +ji = d(j) +d(i) Tr(θ) +for any linear map θ: M(j, Ui) −→ M(j, Ui). +Let η: Φ(U) −→ Φ(U) be a positive natural transformation. Then it induces +a positive operator ηji : M(j, Ui) −→ M(j, Ui) for any i, j ∈ I. Then we have +ωM(Φ(RU)∗ϕU,U(id ⊗η)ϕ∗ +U,UΦ(RU)) = +� +i∈I +d(i)2 � +j∈I +R∗ +ji(id ⊗ηji)Rji += +� +i,j∈I +d(i)d(j) Tr(ηji). +For the right hand side of the equality in the statement, we also can show that +ωM(Φ(RU)∗ϕU,U(η ⊗ id)ϕ∗ +U,UΦ(RU)) = +� +i,j∈I +d(i)d(j) Tr(ηji). +Hence the statement holds. +□ +Based on Proposition 4.14, we introduce a notion of a standard solution in +[M, M]b. +Definition 4.16. Let F : M −→ M be a C*-functor. We say that a solution +(F, F, R, R) is standard if it satisfies the following identity for any positive natural +transformation η: F −→ F: +ωM(R∗(id ⊗η)R) = ωM(R +∗(η ⊗ id)R). +In this case, we define a weight TrF on Endb(F) by η �−→ ωM(R∗(id ⊗η)R). +Proposition 4.14 states that (Φ(U), Φ(U), ϕ∗ +U,UΦ(RU), ϕ∗ +U,UΦ(RU)) is standard +when (U, U, R, R) is standard. +Lemma 4.17. Let F ∈ [M, M]b be a rigid object. +(i) A standard solution for F exists and is unique up to unitary equivalence. +(ii) TrF is tracial and independent of a choice of a standard solution. +(iii) For a standard solution (F, F, R, R), the following map is an anti-∗- +isomorphism from Endb(F) to Endb(F): +η �−→ (id ⊗R +∗)(id ⊗η ⊗ id)(R ⊗ id). +Proof. If we regard F as an object HF ∈ Hilbf +I×I, a solution (F, F, R, R) of con- +jugate equation is decomposed into a family {(M(j, F(i)), M(i, F(j)), Rji, Rji)} + +24 +MAO HOSHINO +of solutions in Hilbf. Then we have the following for any positive natural trans- +formation η: F −→ F: +ωM(R∗(id ⊗η)R) = +� +i,j∈I +d(i)2R∗ +ji(id ⊗ηji)Rji, +ωM(R +∗(η ⊗ id)R) = +� +i,j∈I +d(j)2R +∗ +ji(ηji ⊗ id)Rji, +where ηji is a corresponding positive operator on M(j, F(i)). +Hence (F, F, R, R) is standard if and only if the following quadruple is a stan- +dard solution for each i, j ∈ I: +� +M(j, F(i)), M(i, F(j)), d(j)1/2d(i)−1/2Rji, d(j)−1/2d(i)1/2Rji +� +. +Now all of the statements follow from this characterization. +□ +We would like to come back to our main interest: a comparison of module +functors. Let D be another rigid strict C*-tensor category and (Θ, θ): D −→ C +be a C*-tensor functor. +Definition 4.18. The modular natural transformation aΘ of (Θ, θ) is a natural +transformation from Θ to Θ given by the following: +aΘ,U = (id ⊗ TrΘ(U))(θ∗ +U,UΘ(rUr∗ +U)θU,U). +Here (U, U, rU, rU) is a stardard solution in D. +Since all standard solutions are mutually unitary equivalent, aΘ,U does not +depend on the choice of (U, U, rU, rU). +We collect some fundamental properties of aΘ in the following lemma. +Lemma 4.19. The modular natural transformation aΘ is monoidal and invert- +ible. Its inverse is as follows: +a−1 +Θ,U = (TrΘ(U) ⊗ id)(θ∗ +U,UΘ(rUr∗ +U)θU,U). +Moreover, the following quadruple is a standard solution in C for every U ∈ D: +(Θ(U), Θ(U), (1 ⊗ a1/2 +Θ,U)θ∗ +U,UΘ(rU), (a−1/2 +Θ,U ⊗ id)θ∗ +U,UΘ(rU)). +Proof. One can check the statement by direct calculations. +□ +Example 4.20. If (Θ, θ) is the fiber functor from Repf G to Hilbf, each compo- +nent aΘ,π is precisely ρπ in [NT13, Proposition 1.4.4]. See also [NT13, Example +2.2.13]. +We regard a D-module functor (F, f) as a pair of a C*-functor F and a collec- +tion of unitary morphisms fU : F ⊗ Φ(Θ(U)) −→ Φ(Θ(U)) ⊗ F. +Proposition 4.21. Let (F, f): M −→ M be a D-module functor. If F is rigid +in [M, M]b, the following are equivalent for each U ∈ D. +(i) We have fU(idF ⊗Φ(aΘ,U)) = (Φ(aΘ,U) ⊗ idF)fU. +(ii) For a standard solution (Φ(Θ(U)), Φ(Θ(U)), RΦ(Θ(U)), RΦ(Θ(U))) in [M, M]b, +the following natural transformation is unitary: +(id ⊗ id ⊗R +∗ +Φ(Θ(U)))(id ⊗f ∗ +U ⊗ id)(RΦ(Θ(U)) ⊗ id ⊗ id). + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +25 +(iii) For a standard solution (F, F, R, R), the following natural transformation +f U : F ⊗ Φ(Θ(U)) −→ Φ(Θ(U)) ⊗ F is unitary: +f U = (R∗ ⊗ id ⊗ id)(id ⊗f ∗ +U ⊗ id)(id ⊗ id ⊗R). +Proof. Let (U, U, rU, rU) be a standard solution in D and define rΦ(Θ(U)) and +rΦ(Θ(U)) as follows: +rΦ(Θ(U)) = ϕ∗ +Θ(U),Θ(U)Φ(θ∗ +U,UΘ(rU)), +rΦ(Θ(U)) = ϕ∗ +Θ(U),Θ(U)Φ(θ∗ +U,UΘ(rU)). +By using the braiding equation, we have +(id ⊗ id ⊗r∗ +Φ(Θ(U)))(id ⊗f ∗ +U ⊗ id)(rΦ(Θ(U)) ⊗ id ⊗ id)f ∗ +U += (id ⊗ id ⊗r∗ +Φ(Θ(U)))(id ⊗f ∗ +U⊗U)(rΦ(Θ(U)) ⊗ id ⊗ id) += (id ⊗r∗ +Φ(Θ(U)) ⊗ id)(rΦ(Θ(U)) ⊗ id ⊗ id) += id ⊗ id . +Since fU is unitary, this implies that we have +fU = (id ⊗ id ⊗r∗ +Φ(Θ(U)))(id ⊗f ∗ +U ⊗ id)(rΦ(Θ(U)) ⊗ id ⊗ id), +fU = (id ⊗ id ⊗r∗ +Φ(Θ(U)))(id ⊗f ∗ +U ⊗ id)(rΦ(Θ(U)) ⊗ id ⊗ id). +Now we give a proof of (i) +⇐⇒ +(ii). Let (Θ(U), Θ(U), RΘ(U), RΘ(U)) be the +standard solution as in Lemma 4.19. Set +RΦ(Θ(U)) = ϕ∗ +Θ(U),Θ(U)Φ(RΘ(U)), +RΦ(Θ(U)) = ϕ∗ +Θ(U),Θ(U)Φ(RΘ(U)). +Then Proposition 4.14 asserts that (Φ(Θ(U)), Φ(Θ(U)), RΦ(Θ(U))), RΦ(Θ(U))) is a +standard solution. Under the condition (i), we have +(id ⊗ id ⊗R +∗ +Φ(Θ(U)))(id ⊗f ∗ +U ⊗ id)(RΦ(Θ(U)) ⊗ id ⊗ id) += (id ⊗ id ⊗R +∗ +Φ(Θ(U)))(id ⊗(id ⊗Φ(a1/2 +Θ,U))f ∗ +U(Φ(a−1/2 +Θ,U ) ⊗ id) ⊗ id)(RΦ(Θ(U)) ⊗ id ⊗ id) += (id ⊗ id ⊗r∗ +Φ(Θ(U)))(id ⊗f ∗ +U ⊗ id)(rΦ(Θ(U)) ⊗ id ⊗ id) += fU. +Hence (ii) holds for the standard solution. Since all standard solutions are mu- +tually unitary equivalent, we also have (ii) for a general standard solution. To +show the converse direction, one should note that we have +(id ⊗a1/2 +Θ,U)θ∗ +U,UΘ(rU) = (a−1/2 +Θ,U ⊗ id)θ∗ +U,Ua1/2 +Θ,U⊗UΘ(rU) += (a−1/2 +Θ,U ⊗ id)θ∗ +U,UΘ(rU)a1/2 +1 += (a−1/2 +Θ,U ⊗ id)θ∗ +U,UΘ(rU). +Similary we have +(id ⊗a1/2 +Θ,U)θ∗ +U,UΘ(rU) = (a−1/2 +Θ,U ⊗ id)θ∗ +U,UΘ(rU). + +26 +MAO HOSHINO +By using these formulae, we can see that +(Φ(a−1/2 +Θ,U ) ⊗ id)fU(id ⊗Φ(a1/2 +Θ,U)) += (id ⊗ id ⊗r∗ +Φ(Θ(U)))(Φ(a−1/2 +Θ,U ) ⊗ f ∗ +U ⊗ Φ(a1/2 +Θ,U))(rΦ(Θ(U)) ⊗ id ⊗ id) += (id ⊗ id ⊗R +∗ +Φ(Θ(U)))(id ⊗f ∗ +U ⊗ id)(RΦ(Θ(U)) ⊗ id ⊗ id). +Hence (Φ(a−1/2 +Θ,U ) ⊗ id)fU(id ⊗Φ(a1/2 +Θ,U)) is unitary. Now we have +(Φ(a−1/2 +Θ,U ) ⊗ id)fU(id ⊗Φ(aΘ,U))f ∗ +U(Φ(a−1/2 +Θ,U ) ⊗ id) = 1. +Then we can see that (i) holds for U since fU is unitary. We already have shown +(i) =⇒ (ii) and (ii) for U =⇒ (i) for U, hence (ii) =⇒ (i) also has been shown. +The equivalence of (ii) and (iii) can be seen from Lemma 4.17 (iii) and the +standardness of the following quadruples: +(F ⊗ Φ(Θ(U)), Φ(Θ(U)) ⊗ F, (id ⊗R ⊗ id)RΦ(Θ(U)), (id ⊗R ⊗ id)RΦ(Θ(U))), +(Φ(Θ(U)) ⊗ F, F ⊗ Φ(Θ(U)), (id ⊗RΦ(Θ(U)) ⊗ id)R, (id ⊗RΦ(Θ(U)) ⊗ id)R). +□ +By considering the case of idC : C −→ C, we obtain the following corollary. +Corollary 4.22. For a C-module functor (F, f), it is rigid in [M, M]C +b if and +only if F is rigid in [M, M]b. In this case, the conjugate object (F, f) of (F, f) +can be obtained from a standard solution (F, F, R, R) as follows: +• F: a conjugate object of F in [M, M]b. +• f U = (R∗ ⊗ id ⊗ id)(id ⊗f ∗ +U ⊗ id)(id ⊗ id ⊗R). +Corollary 4.23. If M is connected as a tracial C-module category, a standard +solution in [M, M]C +b is also standard in [M, M]b. +For a C-module category, we can consider an equivalence relation ∼C on Irr M +as follows: +i ∼C j +def +⇐⇒ there exists U ∈ C such that M(i, U ⊗ j) ̸= 0. +For a C*-functor F : M −→ M, we define a function dF : Irr M −→ R as +follows: +dF(i) = d(F(i)) +d(i) +. +Proposition 4.24. If (F, f) ∈ [M, M]D +b satisfies the conditions in Proposition +4.21 for all U ∈ D, the function dF is constant on each equivalence class of ∼D. +If F is an equivalence of categories, the converse direction also holds. +Proof. Let (F, F, R, R) be a standard solution in [M, M]b. Then the condition +(iii) of Proposition 4.21 implies that we have a D-module functor (F, f): M −→ +M given by +f U = (R∗ ⊗ id ⊗ id)(id ⊗f ∗ +U ⊗ id)(id ⊗ id ⊗R). + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +27 +Then R and R are morphisms in [M, M]D +b , hence R∗R is also a morphism from +idM to idM in [M, M]D +b . This implies that we have +(R∗R)Θ(U)⊗i = idΘ(U) ⊗(R∗R)i +for any U ∈ D and i ∈ Irr M. In particular i ∈ Irr M �−→ (R∗R)i is constant on +each equivalence class of ∼D. On the other hand, by the proof of Lemma 4.17, +we have +(R∗R)i = +� +j∈Irr M +d(j) dim M(j, F(i)) +d(i) += d(F(i)) +d(i) +. +This completes a proof of the first statement. +To show the second statement, we will check the condition (iii) of Proposition +4.21. Let (F, F, R, R) be a standard solution. Our assumption implies that F(i) is +irreducible for any i ∈ Irr M, hence d(F(i))−1/2d(i)1/2Ri = dF(i)−1/2Ri is unitary. +Moreover dF(i)−1/2RΘ(U)⊗i is also unitary for any U ∈ D by the assumption on +dF. Since we have +dF(F(i)) = d(F(F(i)))/d(F(i)) = dF(i)−1, +the similar argument shows that dF(i)−1/2RΘ(U)⊗i is also unitary. +Then we have +(R ⊗ id ⊗ id)(id ⊗f ∗ +U ⊗ id)(id ⊗ id ⊗R)i += dF(F(i))−1/2RU⊗F (i)F(f ∗ +U,F(i))F(idΘ(U) ⊗dF(i)−1/2Ri). +This implies that (F, f) fulfills the condition (iii). +□ +We end this subsection by showing that the condition (iii) is automatically +satisfied under a finiteness condition. +Theorem 4.25. Let M be a tracial C-module category with |Irr M| < ∞. Then +[M, M]D +b is rigid. Moreover a standard solution in [M, M]D +b is standard again +in [M, M]b. +We need the following lemma to show the theorem. +Lemma 4.26. Let (F, f) be a D-module functor from M to M and F be a +conjugate of F in [M, M]b. Then the function dF and dF are constant on each +equivalence class of ∼D. +Proof. Let I be an equivalence class of ∼D. Since Irr M is finite, I is also finite. +Hence we can take an object U ∈ D such that M(i, Θ(U)⊗j) ̸= 0 for any i, j ∈ I. +This means the matrix M = (dim M(j, Θ(U) ⊗ i))ij∈I is an irreducible matrix +with positive entries. Now d = (d(i))i∈I satisfies Md = d(Θ(U))d i.e. d is a +Perron-Frobenius eigenvector of M. On the other hand, we have +d(Θ(U))d(F(i)) = d(Θ(U) ⊗ F(i)) += d(F(Θ(U) ⊗ i)) = +� +j∈I +d(F(j)) dim M(j, Θ(U) ⊗ i). + +28 +MAO HOSHINO +Hence (d(F(i)))i∈I is also a Perron-Frobenius eigenvector of M, which must be a +scalar multiple of d. Now we get the statement for dF. For F, one should note +that F is an adjoint functor of F. Hence we have +M(i, F(Θ(U) ⊗ j)) ∼= M(F(i), Θ(U) ⊗ j) +∼= M(Θ(U) ⊗ F(i), j) +∼= M(F(Θ(U) ⊗ i), j) +∼= M(Θ(U) ⊗ i, F(j)) ∼= M(i, Θ(U) ⊗ F(j)). +for any i, j ∈ Irr M. This implies F(Θ(U) ⊗ i) ∼= Θ(U) ⊗ F(i) and we can use +the argument above to show the statement for F. +□ +Proof of Theorem 4.25. Take (F, f) ∈ [M, M]D +b and a standard solution (F, F, R, R) +in [M, M]b. Since (R∗R)i = dF(i)−1 and (R +∗R)i = dF(i)−1, the previous lemma +implies that +R∗R ⊗ idΦ(Θ(U)) = idΦ(Θ(U)) ⊗R∗R, +R +∗R ⊗ idΦ(Θ(U)) = idΦ(Θ(U)) ⊗R +∗R +for any U ∈ D. +Set f U as in Proposition 4.21: +f U = (R∗ ⊗ id ⊗ id)(id ⊗f ∗ +U ⊗ id)(id ⊗ id ⊗R). +It suffices to show that f U is unitary. Take a standard solution (U, U, rU, rU) in +D and set rΦ(Θ(U)) and rΦ(Θ(U)) as in the proof of Proposition 4.21. Now consider +ωl ∈ Endb(F ⊗ Φ(Θ(U)))∗ given by +ωl(η) = ωM(r∗ +Φ(Θ(U))(id ⊗R +∗ ⊗ id)(id ⊗ id ⊗η)(id ⊗R ⊗ id)rΦ(Θ(U))). + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +29 +Then we have +ωl(id) = ωM(r∗ +Φ(Θ(U))(id ⊗R +∗R ⊗ id)rΦ(Θ(U))) += ωM(r∗ +Φ(Θ(U))rΦ(Θ(U)) ⊗ R +∗R) += d(U) TrF(idF), +ωl(f +∗ +Uf U) = ωM((r∗ +Φ(Θ(U)) ⊗ R +∗)(id ⊗fUf ∗ +U ⊗ id)(rΦ(Θ(U)) ⊗ R)) += d(U) TrF(idF), +ωl(f +∗ +Uf Uf +∗ +Uf U) = ωM((r∗ +Φ(Θ(U)) ⊗ R +∗)(id ⊗fU ⊗ id)(id ⊗ id ⊗f Uf +∗ +U) +(id ⊗f ∗ +U ⊗ id)(rΦ(Θ(U)) ⊗ R)) += ωM((r∗ +Φ(Θ(U)) ⊗ R +∗)(id ⊗fU ⊗ id)(fUf ∗ +U ⊗ f Uf +∗ +U) +(id ⊗f ∗ +U ⊗ id)(rΦ(Θ(U)) ⊗ R)) += ωM(R +∗(id ⊗r∗ +Φ(Θ(U)) ⊗ id)(id ⊗ id ⊗f Uf +∗ +U)(id ⊗rΦ(Θ(U)) ⊗ id)R) += ωM((r∗ +Φ(Θ(U)) ⊗ R∗)(id ⊗f Uf +∗ +U ⊗ id)(rΦ(Θ(U)) ⊗ R)) += ωM(r∗ +Φ(Θ(U))(id ⊗R∗ ⊗ id)(id ⊗ id ⊗f ∗ +UfU)(id ⊗R ⊗ id)rΦ(Θ(U))) += ωl(id) += d(U) TrF(idF). +Hence ωl((1 − f +∗ +Uf U)2) = 0 and f +∗ +Uf U = 1 since ωl is faithful. +To show f Uf +∗ +U = 1, one can use the following ωr ∈ Endb(Φ(Θ(U)) ⊗ F)∗: +ωr(η) = ωM(r∗ +Φ(Θ(U))(id ⊗R∗ ⊗ id)(η ⊗ id ⊗ id)(id ⊗R ⊗ id)rΦ(Θ(U))). +Then a similar argument shows f Uf +∗ +U = 1. +□ +4.3. Imprimitivity-type result for the maximal Kac quantum subgroup. +In this subsection, we apply results in the previous subsection to module cate- +gories arising from quantum homogeneous spaces. +At first, we give a characterization of a quantum homogeneous space whose +associated module category has a module trace. +Proposition 4.27. Let G be a compact quantum group of Kac type and A be a +quantum homogeneous space of G. Then the following conditions are equivalent: +(i) G- Modf +A has a Repf G-module trace. +(ii) There is a tracial state on A. +(iii) The G-invariant state on A is tracial. +Proof. The equivalence of (ii) and (iii) follows from the G-invariance of (ϕ ⊗ h)α +for a state ϕ on A. +We show (ii) =⇒ (i) by constructing a Repf G-module trace from a given tracial +state τ on A. Let E be a finitely generated G-equivariant Hilbert A-module. Then +there exists a finite dimensional unitary representation π of G such that there is + +30 +MAO HOSHINO +an isometry V ∈ LG +A(E, Hπ ⊗ A). Now we define a tracial state TrE on LG +A(E) as +follows: +TrE(T) = (Tr ⊗τ)(V TV ∗). +Since Tr ⊗τ is tracial, this definition does not depend on the choice of V and +π. Moreover the corner trick shows that TrE satisfies (i) of Definition 4.9. To +prove the compatibility with the left action of Repf G, one should note that +the standard solution in Repf G is also standard in Hilbf. +Hence the partial +trace Trπ ⊗ id: LG +A(Hπ ⊗ E) −→ LG +A(E) coincides with Tr ⊗ id. This implies the +compatibility. +Next we move to the proof of (i) =⇒ (ii). Fix a Repf G-module trace {TrE}E on +G- Modf +A with TrA(1) = 1. By using the spectral decomposition of the algebraic +core A of A, the G-invariant state ϕ on A can be seen as the projection from +A to A1G. We show that this is tracial by using {TrE}E. Since the spectral +decomposition of A is orthogonal with respect to the inner product coming from +ϕ, it suffices to show that the A1G-components of (T⊗ξ)∗(S⊗η) and (S⊗η)(T⊗ξ)∗ +are same for T ⊗ξ, S ⊗η ∈ Aπ. For the first one, its A1G-component is calculated +as follows: +⟨ξ, η⟩ +dim π(R∗ +π ⊗ idA)(idπ ⊗T ∗S)(Rπ ⊗ idA) += ⟨ξ, η⟩ +dim π TrA((R∗ +π ⊗ idA)(idπ ⊗T ∗S)(Rπ ⊗ idA))1A += ⟨ξ, η⟩ +dim π TrHπ⊗A(T ∗S)1A += ⟨ξ, η⟩ +dim π TrA(ST ∗)1A += ⟨ξ, η⟩ +dim πST ∗. +The last formula is precisely the A1G-component of the second one. Now we get +the conclusion. +□ +Let G be a compact quantum group and K be a its maximal Kac quantum +subgroup. The restriction functor from Repf G to Repf K is denoted by Θ. Fix a +set Irr G of representatives of all equivalence classes of irreducible representations +of G. +For π, ρ ∈ Repf G, we defines L0(π, ρ) as a subset of HomK(π|K, ρ|K) consisting +of intertwiners of the following form: +T ∗ ◦ (an1 +Θ,σ1 ⊗ an2 +Θ,σ2 ⊗ · · · ⊗ ank +Θ,σk) ◦ S +where k ∈ Z>0, ni ∈ +1 +2Z, σi ∈ Irr G, T ∈ HomG(ρ, σ1 ⊗ σ2 ⊗ · · · ⊗ σk), S ∈ +HomG(π, σ1⊗σ2⊗· · ·⊗σk). Then we define a subspace L(π, ρ) of HomK(π|K, ρ|K) +as the linear span of intertwiners of the form TlTl−1 · · · T1 where Tj ∈ L0(σj−1, σj), +σ0 = π, σl = ρ, σj ∈ Irr G for 1 ≤ j ≤ l − 1. +Lemma 4.28. For any π, ρ ∈ Repf G, we have L(π, ρ) = HomK(π|K, ρ|K). +Proof. Consider a C*-category with Obj Repf G as the collection of object and +L(π, ρ) as the morphism set from π to ρ. Let L be its Karoubi envelope. Since + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +31 +T ⊗ S ∈ L(π ⊗ π′, ρ ⊗ ρ′) for any T ∈ L(π, ρ) and S ∈ L(π′, ρ′), L becomes +a C*-tensor category. Moreover the restriction functor from Repf G to Repf K +induces a C*-tensor functor from Repf G to L. In particular π ∈ Repf G is rigid +in L, hence L is rigid. +Now, by composing the inclusion L −→ Repf K and the fiber functor Repf K −→ +Hilbf, we obtain a fiber functor L −→ Hilbf. Then the Woronowicz’s Tannaka- +Krein duality ([NT13, Theorem 2.3.2]) implies that there is a compact qunatum +group H and morphisms K −→ H and H −→ G which correpond to L and the +functors. Since every object of L appears as a direct summand of the image of +π ∈ Repf K, H can be considered as a quantum subroup of G which contains K. +On the other hand, (π, π, (id ⊗a1/2 +Θ,π)Rπ, (a−1/2 +Θ,π +⊗ id)Rπ) is a solution of the +conjugate equation in L. Moreover it satisfies +∥(id ⊗a1/2 +Θ,π)Rπ∥ = ∥(a−1/2 +Θ,π ⊗ id)Rπ∥ = dim Hπ. +This implies that the fiber functor of Repf K = L is dimension-preserving, hence +H is of Kac type. Then H must be contained in K, so we have H = K. Hence the +dimension of L(π, ρ) and HomK(π|K, ρ|K) is same. Now we get the conclusion. +□ +Theorem 4.29. Let A be a quantum homogeneous space of K and �A be its induced +G-C*-algebra. If A has a tracial state and Irr K- Modf +A is finite, G- Corrrf +� +A is rigid +and Ind G +K gives an equivalence K- Corrrf +A ∼= G- Corrrf +� +A as C*-tensor categories. +Proof. Set M = K- Modf +A. Then M is a tracial Repf K-module category with +|Irr M| < ∞. +Since we have Theorem 4.8, it suffices to show the canonical inclusion from +[M, M]Repf K +b +to [M, M]Repf G +b +is an equivalence. It can be easily seen that this +inclusion is fully faithful, hence the remaining part is the essential surjectivity. +Take (F, f) ∈ [M, M]Repf G +b +. Theorem 4.25 and Proposition 4.21 assert that +we have +(Φ(aΘ,π) ⊗ idF)fπ = fπ(idF ⊗Φ(aΘ,π)) +for any π ∈ Repf G, where (Φ, ϕ): Repf K −→ [M, M]b is the canonical C*- +tensor functor. +By using the braiding equation on u and the previous lemma, we also have +(Φ(T) ⊗ idF)fπ = fπ(idF ⊗Φ(T)) +for any π ∈ Repf G and any T ∈ HomK(π|K, π|K). Hence we can define fρ for +ρ ∈ Repf K as follows: +�fρ = (Φ(V )∗ ⊗ idF)fπ(idF ⊗Φ(V )) +where π ∈ Repf G and V ∈ HomK(ρ, π|K) is an isometry. By the above equality, +the definition of �fρ does not depend on the choice of π and V . It can be easily seen +that (F, �f) defines a Repf K-module functor, which is (F, f) as a Repf G-module +functor. +□ +Example 4.30. Set A = C(K). Then the corresponding Repf K-module category +is Hilbf with the action via the fiber functor. This Repf K-module category has a + +32 +MAO HOSHINO +Repf K-module trace and only one irreducible object, hence we can apply Theo- +rem 4.29. Since the induction of C(K) is C(G), we can conclude that G- Corrrf +C(G) +is rigid and we have an equivalence of C*-tensor categories between K- Corrrf +C(K) +and G- Corrrf +C(G). +On the other hand, we have K- Corrrf +C(K) ∼= Repf O(K) and +G- Corrrf +C(G) ∼= Repf O(G) by Proposition 3.10). Combining these equivalences, +we can see that Repf O(G) is rigid and Repf O(K) ∼= Repf O(G). +A direct +calculation shows that this equivalence is induced by the canonical morphism +q: O(G) −→ O(K). +By these obserbation, Theorem 4.29 can be regarded as a generalization of +[CKS, Theorem 2.3] and b�K ∼= b�G, which is essentailly proved in [So05, Section +4.3]. +For a quantum homogeneous space A of G, we define its Picard group PicG(A) +as a group of all equivalence classes of invertible objects of G- Modf +A (c.f [DC12, +Corollary 2.5]). Its product is given by the tensor product. +By using Proposition 4.24 instead of Theorem 4.25 in the proof of Theorem +4.29, we can obtain the following theorem. +Theorem 4.31. Let �Γ be a cocommutative quantum subgroup of K. Then for +any quantum homogeneous space A of �Γ, the induction functor gives a group +isomorphism PicK(Ind K +�ΓA) ∼= PicG(Ind G +�ΓA). +Moreover we can also see that the induction functor gives an isomorphism +between automorphism groups. For a quantum homogeneous space A of G, the +group of G-equivariant automorphisms of A is denoted by AutG(A). +If θ ∈ +AutG(A) is given, we can construct an invertible G-equivariant correspondence +Aθ over A defined as follows: +• As a G-equivariant Hilbert A-module, Aθ = A. +• The left multiplication of x ∈ A on Aθ is given by the left multiplication +of θ−1(x). +Then we have Aθ ⊗A Aθ′ ∼= Aθθ′, so we have a group homomorphism θ ∈ +AutG(A) �−→ [Aθ] ∈ PicG(A). This is injective since AG = C1A. +Lemma 4.32. Let M be an invertible G-equivariant correspondence over A. +Then there is θ ∈ AutG(A) such that M ∼= Aθ if and only if M has a non-zero +G-invariant vector. +Proof. It is trivial that there exists a non-zero G-invariant vector in Aθ. Hence +it suffices to show the existence of a non-zero G-invariant vector implies the exis- +tence of θ. Since M is irreducible as a G-equivariant Hilbert A-module, it must +be isomorphic to A. Hence there is a left A-action on A by which M and A are +isomorphic as G-equivariant correspondence over A. But such a correspondence +must be of the form Aθ where θ: A −→ A is a G-equivariant ∗-homomorphism. +Since M is invertible, this θ must be an automorphism. Now we get the conclu- +sion. +□ +Theorem 4.33. In the same setting as Theorem 4.31, the induction of equivari- +ant automorphism gives a group isomorphism AutK(Ind K +�ΓA) ∼= AutG(Ind G +�ΓA). + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +33 +Proof. One should note that Ind G +KE has a non-zero G-invariant vector if and only +if E has a non-zero K-invariant vector for any E ∈ K- Modf +A. +Then the statement follows from Theorem 4.31 and the previous lemma. +□ +Example 4.34. Let Q ∈ GLn(C) be a positive invertible matrix with multiplicity- +free eigenvalues. Then the maximal Kac quantum subgroup of the free unitary +quantum group U+ +Q ([DW96, Theorem 1.3]) is isomorphic to the discrete dual of +the free group Fn ([DFS, Theorem 2.6]). In particular it is cocommutative, hence +we have group isomorphisms +Pic� +Fn(A) ∼= PicU+ +Q(Ind +U+ +Q +� +Fn A), +Aut� +Fn(A) ∼= AutU+ +Q(Ind +U+ +Q +� +Fn A) +for any quantum homogeneous space A of � +Fn. +5. Classification results for Gq +5.1. Imprimitivity-type results. Let G be a simply-connected compact Lie +group and T be its maximal torus. +The Weyl group of G is denoted by W. +Consider the Drinfeld-Jimbo deformation Gq for a fixed q ∈ (−1, 1) \ {0}. We +use the symbol εt for the character on C(Gq) defined as the evaluation at t ∈ T. +Let us recall the representation theories of C(Gq) and C(T\Gq), developed in +[DS99, NY12, So91]. +For each w ∈ W, we have an irreducible ∗-representation (πw, Hw) of C(Gq). +It coincides with the counit ε if w = e, and is infinite dimensional otherwise. +By using these representations, � +C(Gq) can be identified with W × T as a set via +(w, t) ∈ W × T �−→ (εt ⊗ πw)∆. If we consider a Borel structure on W × T +induced from � +C(Gq) via this identification, each {w} × T ⊂ W × T is a Borel +subset of � +C(Gq). +For C(T\Gq), the restriction of πw on C(T\Gq) is still irreducible and w ∈ +W �−→ πw|C(T\Gq) is a bijection. The induced Borel structure of W is discrete. +The following proposition is what we would like to use in this subsection +Proposition 5.1. The double dual C(T\Gq)∗∗ is isomorphic to � +w∈W B(Hw). +Moreover, its center is contained in Z(C(Gq)∗∗). +Proof. One shoule note that C(T\Gq) is separable type I. Hence we can disinte- +grate any separable ∗-representation of C(T\Gq) on +� +C(T\Gq) ∼= W. Since W is +discrete as a Borel space, this implies that a separable ∗-representation admits an +irreducible decomposition. By using a decomposition into cyclic representations, +we also have an irreducible decomposition for a non-separable ∗-representation. +This implies the former half of the statement. +Let pw ∈ C(T\Gq)∗∗ be the projection corresponding to B(Hw). +To show +the latter half of the statement, it suffices to show π(pw) ∈ π(C(Gq))′ for any +separable ∗-representation (π, H) of C(Gq). Since C(Gq) is type I, we can disin- +tegrate (π, H) on � +C(Gq). Then each w ∈ W defines a projection Pw ∈ π(C(Gq))′ +corresponding to the characteristic function of {w} × T ⊂ W × T ∼= � +C(Gq). +This Pw must be π(pw), since we have (εt ⊗ πw)∆|C(T\Gq) = πw for any t ∈ T. +In particular we have π(pw) = Pw ∈ π(C(Gq))′. +□ + +34 +MAO HOSHINO +Remark 5.2. Let {pw}w∈W as in the proof. +Then the canonical map from +C(Gq)∗∗ to C(T)∗∗ factors though x ∈ C(Gq)∗∗ �−→ pex ∈ peC(Gq)∗∗. More- +over we have peC(Gq)∗∗ ∼= C(T)∗∗. +Corollary 5.3. Let A be a T-C*-algebra and �A be its induced Gq-C*-algebra. +For any tracial positive linear functional τon �A, there is a tracial positive linear +functional τ ′ on A such that τ = τ ′ ◦ (id ⊗ε)| � +A. +Proof. Take a tracial positive linear functional τ on �A. If τ = 0, there is nothing +to prove. Hence we may assume τ ̸= 0, moreover τ is a tracial state by taking a +normalization. +Now consider the GNS triple (πτ, Hτ, ξτ) with respect to τ. +Then πτ( �A)′′ +is a finite von Neumann algebra. +In particular πτ(C(T\Gq))′′ is finite, hence +πτ(pe) = 1 and πτ(pw) = 0 for w ̸= e. +Take a state ϕ on A⊗C(Gq)∗ such that ϕ| � +A = τ and its GNS triple (πϕ, Hϕ, ξϕ). +Then we have an isometry V : Hτ −→ Hϕ such that V πτ(x)ξτ = πϕ(x)τ for every +x ∈ �A. Hence, for any w ∈ W \ {e}, we have +πϕ(1 ⊗ pw)ξϕ = V πτ(pw)ξτ = 0. +Since 1 ⊗ pw is central in (A ⊗ C(Gq))∗∗ and ξϕ is cyclic, this implies πϕ(pw) = 0 +for w ̸= e. Then the remark after Proposition 5.1 implies that πϕ factors though +A ⊗ C(Gq) −→ A ⊗ C(T), in particular πτ factors through �A −→ Ind T +TA. Since +(id ⊗ε)|Ind T +T A gives an isomorphism Ind T +TA ∼= A, our statement follows. +□ +The following is a main theorem of this subsection. +Theorem 5.4. Let A and B be T-C*-algebras. If B has a faithful family of +tracial states, any Gq-equivariant ∗-homomorphism from Ind Gq +T A to Ind Gq +T B is +induced from a T-equivariant ∗-homomorphism from A to B. +Proof. Let ϕ: Ind Gq +T A −→ Ind Gq +T B be an Gq-equivariant ∗-homomorphism. For +any tracial state τ on B, τ ◦ (id ⊗ε) ◦ ϕ is also a tracial state on Ind Gq +T B. Hence +it factors through A. Since B has a faithful family of tracial states, the map +(id ⊗ε) ◦ ϕ also factors through A, i.e. we have a ∗-homomorphism ϕ0 : A −→ B +such that (id ⊗ε) ◦ ϕ = ϕ0 ◦ (id ⊗ε). The T-equivariance of ϕ0 follows from the +Gq-equivariance of ϕ. +We would like to show that ϕ0 is the required homomorphism. For an element +a ∈ Ind Gq +T A, we have +ϕ(a) = (id ⊗ε ⊗ id)(�β(ϕ(a))) += ((id ⊗ε) ◦ ϕ ⊗ id)(�α(a)) += (ϕ0 ◦ (id ⊗ε) ⊗ id)(�α(a)) += (ϕ0 ⊗ id)(a). +Hence we have ϕ = Ind Gq +T ϕ0. +□ +Corollary 5.5. Let A, B be quantum homogeneous spaces of T and �A, �B be its +induced Gq-C*-algebras. Then Ind Gq +T : T- Corrrf +A,B −→ Gq- Corrrf +� +A, � +B is an equiva- +lence of C*-categories. + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +35 +Proof. What we have to show is the essential surjectivity of Ind Gq +T . Take � +M ∈ +Gq- Corrrf +� +A, � +B. By Theorem 4.1, we can take M ∈ T- Modf +B such that � +M ∼= Ind Gq +T M +as a Gq-equivariant Hilbert �B-module. Then the right action of �A on � +M induces +a Gq-equivariant ∗-homomorphism from �A to L � +B(� +M) ∼= Ind Gq +T LB(M). +If we +can show that LB(M) has a faithful trace, the previous theorem implies that +this ∗-homomorphism is induced by a T-equivariant ∗-homomorphism from A +to LB(M). This defines a T-equivariant (A, B)-correspondence M and we have +Ind Gq +T M ∼= � +M. +Now we show the existence of a faithful trace on LB(M). By taking an isometry +from M to Hπ ⊗ B for some π ∈ Repf T, we may assume M = Hπ ⊗ B. Then +LB(Hπ ⊗B) ∼= B(Hπ) ⊗B, and the existence follows from the finiteness theorem +[HKLS, Theorem 4.1], which asserts that there is a faithful trace on B. +□ +Corollary 5.6. Let �A be a quantum homogeneous space of Gq. If �A contains +C(T\Gq) as a unital Gq-C*-subalgebra and has a tracial state, it must be induced +from a quantum homogeneous space of T. +Proof. Let J be an ideal of �A defined as follows: +J = {x ∈ �A | τ(x∗x) = 0 for any tracial state τ ∈ �A∗}. +By our assumption, this is a T-invariant proper closed ideal of �A. +Set A = �A/J. +The quotient map from �A to A is denoted by q. +The T- +invariance of J allows us to induce a T-action on A. We will show that (q ⊗ id)�α +gives a Gq-equivariant ∗-isomorphism from �A to Ind Gq +T A. The well-definedness +and Gq-equivariance of this map are easy to see. Since �AGq = C1 � +A, the injectivity +is automatic. Hence it suffices to show the surjectivity. By Theorem 5.4, the re- +striction (q⊗id)�α|C(T\Gq) is induced from C ⊂ A i.e. coincides with the canonical +inclusion C(T\Gq) ⊂ Ind Gq +T A. Hence the image of this inclusion is contained in +the image of (q ⊗ id)�α. On the other hand, for any a ∈ �A and x ∈ C(Gq), we +have +PA((q ⊗ id)�α(a)(1 ⊗ x)) = (q ⊗ id)�α(a)PA(1 ⊗ x) += (q ⊗ id)�α(a)(1 ⊗ (hT ◦ qT ⊗ id)∆(x)), +where PA is the expectation as in Lemma 4.3, hT is the Haar state on C(T) and +qT is the canonical quotient map from C(Gq) to C(T). Since (hT ◦ qT ⊗ id)∆(x) +is an element of C(T\Gq), we can conclude the surjectivity of (q ⊗ id)�α from this +equality. +□ +Corollary 5.7. Let �A ⊂ �B be a finite quantum Gq-covering space over a quantum +homogeneous space. If �A contains C(T\Gq) and has a tracial state, �A ⊂ �B is +isomorphic to Ind Gq +T A ⊂ Ind Gq +T B, where A ⊂ B is a finite quantum T-covering +space. +5.2. Discrete quantum subgroups. In this subsection, we develop some ap- +plications to discrete quantum groups and give a classification of finite index +discrete quantum subgroup of � +Gq. + +36 +MAO HOSHINO +Let G be a compact quantum group and H be its quotient. Then we have an in- +clusion C(H) ⊂ C(G). In this case we have a unique G-expectation E : C(G) −→ +C(H), namely the Haar state-preserving conditional expectation. +Definition 5.8. Let �G be a discrete quantum group and �H be a quantum sub- +group of G. The scalar index of E : C(G) −→ C(H) is called the quantum group +index and denoted by [�G: �H]. The quantum subgroup �H is said to be finite index +if [�G: �H] is finite. +Remark 5.9. The index [�G: �H] must be either a positive integer or ∞. This +can be seen from Corollary 3.13 if one consider the basic construction of C(H) ⊂ +C(G). +Beford giving a classification result of finite index quantum subgroup of � +Gq, +we prepare the following lemma. Let G be a compact quantum group and K be +its maximal Kac quantum subgroup. The canonical map from O(G) to O(K) is +denoted by q. +Lemma 5.10. Let A be a right coideal of G and A be its algebraic core. If the +canonical inclusion A ⊂ C(G) is a finite quantum G-covering space, we have +q−1(q(A)) = A. +Proof. Let E : C(G) −→ A be a G-expectation. Such a expectation is unique +and finite index by our assumption. +Now consider the G-equivariant Hilbert +A-module C(G)E. +This gives a G-equivariant imprimitivity bimodule betwee +A and B = LA(C(G)E). +In particular we have an equivalence G- Modf +A ∼= +G- Modf +B of Repf G-module category. Moreover, Irr G- Modf +A is finite since –⊗C(G) +C(G)E : G- Modf +C(G) −→ G- Modf +A is adjointable and Irr G- Modf +C(G) consists of +only one element. +On the other hand, since C(G) ⊂ B is a finite quantum G-covering space, +Theorem 4.29 implies that there is a finite quantum K-covering space C(K) ⊂ B0 +such that C(G) ⊂ B is isomorphic to Ind G +KC(K) ⊂ Ind G +KB0. In particular we +have an equivalence G- Modf +B ∼= K- Modf +B0 as Repf G-module categories. Hence +we have a structure of Repf K-module category on G- Modf +A. Since Irr G- Modf +A +is finite, Theorem 4.25 implies that – ⊗A O(G) is a Repf K-module functor from +G- Modf +A to G- Modf +C(G) ∼= K- Modf +C(K). +Take a quantum homogeneous space A0 of K corresponding to an irreducible +object A of a Repf K-module category G- Modf +A. +If we fix an identification +G- Modf +A ∼= K- Modf +A0, we have a corresponding G-equivariant isomorphism A ∼= +Ind G +KA0. Moreover – ⊗A O(G) induces a Repf K-module functor from K- Modf +A0 +to K- Modf +C(K). By Theorem 3.14, we have a corresponding M ∈ K- Corrrf +A0,C(K), +which satisfies A0 ⊗A0 M ∼= C(K). Hence M can be thought as C(K) as a K- +equivariant Hilbert C(K)-module. Then we have a K-equivariant ∗-homomorphism +ϕ: A0 −→ C(K). By our construction, this map makes the following diagram + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +37 +commutative: +Ind G +KA0 +Ind G +Kϕ� +∼ += +� +Ind G +KC(K) +ε⊗id +� +A +� C(G). +This implies that q(A) = ϕ(A0) and q−1(ϕ(A0)) = A. +□ +Now we give a classification theorem of finite index discrete quantum subgroup +of � +Gq. Let P (resp. Q) be the weight (resp. root) lattice. We naturally identifies +P with the Pontrjagin dual of T. +Theorem 5.11. There is a one-to-one correspondence between the set of finite +index discrete quantum subgroups of � +Gq and the set of subgroups of P/Q. +Proof. Let q: C(Gq) −→ C(T) be the canonical map. Take a finite index discrete +quantum subgroup �H ⊂ � +Gq. The representation category Repf H can be thought +as a full subcategory of Repf Gq. +Consider the image q(C(H)) ⊂ C(T). This defines a compact quantum group +H if it is equipped with the restriction of the coproduct on C(T). Then H is +a quotient compact group of T and gives a full subcategory Repf H of Repf T. +The property q(C(H)) = C(H) implies that Repf H consists of π ∈ Repf T which +appears as a direct summand of ρ|T for some ρ ∈ Repf H ⊂ Repf Gq. Combining +this with the highest weight theory, we can see that �H ⊂ �T = P must contains +the root lattice Q. +On the other hand, we have q−1(C(H)) = C(H). This implies that Repf H +consists of π ∈ Repf Gq such that π|T ∈ Repf H ⊂ Repf T. Hence Repf H is the +full subcategory of Repf Gq consisting of π ∈ Repf Gq whose weights are elements +of �H. These obserbations imply that �H �−→ �H/Q is an injective map from the +set of finite index discrete quantum subgroups of � +Gq to the set of subgroups of +P/Q. +To show the surjectivity, take a subgroup Λ ⊂ P containing Q. Let C be the +full subcategory of Repf Gq consisting of π ∈ Repf Gq whose weights are elements +of Λ. Then this is a rigid C*-tensor category, hence we have a discrete quantum +group �H of � +Gq such that Repf H ∼= C. Moreover the inclusion C(H) ⊂ C(Gq) +is induced from C(�Λ) ⊂ C(T). Since Q ⊂ P is finite index, this inclusion has +a T-expectation with finite index. Hence C(H) ⊂ C(Gq) has a Gq-expectation +with finite index i.e. �H is finite index � +Gq. Since the image of �H under the map +defined in the previous paragraph is Λ, we have the surjectivity. +□ +Appendix A. +A.1. Proof of Propostion 2.8. +Lemma A.1. Let A be a G-C*-algebra. Then its fixed point subalgebra AG is +non-degenerate in A. Moreover, for any a ∈ A, we have x ∈ AG and b ∈ A such +that a = xb. + +38 +MAO HOSHINO +Proof. Let A be the algebraic core of A. For the non-degeneracy of AG ⊂ A, it +suffices to show that any a ∈ A can be approximated by an element of the form +xb with x ∈ AG and b ∈ A. Take an arbitrary ε > 0 and set α(a) = �n +i=1 ai +0 ⊗ai +1. +Then we can find e ∈ A such that ∥eai +0 − ai +0∥ < ε(n∥S(ai +1)∥)−1 for 1 ≤ i ≤ n. +Now we have +α(e)(a ⊗ 1) − a ⊗ 1 = +n +� +i=1 +α(eai +0 − ai +0)(1 ⊗ S(ai +1)). +Hence we also have +∥(id ⊗h)(α(e))a − a∥ ≤ ∥α(e)(a ⊗ 1) − a ⊗ 1∥ ≤ +n +� +i=1 +∥eai +0 − ai +0∥∥S(ai +1)∥ < ε. +Since (id ⊗h)(α(e)) is in AG, this inequality completes the proof of the first half. +Next we show tha latter half of our statement. Take an irreducible representa- +tion π of G and let C(G)π be the linear span of matrix coefficients of π. Moreover +set Aπ as follows: +Aπ = {x ∈ A | α(x) ∈ A ⊗ C(G)π}. +Then the following properties hold as shown in [DC16, Section 3]: +(i) A = � +π∈Irr G Aπ. +(ii) Each Aπ is a closed AG-subbimodule of A. +(iii) The projection onto Aπ with respect to the decomposition in (i) extends +to a continuous projection Eπ : A −→ Aπ. +By using (i), it can be seen that (ii) and (iii) hold for AF = � +π∈F Aπ, where +F is a finite subset of Irr G. +Take an arbitrary a ∈ A and a finite subset F ⊂ Irr G such that a ∈ AF. Since +AG is non-degenerate in A, an approximate unit of AG acts as an approximate +unit of the left Banach AG-module AF. Then we can take x ∈ AG and b ∈ AF such +that a = xb by using the Cohen-Hewitt factorization theorem ([BD73, Chapter +I, Section 11, Theorem 10]). Since AF ⊂ A, this completes the proof. +□ +Proposition A.2 (Proposition 2.8). Let A and B be G-C*-algebras with the +algebraic cores A and B respectively. If ϕ: A −→ B is G-equivariant and com- +pletely positive as a map from A to B, it extends to a G-equivariant c.p. map +from A to B. +Proof. By replacing ϕ by ϕ ⊕ id: A −→ B ⊕ A, we may assume ϕ is faithful in +the sense that ϕ(x∗x) = 0 if and only if x = 0. +By our assumption, we can define a B-valued semi-inner product on A ⊗ B as +follows: +⟨a ⊗ b, a′ ⊗ b′⟩B = b∗ϕ(a∗a′)b′. +Let E be a Hilbert B-module obtained by taking a completion of A⊗B. We show +that a ∈ A defines a adjointable right B-module map La on E by La(a′ ⊗ b′) = +aa′ ⊗ b′. Take an arbitrary element �n +i=1 ai ⊗ bi ∈ A ⊗ B. Then Lemma A.1 +implies that we have a presentation ai = xia′ +i with xi ∈ AG and a′ +i ∈ A. Since +AG is a C*-algebra, for any a ∈ AG we can take X ∈ AG ⊗ Mn(C) such that +x∗a∗ax + X∗X = ∥a∥2x∗x, + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +39 +where x = (x1 x2 . . . xn). Then the complete positivity of ϕ implies that +� n +� +i=1 +aai ⊗ bi, +n +� +j=1 +aaj ⊗ bj +� +B += +n +� +i,j=1 +b∗ +i ϕ(a′∗ +i x∗ +i a∗axja′ +j)bj +≤ ∥a∥2 +n +� +i,j=1 +biϕ(a′∗ +i x∗ +i xja′ +j)bj += ∥a∥2 +� n +� +i=1 +ai ⊗ bi, +n +� +j=1 +aj ⊗ bj +� +B +. +Hence La is well-defined and bounded. +For general a ∈ A, we use the argument in the proof of [DC16, Proposition +4.1]. Then we can take a family (vi)n +i=1 ⊂ A such that a = v1 and � +i v∗ +i vi ∈ AG, +which gives an estimate from above as follows: +⟨aξ, aξ⟩B ≤ +� +ξ, +n +� +i=1 +v∗ +i viξ +� +B +≤ ∥ +n +� +i=1 +v∗ +i vi∥ ⟨ξ, ξ⟩B . +This proves the boundedness of La. +The adjointability can be seen from the +following: +⟨aa1 ⊗ b1, a2 ⊗ b2⟩B = ⟨a1 ⊗ b1, a∗a2 ⊗ b2⟩B . +In particular L∗ +a = La∗ holds, hence a �−→ La defines a ∗-homomorphism L: A −→ +LB(E). This map is faithful since ϕ is faithful. +Next we show that there is a unitary V : E ⊗β (B ⊗C(G)) −→ E ⊗C(G) which +satisfies the following for any a ∈ A, b, c ∈ B and x ∈ C(G). +V ((a ⊗ b) ⊗ (c ⊗ x)) = α(a)13β(b)23(1 ⊗ c ⊗ x). +Tha well-definedness and the boundedness of this map can be seen as follows: +⟨V ((a ⊗ b) ⊗ (c ⊗ x)), V ((a′ ⊗ b′) ⊗ (c′ ⊗ x′))⟩B⊗C(G) += (c∗ ⊗ x∗)β(b∗)(ϕ ⊗ id)(α(a∗a′))β(b′)(c′ ⊗ x′) += (c∗ ⊗ x∗)β(b∗ϕ(a∗a′)b′)(c′ ⊗ x′) += ⟨(a ⊗ b) ⊗ (c ⊗ x), (a′ ⊗ b′) ⊗ (c′ ⊗ x′)⟩B⊗C(G) . +In particular V is an isometry, hence it suffices to show that the range of V is +dense in E ⊗ C(G). This follows from span α(A)(C ⊗ O(G)) = A ⊗ O(G) and +span β(B)(C ⊗ O(G)) = B ⊗ O(G). +Let λ: C(G) −→ LC(G)(C(G)) be a ∗-homomorphism defined by the left mul- +tiplication. Then we have V (La ⊗β 1)V ∗ = (L ⊗ λ)α(a) for any a ∈ A. This +implies the following: +∥La∥ = ∥La ⊗β 1∥ = ∥(L ⊗ λ)(α(a))∥ = ∥(L ⊗ id)(α(a))∥. +We use the faithfullness of β and λ at the first and third equality respectively. +Since L is faithful, a ∈ A �−→ ∥La∥ defines a C*-norm on A. The above equality +implies the completion of A with respect to this norm has an action of G induced +by α. Then [DY13, Proposition 4.4] implies that this norm coincides with the +original norm of A. Hence L extends to a ∗-homomorphism L: A −→ LBG(E). + +40 +MAO HOSHINO +Then we have the following inequality for x, a ∈ A, b ∈ B: +∥b∗ϕ(a∗xa)b∥ = ∥⟨a ⊗ b, Lx(a ⊗ b)⟩B∥ +≤ ∥x∥∥a ⊗ b∥2 ≤ ∥x∥∥b∥2∥ϕ(a∗a)∥. +By using an approximate unit of B, we also have the following inequality for +x, a ∈ A: +∥ϕ(a∗xa)∥ ≤ ∥x∥∥ϕ(a∗a)∥. +Hence x �−→ ϕ(a∗xa) is continuous for any a ∈ A. By using the polarization +identity, we also see that ϕa,a′ : x �−→ ϕ(axa′) is continuous for any a, a′ ∈ A. +To show the continuity of ϕ, take a null sequence (xn)n in A. Then we can find +a sequence (Fn)n of finite subsets of Irr G such that xn ∈ AFn, here AFn is the +subspace of A as in the proof of Lemma A.1. Set M = c0- � +n AFn. Then AG acts +on M from the left and the right. Moreover an approximate unit of AG acts as a +bounded approximate unit on the both sides of M, since AG is non-degenerate in +A. Hence we can apply the Cohen-Hewitt factorization theorem ([BD73, Chapter +I, Section 11, Theorem 10]) to M as a AG-bimodule. +Now consider (xn)n as an element x ∈ M. Then we can find a, a′ and y ∈ M +such that x = aya′ i.e. we can find a null sequence (yn)n ⊂ A such that xn = ayna′ +for all n. +The continuity of ϕa,a′ implies that ϕ(xn) = ϕ(ayna′) = ϕa,a′(yn) +converges to 0 in norm. Hence ϕ is continuous with respect to the norm-topology +and extends to A. The complete positivity and G-equivariance follows from the +corresponding property for ϕ. +□ +A.2. Proof of Proposition 2.13. +Proposition A.3 (Proposition 2.13). Let E : B −→ A be a conditional expecta- +tion. Then E is with finite index if and only if it satisfies both of the following +two conditions: +(i) The probabilistic index of E is finite. +(ii) The Hilbert A-module BE can be decomposed into a direct sum of finitely +generated projective Hilbert A-modules. +In this case the scalar index of E coincides with ∥Index E∥. +Proof. Assume E is with finite index and take a quasi-basis (vi)n +i=1 ⊂ B. Then +we have a linear map V : BE −→ An defined by V (x) = (E(v∗ +i x))n +i=1. Then we +have +⟨V (y), V (x)⟩A = +n +� +i=1 +E(v∗ +i y)∗E(v∗ +i x) = E +� +y∗ +n +� +i=1 +viE(v∗ +i x) +� += E(y∗x) = ⟨y, x⟩A . +Hence V ia an isometrical A-module map, so BE is finitely generated projective +Hilbert A-module. The finiteness of the probabilistic index of E follows from the +inequality ∥Index E∥E(x) − x ≥ 0, which was shown in [Wa90, Lemma 2.6.2.]. +Moreover we can also see that ∥Index E∥ is equal to or greater than the scalar +index of E by replacing E by E ⊗ id: B ⊗ Mn(C) −→ A ⊗ Mn(C), where n is an +arbitrary positive integer. + +EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES +41 +Next we prove the converse direction. We firstly remark that the original C*- +norm on B is equivalent to the L2-norm with respect to E, hence BE = B as +C-vector space. +By the previous proposition, the scalar index c of E is finite. Then the product +map on B induces a bounded B-module map T : BE ⊗A B −→ B with ∥T∥2 ≤ c, +since we have +� n +� +i=1 +xiyi, +n +� +i=1 +xiyi +� +B += ty∗x∗xty +≤ cty∗(E ⊗ id)(x∗x)ty = c +� n +� +i=1 +xi ⊗ yi, +n +� +i=1 +xi ⊗ yi +� +B +, +where x = (x1 x2 . . . xn), y = (y1 y2 . . . yn). Now we can find η ∈ BE ⊗A B with +Tξ = ⟨η, ξ⟩B for any ξ ∈ BE ⊗A B since BE is a direc sum of finitely generated +projective Hilbert A-module. Moreover we can find sequences {xi}∞ +i=1 ⊂ BE and +{yi}∞ +i=1 ⊂ B such that �n +i=1 xi ⊗ yi converges to η. Let Sn be an A-module map +on BE defined by +Sn(z) = +n +� +i=1 +y∗ +i ⟨xi, z⟩A = +� n +� +i=1 +xi ⊗ yi, z ⊗ 1 +� +. +Then Sk converges to idB in the operator norm, hence Sk is invertible for some k. +This fact enables us to find (ui)n +i=1 and (vi)n +i=1 in B such that x = �n +i=1 uiE(v∗ +i x) +for any x ∈ B. Then we can find a quasi-basis of E by [Wa90, Lemma 2.1.6.]. +For tha last statement, it suffices to show ∥Index E∥ ≤ c. But this can be +easily seen since T ∗(1B) = �n +i=1 vi ⊗ v∗ +i and TT ∗(1B) = Index E. +□ +Acknowlegements. The author appreciates to Yasuyuki Kawahigashi for help- +ful comments and pointing out mistakes and typos on this paper. He is also +grateful to Reiji Tomatsu for doing seminars with the author. 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Anal. 197 +(2003), no. 2, 323–346. +Department of Mathematical Sciences, The University of Tokyo, Komaba 3- +8-1, Tokyo 153-8914, Japan +Email address: mhoshino@ms.u-tokyo.ac.jp + diff --git a/cNE4T4oBgHgl3EQfPwwb/content/tmp_files/load_file.txt b/cNE4T4oBgHgl3EQfPwwb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cec7a7e1796eb650a4ce5c74ab1f2e449987f31a --- /dev/null +++ b/cNE4T4oBgHgl3EQfPwwb/content/tmp_files/load_file.txt @@ -0,0 +1,1625 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf,len=1624 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='04975v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='OA] 12 Jan 2023 EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES MAO HOSHINO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We develop a fundamental theory of compact quantum group equivariant finite extensions of C*-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In particular we focus on the case of quantum homogeneous spaces and give a Tannaka-Krein type result for equivariant correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' As its application, we show that every Jones’ value appears as the index of an equivariant conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In the latter half of this paper, we give an imprimitivity theorem in some cases: for general compact quantum groups under a finiteness conditions, and for the Drinfeld-Jimbo deformation Gq of a simply-connected compact Lie group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' As an application, we give a complete classification of finite index discrete quantum subgroups of � Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Introduction The notion of a C*-tensor category has played an important role in recent developments of theories of operator algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' It firstly appeared in the subfactor theory, as a tool of classification of subfactors: Actually every inclusion of factors gives a C*-tensor category as an invariant, which is complete in the amenable case ([Po94, Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover we also can use it to understand V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Jones’ celebrated work on the range of indices of subfactors ([Jo83, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' C*-tensor categories also appear in the theory of compact quantum groups, as their representation categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover the notion of a module category over a C*-tensor category is also useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using this, the classical Tannaka-Krein duality is generalized not only to a compact quantum group itself, but also to its action on C*-algebra ([DY13, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' De Commer and Yamashita use this duality theorem to obtain a one-to-one correspondence between quantum ho- mogeneous spaces of SUq(2) and concrete combinatorial datum ([DY15, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The purpose of this paper is to connect these C*-tensor categorical approaches and study an inclusion of C*-algebras with actions of a compact quantum group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this paper, we call it as an equivariant finite quantum covering spaces, since it can be regarded as a genuine finite covering space in the case of commutative C*-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We also call its minimal indices as its covering degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' As in the non-equivariant case, an equivariant finite quantum covering space can be un- derstood as a Q-system in a suitable C*-tensor category, namely the category of equivariant correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The following theorem on this category, which is a generalization of the observation due to De Commer and Yamashita ([DY13, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1]), is fundamental throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Primary 46L67, Secondary 46L08, 17B37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' operator algebra, quantum group, tensor category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 1 2 MAO HOSHINO Theorem 1 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For quantum homogeneous spaces A and B of a compact quantum group G, we have the following canonical equivalence of C*- categories: G- Corrrf A,B ∼= [G- Modf A, G- Modf B]Repf G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The first application of this result is the following existence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Theorem 2 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For any d ∈ {4 cos2(π/n) | n ≥ 3} ∪ [4, ∞), we have a compact quantum group G and a finite quantum G-covering space A ⊂ B with its covering degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In Section 4, we focus on the G-actions induced from the maximal Kac quantum subgroup K of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Instead of treating with them directly, we work on Repf G- module categories and module functors between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This enables us to trans- late the imprimitivity theorem to the comparison theorem of Repf G-module func- tors and Repf K-module functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we develop a general theory of module categories admitting a module trace, concluding the following theorem for actions of compact quantum groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Theorem 3 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='29, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='31, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A be a quantum homogeneous space of K and �A be its induced G-C*algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (i) If A has a tracial state and Irr K- Modf A is finite, the induction func- tor gives an equivalence of C*-tensor categories: K- Corrrf A ∼= G- Corrrf � A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover both of them are rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (ii) If A is induced from a quantum homogeneous space of a cocommutative quantum subgroup of K, the induction functor gives group isomorphisms PicK(A) ∼= PicG( �A) and AutK(A) ∼= AutG( �A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The statement (i) contains the proceeding research [CKS, So05] on the quantum Bohr compactification as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In Section 5, we deal with the Drinfeld-Jimbo deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using the well- developed representation theory of C(Gq) and C(T\\Gq), we show the following theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Theorem 4 (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A, B be quantum homogeneous spaces of T and �A, �B be its induced Gq-C*algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the induction functor gives an equivalence of C*-categories: T- Corrrf A,B ∼= Gq- Corrrf � A, � B Theorem 5 (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let �A be a quantum homogeneous space containing C(T\\Gq) and admitting a tracial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then it is of the form Ind Gq T A, where A is a quantum homogeneous space of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' At the last of this paper, we use the result in Section 4 to give a complete classification result of finite index discrete quantum subgroups of � Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Theorem 6 (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Q) be the weight (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' root) lattice of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' There is a canonical one-to-one correspondece between finite index discrete quantum subgroups of � Gq and subgroups of P/Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Preliminaries In this paper, the symbol – ⊗ – denotes the spatial tensor product of C*- algebras, the algebraic tensor product of C-vector spaces and the external tensor product of Hilbert C*-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For a Hilbert space H, its inner product ⟨–, –⟩ is C-linear with respect to the second argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' An element ξ ∈ H is considered as an operator from C to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If H is finite dimensional, the unnormalized trace on B(H) is denoted by Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Compact quantum groups and their dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this subsection, we give a brief review on compact quantum groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' See [NT13] for detailed discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A compact quantum group is a pair G = (A, ∆) of a unital C*-algebra A and ∗-homomorphism ∆: A −→ A ⊗ A which satisfies the following conditions: the coassociativity (∆ ⊗ id)∆ = (id ⊗∆)∆, the cancellation property (A ⊗ C)∆(A) = (C ⊗ A)∆(A) = A ⊗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this case A is denoted by C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A Haar state on G is a state h on C(G) satisfying the bi-invariance property (h⊗id)∆(x) = (id ⊗h)∆(x) = h(x)1G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Such a state always exists and is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' It is said that G is reduced when h is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We only consider reduced compact quantum groups unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A unitary representation of G is a pair π = (Hπ, Uπ) of a Hilbert space Hπ and a unitary Uπ ∈ M(K(Hπ) ⊗ C(G)) which satisfies (id ⊗∆)(Uπ) = Uπ,12Uπ,13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The category of unitary representations of G is denoted by Rep G, and its full subcategory of the finite dimensinal ones is denoted by Repf G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let π be a finite dimensional unitary representation of G and ξ, η be elements of Hπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then (ξ∗ ⊗ 1)Uπ(η ⊗ 1) defines an element of C(G), called a matrix coefficient of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The set of all matrix coefficients of all finite dimensional unitary representations is called the algebraic core of G and denoted by O(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We can make O(G) into a Hopf ∗-algebra by using the product and the coproduct of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Its counit and antipode are denoted by ε and S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In general S satisfies S(S(x)∗)∗ = x for any x ∈ O(G), but does not S2 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' It is said that G is of Kac type when it holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let H be another compact quantum group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A homomorphism from H to G is a homomorphism of Hopf ∗-algebras from O(G) to O(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' One should note that ϕ need not be defined on C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' On the other hand we can still define (ϕ ⊗ id)∆ as a ∗-homomorphism from C(G) to C(H) ⊗ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' More generally, we have the following lemma, which is a quantum analogue of the Fell’s absorption principle for �G: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A be a C*-algebra and ϕ: O(G) −→ A be a ∗-homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then (ϕ ⊗ id)∆ extends to a ∗-homomorphism from C(G) to A ⊗ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let (π, L2(G)) be the GNS representation of C(G) with respect to the Haar state of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since G is reduced, this representation is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' It suffices to show in the case of A = B(H) for a Hilbert space H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let V be an operator on H ⊗ L2(G) satisfying V (ξ ⊗ Λ(x)) = (ϕ ⊗ π)(∆(x))(ξ ⊗ Λ(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 4 MAO HOSHINO Then V is unitary and satisfies V (1 ⊗ π(x))V ∗ = (ϕ ⊗ π)(∆(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the statement follows from the faithfulness of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ If we are given a Hopf ∗-algebra A generated by matrix coefficients of its finite dimensional unitary representations, we can construct a compact quantum group whose algebraic core coincides with A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This fact allows us to construct the maximal Kac quantum subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The maximal Kac quantum subgroup of G is the compact quan- tum group corresponding to a Hopf ∗-algebra obtained as the quotient of O(G) divided by a two-sided ideal generated with S2(x) − x for all x ∈ O(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Actually K is of Kac type and has the following universal property: Let H be a compact quantum group of Kac type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then any homorphism from H to G factors through K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This follows from that any homomorphism of Hopf ∗-algebra preserves antipodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let Gu be the universal form of G and �K be its canonical Kac sub- group, introduced in [So05, Appendix].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have a canonical ∗-homomorphism �q: O(G) −→ O(�K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This map satisfies �q ◦ S2 = �q, hence �q factors through O(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' On the other hand, the universal property of Gu implies that a canonical map q: O(G) −→ O(K) extends to ∗-homomorphism from Cu(G) to Cr(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since the Haar state of K is faithful on Cr(K), this map factors through C(�K) by its construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This implies that q factors through O(�K) and gives an isomorphism from O(�K) to O(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let G be a semisimple compact Lie group and q ∈ (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then there is a compact quantum group Gq called the Drinfeld-Jimbo q-deformation of G, see [NT13, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='] for a precise definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The representation theory of Gq is quite similar to that of G, which means that they have the common set of equivalence classes of irreducible representations and a common fusion rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' But actually Repf Gq is not equivalent to Repf G as a C*-tensor category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' It is shown that the maximal Kac quantum subgroup of Gq coincides with the maximal torus T of G ([To07, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We end this subsection with the notion of discrete quantum group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Its definition is given by the Pontrjagin dual of a compact quantum group, hence the reduced C*-algebra of discrete quantum group Γ is nothing but C(G) when Γ is the Pontrjagin dual of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this case Γ is denoted by �G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For the notion of closed quantum subgroup, see [DKSS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' There are several approaches, but all of them are equivalent in the discrete case ([DKSS, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' As a result, we can take the following definition: a closed quantum subgroup of G is a pair of discrete quantum group �H and a coproduct-preserving unital inclusion C(H) ⊂ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this case a unitary representation of H gives a unitary repsentation of G with the same intetwiner space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Combining this with the Woronowicz’s Tannaka-Krein duality ([NT13, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2]), we have a one- to-one correspondence between discrete quantum subgroups of �G and C*-tensor full subcategories of Repf G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Actions of compact quantum groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For detailed discussions and ref- erences, see [DC16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let G be a compact quantum group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A (reduced) G-C*- algebra or G-action on a C*-algebra is a pair (A, α) of a C*-algebra A and faithful ∗-homomorphism α: A −→ A ⊗ C(G) with the following conditions: (id ⊗∆)α = (α ⊗ id)α, span (C ⊗ C(G))α(A) = A ⊗ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Its fixed point subalgebra {x ∈ A | α(x) = x ⊗ 1} is denoted by AG or Aα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We also define the algebraic core of A as A = {x ∈ A | α(x) ∈ A ⊗ O(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have a right coaction of the Hopf ∗-algebra O(G) on A given by the restriction of α to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We say that G-C*-algebra A is a quantum homogeneous space of G if it is unital and AG = C1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The C*-algebra C(G) has a canonical G-action defined by ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover, every G-invariant C*-subalgebra of C(G) also has a G-action and actually is a quantum homogeneous space of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We call it a right coideal of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let H be a compact quantum subgroup of G and q: O(G) −→ O(H) be the canonical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then (q ⊗ id)∆ extends to a ∗-homomorphism ℓH on C(G) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since ℓH is G-equivariant with respect to the canonical G-action on C(G), we have a right coideal C(H\\G) = {x ∈ C(G) | ℓH(x) = 1⊗x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' More generally, for a H-C*-algebra, the induced G-C*-algebta of B is defined as a pair of the following: Ind G HA = {x ∈ A ⊗ C(G) | (α ⊗ id)(x) = (id ⊗ℓH)(x)}, �α = the restriction of id ⊗∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For example, the induced G-C*-algebra of the trivial H-C*-algebra C is isomor- phic to C(H\\G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If A and B are G-C*-algebras with the algebraic cores A and B respectively, we say that a linear map ϕ: A −→ B is G-equivariant when (ϕ ⊗ id)α = βϕ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Similarly the equivariance for ψ: A −→ B is also defined when ψ ⊗ id: A ⊗ C(G) −→ B ⊗ C(G) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The following generalization of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1 is useful to get a G-equivariant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If A is ∗-algebra and B is C*-algebra, we say that a linear map ϕ: A −→ B is completely positive if (ϕ ⊗ id)(X∗X) is positive in B ⊗ Mn(C) for any X ∈ A ⊗ Mn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A and B be G-C*-algebras with the algebraic cores A and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If ϕ: A −→ B is G-equivariant and completely positive, it extends to a G-equivariant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' map from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For a proof of this proposition, see Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Next we move on the notion of modules over G-C*-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A be a G-C*-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A G-equivariant Hilbert A-module is a pair (E, αE) of a Hilbert A-module E and a linear map αE : E −→ E ⊗ C(G) with the following conditions: α(⟨ξ, η⟩A) = ⟨αE(ξ), αE(η)⟩A⊗C(G) for any ξ, η ∈ E, span αE(E)(C ⊗ C(G)) = E ⊗ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 6 MAO HOSHINO Here E ⊗ C(G) means the external tensor product of the Hilbert A-module E and the Hilbert C(G)-module C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If A is a quantum homogeneous space and E is finitely generated, E must be projective ([DC16, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The category of G-equivariant Hilbert A-module is denoted by G- ModA, and its full subcategory of finitely generated ones is denoted by G- Modf A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We use the symbol LA(–, –) for the spaces of adjointable right A-module maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We also use LG A(–, –) for the spaces of G-equvariant ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For E ∈ G- Modf A, we have a G-action �α: G ↷ KA(E) on the algebra of com- pact A-module maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover we can extend this action to a ∗-homomorphism �α: LA(E) −→ LA⊗C(G)(E ⊗ C(G)), and this satisfies (�α ⊗ id)�α = (id ⊗∆)�α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For any T ∈ LA(E), the operator �α(T) is the only operator on E ⊗ C(G) satisfying �α(T)αE(ξ) = αE(Tξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We can also define the algebraic core and the induction of equivariant Hilbert C*-modules as follows: E = {ξ ∈ E | αE(ξ) ∈ E ⊗ O(G)}, Ind G HF = {x ∈ F ⊗ C(G) | (αF ⊗ id)(x) = (id ⊗ℓH)(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' It can be easily seen that E has a right A-action and a right coaction of O(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' C*-tensor categories and their module categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For detailed de- scriptions, see [NT13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A C*-category is a C-linear category with a norm on each Hom space and an anti-linear involution ∗: C(X, Y ) −→ C(Y, X) satisfying the C*-identity: ∥T ∗T∥ = ∥T∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this paper, we also assume that C*-categories are closed under taking direct sums and subobjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let C and D be C*-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A functor F : C −→ D is called a C*-functor if it preserves the adjoints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' F(T ∗) = F(T)∗ for any morphism T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For a natural transformation between C*-functors, we can define its adjoint by taking the ad- joint of each components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We use the symbol [C, D]b to denotes the category of C*-functors from C to D and bounded natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Here the bound- edness of a natural transformation is defined by the uniform norm-boudedness of its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Obviously [C, D]b also has a structure of C*-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A strict C*-multitensor category is a triple (C, ⊗, 1) of C*-category C, a C*- bifunctor –⊗–: C ×C −→ C and an object 1 called the unit object which satisfies U ⊗(V ⊗W) = (U ⊗V )⊗W and 1⊗U = U = U ⊗1 for any objects U, V, W ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If C(1, 1) = C id1 holds, C is said to be a strict C*-tensor category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let X be an object of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We say that X is rigid if there is a quadruple (X, X, R, R) with an object X of C, R ∈ C(1, X ⊗ X) and R ∈ C(1, X ⊗ X) which satisfies the conjugate equations (idX ⊗R∗)(R ⊗ idX) = idX, (idX ⊗R ∗)(R ⊗ idX) = idX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This quadruple is called a solution of the conjugate equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The full subcate- gory of C consisting of all rigid objets of C is denoted by Cf, and C is said to be a rigid when C = Cf, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' all objects of C are rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In such a C*-tensor category, each Hom space is finite dimensional ([NT13, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let X be a rigid object of a strict rigid C*-tensor category C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we can minimize the value ∥R∥∥R∥ for a solution of conjugate equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This minimum EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 7 value is called the categorical dimension of X, denoted by d(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A solution (X, X, R, R) is said to be standard when ∥R∥2 = ∥R∥2 = d(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We remark here that any two standard solutions with fixed X are unitary equivalent to each other, which means that we can get all standard solutions by considering (X, X ′, (u ⊗ id)R, (id ⊗u)R) with a fixed standard solution (X, X, R, R) and an arbitrary unitary u ∈ C(X, X ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This fact enables us to define the categorical trace TrX : C(X, X) −→ C given by the formula TrX(T) id1 = R∗(id ⊗T)R using a standard solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This is independent of the choice of standard solutions and equal to R ∗(T ⊗ id)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In a similar way, we can also define the partial traces TrX ⊗ id: C(X ⊗ Y, X ⊗ Z) −→ C(Y, Z) and id ⊗ TrX : C(Y ⊗ X, Z ⊗ X) −→ C(Y, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The categorical traces are actually tracial in the following sense: For any morphisms S, T ∈ C(X, Y ) we have TrX(S∗T) = TrY (TS∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let C, D be strict C*-multitensor categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A C*-tensor functor from C to D is a pair (Θ, θ) of a C*-functor Θ: C −→ D and a unitary natural transformation θ: Θ(–) ⊗ Θ(–) −→ Θ(– ⊗ –) with the conditions Θ(1C) = 1D, The following diagram commutes for U, V, W ∈ C: Θ(U) ⊗ Θ(V ) ⊗ Θ(W) id ⊗θV,W � θU,V ⊗id � Θ(U) ⊗ Θ(V ⊗ W) θU,V ⊗W � Θ(U ⊗ V ) ⊗ Θ(W) θU⊗V,W � Θ(U ⊗ V ⊗ W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If (Θ′, θ′) is also a C*-tensor functor from C to D, then a natural transformation η: F −→ G is said to be a monoidal if it satisfies θ′ U,V ◦ (ηU ⊗ ηV ) = ηU⊗V ◦ θU,V for U, V ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='10 ([DY13, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=']).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let C be a strict rigid C*-tensor category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A C-module category is a triple (M, ⊗, a) of a C*-category M, a C*- bifunctor – ⊗–: C ×M −→ M and a unitary natural transformation a: – ⊗(– ⊗ –) −→ (– ⊗ –) ⊗ – with the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 1 ⊗ X = X for any X ∈ M, The following diagram commutes for any U, V, W ∈ C and X ∈ M: U ⊗ (V ⊗ (W ⊗ X)) aU,V,W ⊗X� id ⊗aV,W,X � (U ⊗ V ) ⊗ (W ⊗ X) aU⊗V,W,X � U ⊗ ((V ⊗ W) ⊗ X) aU,V ⊗W,X � (U ⊗ V ⊗ W) ⊗ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We remark that a C-module structure on a fixed C*-category M gives rise to a C*-tensor functor from C to [M, M]b and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' It is said that M is semisimple if M(X, Y ) is finite dimensional for any X, Y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We assume that all C-module categories in this paper are semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A C-module category M is said to be connected if, for any X, Y ∈ M, there is U ∈ C and an isometry from X to U ⊗ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let G be a compact quantum group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then its representaiton category Rep G is a C*-tensor category and Repf G is a rigid C*-tensor category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 8 MAO HOSHINO Take a quantum homogeneous space A of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then G- Modf A has a structure of Repf G-module category: For π ∈ Repf G and E ∈ G- Modf A, Hπ⊗E can be made into a G-equivariant Hilbert A-module with a G-action v⊗ξ �−→ Uπ,13(v⊗αE(ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This RepG-module category is connected and semisimple ([DY13, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If N is another C-module category, a C-module functor from M to N is a pair of a C*-functor F : M −→ N and a unitary natural transformation f : F(–⊗–) −→ – ⊗ F(–) with the commutativity of the following diagram for any U, V ∈ C and X ∈ M: F(U ⊗ (V ⊗ X)) fU,V ⊗X� F (aU,V,X) � U ⊗ F(V ⊗ X) id ⊗fV,X� U ⊗ (V ⊗ F(X)) aU,V,F (X) � F((U ⊗ V ) ⊗ X) fU⊗V,X � (U ⊗ V ) ⊗ F(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The category of C-module functors from M to N is denoted by [M, N ]C b, here a morphism from (F, f) to (G, g) is given by a bounded natural transformation η: F −→ G making the following diagram commutative for any U ∈ C and X ∈ M: F(U ⊗ X) fU,X � ηU⊗X � U ⊗ F(X) id ⊗ηX � G(U ⊗ X) gU,X � U ⊗ G(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If we regard a C-module structure on M as a C*-tensor functor (Φ, ϕ): C −→ [M, M]b, then [M, M]C b can be thought as something like a Drinfeld center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Actually, if we are given a C-module functor (F, f) from M to M, then f defines a unitary natural transformation c from F ◦Φ(–) to Φ(–)◦F satisfying the braiding equation cU⊗V = (ϕU,V ⊗ idF)(id ⊗cV )(cU ⊗ id)(id ⊗ϕU,V ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The converse also holds, and then a morphism of C-module functor can be considered as a morphism of unitary half-braiding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Index of conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this subsection, we collect defini- tions and facts on indices of conditional expectations, introduced in [Wa90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let B be a unital C*-algebra and A be a unital C*-subalgebra of B with a conditional expectation E : B −→ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we say that E is with finite index if it admits a quasi-basis i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' a finite family (ui)n i=1 ⊂ A satisfying a = n � i=1 uiE(u∗ i a) for any a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this case we can define the index of E by the formula Index E = �n i=1 uiu∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This is independent of the choice of a quasi-basis, and Index E is an element of Z(A)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' On the other hand, we also have a more classical notion of index based on the Pimsner-Popa inequality [PP86, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The following value is called the probabilistic index of E ([Po95, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1]): Indexp E = min{c > 0 | cE − idA is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 9 We also define the scalar index of E by replacing the positivity by the complete positivity: Indexs E = min{c > 0 | cE − idA is completely positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Obviously we have Indexp E ≤ Indexs E ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover the following proposi- tion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='12 ([FK00, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=']).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let E : B −→ A be a conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the probabilistic index of E is finite if and only if the scalar index of E is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We use the symbol BE for a Hilbert A-module obtained by taking the com- pletion of A with respect to a B-valued inner product ⟨x, y⟩B = E(x∗y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The following generalization of [Wa90, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='] may be well-known for experts (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' [BDH, Th´eor`eme 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=']).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For its proof, see Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let E : B −→ A be a conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then E is with finite index if and only if it satisfies both of the following two conditions: (i) Indexp E < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (ii) The Hilbert A-module BE can be decomposed into a direct sum of finitely generated projective Hilbert A-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this case we have Indexs E = ∥Index E∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Equivariant finite quantum covering spaces 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Definition and characterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let G be a compact quantum group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If A ⊂ B is an inclusion of G-C*-algebras, we say that E : B −→ A is a G- expectation when E is a G-equivariant conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A ⊂ B be a unital inclusion of unital G-C*-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We say that A ⊂ B is a finite quantum G-covering space over A if it admits a G- expectation E : B −→ A with finite index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this paper, we only treat with finite quantum G-covering spaces over quan- tum homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In such cases, we can replace the finiteness of the index by the finiteness of the probabilistic index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A be a quantum homogeneous space of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A ⊂ B be a unital inclusion of unital G-C*-algebras with a G-expectation E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the following conditions are equivalent: (i) The index of E is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (ii) The probabilistic index of E is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover, under these conditions, we can take a quasi-basis of E in the algebraic core B of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' At first one should note that BE can be made into a G-equivariant Hilbert A-module by the action of G on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then BE decomposes into a direct sum of finitely generated projective Hilbert A-modules by [DC16, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' ], hence the equivalence of (i) and (ii) follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Next we show the last statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since BE is a finitely generated G-equivariant Hilbert A-module, it has a irreducible decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we can take a finite 10 MAO HOSHINO dimensional unitary representation π of G and a embedding V : BE −→ Hπ ⊗ A of G-equivariant Hilbert A-module by [DC16, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then V and its adjoint V ∗ preserve their algebraic core, hence we can obtain a quasi-basis (V ∗(ei ⊗ 1))n i=1 in B where (ei)n i=1 is an orthonormal basis of Hπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A ⊂ B be a finite quantum G-covering space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then for any intermediate unital G-C*-subalgebra C of A ⊂ B, A ⊂ C is also a finite quantum G-covering space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Fix a G-expectation E : B −→ A with finite index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we can easily check the finiteness of the restriction of E on C by using the condition (ii) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ If G is of Kac type, we can drop the equivariance of a conditional expectation by the averaging procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A, B be G-C*-algebras and A, B be their al- gebraic cores respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If we are given a C-linear map ϕ: B −→ A, then its equivariantization is a C-linear map �ϕ: B −→ A given by the following: �ϕ(x) = (id ⊗h)(α(ϕ(x(0)))(1 ⊗ S(x(1)))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Here h and S denote the Haar state and the antipode of G respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using the strong bi-invariance of h ([KV00, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='24, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='35]), we can see that �ϕ is a G-equivariant map from B to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let ϕ: B −→ A be a c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If G is of Kac type, the equivari- antization of ϕ extends to a G-equivariant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' map from B to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since G is of Kac type, we have �ϕ(y∗x) = (id ⊗h)((1 ⊗ S(y(1))∗)α(ϕ(y∗ (0)x(0)))(1 ⊗ S(x(1)))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This equality implies the complete positivity of �ϕ, hence we can apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Assume G is of Kac type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For a unital inclusion A ⊂ B of unital G-C*-algebras with Aα = C1A, the following conditions are equivalent: (i) The inclusion A ⊂ B is a finite quantum G-covering space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (ii) There is an expectation E : B −→ A with Indexp E < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let E be a conditional expection from B to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' At first we have to show that the equivariantization �E of E is again a conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since �E is a u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' map by the previous proposition, it suffices to show �E is A-bimodule map on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take a ∈ A and x ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have �E(xa) = (id ⊗h)(α(E(x(0)a(0)))(1 ⊗ S(x(1)a(1)))) = (id ⊗h)(α(E(x(0)))α(a(0))(1 ⊗ S(a(1)))(1 ⊗ S(x(1)))) = (id ⊗h)(α(E(x(0)))(a ⊗ S(x(1)))) = �E(x)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The equality �E(ax) = a �E(x) can be seen as follows: �E(ax) = �E(x∗a∗)∗ = ( �E(x)∗a∗)∗ = a �E(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 11 Now the probabilistic index of E is finite, hence the scalar index c of E is also finite by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the equivariantization of cE − idB is a c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' map and coincides with c �E − idB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This implies that �E has the finite scalar index, hence A ⊂ B is a finite quantum G-covering space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Covering degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Next we consider a notion which is analogous to the covering degree of covering space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A ⊂ B be a finite quantum G-covering space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the cov- ering degree of A ⊂ B is the infimum of the scalar indices over all G-expectations from B to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If G is of Kac type, the covering degree coincides with the infimum of the scalar indices of all conditional expectations of the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This fact can be seen from the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To calculate the covering degree of a given finite quantum G-covering space, it is usuful to regard it as the dimension of a rigid object of a suitable C*-tensor category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A and B be quantum homogeneous spaces of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A G-equivariant (A, B)- correspondence is a pair of G-equivariant Hilbert B-module E and a unital G- equivariant ∗-homomorphism from A to LB(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The category of G-equivariant (A, B)-correspondences is denoted by G- CorrA,B, and its full subcategory con- sisting of right-finitely generated correspondences is denoted by G- Corrrf A,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We also use the symbols G- CorrA and G- Corrf A when A = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A ⊂ B be an inclusion of unital G-C*-algebras with Aα = C1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If we are given a G-expectation E : B −→ A with finite index, then we have a G- correspondence BE over A, on which the left action is given by the left multipli- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover BE can be made into a C*-Frobenius algebra ([BKLR, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1]) in G- Corrf A with the following maps: m: BE ⊗A BE −→ BE, induced by the multiplication of m, ι: A −→ BE, induced by the inclusion A ⊂ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using a quasi-basis (vi)n i=1 of E, m∗ : BE −→ BE ⊗A BE can be calculated as follows: m∗(x) = n � i=1 vi ⊗ v∗ i x Hence we have mm∗(1B) = Index E and ι∗mm∗ι(a) = E(Index E)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since ι∗(b) = E(b) for any b ∈ B, we also have d(BE) ≤ ∥ι∗mm∗ι∥ = E(Index E) ≤ ∥Index E∥, here the left-hand side is the dimension of BE as an object of G- Corrf A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' More- over, if (BE, m, ι) is a Q-system, these inequalities turns out to be equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Conversely any C*-Frobenius algebra in G- Corrf A gives a pair of a finite quan- tum G-covering space A ⊂ B and G-expectation E : B −→ A with finite index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we can get the following proposition since any Frobenius C*-algebra is isomorphic to a Q-system, as shown in [NY18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='9] Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A be a quantum homogeneous space of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 12 MAO HOSHINO (i) There is a one-to-one correspondence between finite quantum G-covering spaces of A and Q-systems in G- Corrf A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (ii) Let A ⊂ B be a finite quantum G-covering space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Its covering degree d coincides with the dimension of the corresponding Q-system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover, there is a G-expectation E : B −→ A with Index E = d1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' When A = C, the trivial G-C*-algebra, then G- Corrf C is nothing but Repf G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence the covering degree of a finite quantum G-space over C coincides with the quantum dimension as a unitary representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In particular, the covering degree of C ⊂ B(Hπ) is (dimq π)2 if we consider the adjoint action on B(Hπ) As an application of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='8, we show that the covering degree of a finite quantum G-spaces over C(G) must be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let Rep O(G) be the category of unital ∗-representations of O(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using the coproduct on O(G), we can make Rep O(G) into a C*-tensor category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For any (π, Hπ) ∈ Rep O(G), we can construct a G-equivariant correspondence Eπ over C(G) as follows: As a Hilbert C(G)-module, Eπ = Hπ ⊗ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The left action of G is given by id ⊗∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The left action of C(G) is given by (π ⊗ λ)∆, where λ is the left multi- plication of C(G) on C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Here (π ⊗ λ)∆ can be defined on C(G) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The following proposition is a quantum analogue of [AV16, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The functor (π, Hπ) �−→ Eπ gives an equivalence of C*- tensor categories from Rep O(G) to G- CorrC(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Before proving this, we should prepare the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let E be a G-equivariant Hilbert C(G)-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The the following formula gives an semi-inner product on the algebraic core E of E: ⟨ξ, η⟩ = ε(⟨ξ, η⟩C(G)) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The only non-trivial part is the positivity of ⟨ξ, ξ⟩ for each ξ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By replacing E with its G-equivariant Hilbert C(G) submodule generated by ξ and η, we may assume E is finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If we are given another F ∈ G- Modf C(G) and an isometry V ∈ LG C(G)(E, F), we have ⟨V ξ, V η⟩ = ε(⟨V ξ, V η⟩C(G)) = ε(⟨ξ, η⟩C(G)) = ⟨ξ, η⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since G- Modf C(G) is connected as a Repf G-module category, the equality above implies that it suffices to show the statement for Hπ ⊗ C(G) with an arbitrary π ∈ Repf G But this is trivial since we have ⟨v ⊗ x, u ⊗ y⟩ = ⟨v, u⟩ ε(x∗y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 13 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' At first we make the functor in the statement into a C*-tensor functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This can be completed by associating the following unitary: (Hπ ⊗ C(G)) ⊗C(G) (Hρ ⊗ C(G)) Uπ,ρ � (Hπ ⊗ Hρ) ⊗ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (ξ ⊗ x) ⊗ (η ⊗ y) ✤ � (ξ ⊗ ρ(x(1))η) ⊗ x(2)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' It can be easily seen that this unitary satisfies the conditions in the definition of C*-tensor functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To show that this functor gives the equivalence, we construct a quasi-inverse functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let E be a G-equivariant corrrespondence over C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Set HE as a completion of the algebraic core E of E, with respect the inner product in the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then each element x ∈ O(G) acts on HE from the left, since O(G) is a linear span of matrix coefficients of O(G)-valued unitary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now we have a ∗-representation (πE, Hπ) of O(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To complete the proof, we will check that these functors give a quasi-inverse of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For (π, Hπ) ∈ Repf O(G), the algebraic core of Eπ is precisely Hπ ⊗ O(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we have a unitary from HEπ to Hπ given by ξ ⊗ x �−→ ε(x)ξ and this is an intertwiner of representations of O(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' On the other hand, for any E ∈ G- Corrrf C(G), we have a map from E to HE ⊗ C(G) given as follows: ξ ∈ E �−→ ξ(0) ⊗ ξ(1) ∈ HE ⊗ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This map is a G-equivariant isometry and intertwines with left C(G)-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To show that this is a right C(G)-module map, one should note the following: For any x ∈ O(G) and ξ ∈ E, we have ∥ε(x)ξ − ξx∥2 HE = ε(⟨ε(x)ξ − ξx, ε(x)ξ − ξx⟩C(G)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we have ξx = ε(x)ξ in HE and (ξx)(0) ⊗ (ξx)(1) = ε(x(0))ξ(0) ⊗ ξ(1)x(1) = ξ(0) ⊗ ξ(1)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This implies that the map above is an embedding of G-equivariant correspondence from E to HE ⊗ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence the range of this map contains ξ(0) ⊗ ξ(1)x for any ξ ∈ E and x ∈ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Combining this with span αE(E)(C ⊗ C(G)) = E ⊗ C(G), we can see that this map is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We also have an equivalence Repf O(G) ∼= G- Corrrf C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let us recall the quantum Bohr compactification due to So�ltan [So05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For the dual discrete quantum group �G of G, its quantum Bohr compactification b�G is of Kac type ([So05, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover every finite dimensional ∗- representation of O(G) gives rise to a finite dimensional unitary representation of b�G (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' [CKS, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' These facts imply that Repf O(G) is rigid and the forgetful functor Repf O(G) �−→ Hilbf is dimension-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='10 yields the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For a finite quantum G-covering space over C(G), its covering degree must be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 14 MAO HOSHINO 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A module categorical description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let us recall the Tannaka-Krein type result for quantum homogeneous spaces [DY13, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='4], [Ne14, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This suggests us the possibility to understand finite quantum G-covering space by the language of module categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let M be a right-finitely generated G-equivariant (A, B)-correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have a Repf G-module functor from G- Modf A to G- Modf B given by the follow- ing: As a C*-functor, it is given by taking the internal tensor product with E over A i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' the functor sends E to E⊗AM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Here E⊗AM is considered as a G-equivariant Hilbert B-module by ξ ⊗ m �−→ αE(ξ)13βM(m)23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since E and M is finitely generated, this Hilbert B-module is also finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The unitary from (Hπ ⊗ E) ⊗A M to Hπ ⊗ (E ⊗A M) is given by (v ⊗ ξ) ⊗ m �−→ v ⊗ (ξ ⊗ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This Repf G-module functor is denoted by – ⊗A M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The following is a main theorem of this subsection, which is a generalization of [DY13, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1] Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A and B be quantum homogeneous spaces of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the C*-functor M �−→ – ⊗A M gives the following equivalence of C*-categories: G- Corrrf A,B ∼= [G- Modf A, G- Modf B]Repf G b We would like to use an argument used in [DY13, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1] for the proof of the above theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' At first, we fix a complete set Irr G of representatives of all equivalence classes of irreducible unitary representations of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For any π ∈ Irr G, π denotes an element of Irr G unitary equivalent to the conjugate representation of π, which is uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let E be a G-equivariant Hilbert A-module and E be its algebraic core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For each π ∈ Irr G, Eπ denotes the space LG A(Hπ ⊗ A, E) ⊗ Hπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we can regard Eπ as a subspace of E via T ⊗ ξ �−→ T(ξ ⊗ 1) and have the spectral decompo- sition E = � π Eπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By considering A as a G-equivariant Hilbert A-module, this decomposition also can be applied to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now we can present the right A-action, the A-valued inner product and the G-action on E as follows: For T ⊗ ξ ∈ Eπ, S ⊗ η ∈ Eρ and s ⊗ v ∈ Aρ, we have (T ⊗ ξ)(s ⊗ v) = � σ∈Irr G � V ∈(σ,πρ) T(idπ ⊗s)(V ⊗ idA) ⊗ V ∗(ξ ⊗ v), ⟨T ⊗ ξ, S ⊗ η⟩A = � σ∈Irr G � V ∈(σ,πρ) (R∗ π ⊗ idA)(idπ ⊗T ∗S)(V ⊗ idA) ⊗ (ξ∗ ⊗ V ∗)(Rπ(1) ⊗ η), αE(T ⊗ ξ) = T ⊗ Uπ(ξ ⊗ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Here we use the following notations: The quadruple (π, π, Rπ, Rπ) denotes a standard solution in Repf G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 15 The symbol � V ∈(σ,πρ) means the summation over all elements of a fixed orthonormal basis of HomG(σ, π ⊗ ρ), equipped with an inner product ⟨V, W⟩ = V ∗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' These formulae do not depend on the choices of the standard solutions and the orthonormal bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We also have the following presentation of the involution of A under these notations: (s ⊗ v)∗ = (R∗ π ⊗ id)(idπ ⊗s∗) ⊗ (v∗ ⊗ id)Rπ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now we finishes the preparation for the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' At first we construct a functor from [G- Modf A, G- Modf B]Repf G b to G- Corrrf A,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let (F, f) be a Repf G-module functor from G- Modf A to G- Modf B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we can associate a left action of A on F(A) as follows: For s ⊗ v ∈ Aπ and η ∈ F(A), the left multiplication by s ⊗ v is given by (s ⊗ v)η = F(s)f ∗ π,A(v ⊗ η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then some direct calculations show that this defines an adjointable B-module map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover we can see the following relations: ⟨η, (s ⊗ v)η′⟩B = ⟨(s ⊗ v)∗η, η′⟩B , βF (A)((s ⊗ v)η) = α(s ⊗ v)βF (A)(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='8 shows that this left A-action extends to a left A-action on F(A) compatible with the G-actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now F(A) is equipped with a struc- ture of G-equivariant (A, B)-correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover if we are given another Repf G-module functor (G, g) and a morphism θ: (F, f) −→ (G, g), its compo- nent θA : F(A) −→ G(A) is a morphism of G-equivariant (A, B)-correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' These constructions give rise to a functor we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We will complete the proof by showing that these functors give a quasi-inverse of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Fix an right-finitely generated G-equivariant (A, B)-correspondence M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then A ⊗A M is isomorphic to M as a G-equivariant Hilbert B-module map via x ⊗ ξ �−→ xξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence it suffices to show the compatibility with the left A- actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For s ⊗ v ∈ Aπ and a ⊗ ξ ∈ A ⊗A M, the action as above is calculated as follows: (s ⊗ v)(a ⊗ ξ) = (s ⊗ idM)((v ⊗ a) ⊗ ξ) = s(v ⊗ a) ⊗ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since s ⊗v corresponds to s(v ⊗1) ∈ A, this equality shows the compatibility we required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Next we take a Repf G-module functor (F, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For any finitely generated G- equivariant Hilbert A-module E, we have a map from E ⊗ F(A) to F(E) which sends (T ⊗ v) ⊗ ξ to F(T)f ∗ π,A(v ⊗ ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have � F(T)f ∗ π,A(v ⊗ ξ), F(S)f ∗ ρ,A(w ⊗ η) � B = � v ⊗ ξ, fπ,AF(T ∗S)f ∗ ρ,A(w ⊗ η) � B = � v ⊗ ξ, (idπ ⊗(R∗ π ⊗ idF (A))(idπ ⊗fπ,AF(T ∗S)f ∗ ρ,A))(Rπ ⊗ idρ ⊗ idF (A))(w ⊗ η) � B 16 MAO HOSHINO By using (R∗ π ⊗ idF (A))fπ⊗π,A = F(R∗ π ⊗ idA) and fπ⊗π,A = (idπ ⊗fπ,A)fπ,π⊗A, we also have (R∗ π ⊗ idF (A))(idπ ⊗fπ,AF(T ∗S)f ∗ ρ,A) = F((R∗ π ⊗ idA)(idπ ⊗T ∗S))f ∗ π⊗ρ,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence the map induces an isometry from E ⊗A F(A) to F(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' It can be easily seen that this isometry is a G-equivariant Hilbert B-module map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To prove the unitarity, one should note that F preserves the range projection i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' for any T ∈ LG A(Hπ ⊗ A, E) with the range projection p ∈ LG A(E), the range projection of F(T): F(Hπ ⊗ A) −→ F(E) is F(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This fact implies that F(E) is contained the sum space of the range of F(T) over all T ∈ � π∈Irr G LG A(Hπ ⊗ A, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence the map from E ⊗A F(A) to F(E) is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ As an application of this theorem, we can see that each Jones’ value appears as the covering degree of a finite quantum G-covering space for some G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The covering degree of a finite quantum G-covering space is contained in {4 cos2(π/n) | n ≥ 3} ∪ [4, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Conversely, for any element d of this set, we have a compact quantum group G and a finite quantum G-covering space A ⊂ B with its covering degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In the following proof, we use the monoidal opposite Cop of a given C*-tensor category C (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' [EGNO, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' As a C*-category, Cop = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' But its tensor product – ⊗ –: Cop × Cop −→ Cop is given by (X, Y ) �−→ Y ⊗ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The first half of the statement follows from a general theory on Q-system: Take a finite quantum G-covering space A ⊂ B and let d be its covering degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have a Q-system X in G- Corrf A correspoinding to A ⊂ B with d(X) = d (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let C be a C*-tensor category generated by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we can find a finite factor N and a fully faithful embedding i: C −→ Bimodf N by [Ya03, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since a Q-system in Bimodf N corresponds to a finite extension of von Neumann algebra [Lo94, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1], we have a finite extension N ⊂ M whose minimum index is d(i(X)) = d(X) = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now our assertion follows from the Jones’ result on the value of index ([Jo83, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To prove the second half, take an arbitrary d from the set in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If d ≥ 4, we have a q ∈ (0, 1) such that the fundamental representaion π1/2 of SUq(2) satisfies dimq π1/2 = √ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='9 gives a finite quantum SUq(2)-covering space with the covering degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now consider the case d < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we can take a fusion category C and its object X with d(X) = √ d (For example, let C be the Temperley-Lieb-Jones category [NT13, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We may assume C is generated by X and its conjugate object by replacing C with its full subcateogry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take a q ∈ [−1, 1] \\ {0} and a C*-tensor functor Φ: Repf SUq(2) −→ C with Φ(π1/2) = X ⊕X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The existence of those is guaranteed by the universal property of Repf SUq(2) ([NT13, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then C can be thought as a Repf SUq(2)- module category by π ⊗Y = Φ(π)⊗Y for π ∈ Repf SUq(2) and Y ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover our assumption implies that this is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we can use the duality theorem [DY13, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='4] to find a quantum homogeneous space A of SUq(2) EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 17 for which there is an equivalence C ∼= SUq(2)- Modf A of Repf SUq(2)-module cate- gories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' On the other hand, by using the right multiplication of C on C, we have a C*-tensor functor from Cop to [SUq(2)- Modf A, SUq(2)- Modf A]Repf SUq(2) b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since any C*-tensor functor from a fusion category is dimension- preserving, this implies we have an object Y in [SUq(2)- Modf A, SUq(2)- Modf A]Repf SUq(2) b with the dimension √ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we have a Q-system Y ⊗Y ∈ [SUq(2)- Modf A, SUq(2)- Modf A]Repf SUq(2) b , and we have a finite quantum SUq(2)-covering space with the covering degree d by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Induction from actions of the maximal Kac quantum subgroup 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Module categorical interpretation of induced actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let G be a compact quantum group and H be its quantum subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For an H-C*algebra A, we use the symbol �A to present its induced G-C*-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take π ∈ Repf G and E ∈ H- Modf A arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the unitary U−1 π,13 on Hπ ⊗ E ⊗ C(G) is restricted to an unitary operator from Ind G H(Hπ|H ⊗ E) to Hπ ⊗ Ind G HE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If we regard H- Modf A as a Repf G-module category via the re- striction functor Repf G −→ Repf H, the collection of such unitaries makes the induction functor Ind G H : H- Modf A −→ G- Modf � A into a Repf G-module functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The following theorem is essentially proved in [Va05, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='] Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The functor Ind G H gives an equivalence H- ModA ∼= G- Mod � A as Repf G-module categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This equivalence also gives an equivalence H- Modf A ∼= G- Modf � A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Instead of relying on the Vaes’ generalization of Green imprimitivity, we will give a quasi-inverse of Ind G H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The algebraic core of �A is denoted by � A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The following is the explict presen- tation of � A: � A = {x ∈ A ⊗ O(G) | (α ⊗ id)(x) = (id ⊗ℓH)(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Here q is the canonical surjection from O(G) to O(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let �E be a G-equivariant Hilbert �A-module and �E be its algebraic core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the following gives an A-valued semi-inner product on �E: ⟨ξ, η⟩A = (id ⊗ε)(⟨ξ, η⟩ � A) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By replacing �E with its G-equivariant Hilbert �A-submodule generated by ξ and η, we may assume that �E is countably generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we can find a unitary representation π of G such that there is an isometry V ∈ LG � A( �E, Hπ⊗ �A), by using the equivariant Kasparov stabilization theorem ([Ve02, Th´eor`eme 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence it suffices to show the statement for Hπ ⊗ �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this case we have E = Hπ ⊗ � A and ⟨v ⊗ x, w ⊗ y⟩A = ⟨v, w⟩ (id ⊗ε)(x)∗(id ⊗y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' for v, w ∈ Hπ and x, y ∈ � A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now the statement follows from this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ 18 MAO HOSHINO Let �E0 be a Hilbert A-module obtained as the completion of �E with respect to this A-valued semi-inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the map (id ⊗q)�α � E : �E −→ �E ⊗ O(H) induces a H-action on �E0, making �E into an H-equivariant Hilbert A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let E be an H-equivariant Hilbert A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the following map defined PE on E ⊗O(G) is a surjection onto �E, the algebraic core of Ind G HE: PE(ξ ⊗ x) = (id ⊗hH)((id ⊗S)αE(ξ)(1 ⊗ q((x(1))))) ⊗ x(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Here hH is the Haar state of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since we have �E = {ξ ∈ E ⊗ O(G) | (αE ⊗ id)(ξ) = (id ⊗ℓH)(ξ)}, it can be easily seen that PE is identical on �E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The remaining part is to show PE(E ⊗ O(G)) ⊂ �E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take arbitrary ξ ∈ E and x ∈ O(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have the following: (αE ⊗ id)(PE(ξ ⊗ x)) = (id ⊗ id ⊗hH)((id ⊗(id ⊗S)∆)(αE(ξ))(1 ⊗ 1 ⊗ q(x(1)))) ⊗ x(2) = (id ⊗hH ⊗ S−1)((id ⊗∆ ◦ S)(αE(ξ))(1 ⊗ q(x(1)) ⊗ 1)) ⊗ x(2) (id ⊗ℓH)(PE(ξ ⊗ x)) = (id ⊗hH ⊗ id)(((id ⊗S)(αE(ξ)) ⊗ 1)(1 ⊗ ∆(q(x1)))) ⊗ x(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence the statement follows from the strong bi-invariance of hH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If H is of Kac type, PE extends to a G-equivariant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' map from E ⊗ C(G) to Ind G HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='8 by using the corner trick to E ⊂ KA(A ⊕ E) and �E ⊂ K � A( �A ⊕ �E) ∼= Ind G HKA(A ⊕ E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We have two functors: The first one is Ind G H, and the second one is EvH G : G- Mod � A −→ H- ModA given by �E �−→ �E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The remaining is to show that these functors are quasi-inverses of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take E ∈ H- ModA arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have a linear map (id ⊗ε)| �E : �E −→ E, for which we have ⟨(id ⊗ε)(ξ), (id ⊗ε)(η)⟩A = ⟨ξ, η⟩A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence this map induces an isometry VE ∈ LG A((Ind G HE)0, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To see the surjec- tivity of VE, we use PE as in the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For any ξ ∈ E and x ∈ O(G), we have (id ⊗ε) ◦ PE(ξ ⊗ x) = (id ⊗hH)((id ⊗S)(αE(ξ))(1 ⊗ q(x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since span αE(E)(C ⊗ O(H)) = E ⊗ O(H), the above equality implies the surjec- tivity of VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Next take �E ∈ G- Mod � A arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then �α � E defines a linear map from �E to �E0 ⊗ O(G), for which we have � �α � E(ξ), �α � E(η) � � A = ⟨ξ, η⟩ � A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover we also have � �α � E(ξ), �α � E(ηx) � � A = (id ⊗ε ⊗ id)(�α(⟨ξ, η⟩ � A x)) = (id ⊗ε ⊗ id)(�α(⟨ξ, η⟩ � A))x = � �α � E(ξ), �α � E(η)x � � A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 19 Hence �α � E induces an isometry U � E ∈ LG � A( �E, �E0 ⊗ C(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' It can be easily seen that the image of U � E is contained in Ind G H �E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To see that U � E( �E) = Ind G H �E0, we can use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='6 by considering the case of B = A and M = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The functor EvH G : G- Mod � A −→ H- ModA also has a structure of Repf G-module functor, namely the collection of unitaries v ⊗ ξ ∈ (Hπ ⊗ �E)0 �−→ v ⊗ ξ ∈ Hπ|H ⊗ �E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now let A, B be H-C*-algebras and �A, �B be their induced G-C*-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we also have an induction functor Ind G H : H- Corr(A,B) −→ G- Corr � A, � B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let �E ∈ G- Mod � A and M ∈ H- CorrA,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have the following unitary equivalence �E ⊗ � A Ind G HM ∼= Ind G H(EvH G �E ⊗A M), induced by ξ ⊗ η ⊗ x �−→ �α � E,13(ξ)(1 ⊗ η ⊗ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The only non-trivial part is the surjectivity of the map in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let �E, M and (EvH G �E ⊗A M)alg be the algebraic cores of �E, M and EvH G �E ⊗A M respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let us consider the following diagram: �E ⊗ M ⊗ O(G) � id ⊗PM � �E ⊗ M ⊗ O(G) � (EvH G �E ⊗A M)alg ⊗ O(G) PEvH G � E⊗AM � �E ⊗ � A Ind G HM � Ind G H(EvH G �E ⊗A M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Here the morphism at the top is given by ξ ⊗ η ⊗ x �−→ �α � E,13(ξ)(1 ⊗ η ⊗ x), and the morphism at the bottom is as in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we can see that this diagram is commutative and that the morphisms at the top and at the right have dense ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using these observation, we can see the surjectivity of the morphism at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' One should note that the collection of the unitary equivalences over all �E gives rise to a unitary equivalence of Repf G-module functor from – ⊗ � A Ind G HM to Ind G H(EvH G(–) ⊗A M) Let C be another H-C*-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For M ∈ H- CorrA,B and N ∈ H- CorrB,C, we have a unitary equivalence Ind G H(M ⊗B N) ∼= (Ind G HM) ⊗ � B (Ind G HN), namely the restriction of (ξ ⊗ η) ⊗ x �−→ (ξ ⊗ 1) ⊗ (η ⊗ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover the collection of these unitaries makes Ind G H : H- CorrA −→ G- Corr � A into a C*-tensor functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 20 MAO HOSHINO Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A, B be quantum homogeneous spaces of H and �A, �B be their induced quantum homogeneous spaces of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the following diagram of C*- functors commutes up to a canonical unitary natural transformation: H- Corrrf A,B Ind G H � ∼ = � [H- Modf A, H- Modf B]Repf H � [H- Modf A, H- Modf B]Repf G Ind G H◦–◦EvH G ∼ = � G- Corrrf � A, � B ∼ = � [G- Modf � A, G- Modf � B]Repf G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover, the canonical unitary natural transformation is monoidal if A = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The canonical unitary natural transformation is given by the collection of unitaries as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the statement can be checked by direct calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Tracial module category and its Plancherel weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='8 en- ables us to show an imprimitivity-type result by a comparison of Repf G-module functors and Repf H-module functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The purpose of this subsection is to build a general theory of such a comparison for general module categories with traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' At first we introduce the notion of a trace on a module category, which already appears in [Sc13, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='7] for purely algebraic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let C be a rigid strict C*-tensor category and M be a C-module category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The endomorphism algebra of X ∈ M is denoted by M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For U ∈ C and X ∈ M, we can define the partial trace TrU ⊗ id: M(U ⊗ X) −→ M(X) as follows: (TrU ⊗ id)(T) = (R∗ U ⊗ idX)(idU ⊗X)(RU ⊗ idX) Here (U, U, RU, RU) is a standard solution in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let M be a C-module category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A C-module trace on M is a family {TrX : M(X) −→ C}X∈M of positive linear maps satisfying the following conditions: (i) For any f, g ∈ M(X, Y ), TrX(g∗f) = TrY (fg∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (ii) TrU⊗X = TrX ◦(TrU ⊗ id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We call a pair of C-module category and C-module trace on it as tracial C-module category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this case we define the dimension of X ∈ M as d(X) = TrX(idX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By (ii), we have d(U ⊗ X) = d(U)d(X) for U ∈ C and X ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If M is indecomposable, there exists at most one C-module trace on M up to scalar multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This can be seen by using the compatibility with the left C-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If we consider C as a C-module category in usual way, then the family of categorical traces defines a C-module trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover, if another rigid strict C*-tensor category D and dimension-preserving C*-tensor functor F : C −→ D are given, D can be considered as a tracial C-module category by U ⊗ X = F(U) ⊗ X and the family of categorical traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 21 Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A be a Q-system in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the category Mod -A of right A-modules in C has a C-module trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If C is a fusion category, every C-module category has a C-module trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This follows from the fact that every indecomposable C-module category arises as a corner of a rigid C*-2-category with C in its diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let M be a tracial C-module category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Fix a set of representatives of all equivalence classes of irreducible objects of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' It is denoted by Irr M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For F ∈ [M, M]b, the algebra of bounded natural transformations from F to F is denoted by Endb(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then Endb(idM) is isomorphic to ℓ∞(Irr M) via a map a �−→ (ai)i∈Irr M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using the dimension of objects in M, we can define a weight ωM on Endb(idM) as follows: ωM(a) = � i∈Irr M d(i)2ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We call ωM the Plancherel weight of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This defines a linear functional if and only if Irr M is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Recall we have a C*-tensor functor Φ: C −→ [M, M]b induced by the left action of C on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Namely Φ(U) is a C*-fuctor from M to M which sends X ∈ M to U ⊗ X ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We use the symbol ϕU,V to denote the unitary from Φ(U) ⊗ Φ(V ) to Φ(U ⊗ V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We can show the following nice property of ωM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let M be a tracial C-module category and ωM be the Plancherel weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have the following equalities for any U ∈ C and a positive natural transformation η: Φ(U) −→ Φ(U): ωM(Φ(RU)∗ϕU,U(id ⊗η)ϕ∗ U,UΦ(RU)) = ωM(Φ(RU)∗ϕU,U(η ⊗ id)ϕ∗ U,UΦ(RU)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To prove this proposition, it is convenient to replace [M, M]b by the category of column-finite bi-graded Hilbert spaces introduced in [DY13, Notation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Set I = Irr M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the category HilbI×I of I × I-graded Hilbert spaces has a canonical structure of a C*-multitensor category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Namely, for H = (Hij)ij and K = (Kij)ij, their tensor product H ⊗ K is a I × I-graded Hilbert space whose (i, j)-component is given by � k∈I Hik ⊗ Kkj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A I × I-graded Hilbert space H is said to be column-finite if, for each j ∈ I, there are finitely many i ∈ I such that Hij ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We also say that H is uniformly finite if it satisfies the following: sup i∈I � j∈I dim Hij < ∞, sup j∈I � i∈I dim Hij < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The full subcategory consisting of column-finite (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' uniformly finite) I × I- graded Hilbert spaces is denoted by Hilbcf I×I (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hilbf I×I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the rigid part of HilbI×I is precisely Hilbf I×I ([DY13, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now we compare [M, M]b and Hilbcf I×I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For F ∈ [M, M]b, we have a column- finite I×I-graded Hilbert space HF whose (i, j)-component is given by M(i, F(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Here M(i, F(j)) is regarded as a Hilbert space by the inner product ⟨S, T⟩ = S∗T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 22 MAO HOSHINO Then this construction gives an equivalence of C*-multitensor categories from [M, M]b to Hilbcf I×I ([DY13, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover we also have an equivalence [M, M]f b ∼= Hilbf I×I ([DY13, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' From now on, we identify these categories by this equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let (U, U, R, R) be a standard solution in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then ϕ∗ U,UΦ(R) gives a morphism from 1 to HΦ(U) ⊗ HΦ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By the definition of a tensor product of I × I-graded Hilbert spaces, this morphism can be presented as a direct sum of Rji : C −→ M(i, U ⊗ j) ⊗ M(j, U ⊗ i), where i, j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Similarly ϕ∗ U,UΦ(R) is presented as a direct sum of Rji : C −→ M(j, U ⊗ i) ⊗ M(i, U ⊗ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In the following lemma, U ⊗ i is denoted by Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For any i, j ∈ I, (M(j, Ui), M(i, Uj), Rji, Rji) is a solution of a conjugate equation in Hilbf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover, if we replace Rji and Rji with d(i)1/2d(j)−1/2Rji, d(i)−1/2d(j)1/2Rji respectively, it gives a standard solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since (Φ(U), Φ(U), ϕ∗ U,UΦ(R), ϕU,UΦ(R)) is a solution of conjugate equa- tion in [M, M]b, the following composition of maps is the identity: � j,i∈I M(j, Ui) � i,k � j=l Rjk⊗id −−−−−−−−−−→ � j,i∈I � k,l∈I M(j, Uk) ⊗ M(k, Ul) ⊗ M(l, Ui) � j,l � i=k id ⊗R∗ li −−−−−−−−−−→ � j,i∈I M(j, Ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we have (id ⊗R∗ ji)(Rji ⊗ id) = id for any i, j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The same argument also shows the other equality (id ⊗Rji∗)(Rji ⊗ id) = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now we complete the proof of the former half of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To prove the latter half, we calculate ∥Rji∥ and ∥Rji∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' At first one should note that M(i, Uj)⊗M(j, Ui) can be embedded in M(i, UUi) by T⊗S �−→ (idj ⊗S)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreove the projection pj onto its image is given by the following: pj(X) = � V ∈(j,Ui) (id ⊗V V ∗)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then Rji(1) corresponds to the image by pj of the component of ϕ∗ U,UΦ(R) at i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we have ∥Rji∥2 = � V ∈(j,V i) (R ⊗ idi)∗(id ⊗V V ∗)(R ⊗ idi) = 1 d(i) � V ∈(j,V i) TrUi(V V ∗) = 1 d(i) � V ∈(j,V i) Trj(V ∗V ) = d(j) d(i) dim M(j, Ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 23 Similarly we also have ∥Rji∥2 = d(i) d(j) dim M(i, Uj) = d(i) d(j) dim M(j, Ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we have ∥Rji∥∥Rji∥ = (dim M(j, Ui))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using the characterization of standard solutions, we get the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By the previous lemma, we have Rji(id ⊗θ)R∗ ji = d(j) d(i) Tr(θ) for any linear map θ: M(j, Ui) −→ M(j, Ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let η: Φ(U) −→ Φ(U) be a positive natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then it induces a positive operator ηji : M(j, Ui) −→ M(j, Ui) for any i, j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have ωM(Φ(RU)∗ϕU,U(id ⊗η)ϕ∗ U,UΦ(RU)) = � i∈I d(i)2 � j∈I R∗ ji(id ⊗ηji)Rji = � i,j∈I d(i)d(j) Tr(ηji).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For the right hand side of the equality in the statement, we also can show that ωM(Φ(RU)∗ϕU,U(η ⊗ id)ϕ∗ U,UΦ(RU)) = � i,j∈I d(i)d(j) Tr(ηji).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence the statement holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Based on Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='14, we introduce a notion of a standard solution in [M, M]b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let F : M −→ M be a C*-functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We say that a solution (F, F, R, R) is standard if it satisfies the following identity for any positive natural transformation η: F −→ F: ωM(R∗(id ⊗η)R) = ωM(R ∗(η ⊗ id)R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this case, we define a weight TrF on Endb(F) by η �−→ ωM(R∗(id ⊗η)R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='14 states that (Φ(U), Φ(U), ϕ∗ U,UΦ(RU), ϕ∗ U,UΦ(RU)) is standard when (U, U, R, R) is standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let F ∈ [M, M]b be a rigid object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (i) A standard solution for F exists and is unique up to unitary equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (ii) TrF is tracial and independent of a choice of a standard solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (iii) For a standard solution (F, F, R, R), the following map is an anti-∗- isomorphism from Endb(F) to Endb(F): η �−→ (id ⊗R ∗)(id ⊗η ⊗ id)(R ⊗ id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If we regard F as an object HF ∈ Hilbf I×I, a solution (F, F, R, R) of con- jugate equation is decomposed into a family {(M(j, F(i)), M(i, F(j)), Rji, Rji)} 24 MAO HOSHINO of solutions in Hilbf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have the following for any positive natural trans- formation η: F −→ F: ωM(R∗(id ⊗η)R) = � i,j∈I d(i)2R∗ ji(id ⊗ηji)Rji, ωM(R ∗(η ⊗ id)R) = � i,j∈I d(j)2R ∗ ji(ηji ⊗ id)Rji, where ηji is a corresponding positive operator on M(j, F(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence (F, F, R, R) is standard if and only if the following quadruple is a stan- dard solution for each i, j ∈ I: � M(j, F(i)), M(i, F(j)), d(j)1/2d(i)−1/2Rji, d(j)−1/2d(i)1/2Rji � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now all of the statements follow from this characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ We would like to come back to our main interest: a comparison of module functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let D be another rigid strict C*-tensor category and (Θ, θ): D −→ C be a C*-tensor functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The modular natural transformation aΘ of (Θ, θ) is a natural transformation from Θ to Θ given by the following: aΘ,U = (id ⊗ TrΘ(U))(θ∗ U,UΘ(rUr∗ U)θU,U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Here (U, U, rU, rU) is a stardard solution in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since all standard solutions are mutually unitary equivalent, aΘ,U does not depend on the choice of (U, U, rU, rU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We collect some fundamental properties of aΘ in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The modular natural transformation aΘ is monoidal and invert- ible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Its inverse is as follows: a−1 Θ,U = (TrΘ(U) ⊗ id)(θ∗ U,UΘ(rUr∗ U)θU,U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover, the following quadruple is a standard solution in C for every U ∈ D: (Θ(U), Θ(U), (1 ⊗ a1/2 Θ,U)θ∗ U,UΘ(rU), (a−1/2 Θ,U ⊗ id)θ∗ U,UΘ(rU)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' One can check the statement by direct calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If (Θ, θ) is the fiber functor from Repf G to Hilbf, each compo- nent aΘ,π is precisely ρπ in [NT13, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' See also [NT13, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We regard a D-module functor (F, f) as a pair of a C*-functor F and a collec- tion of unitary morphisms fU : F ⊗ Φ(Θ(U)) −→ Φ(Θ(U)) ⊗ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let (F, f): M −→ M be a D-module functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If F is rigid in [M, M]b, the following are equivalent for each U ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (i) We have fU(idF ⊗Φ(aΘ,U)) = (Φ(aΘ,U) ⊗ idF)fU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (ii) For a standard solution (Φ(Θ(U)), Φ(Θ(U)), RΦ(Θ(U)), RΦ(Θ(U))) in [M, M]b, the following natural transformation is unitary: (id ⊗ id ⊗R ∗ Φ(Θ(U)))(id ⊗f ∗ U ⊗ id)(RΦ(Θ(U)) ⊗ id ⊗ id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 25 (iii) For a standard solution (F, F, R, R), the following natural transformation f U : F ⊗ Φ(Θ(U)) −→ Φ(Θ(U)) ⊗ F is unitary: f U = (R∗ ⊗ id ⊗ id)(id ⊗f ∗ U ⊗ id)(id ⊗ id ⊗R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let (U, U, rU, rU) be a standard solution in D and define rΦ(Θ(U)) and rΦ(Θ(U)) as follows: rΦ(Θ(U)) = ϕ∗ Θ(U),Θ(U)Φ(θ∗ U,UΘ(rU)), rΦ(Θ(U)) = ϕ∗ Θ(U),Θ(U)Φ(θ∗ U,UΘ(rU)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using the braiding equation, we have (id ⊗ id ⊗r∗ Φ(Θ(U)))(id ⊗f ∗ U ⊗ id)(rΦ(Θ(U)) ⊗ id ⊗ id)f ∗ U = (id ⊗ id ⊗r∗ Φ(Θ(U)))(id ⊗f ∗ U⊗U)(rΦ(Θ(U)) ⊗ id ⊗ id) = (id ⊗r∗ Φ(Θ(U)) ⊗ id)(rΦ(Θ(U)) ⊗ id ⊗ id) = id ⊗ id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since fU is unitary, this implies that we have fU = (id ⊗ id ⊗r∗ Φ(Θ(U)))(id ⊗f ∗ U ⊗ id)(rΦ(Θ(U)) ⊗ id ⊗ id), fU = (id ⊗ id ⊗r∗ Φ(Θ(U)))(id ⊗f ∗ U ⊗ id)(rΦ(Θ(U)) ⊗ id ⊗ id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now we give a proof of (i) ⇐⇒ (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let (Θ(U), Θ(U), RΘ(U), RΘ(U)) be the standard solution as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Set RΦ(Θ(U)) = ϕ∗ Θ(U),Θ(U)Φ(RΘ(U)), RΦ(Θ(U)) = ϕ∗ Θ(U),Θ(U)Φ(RΘ(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='14 asserts that (Φ(Θ(U)), Φ(Θ(U)), RΦ(Θ(U))), RΦ(Θ(U))) is a standard solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Under the condition (i), we have (id ⊗ id ⊗R ∗ Φ(Θ(U)))(id ⊗f ∗ U ⊗ id)(RΦ(Θ(U)) ⊗ id ⊗ id) = (id ⊗ id ⊗R ∗ Φ(Θ(U)))(id ⊗(id ⊗Φ(a1/2 Θ,U))f ∗ U(Φ(a−1/2 Θ,U ) ⊗ id) ⊗ id)(RΦ(Θ(U)) ⊗ id ⊗ id) = (id ⊗ id ⊗r∗ Φ(Θ(U)))(id ⊗f ∗ U ⊗ id)(rΦ(Θ(U)) ⊗ id ⊗ id) = fU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence (ii) holds for the standard solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since all standard solutions are mu- tually unitary equivalent, we also have (ii) for a general standard solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To show the converse direction, one should note that we have (id ⊗a1/2 Θ,U)θ∗ U,UΘ(rU) = (a−1/2 Θ,U ⊗ id)θ∗ U,Ua1/2 Θ,U⊗UΘ(rU) = (a−1/2 Θ,U ⊗ id)θ∗ U,UΘ(rU)a1/2 1 = (a−1/2 Θ,U ⊗ id)θ∗ U,UΘ(rU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Similary we have (id ⊗a1/2 Θ,U)θ∗ U,UΘ(rU) = (a−1/2 Θ,U ⊗ id)θ∗ U,UΘ(rU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 26 MAO HOSHINO By using these formulae, we can see that (Φ(a−1/2 Θ,U ) ⊗ id)fU(id ⊗Φ(a1/2 Θ,U)) = (id ⊗ id ⊗r∗ Φ(Θ(U)))(Φ(a−1/2 Θ,U ) ⊗ f ∗ U ⊗ Φ(a1/2 Θ,U))(rΦ(Θ(U)) ⊗ id ⊗ id) = (id ⊗ id ⊗R ∗ Φ(Θ(U)))(id ⊗f ∗ U ⊗ id)(RΦ(Θ(U)) ⊗ id ⊗ id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence (Φ(a−1/2 Θ,U ) ⊗ id)fU(id ⊗Φ(a1/2 Θ,U)) is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now we have (Φ(a−1/2 Θ,U ) ⊗ id)fU(id ⊗Φ(aΘ,U))f ∗ U(Φ(a−1/2 Θ,U ) ⊗ id) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we can see that (i) holds for U since fU is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We already have shown (i) =⇒ (ii) and (ii) for U =⇒ (i) for U, hence (ii) =⇒ (i) also has been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The equivalence of (ii) and (iii) can be seen from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='17 (iii) and the standardness of the following quadruples: (F ⊗ Φ(Θ(U)), Φ(Θ(U)) ⊗ F, (id ⊗R ⊗ id)RΦ(Θ(U)), (id ⊗R ⊗ id)RΦ(Θ(U))), (Φ(Θ(U)) ⊗ F, F ⊗ Φ(Θ(U)), (id ⊗RΦ(Θ(U)) ⊗ id)R, (id ⊗RΦ(Θ(U)) ⊗ id)R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ By considering the case of idC : C −→ C, we obtain the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For a C-module functor (F, f), it is rigid in [M, M]C b if and only if F is rigid in [M, M]b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this case, the conjugate object (F, f) of (F, f) can be obtained from a standard solution (F, F, R, R) as follows: F: a conjugate object of F in [M, M]b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' f U = (R∗ ⊗ id ⊗ id)(id ⊗f ∗ U ⊗ id)(id ⊗ id ⊗R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If M is connected as a tracial C-module category, a standard solution in [M, M]C b is also standard in [M, M]b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For a C-module category, we can consider an equivalence relation ∼C on Irr M as follows: i ∼C j def ⇐⇒ there exists U ∈ C such that M(i, U ⊗ j) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For a C*-functor F : M −→ M, we define a function dF : Irr M −→ R as follows: dF(i) = d(F(i)) d(i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If (F, f) ∈ [M, M]D b satisfies the conditions in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='21 for all U ∈ D, the function dF is constant on each equivalence class of ∼D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If F is an equivalence of categories, the converse direction also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let (F, F, R, R) be a standard solution in [M, M]b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the condition (iii) of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='21 implies that we have a D-module functor (F, f): M −→ M given by f U = (R∗ ⊗ id ⊗ id)(id ⊗f ∗ U ⊗ id)(id ⊗ id ⊗R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 27 Then R and R are morphisms in [M, M]D b , hence R∗R is also a morphism from idM to idM in [M, M]D b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This implies that we have (R∗R)Θ(U)⊗i = idΘ(U) ⊗(R∗R)i for any U ∈ D and i ∈ Irr M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In particular i ∈ Irr M �−→ (R∗R)i is constant on each equivalence class of ∼D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' On the other hand, by the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='17, we have (R∗R)i = � j∈Irr M d(j) dim M(j, F(i)) d(i) = d(F(i)) d(i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This completes a proof of the first statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To show the second statement, we will check the condition (iii) of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let (F, F, R, R) be a standard solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Our assumption implies that F(i) is irreducible for any i ∈ Irr M, hence d(F(i))−1/2d(i)1/2Ri = dF(i)−1/2Ri is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover dF(i)−1/2RΘ(U)⊗i is also unitary for any U ∈ D by the assumption on dF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since we have dF(F(i)) = d(F(F(i)))/d(F(i)) = dF(i)−1, the similar argument shows that dF(i)−1/2RΘ(U)⊗i is also unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have (R ⊗ id ⊗ id)(id ⊗f ∗ U ⊗ id)(id ⊗ id ⊗R)i = dF(F(i))−1/2RU⊗F (i)F(f ∗ U,F(i))F(idΘ(U) ⊗dF(i)−1/2Ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This implies that (F, f) fulfills the condition (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ We end this subsection by showing that the condition (iii) is automatically satisfied under a finiteness condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let M be a tracial C-module category with |Irr M| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then [M, M]D b is rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover a standard solution in [M, M]D b is standard again in [M, M]b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We need the following lemma to show the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let (F, f) be a D-module functor from M to M and F be a conjugate of F in [M, M]b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the function dF and dF are constant on each equivalence class of ∼D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let I be an equivalence class of ∼D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since Irr M is finite, I is also finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we can take an object U ∈ D such that M(i, Θ(U)⊗j) ̸= 0 for any i, j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This means the matrix M = (dim M(j, Θ(U) ⊗ i))ij∈I is an irreducible matrix with positive entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now d = (d(i))i∈I satisfies Md = d(Θ(U))d i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' d is a Perron-Frobenius eigenvector of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' On the other hand, we have d(Θ(U))d(F(i)) = d(Θ(U) ⊗ F(i)) = d(F(Θ(U) ⊗ i)) = � j∈I d(F(j)) dim M(j, Θ(U) ⊗ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 28 MAO HOSHINO Hence (d(F(i)))i∈I is also a Perron-Frobenius eigenvector of M, which must be a scalar multiple of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now we get the statement for dF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For F, one should note that F is an adjoint functor of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we have M(i, F(Θ(U) ⊗ j)) ∼= M(F(i), Θ(U) ⊗ j) ∼= M(Θ(U) ⊗ F(i), j) ∼= M(F(Θ(U) ⊗ i), j) ∼= M(Θ(U) ⊗ i, F(j)) ∼= M(i, Θ(U) ⊗ F(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' for any i, j ∈ Irr M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This implies F(Θ(U) ⊗ i) ∼= Θ(U) ⊗ F(i) and we can use the argument above to show the statement for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take (F, f) ∈ [M, M]D b and a standard solution (F, F, R, R) in [M, M]b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since (R∗R)i = dF(i)−1 and (R ∗R)i = dF(i)−1, the previous lemma implies that R∗R ⊗ idΦ(Θ(U)) = idΦ(Θ(U)) ⊗R∗R, R ∗R ⊗ idΦ(Θ(U)) = idΦ(Θ(U)) ⊗R ∗R for any U ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Set f U as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='21: f U = (R∗ ⊗ id ⊗ id)(id ⊗f ∗ U ⊗ id)(id ⊗ id ⊗R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' It suffices to show that f U is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take a standard solution (U, U, rU, rU) in D and set rΦ(Θ(U)) and rΦ(Θ(U)) as in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now consider ωl ∈ Endb(F ⊗ Φ(Θ(U)))∗ given by ωl(η) = ωM(r∗ Φ(Θ(U))(id ⊗R ∗ ⊗ id)(id ⊗ id ⊗η)(id ⊗R ⊗ id)rΦ(Θ(U))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 29 Then we have ωl(id) = ωM(r∗ Φ(Θ(U))(id ⊗R ∗R ⊗ id)rΦ(Θ(U))) = ωM(r∗ Φ(Θ(U))rΦ(Θ(U)) ⊗ R ∗R) = d(U) TrF(idF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' ωl(f ∗ Uf U) = ωM((r∗ Φ(Θ(U)) ⊗ R ∗)(id ⊗fUf ∗ U ⊗ id)(rΦ(Θ(U)) ⊗ R)) = d(U) TrF(idF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' ωl(f ∗ Uf Uf ∗ Uf U) = ωM((r∗ Φ(Θ(U)) ⊗ R ∗)(id ⊗fU ⊗ id)(id ⊗ id ⊗f Uf ∗ U) (id ⊗f ∗ U ⊗ id)(rΦ(Θ(U)) ⊗ R)) = ωM((r∗ Φ(Θ(U)) ⊗ R ∗)(id ⊗fU ⊗ id)(fUf ∗ U ⊗ f Uf ∗ U) (id ⊗f ∗ U ⊗ id)(rΦ(Θ(U)) ⊗ R)) = ωM(R ∗(id ⊗r∗ Φ(Θ(U)) ⊗ id)(id ⊗ id ⊗f Uf ∗ U)(id ⊗rΦ(Θ(U)) ⊗ id)R) = ωM((r∗ Φ(Θ(U)) ⊗ R∗)(id ⊗f Uf ∗ U ⊗ id)(rΦ(Θ(U)) ⊗ R)) = ωM(r∗ Φ(Θ(U))(id ⊗R∗ ⊗ id)(id ⊗ id ⊗f ∗ UfU)(id ⊗R ⊗ id)rΦ(Θ(U))) = ωl(id) = d(U) TrF(idF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence ωl((1 − f ∗ Uf U)2) = 0 and f ∗ Uf U = 1 since ωl is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To show f Uf ∗ U = 1, one can use the following ωr ∈ Endb(Φ(Θ(U)) ⊗ F)∗: ωr(η) = ωM(r∗ Φ(Θ(U))(id ⊗R∗ ⊗ id)(η ⊗ id ⊗ id)(id ⊗R ⊗ id)rΦ(Θ(U))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then a similar argument shows f Uf ∗ U = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Imprimitivity-type result for the maximal Kac quantum subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this subsection, we apply results in the previous subsection to module cate- gories arising from quantum homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' At first, we give a characterization of a quantum homogeneous space whose associated module category has a module trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let G be a compact quantum group of Kac type and A be a quantum homogeneous space of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the following conditions are equivalent: (i) G- Modf A has a Repf G-module trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (ii) There is a tracial state on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (iii) The G-invariant state on A is tracial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The equivalence of (ii) and (iii) follows from the G-invariance of (ϕ ⊗ h)α for a state ϕ on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We show (ii) =⇒ (i) by constructing a Repf G-module trace from a given tracial state τ on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let E be a finitely generated G-equivariant Hilbert A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then there exists a finite dimensional unitary representation π of G such that there is 30 MAO HOSHINO an isometry V ∈ LG A(E, Hπ ⊗ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now we define a tracial state TrE on LG A(E) as follows: TrE(T) = (Tr ⊗τ)(V TV ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since Tr ⊗τ is tracial, this definition does not depend on the choice of V and π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover the corner trick shows that TrE satisfies (i) of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To prove the compatibility with the left action of Repf G, one should note that the standard solution in Repf G is also standard in Hilbf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence the partial trace Trπ ⊗ id: LG A(Hπ ⊗ E) −→ LG A(E) coincides with Tr ⊗ id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This implies the compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Next we move to the proof of (i) =⇒ (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Fix a Repf G-module trace {TrE}E on G- Modf A with TrA(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using the spectral decomposition of the algebraic core A of A, the G-invariant state ϕ on A can be seen as the projection from A to A1G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We show that this is tracial by using {TrE}E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since the spectral decomposition of A is orthogonal with respect to the inner product coming from ϕ, it suffices to show that the A1G-components of (T⊗ξ)∗(S⊗η) and (S⊗η)(T⊗ξ)∗ are same for T ⊗ξ, S ⊗η ∈ Aπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For the first one, its A1G-component is calculated as follows: ⟨ξ, η⟩ dim π(R∗ π ⊗ idA)(idπ ⊗T ∗S)(Rπ ⊗ idA) = ⟨ξ, η⟩ dim π TrA((R∗ π ⊗ idA)(idπ ⊗T ∗S)(Rπ ⊗ idA))1A = ⟨ξ, η⟩ dim π TrHπ⊗A(T ∗S)1A = ⟨ξ, η⟩ dim π TrA(ST ∗)1A = ⟨ξ, η⟩ dim πST ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The last formula is precisely the A1G-component of the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now we get the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Let G be a compact quantum group and K be a its maximal Kac quantum subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The restriction functor from Repf G to Repf K is denoted by Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Fix a set Irr G of representatives of all equivalence classes of irreducible representations of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For π, ρ ∈ Repf G, we defines L0(π, ρ) as a subset of HomK(π|K, ρ|K) consisting of intertwiners of the following form: T ∗ ◦ (an1 Θ,σ1 ⊗ an2 Θ,σ2 ⊗ · · · ⊗ ank Θ,σk) ◦ S where k ∈ Z>0, ni ∈ 1 2Z, σi ∈ Irr G, T ∈ HomG(ρ, σ1 ⊗ σ2 ⊗ · · · ⊗ σk), S ∈ HomG(π, σ1⊗σ2⊗· · ·⊗σk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we define a subspace L(π, ρ) of HomK(π|K, ρ|K) as the linear span of intertwiners of the form TlTl−1 · · · T1 where Tj ∈ L0(σj−1, σj), σ0 = π, σl = ρ, σj ∈ Irr G for 1 ≤ j ≤ l − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For any π, ρ ∈ Repf G, we have L(π, ρ) = HomK(π|K, ρ|K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Consider a C*-category with Obj Repf G as the collection of object and L(π, ρ) as the morphism set from π to ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let L be its Karoubi envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 31 T ⊗ S ∈ L(π ⊗ π′, ρ ⊗ ρ′) for any T ∈ L(π, ρ) and S ∈ L(π′, ρ′), L becomes a C*-tensor category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover the restriction functor from Repf G to Repf K induces a C*-tensor functor from Repf G to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In particular π ∈ Repf G is rigid in L, hence L is rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now, by composing the inclusion L −→ Repf K and the fiber functor Repf K −→ Hilbf, we obtain a fiber functor L −→ Hilbf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the Woronowicz’s Tannaka- Krein duality ([NT13, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2]) implies that there is a compact qunatum group H and morphisms K −→ H and H −→ G which correpond to L and the functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since every object of L appears as a direct summand of the image of π ∈ Repf K, H can be considered as a quantum subroup of G which contains K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' On the other hand, (π, π, (id ⊗a1/2 Θ,π)Rπ, (a−1/2 Θ,π ⊗ id)Rπ) is a solution of the conjugate equation in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover it satisfies ∥(id ⊗a1/2 Θ,π)Rπ∥ = ∥(a−1/2 Θ,π ⊗ id)Rπ∥ = dim Hπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This implies that the fiber functor of Repf K = L is dimension-preserving, hence H is of Kac type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then H must be contained in K, so we have H = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence the dimension of L(π, ρ) and HomK(π|K, ρ|K) is same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now we get the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A be a quantum homogeneous space of K and �A be its induced G-C*-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If A has a tracial state and Irr K- Modf A is finite, G- Corrrf � A is rigid and Ind G K gives an equivalence K- Corrrf A ∼= G- Corrrf � A as C*-tensor categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Set M = K- Modf A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then M is a tracial Repf K-module category with |Irr M| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since we have Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='8, it suffices to show the canonical inclusion from [M, M]Repf K b to [M, M]Repf G b is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' It can be easily seen that this inclusion is fully faithful, hence the remaining part is the essential surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take (F, f) ∈ [M, M]Repf G b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='25 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='21 assert that we have (Φ(aΘ,π) ⊗ idF)fπ = fπ(idF ⊗Φ(aΘ,π)) for any π ∈ Repf G, where (Φ, ϕ): Repf K −→ [M, M]b is the canonical C*- tensor functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using the braiding equation on u and the previous lemma, we also have (Φ(T) ⊗ idF)fπ = fπ(idF ⊗Φ(T)) for any π ∈ Repf G and any T ∈ HomK(π|K, π|K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we can define fρ for ρ ∈ Repf K as follows: �fρ = (Φ(V )∗ ⊗ idF)fπ(idF ⊗Φ(V )) where π ∈ Repf G and V ∈ HomK(ρ, π|K) is an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By the above equality, the definition of �fρ does not depend on the choice of π and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' It can be easily seen that (F, �f) defines a Repf K-module functor, which is (F, f) as a Repf G-module functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Set A = C(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the corresponding Repf K-module category is Hilbf with the action via the fiber functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This Repf K-module category has a 32 MAO HOSHINO Repf K-module trace and only one irreducible object, hence we can apply Theo- rem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since the induction of C(K) is C(G), we can conclude that G- Corrrf C(G) is rigid and we have an equivalence of C*-tensor categories between K- Corrrf C(K) and G- Corrrf C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' On the other hand, we have K- Corrrf C(K) ∼= Repf O(K) and G- Corrrf C(G) ∼= Repf O(G) by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Combining these equivalences, we can see that Repf O(G) is rigid and Repf O(K) ∼= Repf O(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A direct calculation shows that this equivalence is induced by the canonical morphism q: O(G) −→ O(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By these obserbation, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='29 can be regarded as a generalization of [CKS, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3] and b�K ∼= b�G, which is essentailly proved in [So05, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For a quantum homogeneous space A of G, we define its Picard group PicG(A) as a group of all equivalence classes of invertible objects of G- Modf A (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='f [DC12, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Its product is given by the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='24 instead of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='25 in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='29, we can obtain the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let �Γ be a cocommutative quantum subgroup of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then for any quantum homogeneous space A of �Γ, the induction functor gives a group isomorphism PicK(Ind K �ΓA) ∼= PicG(Ind G �ΓA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover we can also see that the induction functor gives an isomorphism between automorphism groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For a quantum homogeneous space A of G, the group of G-equivariant automorphisms of A is denoted by AutG(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If θ ∈ AutG(A) is given, we can construct an invertible G-equivariant correspondence Aθ over A defined as follows: As a G-equivariant Hilbert A-module, Aθ = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The left multiplication of x ∈ A on Aθ is given by the left multiplication of θ−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have Aθ ⊗A Aθ′ ∼= Aθθ′, so we have a group homomorphism θ ∈ AutG(A) �−→ [Aθ] ∈ PicG(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This is injective since AG = C1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let M be an invertible G-equivariant correspondence over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then there is θ ∈ AutG(A) such that M ∼= Aθ if and only if M has a non-zero G-invariant vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' It is trivial that there exists a non-zero G-invariant vector in Aθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence it suffices to show the existence of a non-zero G-invariant vector implies the exis- tence of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since M is irreducible as a G-equivariant Hilbert A-module, it must be isomorphic to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence there is a left A-action on A by which M and A are isomorphic as G-equivariant correspondence over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' But such a correspondence must be of the form Aθ where θ: A −→ A is a G-equivariant ∗-homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since M is invertible, this θ must be an automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now we get the conclu- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In the same setting as Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='31, the induction of equivari- ant automorphism gives a group isomorphism AutK(Ind K �ΓA) ∼= AutG(Ind G �ΓA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 33 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' One should note that Ind G KE has a non-zero G-invariant vector if and only if E has a non-zero K-invariant vector for any E ∈ K- Modf A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the statement follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='31 and the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let Q ∈ GLn(C) be a positive invertible matrix with multiplicity- free eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the maximal Kac quantum subgroup of the free unitary quantum group U+ Q ([DW96, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3]) is isomorphic to the discrete dual of the free group Fn ([DFS, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In particular it is cocommutative, hence we have group isomorphisms Pic� Fn(A) ∼= PicU+ Q(Ind U+ Q � Fn A), Aut� Fn(A) ∼= AutU+ Q(Ind U+ Q � Fn A) for any quantum homogeneous space A of � Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Classification results for Gq 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Imprimitivity-type results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let G be a simply-connected compact Lie group and T be its maximal torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The Weyl group of G is denoted by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Consider the Drinfeld-Jimbo deformation Gq for a fixed q ∈ (−1, 1) \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We use the symbol εt for the character on C(Gq) defined as the evaluation at t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let us recall the representation theories of C(Gq) and C(T\\Gq), developed in [DS99, NY12, So91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For each w ∈ W, we have an irreducible ∗-representation (πw, Hw) of C(Gq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' It coincides with the counit ε if w = e, and is infinite dimensional otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using these representations, � C(Gq) can be identified with W × T as a set via (w, t) ∈ W × T �−→ (εt ⊗ πw)∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If we consider a Borel structure on W × T induced from � C(Gq) via this identification, each {w} × T ⊂ W × T is a Borel subset of � C(Gq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For C(T\\Gq), the restriction of πw on C(T\\Gq) is still irreducible and w ∈ W �−→ πw|C(T\\Gq) is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The induced Borel structure of W is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The following proposition is what we would like to use in this subsection Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The double dual C(T\\Gq)∗∗ is isomorphic to � w∈W B(Hw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover, its center is contained in Z(C(Gq)∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' One shoule note that C(T\\Gq) is separable type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we can disinte- grate any separable ∗-representation of C(T\\Gq) on � C(T\\Gq) ∼= W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since W is discrete as a Borel space, this implies that a separable ∗-representation admits an irreducible decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using a decomposition into cyclic representations, we also have an irreducible decomposition for a non-separable ∗-representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This implies the former half of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let pw ∈ C(T\\Gq)∗∗ be the projection corresponding to B(Hw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To show the latter half of the statement, it suffices to show π(pw) ∈ π(C(Gq))′ for any separable ∗-representation (π, H) of C(Gq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since C(Gq) is type I, we can disin- tegrate (π, H) on � C(Gq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then each w ∈ W defines a projection Pw ∈ π(C(Gq))′ corresponding to the characteristic function of {w} × T ⊂ W × T ∼= � C(Gq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This Pw must be π(pw), since we have (εt ⊗ πw)∆|C(T\\Gq) = πw for any t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In particular we have π(pw) = Pw ∈ π(C(Gq))′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ 34 MAO HOSHINO Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let {pw}w∈W as in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the canonical map from C(Gq)∗∗ to C(T)∗∗ factors though x ∈ C(Gq)∗∗ �−→ pex ∈ peC(Gq)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' More- over we have peC(Gq)∗∗ ∼= C(T)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A be a T-C*-algebra and �A be its induced Gq-C*-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For any tracial positive linear functional τon �A, there is a tracial positive linear functional τ ′ on A such that τ = τ ′ ◦ (id ⊗ε)| � A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take a tracial positive linear functional τ on �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If τ = 0, there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we may assume τ ̸= 0, moreover τ is a tracial state by taking a normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now consider the GNS triple (πτ, Hτ, ξτ) with respect to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then πτ( �A)′′ is a finite von Neumann algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In particular πτ(C(T\\Gq))′′ is finite, hence πτ(pe) = 1 and πτ(pw) = 0 for w ̸= e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take a state ϕ on A⊗C(Gq)∗ such that ϕ| � A = τ and its GNS triple (πϕ, Hϕ, ξϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have an isometry V : Hτ −→ Hϕ such that V πτ(x)ξτ = πϕ(x)τ for every x ∈ �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence, for any w ∈ W \\ {e}, we have πϕ(1 ⊗ pw)ξϕ = V πτ(pw)ξτ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since 1 ⊗ pw is central in (A ⊗ C(Gq))∗∗ and ξϕ is cyclic, this implies πϕ(pw) = 0 for w ̸= e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the remark after Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1 implies that πϕ factors though A ⊗ C(Gq) −→ A ⊗ C(T), in particular πτ factors through �A −→ Ind T TA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since (id ⊗ε)|Ind T T A gives an isomorphism Ind T TA ∼= A, our statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ The following is a main theorem of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A and B be T-C*-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If B has a faithful family of tracial states, any Gq-equivariant ∗-homomorphism from Ind Gq T A to Ind Gq T B is induced from a T-equivariant ∗-homomorphism from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let ϕ: Ind Gq T A −→ Ind Gq T B be an Gq-equivariant ∗-homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For any tracial state τ on B, τ ◦ (id ⊗ε) ◦ ϕ is also a tracial state on Ind Gq T B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence it factors through A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since B has a faithful family of tracial states, the map (id ⊗ε) ◦ ϕ also factors through A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' we have a ∗-homomorphism ϕ0 : A −→ B such that (id ⊗ε) ◦ ϕ = ϕ0 ◦ (id ⊗ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The T-equivariance of ϕ0 follows from the Gq-equivariance of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We would like to show that ϕ0 is the required homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For an element a ∈ Ind Gq T A, we have ϕ(a) = (id ⊗ε ⊗ id)(�β(ϕ(a))) = ((id ⊗ε) ◦ ϕ ⊗ id)(�α(a)) = (ϕ0 ◦ (id ⊗ε) ⊗ id)(�α(a)) = (ϕ0 ⊗ id)(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we have ϕ = Ind Gq T ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A, B be quantum homogeneous spaces of T and �A, �B be its induced Gq-C*-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then Ind Gq T : T- Corrrf A,B −→ Gq- Corrrf � A, � B is an equiva- lence of C*-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 35 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' What we have to show is the essential surjectivity of Ind Gq T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take � M ∈ Gq- Corrrf � A, � B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1, we can take M ∈ T- Modf B such that � M ∼= Ind Gq T M as a Gq-equivariant Hilbert �B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the right action of �A on � M induces a Gq-equivariant ∗-homomorphism from �A to L � B(� M) ∼= Ind Gq T LB(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If we can show that LB(M) has a faithful trace, the previous theorem implies that this ∗-homomorphism is induced by a T-equivariant ∗-homomorphism from A to LB(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This defines a T-equivariant (A, B)-correspondence M and we have Ind Gq T M ∼= � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now we show the existence of a faithful trace on LB(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By taking an isometry from M to Hπ ⊗ B for some π ∈ Repf T, we may assume M = Hπ ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then LB(Hπ ⊗B) ∼= B(Hπ) ⊗B, and the existence follows from the finiteness theorem [HKLS, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1], which asserts that there is a faithful trace on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let �A be a quantum homogeneous space of Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If �A contains C(T\\Gq) as a unital Gq-C*-subalgebra and has a tracial state, it must be induced from a quantum homogeneous space of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let J be an ideal of �A defined as follows: J = {x ∈ �A | τ(x∗x) = 0 for any tracial state τ ∈ �A∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By our assumption, this is a T-invariant proper closed ideal of �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Set A = �A/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The quotient map from �A to A is denoted by q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The T- invariance of J allows us to induce a T-action on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We will show that (q ⊗ id)�α gives a Gq-equivariant ∗-isomorphism from �A to Ind Gq T A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The well-definedness and Gq-equivariance of this map are easy to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since �AGq = C1 � A, the injectivity is automatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence it suffices to show the surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='4, the re- striction (q⊗id)�α|C(T\\Gq) is induced from C ⊂ A i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' coincides with the canonical inclusion C(T\\Gq) ⊂ Ind Gq T A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence the image of this inclusion is contained in the image of (q ⊗ id)�α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' On the other hand, for any a ∈ �A and x ∈ C(Gq), we have PA((q ⊗ id)�α(a)(1 ⊗ x)) = (q ⊗ id)�α(a)PA(1 ⊗ x) = (q ⊗ id)�α(a)(1 ⊗ (hT ◦ qT ⊗ id)∆(x)), where PA is the expectation as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3, hT is the Haar state on C(T) and qT is the canonical quotient map from C(Gq) to C(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since (hT ◦ qT ⊗ id)∆(x) is an element of C(T\\Gq), we can conclude the surjectivity of (q ⊗ id)�α from this equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let �A ⊂ �B be a finite quantum Gq-covering space over a quantum homogeneous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If �A contains C(T\\Gq) and has a tracial state, �A ⊂ �B is isomorphic to Ind Gq T A ⊂ Ind Gq T B, where A ⊂ B is a finite quantum T-covering space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Discrete quantum subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this subsection, we develop some ap- plications to discrete quantum groups and give a classification of finite index discrete quantum subgroup of � Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 36 MAO HOSHINO Let G be a compact quantum group and H be its quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have an in- clusion C(H) ⊂ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this case we have a unique G-expectation E : C(G) −→ C(H), namely the Haar state-preserving conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let �G be a discrete quantum group and �H be a quantum sub- group of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The scalar index of E : C(G) −→ C(H) is called the quantum group index and denoted by [�G: �H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The quantum subgroup �H is said to be finite index if [�G: �H] is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The index [�G: �H] must be either a positive integer or ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This can be seen from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='13 if one consider the basic construction of C(H) ⊂ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Beford giving a classification result of finite index quantum subgroup of � Gq, we prepare the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let G be a compact quantum group and K be its maximal Kac quantum subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The canonical map from O(G) to O(K) is denoted by q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A be a right coideal of G and A be its algebraic core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If the canonical inclusion A ⊂ C(G) is a finite quantum G-covering space, we have q−1(q(A)) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let E : C(G) −→ A be a G-expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Such a expectation is unique and finite index by our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now consider the G-equivariant Hilbert A-module C(G)E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This gives a G-equivariant imprimitivity bimodule betwee A and B = LA(C(G)E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In particular we have an equivalence G- Modf A ∼= G- Modf B of Repf G-module category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover, Irr G- Modf A is finite since –⊗C(G) C(G)E : G- Modf C(G) −→ G- Modf A is adjointable and Irr G- Modf C(G) consists of only one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' On the other hand, since C(G) ⊂ B is a finite quantum G-covering space, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='29 implies that there is a finite quantum K-covering space C(K) ⊂ B0 such that C(G) ⊂ B is isomorphic to Ind G KC(K) ⊂ Ind G KB0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In particular we have an equivalence G- Modf B ∼= K- Modf B0 as Repf G-module categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we have a structure of Repf K-module category on G- Modf A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since Irr G- Modf A is finite, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='25 implies that – ⊗A O(G) is a Repf K-module functor from G- Modf A to G- Modf C(G) ∼= K- Modf C(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take a quantum homogeneous space A0 of K corresponding to an irreducible object A of a Repf K-module category G- Modf A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If we fix an identification G- Modf A ∼= K- Modf A0, we have a corresponding G-equivariant isomorphism A ∼= Ind G KA0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover – ⊗A O(G) induces a Repf K-module functor from K- Modf A0 to K- Modf C(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='14, we have a corresponding M ∈ K- Corrrf A0,C(K), which satisfies A0 ⊗A0 M ∼= C(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence M can be thought as C(K) as a K- equivariant Hilbert C(K)-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have a K-equivariant ∗-homomorphism ϕ: A0 −→ C(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By our construction, this map makes the following diagram EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 37 commutative: Ind G KA0 Ind G Kϕ� ∼ = � Ind G KC(K) ε⊗id � A � C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This implies that q(A) = ϕ(A0) and q−1(ϕ(A0)) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Now we give a classification theorem of finite index discrete quantum subgroup of � Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Q) be the weight (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' root) lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We naturally identifies P with the Pontrjagin dual of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' There is a one-to-one correspondence between the set of finite index discrete quantum subgroups of � Gq and the set of subgroups of P/Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let q: C(Gq) −→ C(T) be the canonical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take a finite index discrete quantum subgroup �H ⊂ � Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The representation category Repf H can be thought as a full subcategory of Repf Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Consider the image q(C(H)) ⊂ C(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This defines a compact quantum group H if it is equipped with the restriction of the coproduct on C(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then H is a quotient compact group of T and gives a full subcategory Repf H of Repf T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The property q(C(H)) = C(H) implies that Repf H consists of π ∈ Repf T which appears as a direct summand of ρ|T for some ρ ∈ Repf H ⊂ Repf Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Combining this with the highest weight theory, we can see that �H ⊂ �T = P must contains the root lattice Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' On the other hand, we have q−1(C(H)) = C(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This implies that Repf H consists of π ∈ Repf Gq such that π|T ∈ Repf H ⊂ Repf T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence Repf H is the full subcategory of Repf Gq consisting of π ∈ Repf Gq whose weights are elements of �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' These obserbations imply that �H �−→ �H/Q is an injective map from the set of finite index discrete quantum subgroups of � Gq to the set of subgroups of P/Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To show the surjectivity, take a subgroup Λ ⊂ P containing Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let C be the full subcategory of Repf Gq consisting of π ∈ Repf Gq whose weights are elements of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then this is a rigid C*-tensor category, hence we have a discrete quantum group �H of � Gq such that Repf H ∼= C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover the inclusion C(H) ⊂ C(Gq) is induced from C(�Λ) ⊂ C(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since Q ⊂ P is finite index, this inclusion has a T-expectation with finite index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence C(H) ⊂ C(Gq) has a Gq-expectation with finite index i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' �H is finite index � Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since the image of �H under the map defined in the previous paragraph is Λ, we have the surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof of Propostion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A be a G-C*-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then its fixed point subalgebra AG is non-degenerate in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover, for any a ∈ A, we have x ∈ AG and b ∈ A such that a = xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 38 MAO HOSHINO Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A be the algebraic core of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For the non-degeneracy of AG ⊂ A, it suffices to show that any a ∈ A can be approximated by an element of the form xb with x ∈ AG and b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take an arbitrary ε > 0 and set α(a) = �n i=1 ai 0 ⊗ai 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we can find e ∈ A such that ∥eai 0 − ai 0∥ < ε(n∥S(ai 1)∥)−1 for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now we have α(e)(a ⊗ 1) − a ⊗ 1 = n � i=1 α(eai 0 − ai 0)(1 ⊗ S(ai 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we also have ∥(id ⊗h)(α(e))a − a∥ ≤ ∥α(e)(a ⊗ 1) − a ⊗ 1∥ ≤ n � i=1 ∥eai 0 − ai 0∥∥S(ai 1)∥ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since (id ⊗h)(α(e)) is in AG, this inequality completes the proof of the first half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Next we show tha latter half of our statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take an irreducible representa- tion π of G and let C(G)π be the linear span of matrix coefficients of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover set Aπ as follows: Aπ = {x ∈ A | α(x) ∈ A ⊗ C(G)π}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the following properties hold as shown in [DC16, Section 3]: (i) A = � π∈Irr G Aπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (ii) Each Aπ is a closed AG-subbimodule of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (iii) The projection onto Aπ with respect to the decomposition in (i) extends to a continuous projection Eπ : A −→ Aπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using (i), it can be seen that (ii) and (iii) hold for AF = � π∈F Aπ, where F is a finite subset of Irr G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take an arbitrary a ∈ A and a finite subset F ⊂ Irr G such that a ∈ AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since AG is non-degenerate in A, an approximate unit of AG acts as an approximate unit of the left Banach AG-module AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we can take x ∈ AG and b ∈ AF such that a = xb by using the Cohen-Hewitt factorization theorem ([BD73, Chapter I, Section 11, Theorem 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since AF ⊂ A, this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2 (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let A and B be G-C*-algebras with the algebraic cores A and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' If ϕ: A −→ B is G-equivariant and com- pletely positive as a map from A to B, it extends to a G-equivariant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' map from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By replacing ϕ by ϕ ⊕ id: A −→ B ⊕ A, we may assume ϕ is faithful in the sense that ϕ(x∗x) = 0 if and only if x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By our assumption, we can define a B-valued semi-inner product on A ⊗ B as follows: ⟨a ⊗ b, a′ ⊗ b′⟩B = b∗ϕ(a∗a′)b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let E be a Hilbert B-module obtained by taking a completion of A⊗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We show that a ∈ A defines a adjointable right B-module map La on E by La(a′ ⊗ b′) = aa′ ⊗ b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Take an arbitrary element �n i=1 ai ⊗ bi ∈ A ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1 implies that we have a presentation ai = xia′ i with xi ∈ AG and a′ i ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since AG is a C*-algebra, for any a ∈ AG we can take X ∈ AG ⊗ Mn(C) such that x∗a∗ax + X∗X = ∥a∥2x∗x, EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 39 where x = (x1 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the complete positivity of ϕ implies that � n � i=1 aai ⊗ bi, n � j=1 aaj ⊗ bj � B = n � i,j=1 b∗ i ϕ(a′∗ i x∗ i a∗axja′ j)bj ≤ ∥a∥2 n � i,j=1 biϕ(a′∗ i x∗ i xja′ j)bj = ∥a∥2 � n � i=1 ai ⊗ bi, n � j=1 aj ⊗ bj � B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence La is well-defined and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For general a ∈ A, we use the argument in the proof of [DC16, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we can take a family (vi)n i=1 ⊂ A such that a = v1 and � i v∗ i vi ∈ AG, which gives an estimate from above as follows: ⟨aξ, aξ⟩B ≤ � ξ, n � i=1 v∗ i viξ � B ≤ ∥ n � i=1 v∗ i vi∥ ⟨ξ, ξ⟩B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This proves the boundedness of La.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The adjointability can be seen from the following: ⟨aa1 ⊗ b1, a2 ⊗ b2⟩B = ⟨a1 ⊗ b1, a∗a2 ⊗ b2⟩B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In particular L∗ a = La∗ holds, hence a �−→ La defines a ∗-homomorphism L: A −→ LB(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This map is faithful since ϕ is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Next we show that there is a unitary V : E ⊗β (B ⊗C(G)) −→ E ⊗C(G) which satisfies the following for any a ∈ A, b, c ∈ B and x ∈ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' V ((a ⊗ b) ⊗ (c ⊗ x)) = α(a)13β(b)23(1 ⊗ c ⊗ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Tha well-definedness and the boundedness of this map can be seen as follows: ⟨V ((a ⊗ b) ⊗ (c ⊗ x)), V ((a′ ⊗ b′) ⊗ (c′ ⊗ x′))⟩B⊗C(G) = (c∗ ⊗ x∗)β(b∗)(ϕ ⊗ id)(α(a∗a′))β(b′)(c′ ⊗ x′) = (c∗ ⊗ x∗)β(b∗ϕ(a∗a′)b′)(c′ ⊗ x′) = ⟨(a ⊗ b) ⊗ (c ⊗ x), (a′ ⊗ b′) ⊗ (c′ ⊗ x′)⟩B⊗C(G) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In particular V is an isometry, hence it suffices to show that the range of V is dense in E ⊗ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This follows from span α(A)(C ⊗ O(G)) = A ⊗ O(G) and span β(B)(C ⊗ O(G)) = B ⊗ O(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let λ: C(G) −→ LC(G)(C(G)) be a ∗-homomorphism defined by the left mul- tiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have V (La ⊗β 1)V ∗ = (L ⊗ λ)α(a) for any a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This implies the following: ∥La∥ = ∥La ⊗β 1∥ = ∥(L ⊗ λ)(α(a))∥ = ∥(L ⊗ id)(α(a))∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We use the faithfullness of β and λ at the first and third equality respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Since L is faithful, a ∈ A �−→ ∥La∥ defines a C*-norm on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The above equality implies the completion of A with respect to this norm has an action of G induced by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then [DY13, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='4] implies that this norm coincides with the original norm of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence L extends to a ∗-homomorphism L: A −→ LBG(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 40 MAO HOSHINO Then we have the following inequality for x, a ∈ A, b ∈ B: ∥b∗ϕ(a∗xa)b∥ = ∥⟨a ⊗ b, Lx(a ⊗ b)⟩B∥ ≤ ∥x∥∥a ⊗ b∥2 ≤ ∥x∥∥b∥2∥ϕ(a∗a)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using an approximate unit of B, we also have the following inequality for x, a ∈ A: ∥ϕ(a∗xa)∥ ≤ ∥x∥∥ϕ(a∗a)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence x �−→ ϕ(a∗xa) is continuous for any a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By using the polarization identity, we also see that ϕa,a′ : x �−→ ϕ(axa′) is continuous for any a, a′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' To show the continuity of ϕ, take a null sequence (xn)n in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we can find a sequence (Fn)n of finite subsets of Irr G such that xn ∈ AFn, here AFn is the subspace of A as in the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Set M = c0- � n AFn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then AG acts on M from the left and the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover an approximate unit of AG acts as a bounded approximate unit on the both sides of M, since AG is non-degenerate in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence we can apply the Cohen-Hewitt factorization theorem ([BD73, Chapter I, Section 11, Theorem 10]) to M as a AG-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now consider (xn)n as an element x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we can find a, a′ and y ∈ M such that x = aya′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' we can find a null sequence (yn)n ⊂ A such that xn = ayna′ for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The continuity of ϕa,a′ implies that ϕ(xn) = ϕ(ayna′) = ϕa,a′(yn) converges to 0 in norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence ϕ is continuous with respect to the norm-topology and extends to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The complete positivity and G-equivariance follows from the corresponding property for ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='3 (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let E : B −→ A be a conditional expecta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then E is with finite index if and only if it satisfies both of the following two conditions: (i) The probabilistic index of E is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' (ii) The Hilbert A-module BE can be decomposed into a direct sum of finitely generated projective Hilbert A-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' In this case the scalar index of E coincides with ∥Index E∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Assume E is with finite index and take a quasi-basis (vi)n i=1 ⊂ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have a linear map V : BE −→ An defined by V (x) = (E(v∗ i x))n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we have ⟨V (y), V (x)⟩A = n � i=1 E(v∗ i y)∗E(v∗ i x) = E � y∗ n � i=1 viE(v∗ i x) � = E(y∗x) = ⟨y, x⟩A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Hence V ia an isometrical A-module map, so BE is finitely generated projective Hilbert A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The finiteness of the probabilistic index of E follows from the inequality ∥Index E∥E(x) − x ≥ 0, which was shown in [Wa90, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover we can also see that ∥Index E∥ is equal to or greater than the scalar index of E by replacing E by E ⊗ id: B ⊗ Mn(C) −→ A ⊗ Mn(C), where n is an arbitrary positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' EQUIVARIANT COVERING SPACES OF QUANTUM HOMOGENEOUS SPACES 41 Next we prove the converse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' We firstly remark that the original C*- norm on B is equivalent to the L2-norm with respect to E, hence BE = B as C-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' By the previous proposition, the scalar index c of E is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then the product map on B induces a bounded B-module map T : BE ⊗A B −→ B with ∥T∥2 ≤ c, since we have � n � i=1 xiyi, n � i=1 xiyi � B = ty∗x∗xty ≤ cty∗(E ⊗ id)(x∗x)ty = c � n � i=1 xi ⊗ yi, n � i=1 xi ⊗ yi � B , where x = (x1 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' xn), y = (y1 y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' yn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Now we can find η ∈ BE ⊗A B with Tξ = ⟨η, ξ⟩B for any ξ ∈ BE ⊗A B since BE is a direc sum of finitely generated projective Hilbert A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Moreover we can find sequences {xi}∞ i=1 ⊂ BE and {yi}∞ i=1 ⊂ B such that �n i=1 xi ⊗ yi converges to η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Let Sn be an A-module map on BE defined by Sn(z) = n � i=1 y∗ i ⟨xi, z⟩A = � n � i=1 xi ⊗ yi, z ⊗ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then Sk converges to idB in the operator norm, hence Sk is invertible for some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This fact enables us to find (ui)n i=1 and (vi)n i=1 in B such that x = �n i=1 uiE(v∗ i x) for any x ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Then we can find a quasi-basis of E by [Wa90, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' For tha last statement, it suffices to show ∥Index E∥ ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' But this can be easily seen since T ∗(1B) = �n i=1 vi ⊗ v∗ i and TT ∗(1B) = Index E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' □ Acknowlegements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' The author appreciates to Yasuyuki Kawahigashi for help- ful comments and pointing out mistakes and typos on this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' He is also grateful to Reiji Tomatsu for doing seminars with the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' This work was supported by Forefront Physics and Mathematics Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' References [AV16] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Arano, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' Vaes, C*-tensor categories and subfactors for totally disconnected groups, Abel Symposia, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' 1–43, Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} +page_content=' [BD73] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfPwwb/content/2301.04975v1.pdf'} 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+violetpeng@cs.ucla.edu +Abstract +While GPT-2 generates sentences that are re- +markably human-like, longer documents can +ramble and do not follow human-like writing +structure. We study the problem of imposing +structure on long-range text. +We propose a +novel controlled text generation task, sequen- +tially controlled text generation, and identify a +dataset, NewsDiscourse as a starting point for +this task. We develop a sequential controlled +text generation pipeline with generation and +editing. We test different degrees of structural +awareness and show that, in general, more +structural awareness results in higher control- +accuracy, grammaticality, coherency and top- +icality, approaching human-level writing per- +formance. 1 +1 +Introduction +Imagine that you are tasked with: Write a “Related +Works” section. Would it help to know the past +structure of the article (e.g. it is coming after the +“Discussion” section)? How about the full struc- +ture of the article (e.g. after the “Introduction” but +before the “Problem Statement”)? +The macro-structure of text (i.e. its discourse +structure (Po¨ ttker, 2003)) impacts both human +and machine comprehension (Emde et al., 2016; +Sternadori and Wise, 2010; Lu et al., 2019; Zhou +et al., 2020). Although naive language models have +made impressive advancements and generate fluent +text (Radford et al., 2019; Brown et al., 2020; Belt- +agy et al., 2020), the text is structurally dissimilar +to human-written text (Figure 2, Section 7). Even +the well-known Ovid’s Unicorn generation, which +seems like a natural news article, exhibits unnatural +structure (see Appendix F). +On the other hand, although numerous works +have focused on content planning using keywords +1This paper has been accepted to Findings of the 2022 +Conference on Empirical Methods in Natural Language Pro- +cessing. +Figure 1: We study the task of sequentially-controlled +generation: generating documents exhibiting structure +given by a sequence of local control codes. Shown is +a news article with it’s Van Dijk structure (Van Dijk, +2013) and headline. Our models take as input the head- +line and discourse tags and generate a sequence of sen- +tences. We explore the degree of structural awareness +(local, past-aware or full-sequence) for controlling each +sentence in the document, with the goal of generating +the most structurally faithful, coherent and topical text. +(Yao et al., 2019), plot-design (Rashkin et al., 2020) +and entity tracking (Peng et al., 2021), macro- +structural control has been relatively understudied. +So, in this work, we study (1) how to impose +macro-structural control on narrative text genera- +tion and (2) how much structural awareness during +generation contributes to well-structured and flu- +ent text. We propose a novel task, sequentially +controlled text generation. In this task, the user +provides a prompt as well as a sequence of local +control codes, each of which guides the genera- +tion of a single sentence. (In our experiments, we +use headlines as prompts and Van Dijk (2013) dis- +course tags as control codes (Figure 1).) +We develop methods to address this task, ex- +panding prior work focused on single control code +generation (Keskar et al., 2019; Dathathri et al., +2019; Yang and Klein, 2021). As in prior work, +the controlled generation problem is decomposed +into a discriminator and a generator. However, in +this work, the discriminator learns to incorporate +an entire sequence of control codes. We hypoth- +esize that information about structural intention +can positively impact generative output (intuition +arXiv:2301.02299v1 [cs.CL] 5 Jan 2023 + +Neo-Nazi murder gang member iailed for life in Germany +MUNiCH (Reuters) - A member of a German +Main Event +neo-Nazi gang was jailed for life on Wednesday +She was part of the National Socialist Underground +Historical Event +(NSU), whose members killed eight Turks +The murders shook a country that believed it had +Expectation +learned the lessons of its past.(a) Structure of human-written articles. (b) Structure of naively generated GPT-2 +articles +(c) Structure of sequentially controlled +GPT-2 articles. +Figure 2: Discourse structure (Van Dijk, 2013) of articles generated according to different processes. The likeli- +hood of a tag in the kth fraction of a news article is shown. Machine-generated structure is labeled by humans. +for this is given in the hypothetical at beginning +of this introduction). We show that our methods +improve structural cohesion and certain aspects of +coherence over naive GPT-2 output. +Next, we hypothesize that more structural aware- +ness improves generation. Again, we refer to the +introduction hypothetical: humans craft text ac- +cording to how it fits into a document’s full struc- +ture (Chenlo et al., 2014), so a generative model +should similarly benefit from having such infor- +mation. We test this hypothesis by varying the +discriminator’s conditional independence assump- +tions. We experiment with three different degrees +of control: local-only (where the discriminator is +only aware of the current sentences’ control code), +past-aware (where the discriminator is aware of +the current sentences’ control code and all previ- +ous control codes), and full-sequence (where the +discriminator is aware of the entire document’s se- +quence of control codes). We show that more struc- +tural awareness, especially of past structure, helps +generate the highest-quality text. Finally, we show +how to re-introduce a degree of local control by +combining structurally-aware generation methods +with a local sentence-level editing technique. +In summary, our novel contributions are: +• We propose a novel task, sequentially con- +trolled text generation and identify a dis- +course schema (Van Dijk, 2013) and dataset +(Choubey et al., 2020) to explore this task +(Sections 2, 4). +• We combine two different approaches in con- +trolled text generation: generation and editing, +and show that the highest-quality text is gener- +ated when both of these approaches are used +(Section 3). +• We use our methods to study the degree +of structural control that yields the highest- +quality text: +local, past-aware and full- +sequence control. We show that overall, full- +sequence produces optimal text over an array +of metrics (Section 7). +We see this work opening the door to a variety +of follow-on directions: giving users control over +the macro-structure of their generated output can +allow users to quickly prototype different struc- +tures for their work. It can allow them to work in +tandem with a generative algorithm to infill miss- +ing structural components in a piece of writing2. +It might even allow them produce different ver- +sions of the same story for readers with different +reading preferences. Finally, we also see macro- +structural control providing a natural complement +to, and being used in tandem with, other forms +of controlled generation, like fact-aware genera- +tion (Logan IV et al., 2019) or creative generation +(Goldfarb-Tarrant et al., 2020; Tian and Peng, 2022; +Peng, 2022) to yield more engaging and useful gen- +erative content. +2 +Problem Statement +We assume, as input, a headline sentence, X0, and +a sequence of control codes ⃗c = c1, ..., cS of length +S (i.e., one for each sentence we wish to generate +in the document. Adjacent codes can be of the same +type.) We wish to produce, as output, a document +X of length S as a sequence of sentences X = +X1, ..., XS, each composed of a sequence of words +Xk = x1, ..., xnk of length nk. +We define the sequentially controlled text gener- +ation objective as: +2Perhaps aiding in human-in-the-loop computational jour- +nalism (Cohen et al., 2011) + +Main Event +0.12 +Consequence +0.10 +Previous Event +Cuent Context +0.08 +Historical Event +0.06 +Anecdotal Event +t0'0 +Evaluation +0.02 +Expectation +0.05] +[st'o +0.25] +0.45] +0.55] +65 +0.75] +[58'0 +[56'0 +d +(0.0, +N +m +0.4, +s'0) +(0.8. +'6'0] +8 +Location in DocumentMain Event +0.12 +Consequence +0.10 +Previous Event +0.08 +Cuent Context +Historical Event +0.06 +Anecdotal Event +0.04 +Evaluation +0.02 +Expectation +0.0-0 +0.05] +[st'o +0.25J +0.45] +.55] +.65] +0.75] +[58'0 +,0.95] +d +(0.0. +N +m +[0.4. +5'0) +(0.6. +(0.8. +'6'0] +Location in Documentp(x|⃗c) += +S +� +k=1 +nk +� +i=1 +p(xi|xk, are conditionally independent. In Equation +1, this results in xi being dependent on ck and the +sequence of control codes, c token, we edit the sentence. We use a discriminator to identify class-salient +words to mask, generating masked sentence M, and infill to boost class likelihood. +For a full description of architecture, see Appendix +A. +We train it to model local-only, past-aware and +full-sequence control variants expressed in Section +2: we train separate prediction heads to make pre- +dictions on ck−w, ...ck, ...ck+w, i.e. labels from +−w, ..., +w steps away from current sentence k5. +For local-only control (Equation 3) we only use +predicted probabilities from the main head, k. In +past-aware control (Equation 4), we multiply pre- +dicted probabilities from heads prior to the current +sentence < k, and for full-sequence control, we +multiply predicted probabilities from all heads.6 +We now describe how we use these predictions. +3.2 +Generation +We combine our discriminator’s predictions with a +naive PTLM to solve Equation 2 in two different +ways: Hidden-State Control, based on (Dathathri +et al., 2019) and Direct Probability, based on +(Yang and Klein, 2021). +Hidden-State Control (HSC): Wolf et al. +(2019)’s +GPT-2 +implementation +caches +hid- +den states H to produce logits approximating +p(xi|x